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# physics
posted by .
I will be greateful to you,if you kindly hep me to solve the below problem:
When a horizontal force of 200N is applied on a body ,he acceleration produced in it is 1.5m/s2.when the force is 300N the acceleration produced in it is 2.5m/s2.find the mass of the body.
i have solved the above problem like this:
F=ma
(200+300)N=m*(1.5+2.5)m/s2
500N=m*(4)m/s2
500N=4m/s2*m
4m/s2*m=500
m=500/4
m=125kg
therefore mass of the body is 125kg.
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# Chapter 2 Lesson Starter
## Presentation on theme: "Chapter 2 Lesson Starter"— Presentation transcript:
Chapter 2 Lesson Starter
Section 3 Using Scientific Measurements Chapter 2 Lesson Starter Look at the specifications for electronic balances. How do the instruments vary in precision? Discuss using a beaker to measure volume versus using a graduated cylinder. Which is more precise? Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter 2 Objectives **Distinguish between accuracy and precision.
Section 3 Using Scientific Measurements Chapter 2 Objectives **Distinguish between accuracy and precision. **Determine the number of significant figures in measurements. Perform mathematical operations involving significant figures. Convert measurements into scientific notation. Distinguish between inversely and directly proportional relationships. Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Measurements and Their Uncertainty
2.3 On January 4, 2004, the Mars Exploration Rover Spirit landed on Mars. Each day of its mission, Spirit recorded measurements for analysis. In the chemistry laboratory, you must strive for accuracy and precision in your measurements. (In August 2012, a new rover landed on Mars to collect data.) Slide of 48 3 End Show © Copyright Pearson Prentice Hall
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error Accuracy, Precision, and Error How do you evaluate accuracy and precision? Slide of 48 4 End Show © Copyright Pearson Prentice Hall
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error Accuracy and Precision Accuracy is a measure of how close a measurement comes to the actual or true value of whatever is measured. Precision is a measure of how close a series of measurements are to one another. Slide of 48 5 End Show © Copyright Pearson Prentice Hall
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error To evaluate the accuracy of a measurement, the measured value must be compared to the correct value. To evaluate the precision of a measurement, you must compare the values of two or more repeated measurements. Slide of 48 6 End Show © Copyright Pearson Prentice Hall
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error The distribution of darts illustrates the difference between accuracy and precision. a) Good accuracy and good precision: The darts are close to the bull’s-eye and to one another. b) Poor accuracy and good precision: The darts are far from the bull’s-eye but close to one another. c) Poor accuracy and poor precision: The darts are far from the bull’s-eye and from one another. Slide of 48 7 End Show © Copyright Pearson Prentice Hall
Accuracy and Precision
Accuracy and Precision
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error Just because a measuring device works, you cannot assume it is accurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. The scale below has not been properly zeroed, so the reading obtained for the person’s weight is inaccurate. There is a difference between the person’s correct weight and the measured value. Calculating What is the percent error of a measured value of 114 lb if the person’s actual weight is 107 lb? Slide of 48 10 End Show © Copyright Pearson Prentice Hall
Accuracy, Precision, and Error
2.3 Measurements and Their Uncertainty > Accuracy, Precision, and Error Determining Error The accepted value is the correct value based on reliable references. The experimental value is the value measured in the lab. The difference between the experimental value and the accepted value is called the error. Slide of 48 11 End Show © Copyright Pearson Prentice Hall
Accuracy and Precision, continued
Section 3 Using Scientific Measurements Chapter 2 Accuracy and Precision, continued Percentage Error Percentage error is calculated by subtracting the accepted value from the experimental value, dividing the difference by the accepted value, and then multiplying by 100. Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter 2 Percentage Error Visual Concept
Section 3 Using Scientific Measurements Chapter 2 Percentage Error Visual Concept Click below for Visual Concept Click for Visual Concept on Percent Error Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
PERCENT ERROR PROBLEM ACTIVITY
Accuracy and Precision, continued
Section 3 Using Scientific Measurements Chapter 2 Accuracy and Precision, continued Problem Example - RECORD in notes before checking. A student measures the mass and volume of a substance and calculates its density as 1.40 g/mL. The correct, or accepted, value of the density is 1.30 g/mL. What is the percentage error of the student’s measurement? Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Accuracy and Precision, continued
Accuracy and Precision, continued
Section 3 Using Scientific Measurements Chapter 2 Accuracy and Precision, continued Error in Measurement Some error or uncertainty always exists in any measurement. skill of the measurer conditions of measurement measuring instruments Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Chapter 2 Significant Figures
Section 3 Using Scientific Measurements Chapter 2 Significant Figures Significant figures in a measurement consist of all the digits known with certainty plus one final digit, which is somewhat uncertain or is estimated. The term significant does not mean certain. Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Section 3 Using Scientific Measurements
Chapter 2 Significant Figures
Significant Figures, continued
Rules for Determining Significant Zeros
Counting Sig Fig Examples
C. Significant Figures Counting Sig Fig Examples 4 sig figs 3 sig figs 3. 5,280 3. 5,280 3 sig figs 2 sig figs C. Johannesson
Significant Figures, continued
Section 3 Using Scientific Measurements Chapter 2 Significant Figures, continued Sample Problem D How many significant figures are in each of the following measurements? a g b cm c. 910 m d L e kg Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Significant Figures, continued
Section 3 Using Scientific Measurements Chapter 2 Significant Figures, continued Sample Problem D Solution a g There are no zeros, so all three digits are significant. b cm By rule 4, the zero is significant because it is immediately followed by a decimal point; there are 4 significant figures. c. 910 m By rule 4, the zero is not significant; there are 2 significant figures. Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
Significant Figures, continued
Section 3 Using Scientific Measurements Chapter 2 Significant Figures, continued Sample Problem D Solution, continued d L By rule 2, the first two zeros are not significant; by rule 1, the third zero is significant; there are 4 significant figures. e kg By rule 2, the first three zeros are not significant; by rule 3, the last three zeros are significant; there are 5 significant figures. Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
SIG. FIG. PRACTICE QUESTIONS
SIG. FIG. ANSWERS - CHECK WORK
Significant Figures, continued
Rules for Rounding Numbers
C. Significant Figures (13.91g/cm3)(23.3cm3) = 324.103g 324 g
Calculating with Sig Figs Multiply/Divide - The # with the fewest sig figs determines the # of sig figs in the answer. (13.91g/cm3)(23.3cm3) = g 4 SF 3 SF 3 SF 324 g C. Johannesson
C. Significant Figures 3.75 mL + 4.1 mL 7.85 mL 3.75 mL + 4.1 mL
Calculating with Sig Figs (con’t) Add/Subtract - The # with the lowest decimal value determines the place of the last sig fig in the answer. 3.75 mL mL 7.85 mL 3.75 mL mL 7.85 mL 224 g + 130 g 354 g 224 g + 130 g 354 g → 7.8 mL → 350 g C. Johannesson
C. Significant Figures Calculating with Sig Figs (con’t)
Exact Numbers do not limit the # of sig figs in the answer. Counting numbers: 12 students Exact conversions: 1 m = 100 cm “1” in any conversion: 1 in = 2.54 cm C. Johannesson
C. Significant Figures Practice Problems f). (15.30 g) ÷ (6.4 mL)
g) g g → 18.1 g 18.06 g C. Johannesson
D. Scientific Notation 65,000 kg → 6.5 × 104 kg
Converting into Sci. Notation: Move decimal until there’s 1 digit to its left. Places moved = exponent. Large # (>1) ⇒ positive exponent Small # (<1) ⇒ negative exponent Only include sig figs. C. Johannesson
D. Scientific Notation Practice Problems h. 2,400,000 μg 2.4 × 106 μg
i kg j. 7 × km k. 6.2 × 104 mm 2.4 × 106 μg 2.56 × 10-3 kg km 62,000 mm C. Johannesson
D. Scientific Notation Calculating with Sci. Notation
(5.44 × 107 g) ÷ (8.1 × 104 mol) = Type on your calculator: EXP EE EXP EE ENTER EXE 5.44 7 ÷ 8.1 4 = = 670 g/mol = 6.7 × 102 g/mol C. Johannesson
Stop here and play BINGO GAME ACTIVITY
HW: Study Guide Questions due on Friday. Get out a sheet of paper and make a 5 x 5 grid. (5 rows and 5 columns) Chapter menu Resources Copyright © by Holt, Rinehart and Winston. All rights reserved.
E. Proportions Direct Proportion Inverse Proportion y x y x
C. Johannesson
Assess students’ understanding of the concepts in Section 2.3.
Section Assessment Assess students’ understanding of the concepts in Section 2.3. Continue to: Launch: -or- Section Quiz Slide of 48 End Show © Copyright Pearson Prentice Hall
2.3 Check up Section Quiz 1. In which of the following expressions is the number on the left NOT equal to the number on the right? × 10–8 = 4.56 × 10–11 454 × 10–8 = 4.54 × 10–6 842.6 × 104 = × 106 × 106 = 4.52 × 109 Slide of 27 End Show © Copyright Pearson Prentice Hall
2. Which set of measurements of a 2.00-g standard is the most precise?
2.3 Section Quiz 2. Which set of measurements of a 2.00-g standard is the most precise? 2.00 g, 2.01 g, 1.98 g 2.10 g, 2.00 g, 2.20 g 2.02 g, 2.03 g, 2.04 g 1.50 g, 2.00 g, 2.50 g Slide of 27 End Show © Copyright Pearson Prentice Hall
2.3 Section Quiz 3. A student reports the volume of a liquid as L. How many significant figures are in this measurement? 2 3 4 5 Slide of 27 End Show © Copyright Pearson Prentice Hall
Online Self-Check Quiz
Additional Videos for Section 2.3 Using Scientific Measurements (4 videoclips) Direct Variation (1:44) Inverse Variation (2:12) Significant Figures (6:04) Scientific Notation (2:26) Slide of 27 End Show © Copyright Pearson Prentice Hall
Additional Videos for Section 2.3 Using Scientific Measurements (4 videoclips) Direct Variation (1:44) Inverse Variation (2:12) Significant Figures (6:04) Scientific Notation (2:26) Slide of 27 End Show © Copyright Pearson Prentice Hall
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Nt YNk
The mean particle diameter, Dp, of the population is
The variance a2, a measure of the spread of the distribution around the mean diameter Dp, is defined by a2 = Ek=iMDk - Dp) = i^Nk{Dk _ Dpf {m
A value of a2 equal to zero would mean that every one of the particles in the distribution has precisely diameter Dp. An increasing a2 indicates that the spread of the distribution around the mean diameter Dp is increasing.
We will usually deal with aerosol distributions in continuous form. Given the number distribution nN(Dp), (8.27) and (8.28) can be written in continuous form to define the mean particle diameter of the distribution by
\$~DpnN{Dp)dDp 1 °> ~ j-nN{Dp)dDp ~ N, L DpHN(8'29)
and the variance of the distribution by a "-rnN(Dp)dDp--NJ0 {Dp ~ Dp) nN{Po)dD^ ^
Table 8.2 presents a number of other mean values that are often used in characterizing an aerosol size distribution.
TABLE 8.2 Mean Values Often Used in Characterizing an Aerosol Size Distribution
Property Defining Relation Description
Number mean diameter Dp
Median diameter
^med
Mean surface area S
Mean volume V
Surface area mean diameter D\$
Volume mean diameter Dv
Surface area median diameter DSm
Volume median diameter DVm
Mode diameter
^mode
î>p = bSDPn»(Dp)dDp J®"0 tin(Pp)dDp = \Nt
V dDP )Dmit
Average diameter of the population
Diameter below which one-half the particles lie and above which one-half the particles lie Average surface area of the population
Average volume of the population
Diameter of the particle whose surface area equals the mean surface area of the population Diameter of the particle whose volume equals the mean volume of the population Diameter below which one-half the particle surface area lies and above which one-half the particle surface area lies Diameter below which one-half the particle volume lies and above which one-half the particle volume lies Local maximum of the number distribution
8.1.6 The Lognormal Distribution
A measured aerosol size distribution can be reported as a table of the distribution values for dozens of diameters. For many applications carrying around hundreds or thousands of aerosol distribution values is awkward. In these cases it is often convenient to use a relatively simple mathematical function to describe the atmospheric aerosol distribution. These functions are semiempirical in nature and have been chosen because they match well observed shapes of ambient distributions. Of the various mathematical functions that have been proposed, the lognormal distribution (Aitchison and Brown 1957) often provides a good fit and is regularly used in atmospheric applications. A series of other distributions are discussed in the next section.
The normal distribution for a quantity u defined from — oc < u < oo is given by where u is the mean of the distribution, a^ is the variance and
The normal distribution has the characteristic bell shape, with a maximum at u. The standard deviation, crH, quantifies the width of the distribution, and 68% of the area below the curve is in the range u ± a„.
A quantity u is lognormally distributed if its logarithm is normally distributed. Either the natural (In u) or the base 10 logarithm (log u) can be used, but since the former is more common, we will express our results in terms of In Dp. An aerosol population is therefore log-normally distributed if u = In Dp satisfies (8.31), or where N, is the total aerosol number concentration, and Dpg and ag are for the time being the two parameters of the distribution. Shortly we will discuss the physical significance of these parameters. The distribution n^{Dp) is often used instead of n^(ln Dp). Combining (8.21) with (8.33)
A lognormal aerosol distribution with Dpg = 0.8 pm and <JX = 1.5 is depicted in Figure 8.7.
FIGURE 8.7 Aerosol distribution functions, iin(Dp), n°N(logDp) and neN{\aDp) for a lognormally distributed aerosol distribution Dpg = 0.8 |im and a,, = 1.5 versus log D„. Even if all three functions describe the same aerosol population, they differ from each other because they use a different independent variable. The aerosol number is the area below the «'v(log Dp) curve.
FIGURE 8.7 Aerosol distribution functions, iin(Dp), n°N(logDp) and neN{\aDp) for a lognormally distributed aerosol distribution Dpg = 0.8 |im and a,, = 1.5 versus log D„. Even if all three functions describe the same aerosol population, they differ from each other because they use a different independent variable. The aerosol number is the area below the «'v(log Dp) curve.
We now wish to examine the physical significance of the two parameters Dpg and <yg. To do so we will use the cumulative size distribution N(DP).
If the aerosol distribution is lognormal, nN(Dp) is given by (8.34) and therefore
To evaluate this integral we let
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# Proportion Playground
Bu benzetimin henüz çevirisi yapılmamıştır. Ancak aşağıdaki İngilizce sürümüne hala ulaşabilirsiniz.
İndir Göm(Embed) Kapat Bu benzetimin çalışır kopyasını göm Bu benzetimin çalışır halini eklemek için bu HTML'i kullan.Genişlik ve boyunu HTML kısmından değiştirebilirsiniz Tıklandığında benzetim başlatacak bir görüntü iliştir
Çalıştırmak için tıklayın
Ekranda görüntülemek için bu HTML kodunu kullanın "Çalıştır için tıklayın". Ratios Proportional Reasoning Unit Rate PhET ekibini destekleyenler Sizin gibi eğitimciler.
### İçerik
• Ratios
• Proportional Reasoning
• Unit Rate
### Description
Play with ratios and proportions by designing a necklace, throwing paint balloons, playing billiards, or shopping for apples! Make predictions about proportions before they are revealed.
### Örnek Öğrenim Hedefleri
• Use a ratio as a descriptive characteristic
• Compare the meaning of a ratio in different contexts
• Use scaling to build proportions or find a missing value in a proportion
• Simplify ratios to build proportions or find a missing value in a proportion
• Use multiplicative reasoning to solve problems
### Standards Alignment
#### Common Core - Math
5.NF.B.4a
Interpret the product (a/b) × q as a parts of a partition of q into b equal parts; equivalently, as the result of a sequence of operations a × q ÷ b. For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)
6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."
6.RP.A.3
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
7.RP.A.2a
Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
Version 1.0.15
### Teacher Tips
Öğretmen el kitabı (pdf) Phet ekibi tarafından oluşturulan ipuçları içerir. ( PDF ).
### Teacher-Submitted Activities
Patterns and Ratios Ian Whitacre, Sebnem Tekin K-5
Ortaokul
Guided Matematik
Bank Shots Daniel Greenberg Ortaokul Guided Matematik
Ortaokul
Guided Matematik
Paint Splat: Comparing Ratios Kathyrn Studer, Heather Politi Ortaokul Guided Matematik
Making Green Paint Michael Matassa Ortaokul Guided
Discuss
Matematik
Mixing paint with ratios Jennifer Knudsen, Teresa Lara-Meloy, Amanda McGarry Ortaokul Discuss Matematik
Middle School Math Sim Alignment Amanda McGarry Ortaokul Diğer Matematik
SECUNDARIA: Alineación PhET con programas de la SEP México (2011 y 2017) Diana López Ortaokul
Lİse
Diğer Matematik
Fizik
Biyoloji
Kimya
PRIMARIA: Alineación con programas de la SEP México (2011 y 2017) Diana López Ortaokul
K-5
Laboratuvar
Ödev
Discuss
Gösteri
Guided
Fizik
Kimya
Matematik
Astronomi
Unit Rate as Slope Cary Hoste Ortaokul
Lİse
Guided
Diğer
Matematik
Almanca All Deutsch Proportionen Spiel
Arapça All العربية ملعب النسب
Arnavutça All shqip Këndi i eksperimentimit për Raportet
Boşnakça All Bosanski Igralište proporcije
Brezilya Portekizcesi All português (Brasil) Parque da Proporção
Çince (Geleneksel) All 中文 (台灣) Proportion Playground_比例遊樂場
Çince (Sadeleştirilmiş) All 中文 (中国) 比例游乐场
Farsça All فارسی زمین بازی نسبت
Flemenkçe All Nederlands Spelen met verhoudingen
Fransızca All français Aire de jeu proportionnelle
Gujarati All Gujarati પ્રમાણ રમતો -Pankajsid34
Hırvatça All hrvatski Igralište proporcija
Hintçe All हिंदी समानुपात का मैदान
İrlandaca All Gaeilge Ionad Súgartha na Comhréire
İspanyolca All español Pista de juego de proporciones
İspanyolca (Meksika) All español (México) Juego de Proporciones
İspanyolca (Peru) All español (Perú) Juegos de Proporciones
İsveçce All svenska Lek med proportioner!
İtalyanca All italiano Area giochi Proporzioni
Japonca All 日本語 数と割合
Kazakça All Kazakh «Пропорция» ойыны
Korece All 한국어 비율 놀이터
Lehçe All polski Zabawa w proporcje
Letonca All Latviešu Darbības ar proporcijām
Macarca All magyar Arányjáték
Moğolca All Монгол (Монгол) Хамаарлын Тоглоомын талбай
Rusça All русский Игра «Пропорциональность»
Sırpça All Српски Игралиште пропорције
Tayca All ไทย เล่นกับสัดส่วน
Ukraynaca All українська Гра в пропорції
Vietnamca All Tiếng Việt Sân chơi tỷ lệ
Yunanca All Ελληνικά Παιχνίδια με αναλογίες
HTML5 sims can run on iPads and Chromebooks, as well as PC, Mac, and Linux systems.
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## Covariant derivative and connections
Required math: algebra, calculus
Required physics: none
Reference: d’Inverno, Ray, Introducing Einstein’s Relativity (1992), Oxford Uni Press. – Section 6.3; Problem 6.3.
The Lie derivative is one way of calculating the derivative of a tensor field in such a way that this derivative is itself a tensor. The problem solved by the Lie derivative is that we cannot define a new tensor as the difference of two other tensors evaluated at different points, since in that case the transformation between coordinate systems of this difference does not follow the equation required of a tensor.
The Lie derivative required the introduction of an auxiliary vector field which defined a congruence of curves, which in turn defined the directions along which the Lie derivative is calculated. Suppose we try to find another formula for a derivative of a vector which does not require this congruence of curves.
A general vector ${\mathbf{V}}$ can be written in terms of the basis vectors ${\mathbf{e}_{a}}$ in some coordinate system as
$\displaystyle \mathbf{V}=V^{a}\mathbf{e}_{a}$
As we vary the position, both the components of ${\mathbf{V}}$ and the basis vectors will, in general, vary. For example, although the basis vectors in rectangular coordinates are constant, those in polar coordinates are not. Thus if we want the derivative of V we have to take into account this change in the basis vectors, so we get
$\displaystyle \frac{\partial\mathbf{V}}{\partial x^{b}}=\frac{\partial V^{a}}{\partial x^{b}}\mathbf{e}_{a}+V^{a}\frac{\partial\mathbf{e}_{a}}{\partial x^{b}}$
The change in a basis vector is itself a vector, so it can be written in terms of the original set of basis vectors:
$\displaystyle \frac{\partial\mathbf{e}_{a}}{\partial x^{b}}=\Gamma_{ab}^{c}\mathbf{e}_{c}$
where the ${\Gamma_{ab}^{c}}$ are defined by this equation, and are called the connections. We can use this definition to write the derivative of V entirely in terms of the original basis vectors:
$\displaystyle \frac{\partial\mathbf{V}}{\partial x^{b}}$ $\displaystyle =$ $\displaystyle \frac{\partial V^{a}}{\partial x^{b}}\mathbf{e}_{a}+V^{a}\Gamma_{ab}^{c}\mathbf{e}_{c}$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\partial V^{a}}{\partial x^{b}}\mathbf{e}_{a}+V^{c}\Gamma_{cb}^{a}\mathbf{e}_{a}$ $\displaystyle$ $\displaystyle =$ $\displaystyle \left(\frac{\partial V^{a}}{\partial x^{b}}+V^{c}\Gamma_{cb}^{a}\right)\mathbf{e}_{a}$
where in the second line, we swapped the dummy indices ${a}$ and ${c}$. The quantity in parentheses is called the covariant derivative of V and is written in a variety of ways in different books. Two of the more common notations are
We can require the covariant derivative to be a tensor, which means we can derive transformation equations for the connections ${\Gamma_{cb}^{a}}$. Since ${V_{\;;b}^{a}}$ is a mixed second-rank tensor, it must transform as
Since ${V^{a}}$ is a contravariant vector, we have
$\displaystyle V^{\prime a}=\frac{\partial x^{\prime a}}{\partial x^{c}}V^{c}$
Taking the derivative of this we get
$\displaystyle \frac{\partial V^{\prime a}}{\partial x^{\prime b}}$ $\displaystyle =$ $\displaystyle \frac{\partial^{2}x^{\prime a}}{\partial x^{\prime b}\partial x^{c}}V^{c}+\frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial V^{c}}{\partial x^{\prime b}}$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{c}}V^{c}+\frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial V^{c}}{\partial x^{d}}$
For the second term in 1, we have
$\displaystyle V^{\prime c}\Gamma_{cb}^{\prime a}=\frac{\partial x^{\prime c}}{\partial x^{d}}V^{d}\Gamma_{cb}^{\prime a}$
Summing these last two results and requiring they give 2 gives
$\displaystyle \frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{c}}V^{c}+\frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial V^{c}}{\partial x^{d}}+\frac{\partial x^{\prime c}}{\partial x^{d}}V^{d}\Gamma_{cb}^{\prime a}$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}V_{\;;d}^{c}$ $\displaystyle$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}\left(\frac{\partial V^{c}}{\partial x^{d}}+V^{e}\Gamma_{ed}^{c}\right)$ $\displaystyle \frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{c}}V^{c}+\frac{\partial x^{\prime c}}{\partial x^{d}}V^{d}\Gamma_{cb}^{\prime a}$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}V^{e}\Gamma_{ed}^{c}$ $\displaystyle \frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{e}}V^{e}+\frac{\partial x^{\prime c}}{\partial x^{e}}V^{e}\Gamma_{cb}^{\prime a}$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}V^{e}\Gamma_{ed}^{c}$ $\displaystyle \frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{e}}+\frac{\partial x^{\prime c}}{\partial x^{e}}\Gamma_{cb}^{\prime a}$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}\Gamma_{ed}^{c}$
In the fourth line we relabelled the dummy index on ${V^{c}}$ and ${V^{d}}$ to give all the ${V}$s the same index. In the last line, we can cancel off ${V^{e}}$ since this equation must be true for all vectors, which means the coefficients of each component ${V^{e}}$ must be equal.
To isolate ${\Gamma_{cb}^{\prime a}}$ we can multiply both sides of this equation by ${\partial x^{e}/\partial x^{\prime f}}$ and sum over ${e}$:
$\displaystyle \frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{d}}{\partial x^{\prime b}}\frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{e}}+\frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{\prime c}}{\partial x^{e}}\Gamma_{cb}^{\prime a}=\frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}\Gamma_{ed}^{c}$
Since
$\displaystyle \frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{\prime c}}{\partial x^{e}}=\delta_{f}^{c}$
we get
The second term can be compressed a little by the calculation:
$\displaystyle \delta_{b}^{a}$ $\displaystyle =$ $\displaystyle \frac{\partial x^{\prime a}}{\partial x^{d}}\frac{\partial x^{d}}{\partial x^{\prime b}}$ $\displaystyle \frac{\partial\delta_{b}^{a}}{\partial x^{\prime f}}$ $\displaystyle =$ $\displaystyle \frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{\prime f}}\frac{\partial x^{d}}{\partial x^{\prime b}}+\frac{\partial x^{\prime a}}{\partial x^{d}}\frac{\partial^{2}x^{d}}{\partial x^{\prime b}\partial x^{\prime f}}$ $\displaystyle$ $\displaystyle =$ $\displaystyle 0$
since ${\frac{\partial\delta_{b}^{a}}{\partial x^{\prime f}}=0}$, as ${\delta_{b}^{a}}$ is a constant tensor.
Thus we get
$\displaystyle \frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{\prime f}}\frac{\partial x^{d}}{\partial x^{\prime b}}$ $\displaystyle =$ $\displaystyle -\frac{\partial x^{\prime a}}{\partial x^{d}}\frac{\partial^{2}x^{d}}{\partial x^{\prime b}\partial x^{\prime f}}$ $\displaystyle \frac{\partial^{2}x^{\prime a}}{\partial x^{d}\partial x^{e}}\frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{d}}{\partial x^{\prime b}}$ $\displaystyle =$ $\displaystyle -\frac{\partial x^{\prime a}}{\partial x^{d}}\frac{\partial^{2}x^{d}}{\partial x^{\prime b}\partial x^{\prime f}}$
Substituting this into 3 we get
$\displaystyle \Gamma_{fb}^{\prime a}=\frac{\partial x^{e}}{\partial x^{\prime f}}\frac{\partial x^{\prime a}}{\partial x^{c}}\frac{\partial x^{d}}{\partial x^{\prime b}}\Gamma_{ed}^{c}+\frac{\partial x^{\prime a}}{\partial x^{d}}\frac{\partial^{2}x^{d}}{\partial x^{\prime b}\partial x^{\prime f}}$
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SAT PHYSICS SUBJECT TEST PRACTICE PAPER 1. For an object traveling in a straight line, its velocity (v, in m/s) as a function of time (t, in s) is given by the following graph. Which graph best depicts the object’s momentum? A. B. C. D. E. 2. For an object traveling in a straight line, its velocity (v, in m/s) as a function of time (t, in s) is given by the following graph. Which graph best illustrates the object’s acceleration? A. B. C. D. E. 3. For an object traveling in a straight line, its velocity (v, in m/s) as a function of time (t, in s) is given by the following graph. Which graph best depicts the object’s kinetic energy? A. B. C. D. E. 4. For an object traveling in a straight line, its velocity (v, in m/s) as a function of time (t, in s) is given by the following graph. Which graph best illustrates the object’s distance from its starting point? A. B. C. D. E. 5. Which one is NOT a vector? A. Displacement B. Velocity C. Acceleration D. Linear momentum E. Kinetic energy 6. If an object’s mass and the net force it feels are both known, then Newton’s second law could be used to directly calculate which quantity? A. Displacement B. Velocity C. Acceleration D. Linear momentum E. Kinetic energy 7. Which quantity can be expressed in the same units as impulse? A. Displacement B. Velocity C. Acceleration D. Linear momentum E. Kinetic energy 8. If an object’s speed is changing, which of the quantities could remain constant? A. Displacem ...
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https://www.teacherspayteachers.com/Product/Money-Counting-Coins-Up-to-100-Practice-Cards-Quarter-Dime-Nickel-Penny-1150067
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# Money: Counting Coins (Up to \$1.00) Practice Cards - Quarter Dime Nickel Penny
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Money: Counting Coins (Up to \$1.00) Practice Cards - Quarter Dime Nickel Penny
Let's make math fun! These practice cards were designed to help students practice counting coins (quarters, dimes, nickels, and pennies) up to \$1.00. Coin cards are included, which can be copied and sent home for extra practice!
Here is how I use these in my classroom:
I give each student a bag of coins (or a copy of the attached coin cards) and 1 practice card. As students use the coins to create the amount listed on the practice card, I circulate the room and check their answers. Students are encouraged to create the amount listed using the least amount of coins. While waiting for the teacher to check their answers, students come up with other ways to create the listed amount. Once I check a student's answer, I collect the practice card and hand them another one.
This set includes:
72 practice cards
72 practice cards (dollars and cents notation)
1 page coin cards
72 practice cards (black and white)
72 practice cards (dollars and cents notation) (black and white)
1 page coin cards (black and white)
Enjoy!
:)
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# Radius of the smallest exoplanet detectable with the transit method
by QuantumX
Tags: detectable, exoplanet, radius, smallest, transit
P: 31 Please help me with this astronomy problem. I am supposed to calculate the smallest planet that is detectable with the transit method, given a signal to noise ratio and a star's radius: Suppose the star is seen at its distance D with a signal to noise ratio of S/N = 10^4. This means that in the light curve for the star (i.e. flux vs. time), the ‘noise’ is 1/10000 the mean of the star’s flux). What is the smallest planet that is detectable? (Hint: assume that the noise has a standard deviation σ and that detection of the transit requires at least a 3σ dip in the flux.) Consider a star exactly like the Sun and evaluate your expression in kilometers by using the known radius of the Sun (5*10^5 km). Any help is appreciated!
Sci Advisor P: 2,847 Do you have any guesses on how to approach this problem? For example, what does the hint tell you?
P: 31 Well, my guess is this - I know how big the star is and the SNR tells me the resolution of the instrument, which is to say how small an object it can distinguish from the background noise - in this case a dip in the flux of 3 standard deviations of the noise. So I need to use this information to come up with the smallest radius of a planet that can be detected, but I just don't know how to do that....
Sci Advisor P: 2,847 Radius of the smallest exoplanet detectable with the transit method Ok, you're definitely going in the right direction. Do you know in what way (mathematically) the planet will reduce the flux coming from the star? The planet basically will just block star-light right? If the planet was the same size as the star, one would expect 100% of the starlight to be blocked during transit right? If the planet was half as big as the star (i.e. radius of the planet is 1/2 radius of the star), how much starlight do you think will be blocked?
P: 31 Well an area of a circle is pi*r^2, therefore the area blocked would decrease with the square of the radius. So if the planet has 1/2 the radius of the star, then it would block 1/4th of the starlight? I just don't know how to convert the S/N ratio with the 3 standard deviation sensitivity into flux blockage
Sci Advisor P: 2,847 Ah, so that's where your question is. Well, the signal to noise is simple: $$SNR=P_{signal}/P_{noise}$$ Which you have given as 10,000. This means if I block 1/10,000 of the signal, then I could attribute this to the noise. But simplistically, if I block 3/10,000 of the signal, then I can no longer attribute this to the noise. Do you know if your instructor is asking specifically about SNR issues dealing with the noise, or if they just want a rough answer? To give a full answer some statistical analysis is needed, but that might not be in the spirit of this class (unless it's a class on signal processing or some such).
P: 31 Ah, I'm starting to understand. No, it's an astronomy class, definitely not a signal processing/engineering or anything like that. However, I do need to use all of the information given. Okay, so how do I get the smallest planet detectable from a 3/10,000 flux blockage minimum. I need to know the flux of the star, don't I? How do I get the flux from the radius? Don't I also need the luminosity? The problem does say assume the star is the same as the Sun, but I really don't think I'm supposed to look up solar luminosity and use it for the problem, I'm pretty sure all I need to use is 3/10,000th and the radius 5*10^5 km.
Sci Advisor P: 2,847 You don't need to know the flux of the star, you just need to know that you have to block 3/10,000 of it. In post #5, you stated that if the radius of the planet is 1/2 the radius of the star, then you block 1/4 the flux. This is true. Now just try the statement, if the radius of the planet is 1/x the radius of the star, then you block 3/10,000 the flux of the star. What is x?
P: 31 Ah, I see. So the relationship is radius^2 ∝ area (or is it radius ∝ area^2? I get confused). So it's x^2 = 3/10,000. x = 0.0173 the area of the Sun, which is 0.0173*5*10^5 km = 8650 km (or slightly bigger than Earth). Is this correct?
Sci Advisor P: 2,847 Looks OK to me, except x is not the area, but the radius (I assume it's just a typo on your part).
P: 31 Yes, I meant radius :) Thank you so much for walking me through it! If I may ask you for a brief advice on the next part of the question: Three different flux levels can be measured (assuming the star’s luminosity does not vary): (1) when neither the star nor the planet block one another; (2) when the planet transits the star; and (3) when the star occults the planet. Sketch the three geometries, showing the line of the sight to the observer. Flux from the planet is reflected or re-radiated starlight. Assume it does not vary with orbital phase. Let F* be the flux incident on the Earth from the star and Fp that from the planet. For the three cases, write expressions for the fluxes using these quantities and, as needed, other quantities. Sketch a time sequence where you show schematically these flux levels vs. time over one orbit of the planet. In some cases the planet’s flux does vary with orbital phase because the planet is tidally locked to the star, presenting the same side to the star. When in the time sequence would you then expect the planet’s observable flux to be maximum? I think I have it all figured out except for the bolded part. I'm confused as to how I'm supposed to write expressions for the fluxes in the three cases. I'm guessing that when they don't block one another the flux is F* + Fp, when the planet transits the star the flux is F* - Fp and when the planet is behind the star the flux is just F*. Are those really "expressions" though, or do I need to write equations? Am I to extrapolate from the flux incident on the Earth to figure out the entire flux of the star-planet system? Nothing else is given, so I am a little confused, what's your take on it? Also, orbital phase is to be ignored, but for the very last part of the question - if the planet is tidally locked, I think it would be brightest when it is right on the edge, right before it goes behind the star, as then you would see half the planet illuminated. Make sense? And for the next part where I am to sketch a time sequence of the flux, I'm thinking it's a straight line when the planet is behind the star, then there's a dip when it transits and then it goes up when both are radiating. Thoughts? Sorry if I overwhelmed you, thanks for your help so far.
Sci Advisor P: 2,847 1) For the fluxes during different periods. When you can see both the star and the planet, it makes sense that the flux is F*+Fp (correct). When you can't see the planet, then the planet is entirely blocked by the star, and therefore the flux is F* only (correct). But when the planet is blocking the star, why do you say the flux is F*-Fp? We just did this problem earlier, and the answer did not have anything to do with Fp! 2) Your answer here makes sense to me. The surface of the planet pointing directly at the star will be much hotter and brighter than the other side. 3) Most of the time will not be transits (of either star blocking planet or planet blocking star). This is because the orbit of a planet is usually much larger than the radius of the star. So most of the time you should have both the planet and the star's light shining on you. That should be your "straight (horizontal) line". Can you figure out what happens when the planet blocks the star, and then when the star blocks the planet? A harder question (which cannot be answered with the variables given to you) would be which transit event gives you a deeper transit depth? Can you make a guess? Lastly, for a bonus question, the flux that we have been considering in this question has always been the integrated flux over all wavelengths. What do you think, though, is the difference between the light emitted by the star, and the light emitted by the planet? And therefore, what do you think we will see in terms of transits if we looked at a spectroscopic image (resolving all the wavelengths) instead of a total light curve?
P: 31 1) Okay, so my logic there was that when the planet is blocking the star, the flux is F* - the area of the planet, and I thought that that would be F* - the planet's flux, but now that you've pointed it out, that doesn't make much sense. So is it just F* - area of the planet? 2) Actually now that I think about it, what difference does it make whether it's tidally locked or not from our perspective? The planet's rotation has nothing to do with what percentage of it is illuminated? The only thing that would make a difference is, as you pointed out, that it would be much hotter, so we'll see more infrared radiation, but as far as visible light, it would be the same whether or not it's tidally locked, correct? 3) Okay, I think I see where you're going here. So it would be a straight line when both are radiating (ignoring orbital phase), then I'm guessing when the planet blocks the star you would see a big dip (big is relative here of course) as now you have the star's radiation - the area of the planet, and when the planet goes behind the star, the line would go up a little as now you only have the star's radiation, and then when the planet comes around again, the line would go up to its original height...? Bonus - I would definitely think the light emitted from the star would be much richer in terms of wavelengths, as in you would have a lot more thermal, x-ray, maybe radio? And the planet would be mostly visible light with some thermal, depending on orbital phase here...?
Sci Advisor P: 2,847 1) Yea, except I definitely wouldn't call it "area of the planet". I'd call it "flux blocked by the planet" which is proportional to the area (but not equal to it of course, the units wouldn't work that way). 2) Most of the light would be reflected light. So really, the light curve should be slightly different depending on orbital phase, all the time (because we'll see phases of the planet, just like phases of the moon, or phases of venus), but this is presumably a small effect. 3) Straight line during both radiating. When the planet is blocking the star, you get a large dip as you said (correct), but why do you think the flux goes up when the planet is behind the star (is that just a typo)? Bonus: The star emits more light at all wavelengths than the planet, of course (it's much much brighter after all). But the planet's emitted light would be mostly in the infrared. This means when you have the star blocking the planet, you see a dip mostly in the infrared portion of the light curve. From the nature of this dip, you can actually derive the temperature of the planet!
P: 1,857 Looks like Matterwave has helped enough, I didn't want to add interference, or confusion. So I held off posting a couple of useful papers. These don't apply directly to your question however related. Figured they may help in your studies. Though both will help relate to your questions. I particularly like this one for the visual correlations. transit detection algorithms http://www.tls-tautenburg.de/researc...ctures/TDA.pdf TRANSITS OF EARTH-LIKE PLANETS http://arxiv.org/ftp/arxiv/papers/0903/0903.3371.pdf
P: 31 1) I see. Is there a way to express the scenario when the planet is in front of the star in terms of F* and Fp? 2) Agreed. 3) Well, I think you have a straight line when both are radiating, and then a large dip means the planet is now not radiating as it is in front of the star AND the star's flux is blocked, and then you go back to a straight line when the 2 are radiating again, then when the planet goes behind the star, only the star is radiating, so you have a smaller dip, and then when both are radiating again, you go back to a straight line. Does that not make sense?
Sci Advisor P: 2,847 1) No, not in general. The only way to do it would be if you knew the temperature of the planet, and obtained the surface area via Fp. But that's really just a round-about way. The essential information you need is the radius of the planet. 3) That's fine. You had a typo in your previous post where you said the light curve goes UP when the planet is behind the star, when in fact it goes down.
P: 31 Okay, well thank you so much for all your help, I really appreciate you taking the time!
Related Discussions Precalculus Mathematics Homework 4 Classical Physics 3 Astronomy & Astrophysics 12 Biology, Chemistry & Other Homework 9 Introductory Physics Homework 12
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# Your intuitions are not magic
post by Kaj_Sotala · 2010-06-10T00:11:30.121Z · LW · GW · Legacy · 36 comments
People who know a little bit of statistics - enough to use statistical techniques, not enough to understand why or how they work - often end up horribly misusing them. Statistical tests are complicated mathematical techniques, and to work, they tend to make numerous assumptions. The problem is that if those assumptions are not valid, most statistical tests do not cleanly fail and produce obviously false results. Neither do they require you to carry out impossible mathematical operations, like dividing by zero. Instead, they simply produce results that do not tell you what you think they tell you. As a formal system, pure math exists only inside our heads. We can try to apply it to the real world, but if we are misapplying it, nothing in the system itself will tell us that we're making a mistake.
Examples of misapplied statistics have been discussed here before. Cyan discussed a "test" that could only produce one outcome. PhilGoetz critiqued a statistical method which implicitly assumed that taking a healthy dose of vitamins had a comparable effect as taking a toxic dose.
Even a very simple statistical technique, like taking the correlation between two variables, might be misleading if you forget about the assumptions it's making. When someone says "correlation", they are most commonly talking about Pearson's correlation coefficient, which seeks to gauge whether there's a linear relationship between two variables. In other words, if X increases, does Y also tend to increase. (Or decrease.) However, like with vitamin dosages and their effects on health, two variables might have a non-linear relationship. Increasing X might increase Y up to a certain point, after which increasing X would decrease Y. Simply calculating Pearson's correlation on two such variables might cause someone to get a low correlation, and therefore conclude that there's no relationship or there's only a weak relationship between the two. (See also Anscombe's quartet.)
The lesson here, then, is that not understanding how your analytical tools work will get you incorrect results when you try to analyze something. A person who doesn't stop to consider the assumptions of the techniques she's using is, in effect, thinking that her techniques are magical. No matter how she might use them, they will always produce the right results. Of course, assuming that makes about as much sense as assuming that your hammer is magical and can be used to repair anything. Even if you had a broken window, you could fix that by hitting it with your magic hammer. But I'm not only talking about statistics here, for the same principle can be applied in a more general manner.
Every moment in our lives, we are trying to make estimates of the way the world works. Of what causal relationships there are, of what ways of describing the world make sense and which ones don't, which plans will work and which ones will fail. In order to make those estimates, we need to draw on a vast amount of information our brains have gathered throughout our lives. Our brains keep track of countless pieces of information that we will not usually even think about. Few people will explicitly keep track of the amount of different restaurants they've seen. Yet in general, if people are asked about the relative number of restaurants in various fast-food chains, their estimates generally bear a close relation to the truth.
But like explicit statistical techniques, the brain makes numerous assumptions when building its models of the world. Newspapers are selective in their reporting of disasters, focusing on rare shocking ones above common mundane ones. Yet our brains assume that we hear about all those disasters because we've personally witnessed them, and that the distribution of disasters in the newspapers therefore reflects the distribution of disasters in the real world. Thus, people asked to estimate the frequency of different causes of death underestimate the frequency of those that are underreported in the media, and overestimate the ones that are overreported.
On this site, we've also discussed a variety of other ways by which the brain's reasoning sometimes goes wrong: the absurdity heuristic, the affect heuristic, the affective death spiral, the availability heuristic, the conjunction fallacy... the list goes on and on.
So what happens when you've read too many newspaper articles and then naively wonder about how frequent different disasters are? You are querying your unconscious processes about a certain kind of statistical relationship, and you get an answer back. But like the person who was naively misapplying her statistical tools, the process which generates the answers is a black box to you. You do not know how or why it works. If you would, you could tell when its results were reliable, when they needed to be explicitly corrected for, and when they were flat-out wrong.
Sometimes we rely on our intuitions even when they are being directly contradicted by math and science. The science seems absurd and unintuitive; our intuitions seem firm and clear. And indeed, sometimes there's a flaw in the science, and we are right to trust our intuitions. But on other occasions, our intuitions are wrong. Yet we frequently persist in holding onto our intuitions. And what is ironic is that we persist on holding onto them exactly because we do not know how they work, because we cannot see their insides and all the things inside them that could go wrong. We only get the feeling of certainty, a knowledge of this being right, and that feeling cannot be broken into parts that could be subjected to criticism to see if they add up.
But like statistical techniques in general, our intuitions are not magic. Hitting a broken window with a hammer will not fix the window, no matter how reliable the hammer. It would certainly be easy and convenient if our intuitions always gave us the right results, just like it would be easy and convenient if our statistical techniques always gave us the right results. Yet carelessness can cost lives. Misapplying a statistical technique when evaluating the safety of a new drug might kill people or cause them to spend money on a useless treatment. Blindly following our intuitions can cause our careers, relationships or lives to crash and burn, because we did not think of the possibility that we might be wrong.
That is why we need to study the cognitive sciences, figure out the way our intuitions work and how we might correct for mistakes. Above all, we need to learn to always question the workings of our minds, for we need to understand that they are not magical.
comment by GreenRoot · 2010-06-10T16:46:20.551Z · LW(p) · GW(p)
Thanks for the well-written article. I enjoyed the analogy between statistical tools and intuition. I'm used to questioning the former, but more often than not I still trust my intuition, though now that you point it out, I'm not sure why.
Replies from: xv15, Jayson_Virissimo
comment by xv15 · 2010-06-11T10:42:15.443Z · LW(p) · GW(p)
You shouldn't take this post as a dismissal of intuition, just a reminder that intution is not magically reliable. Generally, intuition is a way of saying, "I sense similarities between this problem and other ones I have worked on. Before I work on this problem, I have some expectation about the answer." And often your expectation will be right, so it's not something to throw away. You just need to have the right degree of confidence in it.
Often one has worked through the argument before and remembers the conclusion but not the actual steps taken. In this case it is valid to use the memory of the result even though your thought process is a sort of black box at the time you apply it. "Intuition" is sometimes used to describe the inferences we draw from these sorts of memories; for example, people will say, "These problems will really build up your intuition for how mathematical structure X behaves." Even if you cannot immediately verbalize the reason you think something, it doesn't mean you are stupid to place confidence in your intuitions. How much confidence depends on how frequently you tend to be right after actually trying to prove your claim in whatever area you are concerned with.
comment by Jayson_Virissimo · 2010-06-10T21:02:25.625Z · LW(p) · GW(p)
I do know why I trust my intuitions as much as I do. My intuitions are partly the result of natural selection and so I can expect that they can be trusted for the purposes of surviving and reproducing. In domains that closely resemble the environment where this selection process took place I trust my intuition more, in domains that do not resemble that environment I trust my intuition less.
Black box or not, the fact that we are here is good evidence that they (our intuitions) work (on net).
Replies from: diegocaleiro, therufs, tommyjohn
comment by diegocaleiro · 2010-06-12T08:18:54.954Z · LW(p) · GW(p)
How sexy is that?
If you are evaluating intuitions, there are two variables you should account for. The similarity with evolutionary environment, indeed. AND your current posterior belief of the importance of this kind of act in the variance of offspring production.
We definitely evolved in an environment full of ants. Does that mean my understanding of ant-colony intelligence is intuitive?
comment by therufs · 2012-09-29T17:22:40.120Z · LW(p) · GW(p)
I'm very curious how you decide what constitutes a similar environment to that of natural selection, and what sorts of decisions your intuition helps make in such an environment.
comment by tommyjohn · 2011-11-18T22:27:32.188Z · LW(p) · GW(p)
So then anything that has evolved may be relied upon for survival? It is impossible to rationalize faith in an irrational cognitive process. In the book Blink, the author asserts that many instances of intuition are just extremely rapid rational thoughts, possibly at a sub-conscious level.
comment by fool_hill · 2010-06-10T18:20:11.033Z · LW(p) · GW(p)
i don't know why we prefer to hold on to our intuitions. your claim, that " we persist on holding onto them exactly because we do not know how they work" has not been proven, as far as I can tell, and seems unlikely. I also don't know why our own results seem sharper than what we learn from the outside [although about this later point, i bet there's some story about lack of trust in homo hypocritus societies or something] .
As somebody who fits into the "new to the site" category, I enjoyed your article.
Replies from: RobinZ, JDM
comment by RobinZ · 2010-06-10T19:15:23.729Z · LW(p) · GW(p)
Welcome to Less Wrong! Feel free to post an explicit introduction on that thread, if you're hanging around.
I think the critical point is in the next sentence:
We only get the feeling of certainty, a knowledge of this being right, and that feeling cannot be broken into parts that could be subjected to criticism to see if they add up.
Yes, we don't know what the interiors are - but the original source of our confidence is our (frequently justified) trust in our intuitions. I think another related point is made in How An Algorithm Feels From Inside, which talks about an experience which is illusory, merely reflecting an artifact of the way the brain processes data. The brain usually doesn't bother flagging a result as a result, it just marks it as true and charges forward. And as a consequence we don't observe that we are generalizing from the pattern of news stories we watched, and therefore don't realize our generalization may be wrong.
comment by JDM · 2012-11-05T13:18:55.680Z · LW(p) · GW(p)
I think it's a combination of not understanding the process with a lifetime of experience where's it's far more right than wrong (Even for younger people, if they have 10-15 years of instinctive behavior being rewarded on some level, it's hard to accept there are situations it doesn't work as well). Combine that with the tendency of positive outcomes to be more memorable than others, and it's not too difficult to understand why people trust their intuition as much as they do.
your claim, that " we persist on holding onto them exactly because we do not know how they work" has not been proven, as far as I can tell, and seems unlikely.
It may not be the only reason, but an accurate understanding of how intuitions work would make it easier to rely less on it in situations it's not as we'll equipped for, just as an understanding of different biases makes it easier to fight them in our own thought processes.
comment by Wei_Dai · 2010-06-11T14:45:47.444Z · LW(p) · GW(p)
Intuition seems to be one of the least studied areas of cognitive science, at least until very recently. The Wikipedia entry on cognitive sciences that the post links to has no mention of "intuition", and one paper I found said that the 1999 MIT Encyclopedia of Cognitive Sciences doesn't even have a single index entry for it (while "logic" has almost 100 references).
After a bit more searching, I found a 2007 book titled Intuition in Judgment and Decision Making, which apparently represents the current state of the art in understanding the nature of intuition.
comment by Louie · 2010-06-11T00:59:14.054Z · LW(p) · GW(p)
Having just pressed "Send" on an email that estimates statistics based on my intuitions, this feels particularly salient to me.
Really well written. Great work Kaj.
Thanks for reminding me that my thoughts aren't magic.
comment by Douglas_Knight · 2010-06-10T22:40:08.507Z · LW(p) · GW(p)
People who know a little bit of statistics - enough to use statistical techniques, not enough to understand why or how they work - often end up horribly misusing them.
How often do people harm themselves with statistics, rather than further their goals through deception? Scientists data-mining get publications; financiers get commissions; reporters get readers.
ETA: the people who are fooled are harming themselves with statistics. But I think the people want to understand for themselves generally only use statistics that they understand.
Replies from: SilasBarta, Dre
comment by SilasBarta · 2010-06-10T22:44:02.780Z · LW(p) · GW(p)
True, but many of those scientists and reporters really do want to unravel the actual truth, even if it means less material wealth or social status. These people would enjoy being corrected.
comment by Dre · 2010-06-11T01:25:21.908Z · LW(p) · GW(p)
There is also an opportunity cost to the poor use of statistics instead of proper use. This may be only externalities (the person doing the test may actually benefit more from deception), but overall the world would be better if all statistics were used correctly.
comment by RobinZ · 2010-06-10T00:43:20.355Z · LW(p) · GW(p)
Elegantly done - clear and informative.
comment by Franco Vairoletti (franco-vairoletti) · 2019-08-05T19:05:20.517Z · LW(p) · GW(p)
Hi! I'm new here and I'd like to thank for the site and for this instructive article in particular. I'm quite convinced that the overconfidence in our own intuition, even without knowing its underlying mechanisms, is one of the main obstacles to rational thinking. Maybe this blackbox working of our intuition also plays a role in communication of our ideas to others. How we can change someone's opinions if we don't know how they got them in first place?
Thanks again and congrats for your work!
comment by michael_b · 2015-01-29T13:02:15.917Z · LW(p) · GW(p)
The immediately available example supporting your article for me is the relationship between dietary cholesterol and blood cholesterol. There's high general confusion around this health claim.
What no doubt compounds the confusion on the issue is that intuitively you might infer that eating zero cholesterol should lower blood cholesterol, or that eating high cholesterol should raise blood cholesterol. Evidence shows this often happens, but not always. There are enough notable outliers that the claim has been defeated in the general mind because it doesn't support the intuitive story.
That is, vegans who eat almost no cholesterol containing foods, can have high blood cholesterol. On the flip side, surely everyone has heard of that friend of a friend who eats inf eggs a day and has low blood cholesterol.
There's a reasonably interesting story that fits the evidence for the claim that if dietary cholesterol then blood cholesterol, but the nonlinearity of the relationship and also the incidence of intuition defeating cases cloud the issue.
comment by akshatrathi · 2010-08-13T01:50:57.295Z · LW(p) · GW(p)
I enjoyed your article and as a scientist, I've been interested to understand this: what seems an intuitive method to use to solve a scientific problem is not seen as an intuitive method while solving 'other' problems.
By 'other', I mean things like psychological problems or problems that arise from conflicts amongst people. It may be obvious why it is not 'intuitive' but what goes beyond my understanding is most will not even consider using the scientific method for the latter types of problem ever.
comment by Yuval Maharshak · 2021-02-11T17:44:31.424Z · LW(p) · GW(p)
Your first idea is always in doubt seems like a nice law. But then I think is it a nice law cause my first idea was that it was nice.
comment by Данило Глинський (danilo-glinskii) · 2019-09-17T20:21:04.095Z · LW(p) · GW(p)
Yet in general, if people are asked about the relative number of restaurants in various fast-food chains, their estimates generally bear a close relation to the truth.
comment by toonalfrink · 2017-01-19T14:17:37.635Z · LW(p) · GW(p)
Indeed, intuitions are fallible. Though beware of the other extreme: writing off your intuitions altogether and trying to live solely based on logic. I've seen various people in the LW sphere try this, and it doesn't quite work. In some cases, like nutrition or social life, there is a bottomless pit of complexity. Trying to provably 'solve' such problems will lead to a bottomless pit of thinking, stagnation, and depression.
Logic is not a magic hammer either.
comment by AFinerGrain_duplicate0.4555006182262571 · 2017-10-02T23:52:39.505Z · LW(p) · GW(p)
I originally learned about these ideas from Thinking Fast and Slow, but I love hearing them rephrased and repeated again and again. Thinking clearly often means getting in the cognitive habit of questioning every knee-jerk intuition.
On the other hand, coming from a Bryan Caplan / Michael Huemer perspective, aren't we kind of stuck with some set of base intuitions? Intuitions like; I exist, the universe exists, other people exist, effects have causes, I'm not replaced by a new person with memory implants every time I go to sleep...
You might even call these base intuitions, "magic," in the sense that you have to have faith in them in order to do anything like rationality.
Replies from: TheAncientGeek
comment by TheAncientGeek · 2017-10-03T11:37:19.564Z · LW(p) · GW(p)
Well, we don't know if they work magically, because we don't know that they work at all. They are just unavoidable.
It's not that philosophers weirdly and unreasonably prefer intuition to empirical facts and mathematical/logical reasoning, it is that they have reasoned that they can't do without them: that (the whole history of) empiricism and maths as foundations themselves rest on no further foundation except their intuitive appeal. That is the essence of the Inconvenient Ineradicability of Intuition. An unfounded foundation is what philosophers mean by "intuition". Philosophers talk about intution a lot because that is where arguments and trains of thought ground out...it is away of cutting to the chase. Most arguers and arguments are able to work out the consequences of basic intutitions correctly, so disagrements are likely to arise form differencs in basic intuitions themselves.
Philosophers therefore appeal to intuitions because they can't see how to avoid them...whatever a line of thought grounds out in, is definitiionally an intuition. It is not a case of using inutioins when there are better alternatives, epistemologically speaking. And the critics of their use of intuitions tend to be people who haven't seen the problem of unfounded foundations because they have never thought deeply enough, not people who have solved the problem of finding sub-foundations for your foundational assumptions.
Scientists are typically taught that the basic principles maths, logic and empiricism are their foundations, and take that uncritically, without digging deeper. Empircism is presented as a black bx that produces the goods...somehow. Their subculture encourages use of basic principles to move forward, not a turn backwards to critically relflect on the validity of basic principles. That does not mean the foundational principles are not "there". Considering the foundational principles of science is a major part of philosophy of science, and philosophy of science is a philosophy-like enterprise, not a science-like enterprise, in the sense it consists of problems that have been open for a long time, and which do not have straightforward empirical solutions.
Does the use of empiricism shortcut the need for intuitions, in the sense of unfounded foundations?
For one thing, epistemology in general needs foundational assumptions as much as anything else. Which is to say that epistemogy needs epistemology as much as anything else. -- to judge the validity of one system of epistemology, you need another one. There is no way of judging an epistemology starting from zero, from a complete blank. Since epistemology is inescapable, and since every epistemology has its basic assumptions, there are basic assumptions involved in empiricism.
Empiricism specifically has the problem of needing an ontological foundation. Philosophy illustrates this point with sceptical scenarios about how you are being systematically deceived by an evil genie. Scientific thinkers have closely parallel scenarios in which humans cannot be sure whether you are not in the Matrix or some other virtual reality. Either way, these hypotheses illustrate the point that the empiricists are running on an assumption that if you can see something, it is there.
comment by [deleted] · 2015-02-10T01:10:26.886Z · LW(p) · GW(p)
I think this article doesn't quite appreciate the full role intuitions play in science. It seems to me that intuitions help shape science in large ways. For instance, our intuitions that 'deduction works' and 'induction works' seems to stop all of us from turning into Cartesian sceptics, and preventing any science. Intuitions (and philosophical arguments) about metaphysics shape the basis of acceptable hypotheses within physics. Intuitions about what makes a scientific theory good/explanatory/falsified shape how science proceeds. Intuitions also serve to define concepts we have. If I remember correctly, in the Newtonian era, mass was not analysed in terms of anything else. It was a primitive concept in Newton's physics, and it was defined intuitively. Nowadays, modern physics has analysed concepts in terms of more and increasingly obscure concepts; but nevertheless, there is always a limit to what has been analysed in terms of what, and what remains is held, insofar as we know of it, as known primitively. That is, known intuitively.
I also have a question: Does this site in general take a negative view of heuristics humans have? I've seen various pages complaining about heuristics humans have, and not much about how helpful they are in keeping us all functioning.
comment by MikeDobbs · 2013-03-25T13:28:08.270Z · LW(p) · GW(p)
This was an excellent read- I particularly enjoyed the comparison drawn between our intuition and other potentially "black box" operations such as statistical analysis. As a mathematics teacher (and recreational mathematician) I am constantly faced with, and amused by, the various ways in which my intuition can fail me when faced with a particular problem.
A wonderful example of the general failure of intuition can be seen in the classic "Monty Hall Problem." In the old TV game show Monty Hall would offer the contestant their choice of one of three doors. One door would have a large amount of cash, the other two a non-prize such as a goat. Here's where it got interesting. After the contestant makes their choice, Monty opens one of the "loosing" doors, leaving only two closed (one of which contains the prize), then offers the contestant he opportunity to switch from their original door to the other remaining door.
The question is, should they switch? Does it even matter? For most people (myself included) our intuition tells us it doesn't matter. There are two doors, so there's a 50/50 chance of winning whether you switch or not. However a quick analysis of the probabilities involved shows us that they are in fact TWICE as likely to win the prize if they switch than if they stay with their original choice.
That's a big difference- and a very counterintuitive result when first encountered (at least in my opinion)
Replies from: TheOtherDave
comment by TheOtherDave · 2013-03-25T15:39:49.838Z · LW(p) · GW(p)
I was first introduced to this problem by a friend who had received as a classroom assignment "Find someone unfamiliar with the Monty Hall problem and convince them of the right answer."
The friend in question was absolutely the sort of person who would think it was fun to convince me of a false result by means of plausible-sounding flawed arguments, so I was a very hard sell... I ended up digging my heels in on a weird position roughly akin to "well, OK, maybe the probability of winning isn't the same if I switch, but that's just because we're doing something weird with how we calculate probabilities... in the real world I wouldn't actually win more often by switching, cuz that's absurd."
Ultimately, we pulled out a deck of cards and ran simulated trials for a while, but we got interrupted before N got large enough to convince me.
So, yeah: counterintuitive.
Replies from: None
comment by [deleted] · 2013-03-25T16:40:32.395Z · LW(p) · GW(p)
I remember how my roommates and I drew a game tree for the Monty Hall problem, assigned probabilities to outcomes, and lo, it was convincing.
Replies from: TheOtherDave
comment by TheOtherDave · 2013-03-25T16:59:38.385Z · LW(p) · GW(p)
(nods)
It continues to embarrass me that ultimately I was only "convinced" that the calculated answer really was right, and not some kind of plausible-sounding sleight-of-hand, when I confirmed that it was commonly believed by the right people.
Replies from: MikeDobbs
comment by MikeDobbs · 2013-03-25T17:29:23.233Z · LW(p) · GW(p)
One of my favorites for exactly that reason- if you don't mind, let me take a stab at convincing you absent "the right people agreeing."
The trick is that once Monty removes one door from the contest you are left with a binary decision. Now to understand why the probability differs from our "gut" feeling of 50/50 you must notice that switching amounts to winning IF your original choice was wrong, and loosing IF your original choice was correct (of course staying with your original choice results in winning if you were right and loosing if you were wrong).
So, consider the probability that you original guess was correct. Clearly this is 1/3. That means the probability of your original choice being incorrect is 2/3. And there's the rub. If you will initially guess the wrong door 2/3 of the time, then that means that when you are faced with the option to switch doors you're original choice will be wrong 2/3 of the time, and switching would result in you switching to the correct door. Only 1/3 of the time will your original choice be correct, making switching a loosing strategy.
It becomes more clear if you begin with 10 doors. In this modified Monty Hall problem, you pick a door, then Monty opens 8 doors, leaving only your original choice and on other (one of which contains the prize money). In this case your original choice will be incorrect 9/10 times, which means when faced with the option to switch, switching will result in a win 9/10 times, as opposed to staying with your original choice, which will result in a win only 1/9 times.
Replies from: TheOtherDave
comment by TheOtherDave · 2013-03-25T18:07:25.620Z · LW(p) · GW(p)
(nods) Yah, I'm familiar with the argument. And like a lot of plausible-sounding-but-false arguments, it sounds reasonable enough each step of the way until the absurd conclusion, which I then want to reject. :-)
Not that I actually doubt the conclusion, you understand.
Of course, I've no doubt that with sufficient repeated exposure this particular problem will start to seem intuitive. I'm not sure how valuable that is.
Mostly, I think that the right response to this sort of counterintuitivity is to get seriously clear in my head the relationship between justified confidence and observed frequency. Which I've never taken the time to do.
comment by JamesCole · 2010-06-11T07:07:17.933Z · LW(p) · GW(p)
Yet our brains assume that we hear about all those disasters [we read about in the newspaper] because we've personally witnessed them, and that the distribution of disasters in the newspapers therefore reflects the distribution of disasters in the real world.
Even if we had personally witnessed them, that wouldn't, in itself, be any reason to assume that they are representative of things in general. The representativeness of any data is always something that can be critically assessed.
Replies from: Kutta
comment by Kutta · 2010-06-11T09:25:53.020Z · LW(p) · GW(p)
For many people, representativeness is the primary governing factor in any data analysis, not just a mere facet of reasoning that should be critically assessed. Also, aside from the mentioned media bias that is indeed relatively easily correctable, there are many subtler instances of biasing via representativess, on the level of cognitive processes.
comment by nazgulnarsil · 2010-06-11T02:23:38.490Z · LW(p) · GW(p)
"However, like with vitamin dosages and their effects on health, two variables might have a non-linear relationship."
if we limit our interval we can make a linear approximation within that interval. this is often good enough if we don't much care about data outside that interval. the easy pitfall of course is people wanting to extend the linearization beyond the bounds of the interval.
Replies from: Nanani
comment by Nanani · 2010-06-16T03:10:25.507Z · LW(p) · GW(p)
Voted down because tangential replies that belong elsewhere really get on my nerves. Please comment on the post about the vitamin study, linked in the OP.
Replies from: nazgulnarsil
comment by nazgulnarsil · 2010-06-17T16:04:15.821Z · LW(p) · GW(p)
0_o I was responding directly to the OP.
comment by no_titman · 2014-04-16T03:12:49.576Z · LW(p) · GW(p)
Yudkowsky's intuitions are always right and thus magic.
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# Column Number Of Last Column In A Row Where Cell Contents Greater Than 0
Jun 24, 2014
I have a spreadsheet with rows of data. I need a formula that will return the column number of the last column in a row where there is a value >0.
Let's say that cells A1:F1 contain values. Some have values of 0 while others have values >0. I need a formula in cell G1 that will tell me the column number of the last value >0.
A B C D E F
0 2 5 0 6 0
So the formula in G1 would return a value of 5, which corresponds to column E.
## Highlight Dates In Column H If They Are Greater Than Column G By Certain Number
Sep 5, 2013
I need formatting to highlight the dates in Column H if they are a greater than a week or more from Column G. Tried some different ways of doing this with the conditional formatting but cannot get it to work yet
## Getting Sum Of Cells B-F To Post In Column G Unless Column J Has Number Greater Than Zero
May 9, 2014
In my spreadsheet, I have a column for credit card totals (G) and cash totals (J)
I am adding the amounts for Food (Column B), Liquor (Column C), Wine (Column D), Beer (Column E), and Taxes ( Column F) for a total in Column G.
However, Column G is only for credit card totals and I want to do the same calculations for cash totals (Column J) using the same B through F Column without those numbers being plugged back into Column G, when I have no credit card total.
## Getting A Minimum Number Other/greater Than Zero From A Column
Aug 25, 2009
What is the formula for getting a minimum number other/greater than zero from a column of added numbers?
## How To Convert Column Contents To Number
Jan 4, 2014
Im trying to convert the data in certain columns to number. I need to select the rows in those columns based on the rows counted in another column with a different heading, this is because there can be breaks in the data half way down the columns.
The Code I have so Far is:
Code:
Sub ConvertTonumber()
'Convert Certain Columns to numbers
'Use the "x"column to Calculate how many rows are required to fill the columns.
Dim ColX As Range
[Code]....
## MINIF Function; Find The Minimum Value Of Column A If There Is Anything Greater Than Zero In Column B
Jul 29, 2006
I want to find the minimum value of column A if there is anything greater than zero in column B. I tried this formula (simplified): MINIF(B2:B6,">"&0,A2:A6) And it gives me the # NAME? error
## Copy Row When Cell In Column Has Value Greater Than 0
Feb 24, 2012
I'm creating a POS system for my wife's business. The workbook has three sheets.
On the first sheet are all the articles that her shop sells.
On A1 to D1 is the following text:
Quantity Discription Price Totalprice
B1 to D1 is already filled in. When a customer buys something, a employee only has to fill in A1, how many of what item the customer buys.
So it is possible that only 5 items have a number filled in, in A1.
Now, i have a vba, which has to search all the items which have filled in quantity and then copy row A to D to a sheet called Receipt. (Later I want to print this receipt)
Now this vba work when all the filled in cells are above each other. When there is a gap between the cells, it only copies the first row.
Here is the VBA:
Sub test()
Dim r As Range, c As Range
With Worksheets("Kassa")
Set r = Range(.Range("A1"), .Range("A1").End(xlDown))
For Each c In r
If WorksheetFunction.IsNumber(c) Then
Range(.Cells(c.Row, "A"), .Cells(c.Row, "D")).Copy
[Code] ........
When I change the second A1 in the fourth line to A200. I works, but is extremely slow. It's almost like if Excel is responding anymore. Also, when no quantities are filled in, it's also slow.
## Color Cells In Column Greater Than Corresponding Cell
Dec 8, 2006
I have a list of numbers in two columns. All I want to do is that if the number in column B is larger than the number in column A I want it to be in red font. I know I need to use the Options>View - Zero Values.", "style="background: ...
## Extract Contents Of Cell If Greater Than Zero To A Comment
Dec 30, 2009
I am trying to extract contents of cell to a comment.
Column K is Overtime Hours
Column L is Regular Hours
I have managed to create the following macro that will copy the contents of a cell in Column K and put it as a comment in Column L. But I am needing the macro only to create a comment if the value of the cell in Column K is greater than 0
## Copy Cell Range If Column Cell Greater Than 0
Feb 6, 2014
I’m trying to write a macro to look down two different columns in my work sheet and if the is a value >0 then copy a set range of cells to another sheet,
I need to search column (k) and (x) range (“K2 : K147”) and (“X2 : X147”) in (sheet1) for values >0 if there is, then copy from (“f : m”) if it is found in the (k) column, or copy from (“s : z”) if it is found in the (X) column, and paste value only to the next empty cell in sheet2 . Sheet2 is empty so (A) on down is fine for paste range. There is a spin button in the copy range column (L) and I don’t want that picked up in the copy. Manually I (paste text) only but I think (values only) will do the same thing.
[Code] .........
## Compare Column Number Of Cell With Last Cell Column Number In Loop
Jan 11, 2014
I am trying to save an excel sheet to .csv format with the following macro:
[Code] ......
The following part of the code needs to be modified so that the commas appear correctly in .csv file even for null values for any given column.
[Code] .....
I have attached the Sample Sheets. Source sheet and the Result sheet. The Result sheet doesn't provide the required result. Some data are missing and unwanted commas have been added. How to correct the logic in the above piece of code.
Attached File :
Result Sheet.csv‎
Source.xlsm‎
## Every N-th Cell Value From A Column And Create Another Column Consisting Of Every 4th Number
Nov 5, 2008
I was wondering, is there a function that would take, lets say, every 4th cell value from a column and create another column consisting of every 4th number ?
## Clear Contents Of Column Based On Data In Another Column?
Apr 10, 2014
I need to clear the contents of a columns G and H starting 11th row based on what is there in F column. The macro should check for last non-empty cell in column F starting F11 (assume it finds F30 to be last non-empty cell), then it should clear the contents of G11 to G30, H11 to H30.
## Color Fonts In Column G Based On Contents In Column E
Jan 21, 2009
I have created an Excel spreadsheet teachers schedule for a small school with 8 teachers. I have assigned a number to each teacher (1 - 8) so that a number typed in a cell in Column E will cause a teachers name to appear in a cell in Column G. The ranges are E3:E20 and G3:G20. I hope to find a Macro that will display each teachers name in a different color.
## Macro To Find End Of Column And Paste Contents In Column 1
Mar 11, 2009
I have the file here i work with, basically the first column is a legend and the column to the right of it is a pointer column to help me find out where a legend is located in another file. So i was wondering if a macro could be made to basically find where the "legend column" A, C, E ect ends ( every other column is a legend column , one next to it is a pointer column ). and then combine the ends all of the columns contents and put them into 1 column.
In the file with this question i have showed you what i start off with, i highlighed in yellow where each column legend ends, ( normally these are not highlighted and i find them manually ). In the 2nd tab i show what the end result should be. All the columns are now consolidated into 1 column. 1 after the other.
## Auto Fill Column B Based Contents Of Column A
Apr 19, 2007
I have a list that is streets and addresses. All contained in column A. Cell A1 is the street name and then Column A3 is the street number. This repeats down column A for almost 1000 street names. I need to fill column B3 with the street name, as well as B4, B5, B6, etc until the street name changes. I was trying to do this with an if..then but couldn't get it to work. I also tried to work on a do.. loop looking for the change from a string to number. But my programming is a little rusty. If anyone can help I was be forever grateful. I mean the alternative is to sit here and copy and paste all day.
ABBEY CT
1700
1700
1703
1703
ABERDEEN LN
1305
1313
1321
1321
## How To Return Contents Of Last Non-blank Cell In Column
May 7, 2014
I am trying to figure out how to return the contents of last non blank cell in column B, based on the name in column A. So, if I choose "Sam", the result I am looking for is "blue". If I choose "Pete", the result I am looking for is "orange".
Sam
red
Sam
blue
Sam
Pete
orange
Sam
## Convert Column Number Into Another Column Cell?
Dec 27, 2013
A column:
18
42
55
11
65
72
80
48
..
I want to put/write to B column the some numbers (42,11,72,48..) in A column.
B column:
=A2
=A4
=A6
=A8
..
How can I achieve it?
## Find Highest Value In Column And Then Copy Contents Of Another Cell In That Row?
Aug 18, 2014
I have a worksheet where columns C, F, I, L, O record scores within a league. Each row records a persons score in that league and there are two rows per person recording their score and their handicap score. So Person A would be on rows 3, 4, Person B on 5, 6 etc. The persons name is recorded in Column B.
What I would like to do is to have a cell(s) elsewhere in the worksheet which show the highest score in that league and display that score and the name of the person who achieved it. This ideally would need to be done for the highest score and the highest handicap score.
For the life of me I can't even begin to work out how to do that or even know if it is possible in Excel.
So to clarify, lets say the highest score is in cell L7 and the highest handicap score was in M3. The cell(s) containing this formula should then show the name in B7 and the score in L7 and below it the name in B3 and the score in M3.
## Copying Cell Contents From One Column To Multiple Rows
Mar 20, 2014
I'm sure there is an easier way than copying and pasting each individual cell from one spreadsheet to another. Is there a way I can define a batch of cells (city, state, phone #) and copy them into the other spreadsheet so I don't have to do each cell individually?
Here is a picture to show what exactly I am trying to do.
SS.PNG
Also, the cells that belong in the same column and row on one spreadsheet are equal distance from each other throughout the other spreadsheet that has all the info in 1 column.
## INDIRECT - Sum Contents Of Column On Another Worksheet Up To Certain Cell Reference
Jan 24, 2012
Trying to use INDIRECT to sum the contents of a column on another worksheet upto a certain cell reference which is in another cell on the worksheet.
=SUM(INDIRECT("Sheet1!A4:Sheet1!"&B1))
I have taken it back to simply having sheet1 with numbers in A4 to A23, then sheet2 with A23 in cell B1, and the formula above it C9, but I keep getting #REF!.
The formula works fine if on sheet1 without the worksheet names in it. Formula evaluation gets to =SUM(INDIRECT("Sheet1!A4:Sheet1!A23")) then gives =SUM(#REF!)
## VBA To Loop Column And Copy Cells Contents To Another Cell
Dec 17, 2013
I have a script that copies data to files based on many cells contents but where I am having a problem is creating series numbers for each file.
File-01.txt
File-02.txt
..
File-100.txt
In my current code I copy files to folders by date and each folder I need series of files (Lab Testing series)
In column A1:A100 I have a series of numbers 01, 02, 03 ...100
Column B contains the Files to be saved
C:LAB2012Jan1file-01.txt
C:LAB2012Jan2file-01.txt
C:LAB2012Jan3file-01.txt
C:LAB2012Jan4file-01.txt
...
..
C:LAB2012Dec31file-01.txt
So this works fine
I now need the Script to do is to loop to Column A and select Cell 2 and do the File Copy again on the Next series
C:LAB2012Jan1file-02.txt
C:LAB2012Jan2file-02.txt
C:LAB2012Jan3file-02.txt
C:LAB2012Jan4file-02.txt
...
..
C:LAB2012Dec31file-02.txt
When Complete repeat until it reaches the end of column A
Since my Cells are populated by all the data in the workbook I thought at the end of my copy script I would take the next Cells data in A and put it in Cell H8 where all the constants are for the file names.
Column B is built using
=IF(\$C1="","",\$I\$1&\$G\$4&\$D1&""&\$I\$8&\$H\$8)
=IF(\$C2="","",\$I\$1&\$G\$4&\$D2&""&\$I\$8&\$H\$8)
## IF Statement Within A COUNTIF Statement: Cell In Sheet "Summary" Count The Number Of Cells In Column DX Of Sheet "Analyses" That Are Greater Than 0
Apr 22, 2009
I am trying to have a cell in sheet "Summary" count the number of cells in column DX of sheet "Analyses" that are greater than 0, provided that the value in column A of "Analyses" corresponds with the value in B8 of sheet "Summary."
(In "Analyses," there are 106 subjects, each taking up 64 rows. So, columns 1-64 correspond to Subject 1, columns 65-128 correspond to subject 2, etc. In column DX, each subject has 64 values that are either 0 or greater than 0. In "Summary," each subject has one row that summarizes the 64 trials. I want a single cell in the "Summary," sheet to reflect the number of times each subject produces a value greater than 0 in column DX of "Analyses.") I tried using this formula, but it did not work correctly:
=COUNTIF(IF(Analyses!\$A\$1:\$A\$10000=Summary!B8,Analyses!\$DX\$1:\$DX\$10000,""),">0")
(Summary!B8 = 1, so I am trying to calculate the number of values in DX that are greater than 0 only for subject 1.) When I press enter, this yields a value of 384. This is impossible, given that subject 1 only has 64 possibilities of yielding a value greater than 0. Subject 1 has 2 values in column DX that are greater than 0. I tried making this an array formula by pressing Shift+Ctrl+Enter, and that just gives me a #VALUE! error.
## Move Down Column And For Every Blank Cell Clear Contents Of Cell To The Left?
Jun 1, 2014
i need a code that moves down a column and for every empty cell in the column the cell to the left is cleared and then it moves on to the next cell down. the column is not always the same and will start from a selected cell, and the column will contain no more than 5 rows
## Insert Cell Contents From One Column Into Another Based On Specific Part?
Jul 30, 2014
I have a spreadsheet with four columns of text.
In column A, i have multiple levels followed by a letter (i.e. Level 1A, Level 1B etc).
In column B, i have some other details and then so on and so forth.
In column C/E/G lets say, i want to copy the information from column A to show only items that appear as "Level 1" (not "Level 1A", i only want it to check for things without the letter at the end). Then the same in column E but with "Level 2" and so on and so forth.
Column A...Column B-Column C...Column D--Column E...Column F--Column G...Column H
Level 1A....Metals----Level 1A....Metals ---Level 2A....Integral---Level 3A....Television
Level 1B....Energy----Level 1B....Energy--- Level 2B....Flowers---Level 3B....Kitchen
Level 1C....Synergy---Level 1C...Synergy--Level 2C....Full
Level 2A....Integral---Level 1D....Orders
Level 2B....Flowers
Level 1D....Orders
Level 3A....Television
Level 3B....Kitchen
Level 2C....Full
I also have data in Column B that is to do with column A (i.e "Level 1A" - "Metals") and so on with the following columns. I want the items that are in column B to also move over to column D when the things from Column A move to Column C, so at the end it will appear as below so it appears as above.
## Hide Column And Clear Contents Based On A Cell Value In Another Sheet
Mar 19, 2009
I have a workbook with two sheets of data. I want to hide column B of Sheet2 and clear contents of range B2:B50 if the value in A1 of Sheet1 is "a".
## Excel 2010 :: Copy Contents Of Column Based On Value In Another Cell
Jul 30, 2013
I'm using excel 2010 and windows XP with a moderate amount of experience tinkering with macro programming. I know what I need is very doable but I can't get my head around what the code would look like. I must not be wording my searches correctly because most of what I'm getting for results are iterative programs based on a cells value which isn't what I need.
I'm trying to build a macro that will check a cell (C3) and based on the contents of that cell copy a column (I) to one of 12 different columns (K:V). So if the value in C3 is 1 it should copy I to K, if the value is 2 it should copy I to L, and so on.
## If Column F Has “Out” And Column K Has Any Contents Copy The Row
Dec 9, 2009
I’d like to check each row in Column F and Column K of Sheet Check. If Column F has the contents “Out” and Column K has any contents inside its cell, I’d like to copy that row and insert it into Sheet Alert. As a result the same row will exist in Sheet Check and Sheet Alert. This code will cut the row out of Sheet Check and paste the row into Sheet Alert if the contents “Out” is found in Column F.
## Lookup/Match: Compare A1 For The Values In Column B, Then Return The Corresponding Cell (column C) In Column D
Jan 31, 2008
I've been searching the forums for this problem but I can't seem to find any answers. Anyway, this is the problem. See screenshot.
I want to compare A1 for the values in column B, then return the corresponding cell (column C) in column D.
e.g. D1 = 2, D2 = 1, D3 = 4, D4 = 5 and D5 = 3.
## Multiply Every Integer Less Than The Number In Cell A1 And Greater Than Zero By The Number Cell In A2..
Feb 5, 2009
Cell A1 = 3
Cell A2 = 4
Is there a formula to calculate
(3*4)+(2*4)+(1*4)
I need to multiply every integer less than the number in cell A1 and greater than zero by the number cell in A2. I was thinking factorial, but that's not it... Can't remember from my math days.
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# HackerRank Data Structures Quiz -1
Question 1
Time: 00:00:00
What is the worst time complexity of mergesort?
O(n ^ 2)
O(n ^ 2)
O(nlogn)
O(nlogn)
O(n ^ 3)
O(n ^ 3)
O(n)
O(n)
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Question 2
Time: 00:00:00
What is the output of the following code?
Will print odd numbers
Will print odd numbers
Will print squares of natural numbers
Will print squares of natural numbers
Will print Fibonacci series
Will print Fibonacci series
Will print even numbers
Will print even numbers
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Question 3
Time: 00:00:00
What will be the position of the array after 2 iterations of insertion sort?
arr - {5,4,3,2,1}
1 2 3 4 5
1 2 3 4 5
2 3 4 5 1
2 3 4 5 1
3 4 5 2 1
3 4 5 2 1
4 5 3 2 1
4 5 3 2 1
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Question 4
Time: 00:00:00
A linked list 1 -> 2 -> 3 > 4. What will be the position of the linked list after the head of the linked list is passed to the function?
1 3 2 4
1 3 2 4
1 2 3 4
1 2 3 4
4 3 2 1
4 3 2 1
4 2 3 1
4 2 3 1
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Question 5
Time: 00:00:00
The following elements are inserted in a set - {1,8,9,2,3,2,3,4,5}.
After adding the elements in the set, the size function is called. What is the output of the size function?
9
9
8
8
7
7
6
6
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Question 6
Time: 00:00:00
Quicksort is applied to an array. The position of the array after the iteration is - 4 1 5 10 9 8. Which of the following statements is true?
4 is the pivot
4 is the pivot
10 is the pivot
10 is the pivot
1 is the pivot
1 is the pivot
5 is the pivot
5 is the pivot
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Question 7
Time: 00:00:00
The following linked list is given - 1 -> 2 -> 3. The head is at 1. What is the output of the following code?
'0'
'0'
Runtime Error
Runtime Error
NULL
NULL
Garbage Value
Garbage Value
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Question 8
Time: 00:00:00
What will be the position of the array after 2 iterations of bubble sort?
arr - {5,4,3,2,1}
1 2 3 4 5
1 2 3 4 5
4 3 2 1 5
4 3 2 1 5
3 2 1 4 5
3 2 1 4 5
2 1 3 4 5
2 1 3 4 5
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Question 9
Time: 00:00:00
Average Time Complexity of Quicksort with the first element as the pivot?
O(n ^ 2)
O(n ^ 2)
O(n)
O(n)
O(nlogn)
O(nlogn)
O(n ^ 2logn)
O(n ^ 2logn)
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Question 10
Time: 00:00:00
What is the avg time complexity of Binary Search?
O(n)
O(n)
O(n ^ 2)
O(n ^ 2)
O(nlogn)
O(nlogn)
O(logn)
O(logn)
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# Examples of Non-Associative Algebras with (Non-Associative) Division and with Unity without Classical Division
Let $$\mathbb{K}$$ be a field. By an $$\mathbb{K}$$-algebra I mean a $$\mathbb{K}$$-module $$A$$ equipped with a binary operation $$∗:A\times A\rightarrow A$$ that is $$\mathbb{K}$$-bilinear and denoted by $$(x,y)\mapsto xy$$.
Let $$A$$ be an algebra.
1. $$A$$ is said to have (non-associative) division if $$A\neq0$$ and for any $$a\in A$$ with $$a\neq0$$ the functions $$L_a:x\mapsto ax$$ and $$R_a:x\mapsto xa$$ are both bijective.
2. $$A$$ is said to classical division if $$A$$ has unity and for every $$a\in A$$ with $$a\neq0$$ there is an element $$b\in A$$ such that $$ab=ba=1$$.
I know that, if $$A$$ is alternative, that is, satisfies $$(aa)b=a(ab)$$ and $$(ab)b=a(bb)$$ for any $$a,b\in A$$, then (1) and (2) are equivalent.
What would be examples of non-associative algebras that:
a) Has (non-associative) division and unity but does not have classical division?
b) Has classical division but does not have (non-associative) division?
I already saw the question here, but it does not have a satisfactory answer yet.
• What is $\mathbb{K}$? A field? A commutative ring? Commented Aug 22, 2022 at 16:19
• For a) you can use Baez example on modified Quaternions and for b) you can take (imho) the Sedenions, since nonzero elements are invertible ($a^ {-1}=a^\star/||a||$)and it has zero divisors so $R_a$ and $L_a$ can''t be bijections. All of the above are unital algebras. Commented Aug 22, 2022 at 17:00
Consider the real algebra $$A$$ given by the usual Quaternion multiplication $$\mathbb{H}$$ except that $$i^2=-1+\frac{j}{2}$$. I will denote the product of $$a,b\in A$$ by "$$a.b$$" and the product of $$a,b\in \mathbb{H}$$ by "$$ab$$" and write $$v=v_1+v_2i+v_3j+v_4k$$ for a generic vetor since the underlying vector space structure on $$A$$ and $$\mathbb{H}$$ coincide. Also, consider the Sedenions $$\mathbb{S}$$ with its usual structure.
Af 1.: The sedenions $$\mathbb{S}$$ satisfy $$(2)$$ but not $$(1)$$.
Given $$0\neq a\in \mathbb{S}$$ we have $$a(a^\star/||a||)=1$$ where $$a^\star$$ is the conjugation of $$a$$ (it is an easy exercise to show that $$aa^\star=a^\star a$$). Notice that taking $$e_0=1,\ldots,e_{15}$$ as the usual basis for $$\mathbb{S}$$ we have $$(e_3+e_{10})(e_{6}-e_{15})=0$$ therefore $$L_{e_3+e_{10}}$$ is not bijective.
Af 2.: $$A$$ satisfies $$(1)$$ but not $$(2)$$.
Take $$a,b\in A$$ such that $$a.b=0$$ but $$a,b\neq 0$$.
Notice that $$a.b=ab+\frac{a_2b_2j}{2}$$. Clearly $$a_2=0$$ or $$b_2=0$$ implies $$||ab||=||a||\cdot||b||=0$$ hence $$a=0$$ or $$b=0$$. WLOG $$a,b$$ may satisfy $$a_2=1=b_2$$. If $$0=a.b=ab+\frac{j}{2}$$ then $$||ab||=||a||\cdot ||b||=1/2$$ but this is a contradiction for $$||a||=(a_1^2+1+a_3^2+a_4^2)^{1/2}\geq 1$$ and the same applies to $$b$$. This shows that $$L_x,R_x$$ are indeed bijective in $$A$$. Clearly '$$i$$' has a right inverse named $$(-i-k/2)$$ and a left inverse $$(-i+k/2)$$, also by injectivity of $$R_i,L_i$$ they are unique, hence $$A$$ can't satisfy $$(2)$$.
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1. IMPs. None vulnerable.
♠ A 9 6 5
K Q 7 3
Q
♣ Q 10 9 3
♠ 8 4
A 8 5 4
A
♣ A K J 8 5 4
West North East South Pass 1♣ 1♠ Dbl Pass 2♠ Pass 3♣ Pass 3♥ Pass 3♠ Pass 4♦ Pass 6♣ All Pass
West leads the ♠K. Plan the play.
♠ A 9 6 5 ♥ K Q 7 3 ♦ Q ♣ Q 10 9 3 ♠ K Q J 7 3 ♠ 10 2 ♥ 9 ♥ J 10 6 2 ♦ J 10 7 6 3 ♦ K 9 8 5 4 2 ♣ 7 2 ♣ 6 ♠ 8 4 ♥ A 8 5 4 ♦ A ♣ A K J 8 5 4
The contract is cold if hearts are 3–2, so assume they’re 4–1, the length surely with East given those 11 diamonds roaming around in the East‑West hands.
Your best bet is to win the ♠A, draw trump, cash the K and the A and exit a spade. Assuming hearts are 4–1, you have to find East with ♠10 x or ♠J x and West with a singleton 9, 10 or jack of hearts. If the cards are favorable, whoever wins the spade exit is endplayed.
If West wins, he can’t exit a spade without setting up the ♠9 for a heart discard. A diamond is no better since it gives you a ruff and sluff.
If East wins, he must lead a heart from a presumed J–10–x, J–9–x or 10–9–x, thus losing his natural trick and a diamond exit gives you a ruff and sluff. All roads lead to success.
2. IMPs. None vulnerable.
♠ A Q 10
6 3
Q 4
♣ Q J 9 8 4 2
♠ 9 6 5
A 10
A J 9 8 4 2
♣ A 5
West North East South 1♦ 1♥ 2♣ 3♥(1) 3NT All Pass
(1) Preemptive.
OK, you took the bull by the horns and bid everyone’s favorite contract. Now that you’re there, how do you play it when West leads the K?
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# Gauge Anomaly - Calculation of The Anomaly
Calculation of The Anomaly
In vector gauge anomalies (in gauge symmetries whose gauge boson is a vector), the anomaly is a chiral anomaly, and can be calculated exactly at one loop level, via a Feynman diagram with a chiral fermion running in the loop (a polygon) with n external gauge bosons attached to the loop where where is the spacetime dimension. Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.
Let us look at the (semi)effective action we get after integrating over the chiral fermions. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by the operator corresponding to an infinitesimal gauge transformation by ε, then the Frobenius consistency condition requires that
for any functional, including the (semi)effective action S where is the Lie bracket. As is linear in ε, we can write
where Ω(4) is d-form as a functional of the nonintegrated fields and is linear in ε. Let us make the further assumption (which turns out to be valid in all the cases of interest) that this functional is local (i.e. Ω(d)(x) only depends upon the values of the fields and their derivatives at x) and that it can be expressed as the exterior product of p-forms. If the spacetime Md is closed (i.e. without boundary) and oriented, then it is the boundary of some d+1 dimensional oriented manifold Md+1. If we then arbitrarily extend the fields (including ε) as defined on Md to Md+1 with the only condition being they match on the boundaries and the expression Ω(d), being the exterior product of p-forms, can be extended and defined in the interior, then
The Frobenius consistency condition now becomes
As the previous equation is valid for any arbitrary extension of the fields into the interior,
Because of the Frobenius consistency condition, this means that there exists a d+1-form Ωd+1 (not depending upon ε) defined over Md+1 satisfying
Ωd+1 is often called a Chern-Simons form.
Once again, if we assume Ωd+1 can be expressed as an exterior product and that it can be extended into a d+1 -form in a d+2 dimensional oriented manifold, we can define
in d+2 dimensions. Ωd+2 is gauge invariant:
as d and δε commute.
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https://pt.scribd.com/document/350631444/ubd-lesson-edet-636
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# Ms.
Lucas-Physical Science B
## Stage 1 Desired Results
ESTABLISHED GOALS Transfer
NGSS Science Standards Students will be able to independently use their learning to
HS-PS3-4. Plan and conduct an Locate and use evidence to support a claim
investigation to provide evidence Cite sources
that the transfer of thermal energy Explain how thermal energy keeps a house warm
when 2 components of different Draw conclusions based on observations
temperature are combined within a Calculating costs
closed system results in a more Meaning
uniform energy distribution among UNDERSTANDINGS ESSENTIAL QUESTIONS
the components in the system (2nd Students will understand that Will a cup of hot soup cool down faster,
law of thermodynamics) Understand what heat energy is and what slower, or at an identical rate as a large
temperature is and how they are related bowl of soup that begins at the same
HS-ETS1-2. Design a solution to a to the random motion of molecules temperature?
complex real-world problem by (kinetic energy). Why are there 3 different temperature
breaking it down into smaller, more Understand how a thermometer measures scales in use today?
manageable problems that can be temperature. How is temperature measured? (i.e. how
solved through engineering. Understand the definition of heat and the do thermometers work?)
relationship between heat and thermal If you fill a cup with warm bathwater, will
NETS Student Standard energy. it contain more heat, less heat, or the
3. Research and information fluency There are 3 different temperature scales: same amount of heat?
a.) Plan strategies to guide 1. Fahrenheit How does thermal energy apply in
inquiry 2. Celsius keeping your house warm?
b.) Locate, organize, analyze, 3. Kelvin - Absolute Zero
evaluate, synthesize, and Energy exists in various forms
ethically use information form a Energy can be transferred from one place
variety of sources and media to another.
c.) Evaluate and select Acquisition
information sources and digital Students will know Students will be skilled at
tools based on the The 3 temperature scales and how to Observe, analyze, and report
appropriateness to specific tasks convert between different scales. observations of objects and events
d.) Process data and report The difference between thermal energy Generate appropriate questions (teacher
results and heat and how the 2 are related and student based) in response to
The different forms of energy observations, events, and other
experiences
Observe, collect, organize, and
appropriately record data, then
accurately interpret results
Make predictions based on prior
experiences and/or information
Using research skills to locate evidence
to support a claim
Stage 2 - Evidence
Evaluative Criteria Assessment Evidence
PERFORMANCE TASK(S):
Organization (5 pts.) Lab Report
Lab Report (Evidence)
o Question is clear (5 pts.)
o Hypothesis is testable and measurable (10 pts.)
o Research (25 pts.)
o Procedure (5 pts.)
o Data/Observations (15 pts.)
o Conclusion (20 pts.)
Sources (10 pts.)
Language/Grammar (5 pts.)
Forms to self assess group and self. These will a separate grade Reflection/Self-Assessment
from the report.
OTHER EVIDENCE:
Participation (bubble thinking map) ED puzzle
Kahoot Participation (Thermal Energy Activity)-Daily Reflection
Actively Learn annotations and question responses
Stage 3 Learning Plan
Summary of Key Learning Events and Instruction
Day 1
Types of Energy Review-As a class we will create a summary of the types of energy from the previous chapter. We will create a
bubble thinking map as a class. This will help students to realize we are focusing on just one type of energy.
Actively Learn-Students will read and article about temperature, heat, and thermal energy. Students will annotate the article
and respond to ?s at the end. Some questions will be the essential questions.
Day 2
EDpuzzle-videos to explain temperature, heat, and thermal energy. Students will respond to ?s while watching the videos.
Essential ?s not covered in actively learn will be covered here.
Kahoot-Review of previous information on temperature, heat, and thermal energy
Day 3-4
Thermal energy research-Students will research thermal energy and how it is applied to building a house.
I will show students how to cite sources. Students will need at least 2 sources other than their textbook. This will be part of their
final assessment (lab report)
Self assessment each day
o How did you contribute to your project today? Describe with as much detail as possible?
o Do you think you worked well as a group? Why or why not?
o What grade would you give yourself for the work you did today? Why?
Day 4
Sketch house design and estimate the cost. Groups will have a variety of materials to use and will have to calculate a cost.
Their goal is to create the most efficient house, in terms of keeping heat in, while also keeping cost low.
o Scotch Tape \$0.05 per cm.
o Duct Tape \$0.10 per cm.
o Aluminum Foil \$0.25 per 10 cm. sheet
o Craft Sticks \$0.02 each
o Straws \$0.02 each
o Wax Paper \$0.15 per 10 cm. sheet
o Plastic Wrap \$0.20 per 10 cm. sheet
o String \$0.05 per 10 cm. length
o Cotton Balls \$0.02 each
Self assessment
o How did you contribute to your project today? Describe with as much detail as possible?
o Do you think you worked well as a group? Why or why not?
o What grade would you give yourself for the work you did today? Why?
Days 5-6
https://www.teachingchannel.org/videos/stem-lesson-ideas-heat-loss-project#video-sidebar_tab_video-guide-tab
Thermal Energy Activity-students will construct a house out of paper and other material and will measure the temperature of
various parts of the house using a go temp probe, and collect the data in a table. We will use an ipad app to visualize thermal
energy. (It wont be perfect but I will explain to the students that this is just for visualization.)
Self assessment each day
o How did you contribute to your project today? Describe with as much detail as possible?
o Do you think you worked well as a group? Why or why not?
o What grade would you give yourself for the work you did today? Why?
Days 7-8
Lab Report-Students will either create a written lab report or a video lab report to demonstrate their understanding.
After turning in a lab report each student will complete a self-reflection and group reflection.
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# Questions Asked in SSC CHSL Tier I - 16th Jan 2017 All Slots.
Dear Readers,
In this post we are sharing some of Questions which was asked in today’s exam all Slots. & Also we are updating this section regularly.
Please Note, This is the Questions & Data given by our users, we are stating that here, if anything does not match which came in the exam, we will not be responsible for the same, Please mention in the comments section We will try to rectify that.
UPDATING SOON STAY TUNED OR SUBSCRIBE FOR REGULAR UPDATES...
## Quants Question Asked in SSC CHSL Tier-1 -16 Jan 2017
### Quants Questions asked in 1st Shift
1. Car A and Car B travelled from City A to City B with speed 24 & 32 kmph One Car takes 2.5 hour less than the other. What is the distance between City A to City B?
2. What number should be subtracted from both terms of the ratio 15 : 19 in order to make it 3 : 4?
3. If p : q = r : s = t : u = 2 : 3, then (mp + nr + ot) : (mq + ns + ou) is equal to
4. If a : b = c : d = e : f = 1 : 2, then (pa + qc + re) : (pb + qd + rf) is equal to
Quants Questions asked in 2nd shift
1. √1+tan^2A/tanA
2. Cot 5 pie/3 ?
3. Sec4Ï€/3 =?
4. 5 digit highest number multiplied by 89?
5. 1 quintile wheat in which one part sale @10% and remaining part @20% . while overall profit 14% find quantity of 20% profit part
6. A does 4 time work than B AND A done his work in 45 days less than B . in how many days both can complete work.
7. value of sec 240
8. linear two equations in which x and y are given and in one equation k is also given . find value of k
9. simple interest 28000 and compund interest also given find rate of interest
Quants Questions asked in 3rd shift
## GK Questions Asked in SSC CHSL Exam Tier-1 -16 Jan 2017
### General Awareness Questions Asked SSC CHSL Exam 2017 16-01-2017 (Shift 1)
1. Who was awarded the Bharat Ratna on his 100the birthday?- Dhondo Keshav Karve
2. When was the national anthem sung for the very first time?-27 December 1911
3. Who won the national snooker championship for 2016?- Aditya Mehta
4. In which year was Jahangir born?-1569
5. 2016 Nobel in Economics was given for?-Contract Theory
6. 4 bits equal to?- 1 Nibble
7. A person how many time can become PM of country ?-No limit
8. Allora caves in which state?-Maharastra
9. Arthshathra written By ?-Chanakya
10. Arya samaj Founder-Raja Ram Mohan
11. Barograph made by?-woods and Brass
12. Bhavani dam is located in which state?-Tamil Nadu
13. Deepika Kumari related with which game?-Archery
14. Dwarf planet in solar system?-Pluto
15. Full form of GDP?-Gross Domestic Product
16. Who invented logarithms-John Napier
17. Vehicles exhaust which poisonous gas-Hydrocarbons
18. Largest Union territories in india?-Andabar Nicobar
19. Chlorophyll occurs in which part of the plant?-Chlorophyll absorbs the light energy needed to make photosynthesis happen. It is important to note that not all the color wavelengths of light are absorbed. Plants mostly absorb red and blue wavelengths — they do not absorb light from the green range.
20. Benzene discovered by ?-English scientist Michael Faraday in 1825
21. Who built gateway of India?-British
22. Who wrote Train To Pakistan book?- khushwant Singh (Indian)
23. Who was awarded the Bharat Ratna on his 100th birthday? Dr. Dhondo Keshav Karve
24. When was the national anthem sung for the very first time? 27 December 1911
25. Who won the national snooker championship for 2016? Aditya Mehta
26. Mahatma gandi's Year of birth ?1869
27. Benzene discovered by Faraday
28. Person who was awarded bharat ratna on his 100 birthday? Keshav karve
29. Winner of world snooker championship 2016
30. Sachin tendulkar:cricket::roger federer: ?
### General Awareness Questions Asked SSC CHSL Exam 2017 16-01-2017 (Shift 2)
1. Ryder cup associated with? Golf
2. Kaziranga National park is famous for?- Rhinoceros
3. Lal Bahadur Shastri born in?- October 2, 1901 at Mughalsarai
4. Pushkar fair in which city? Rajasthan
5. Chairman of Rajya sabha? Md.Hamid Ansari
6. Taxe which cannot be transferred?
7. First individual medal for India in Olympic?-Khashaba Dadasaheb Jadhav
8. Which one is not inner planet? -Earth, Venus, Mercury, Saturn? -Saturn
9. The man who know infinity movie director?-Matthew Brown
10. First individual medal for India in Olympic?-Vijender Singh
11. Kajiranga National park is famous for? One Horned Rhinos
12. Lal Bahadur shahtri born in? 2 October 1904, Mughalsarai
13. Taxe which cannot be transferred?
14. First individual medal for India in Olympic? Khashaba Dadasaheb Jadhav
15. Which one is not inner planet? -earth, venus, mercury, saturn ? Saturn
16. The man who know infinity movie director? Matthew Brown
17. Binomial nomenclature given by? Lineaud
18. Magnatite is alloy of?
19. Hawa mahal is located in which city
20. Scientific name "annona squamosa"
21. What is impulse?
22. Change of momentum.
23. 2nd largest gland in human body.
24. Rome was not built in a day?
25. Inventor of sewing machine
### General Awareness Questions Asked SSC CHSL Exam 2017 16-01-2017 (Shift 3)
1. Who built Gate Way of India? British
2. Dwarf planet in solar system?-Pluto
3. Barograph made by?-woods and Brass
4. In which year was Jahangir born?-1569
5. 2016 Nobel in Economics was given for? - Contract Theory
6. 4 bits equal to? - 1 Nibble
7. A person how many time can become PM of country ? - No limit
8. Allora caves in which state?-Maharastra
9. Arthshathra written By ?-Chanakya
10. Arya samaj Founder - Raja Ram Mohan
11. Which of the following districts is on the international border of India?
12. The deposits of the ancient Tethys Sea were folded to form?
13. The largest irrigation canal in India is called ?
## Reasoning Quest Asked in SSC CHSL Exam Tier 1-16 Jan 2017
### Reasoning Questions asked in SSC CHSL Exam 2017 16-01-2017 1st Shift
1. Relationship between Footballer,Cricketer and Indian?
2. Missing series was probably 19,29,40,44,?
3. Flog synonym -19,29,40,44
### Reasoning Questions asked in SSC CHSL Exam 2017 16-01-2017 2nd shift
1. Find the odd one out from the given alterna lives.(a) Mizoram (b) Sikkim (c) Kohima (d) Manipur.
2. Which one of the given responses would be a meaningful order of the following? 1. Book 2. Pulp 3. Timber 4. Jungle 5. Paper
3. Arrange the following words according to dictionary arrangement. 1. Epitaxy 2. Episode 3. Epigene 4. Epitome 5. Epilogue
### English Questions asked in SSC CHSL Exam 2017 16-01-2017 1st Shift
1. Antonym of ‘Abominable’
2. Antonym of ‘Flog’
3. Synonym of ‘Aground’
### English Questions asked in SSC CHSL Exam 2017 16-01-2017 2nd shift
1. Mostly of the (a)/newly recruited officers (b)/have no experience (c) No error (d)
2. The resignation of (a)/one of our directors (b)/ have caused the price (c)/of shares to fall (d).
3. There are many (a)/ways of which (b)/inflation can (c)/ be measured (d)
4. Synonym of extravagant
### English Questions asked in SSC CHSL Exam 2017 16-01-2017 3nd shift
Note: You are all advised to bookmark this page to get an update for the MORE Questions. We will keep on updating this space as soon as we get them.
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2. ### Questions Asked in SSC CHSL Tier I - 14th Jan 2017 All Slots.
3. Questions Asked in SSC CHSL Tier I - 13th Jan 2017 All Slots
4. Questions Asked in SSC CHSL Tier I - 12th Jan 2017 All Slots
5. Questions Asked in SSC CHSL Tier I - 11th Jan 2017 All Slots
6. Questions Asked in SSC CHSL Tier I - 10th Jan 2017 All Slots
7. Questions Asked in SSC CHSL Tier I - 09th Jan 2017 All Slots
8. Questions Asked in SSC CHSL Tier I - 08th Jan 2017 All Slots
9. Questions Asked in SSC CHSL Tier I - 07th Jan 2017 All Slots
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# CONTROL POINT SURVEYING AND TOPOGRAPHIC MAPPING
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## Transcription
1 CONTROL POINT SURVEYING AND TOPOGRAPHIC MAPPING Shoichi Oki Land Bureau, National Land Agency, Japan Keywords: control point, survey, topography, mapping, geodesy Contents 1. Introduction 2. Geometric Background 2.1. Ellipsoid 2.2. Transformation between Geodetic Systems 2.3. Geoid 2.4. Datum Reconstruction with Space Geodesy 2.5. Surveying Instruments for Horizontal Survey Transit EDM Total Station 2.6. Survey Instruments for Vertical Survey 3. Horizontal Control 3.1. Datum Origin 3.2. Geodetic Network 3.3. Control Point surveying Triangulation Trilateration Traversing GPS in Trilateration and Traversing 4. Vertical Control 4.1. Vertical Datum 4.2. Leveling 5. Topographic Mapping 5.1. Photogrammetry 5.2 Ground Method Glossary Bibliography Biographical Sketch Summary Surveying is based on geodesy. With precise survey instrument, the Earth must be treated as an ellipsoid instead of a sphere. The ellipsoid that is chosen for geodetic survey is called reference ellipsoid. The shape of the Earth, so called geoid, is expressed as an equipotential surface of gravity. Usually, position on the ground is measured in longitude, latitude on the reference ellipsoid and height on the geoid. For convenience to measure the three dimensional values, a lot of monuments whose longitude, latitude or height is known are settled as reference point all over the countries. Control point surveying is a surveying to establish the reference point network, and is also to
3 field. Satellite imaging and GPS measuring are rapidly introduced in topographic mapping. 2. Geometric Background The first step of the geodetic control work is usually to choose a reference ellipsoid because the first approximation to the Earth s surface is an ellipsoid. Precise measurements with geodetic instruments can detect small deviations of the Earth s shape from sphere and ellipsoid as Earth s reference, while it seems perfect sphere from thge photo taken from the moon. The undulations such as mountains and hills are measured from the reference ellipsoid Ellipsoid An ellipsoid is a solid shape which is produced by rotating an ellipse around its minor axis. An ellipse is defined as a curved line which has two points geometrically termed as foci and in which the sum of distances from the two focal points to any point on the ellipse is constant as illustrated in Figure.1. If the coordinates of the foci are is 2a, and the point on the ellipse is Figure.1 An ellipse F 1(- X,0) and Px (, y) F 2(- X,0), and the sum of the distances, the ellipse is described by the equation: 2 2 1/ /2 (( x + X) + y ) + (( x - X) + y ) = 2a It is rearranged as: x / a + y / b = 1
4 where a and b respectively are termed semi-major axis and semi-minor axis of the 2 2 1/2 b= ( a - X ) ellipse where. To express the character of an ellipse, flattening f as ( a- b)/ a and eccentricity e as e = ( a - b )/ a are used in geometric geodesy. The geodetic latitude ϕ at a random point P on the ellipse is defined as an angle to be formed between a normal at the point P and the major axis. By differentiating Eq.(1), 2 2 dy / dx = -( b / a ) ( x / y) According to the definition of the geodetic latitude, dy / dx = tan( ϕ + 90) Hence, 2 2 b xsin ϕ = a ycosϕ By substituting this into the formula of an ellipse, 2 1/2 x = acos ϕ/(1- e sin 2 ϕ) y = a(1- e ) sin ϕ 2 An ellipsoid is a figure to be produced by rotating an ellipse. The expression of an ellipsoid using rectangular coordinates is: x / a + y / a + z/ b = 1 The definition of the geodetic latitude of point P in the surface of an ellipsoid is represented by angle between the normal line of the ellipse at point P and the equatorial plane. The definition of the geodetic longitude is represented by angle from the zero meridian to a meridian passing point P, as measured counterclockwise.
5 Figure.2 Ellipsoid 2.2. Transformation between Geodetic Systems Usually geodetic systems of all countries are not same. To compare coordinates between different geodetic systems, relationship between the coordinate systems must be known. Relationship between two geodetic systems is expressed by using components of revolution, ( ε x, εy, εz ) and origin shifting components ( ΔX, ΔY, ΔZ) and taking difference of scale factors (S). The relationship among the rectangular coordinates is respected as: System1 x System 2 x x y Δ = (1 + S) R3εzR2εyR1εx y +ΔY z Δz z where ( R1, R2, R3) represent matrices of revolution for axes x, y, and z :
6 - - - Bibliography Figure 3. Transformation from Geodetic System 1 to Geodetic System 2 TO ACCESS ALL THE 15 PAGES OF THIS CHAPTER, Visit: Heiskanen W. A. & Moritz H.(1967) Physical Geodesy, W. H. Freeman and Company [A textbook of height and geoid] Leick A. (1995): GPS, Satellite Surveying, 2nd edition, John Wiley & Sons, Inc. (ISBN ) [This book presents elements and computations of Global Positioning System and geodetic analysis on the ellipsoid.] Mikhail E. M.(1976): Observations and least squares, IEP Dun-Donnelley, Harper & Row [ This book presents theory of the least square method for observation data analysis] Schwarz C. R.(ed.) (1989): North American Datum of 1983, NOAA Professional Paper NOS 2 [This book presents detailed report how to reconstruct the North American Datum with space geodetic technologies] Takahashi F., Kondo T., Tkahashi Y, and Koyama Y.(2000): Very Long Baseline Interferometer, IOS Press (ISBN ) [This book presents basic concepts, data analysis, and geodetic applications of VLBI.] Torge W.(1991): Geodesy, 2nd ed., de Gruyter [A textbook of height and geoid]
7 Vanicek P. and Christou N. T.(1994): Geoid and its geophysical Interpretations, CRC Press [A textbook of height and geoid] Vanicek P. and Krakiswsky E.(1986): Geodesy : the concept, Elsevier [A textbook of height and geoid] Biographical Sketch Shoichi Oki holds a B.Sc in Science from Tohoku University and an M.Sc in Science from Tohoku University. His master s research was analysis of plasma waves in space by Very Long Baseline Interferometry(VLBI) survey and computer simulation. He was hired after graduation from the university by Geographical Survey Institute (GSI). He conducted geodetic survey, topographic mapping and digital mapping. During 1991 to 1993 he was transferred temporarily to the National Land Agency as a chief of National Land Information Office. During 1996 to 1997 he stayed in Germany as a guest scientist of Technical University Munich to research digital photogrammetry. During 1997 to 1998 he worked to establish International VLBI Service as the VLBI group leader of GSI. He participated in basic survey planning and system reforming of survey as a member of strategic planning team of GSI. In he was prompted to the Deputy Director for cadastral survey of the National Land Agency.
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### Undergraduate Honors Thesis. Presented in Partial Fulfillment of the Requirements for. Graduation with Distinction. at The Ohio State University
Multi-Sensor Calibration of an Integrated Mobile Mapping Platform Undergraduate Honors Thesis Presented in Partial Fulfillment of the Requirements for Graduation with Distinction at The Ohio State University
### Perspective of Permanent Reference Network KOPOS in Kosova
143 Perspective of Permanent Reference Network KOPOS in Kosova Meha, M. and Çaka, M. Kosovo Cadastral Agency, Kosovo Archive Building II nd floor, P.O. 10000, Prishtina, Republic of Kosovo, E-mail: mmeha@yahoo.com,
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CBSE Class 12 Sample Paper for 2021 Boards
Class 12
Solutions of Sample Papers and Past Year Papers - for Class 12 Boards
## Find the coordinates of the point where the line (x+3)/3 = (y-1)/(-1) = (z-5)/(-5) cuts the XY Plane.
Β
Note : This is similar to Example 30 of NCERT β Chapter 11 Class 12 Three Dimensional Geometry
Check the answer here https:// www.teachoo.com /3586/756/Example-30---Find-point-where-line-crosses-the-XY-plane/category/Examples/
Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
### Transcript
Question 14 Find the coordinates of the point where the line (π₯ + 3)/3 = (π¦ β 1)/(β1) = (π§ β 5)/(β5) cuts the XY Plane. If line (π₯ + 3)/3 = (π¦ β 1)/(β1) = (π§ β 5)/(β5) cuts the XY Plane Then z = 0 So, let coordinates of point be (x, y, 0) Now, (π₯ + 3)/3 = (π¦ β 1)/(β1) = (π§ β 5)/(β5) = k Thus, x = 3k β 3, y = βk + 1, z = β5k + 5 Since z = 0 β5k + 5 = 0 5k = 5 k = 5/5 k = 1 Now, x = 3k β 3 = 3(1) β 3 = 0 y = βk + 1 = β1 + 1 = 0 Thus, Coordinates of Point are (0, 0, 0)
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1. ## Angular dimension edit issue , ACAD 2010
Registered forum members do not see this ad.
This is angular dimension of the arc, in my drawing ( picture 1)
I need for my, angular dimension to be identical as here (2) :
I have menaged to reposition right arrow and extension line (3) :
Problem is obvious, extension is sloping to the left, I want this line to overlaps withe red one. How can I achieve this ?
2. If the radius point of the arc doesn't lie on the 'red' line, none of the dimension commands will do what you want... you'll need to 'draw' the dimension (which is poor drafting practice). The extension lines should converge (the apparent intersection) on the radius point.
3. Originally Posted by lpseifert
If the radius point of the arc doesn't lie on the 'red' line, none of the dimension commands will do what you want... you'll need to 'draw' the dimension (which is poor drafting practice). The extension lines should converge (the apparent intersection) on the radius point.
I've looked at the second picture more carefully, neither of the arcs are not centered at dashed red line intersection. Is there any way to set angular dimension, given center point (in my case intersection) , osnap endpoind on the left, and orthogonal dashed line ?
4. You need to draw the centerlines and use those to define the angular dimension. The 55 degree dimension isn't related to either of the arcs, it's there to locate the nose of the hook.
http://screencast.com/t/YMCQo7Asm
5. Originally Posted by nestly
You need to draw the centerlines and use those to define the angular dimension. The 55 degree dimension isn't related to either of the arcs, it's there to locate the nose of the hook.
http://screencast.com/t/YMCQo7Asm
Thank you
I should have paid more attention to the command line
6. Of course, if you were going down the road of poor drafting practice, you could always explode the dimension, and edit the end to suit
7. How about posting the drawing?
8. Here it is :
9. Originally Posted by djole_ri
Here it is :
Does this mean you were successful?
10. Registered forum members do not see this ad.
Originally Posted by ReMark
Does this mean you were successful?
Yes, but there is another thing I come across : picture 1
I follow these instructions, but in the fourth step I'm having this :2
Angle (third item in square barckets), applies not to the extension lines, but to the dimension text, so I been missing : Horizontal, Vertical, Rotated, and Rotated is exactly what I need. Is there an solution for this problem ?
#### Posting Permissions
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# C.test: Cochran's C test of homogeneity of variances In GAD: GAD: Analysis of variance from general principles
## Description
Performs a Cochran's test of the null hypothesis that the largest variance in several sampled variances are the same.
## Usage
`1` ```C.test(object) ```
## Arguments
`object` An object of class lm, containing the specified design.
## Details
The test statistic is a ratio that relates the largest variance to the sum of the sampled variances.
## Value
A list of class htest containing the following components:
`statistic` Cochran's C test statistic `p-value` The p-value of the test `alternative` A character string describing the alternative hypothesis `method` The character string Cochran test of homogeneity of variances `data.name` A character string giving the name of the lm object `estimate` Sample estimates of variances
## Author(s)
Leonardo Sandrini-Neto (leonardosandrini@gmail.com)
`gad`
## Examples
```1 2 3 4 5 6``` ``` library(GAD) data(rohlf95) cg <- as.fixed(rohlf95\$cages) mq <- as.random(rohlf95\$mosquito) model <- lm(wing ~ cg + mq%in%cg, data = rohlf95) C.test(model) ```
### Example output
```Loading required package: matrixStats
R.methodsS3 v1.7.1 (2016-02-15) successfully loaded. See ?R.methodsS3 for help.
Cochran test of homogeneity of variances
data: model
C = 0.30762, n = 2, k = 12, p-value = 0.5899
alternative hypothesis: Group cage1.m2 has outlying variance
sample estimates:
cage1.m1 cage1.m2 cage1.m3 cage1.m4 cage2.m1 cage2.m2 cage2.m3 cage2.m4
0.500 4.805 0.080 1.620 0.000 1.125 0.980 2.000
cage3.m1 cage3.m2 cage3.m3 cage3.m4
0.405 0.980 0.245 2.880
```
GAD documentation built on May 2, 2019, 3:01 a.m.
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# Large Pendulum Wave Question
• Orb Brehs
In summary, the guys building a large pendulum wave ran into some issues with calculating the length of the wire connecting the individual pendulums. They are using mason jar bobs, but are running into problems with the period being shorter than expected. They are also wondering if the jars are empty or full (of what?). They are considering using sand to increase the g force.
#### Orb Brehs
Hello,
I should preface that we are not physicists. We are just a few relatively ordinary guys that committed to building a large pendulum wave (6ft tall, 8ft long). The construction went very well. The frame and the individual pendulum are all strung up. We have a rather ingenious rig that allows us to very easily adjust the length of the wire connecting the pendulum.
However, we are running into some issues that hopefully some of you experts will be able to shed some light on. We are using a calculator to determine our 'length' of each pendulum. So for a time period of 60 seconds and an oscillation value of 30 we are provided with a 'length' for the initial pendulum of aprox 100cm. But when we string it up the oscillation only runs for 29.
Other considerations:
- We are using mason jars as the bobs. So not spherical, roughly cylindrical
- We have no idea what 'length' actually refers to. We assumed (wrongly I suspect) that it was the distance between the top of the jar (bob) to the support bar, and that is what we have measured.
So, I guess the overall question, after such a lengthy preamble, is where do we measure our 'length' from when using mason jar bobs on a large pendulum wave? Top, middle, or bottom of the jar?
And just as an aside, we spent a lot of time eyeballing and counting oscillations by eye, and although this was pretty good in terms of results we believe this exact measuring should be better. Maybe not?
Thank you very much!
Hi and welcome to PF.
It looks as if your period is shorter than you expected so perhaps there is another 'restoring force', apart from gravity, acting on the jars. Could it be due to the wire? How free is the wire at the top? Posh pendulums use a knife edge to eliminate torque as the bob swings.
I wonder if the jars are empty or full (of what?). If they are empty then you could fill them with sand, perhaps and then the g force would be proportionally greater compared with the torque on the wire. You could see what difference the mass of the bob makes on your agreement with theory. (Period =2Π √(l/g) . . . . . yes?) Also, the effective length of the pendulum is really to the Centre of Mass of the bob - and assumes that the length of the bob is small c/w the length of the string. If the bob is not a 'point mass, then the moment of inertia of the bob will slow it down. But I don;t think, on such a long wire, it's relevant. Look up the formula for a swinging bar, pivoting at a point along its length. (A little nerdy diversion for you)
PS aren't Jars likely to smash and ruin your fun? Bean cans would be stronger :).
- We tested the 'free wire' factor and there wasn't any change when we switched methods. So we believe the torque is insignificant.
- The jars are filled with glass beads (the point of this whole thing is for it to be lit up so sand isn't an option)
So you're suggesting that the centre of mass would be the middle of the filled jar?
Formula we used for length is
l(n)=g(Γ/2π(N+n))^2
N is number of oscillations the longest pendula performs
n is number of pendula
Γ is the duration of a cycle
Is this formula not effective for such long lengths? our longest length would be aprox 1meter. Is our formula no good? Remember, we are just amateurs here!
There are several steps in this problem.
My formula is just for the period of a single pendulum of length l (when n=0, in your formula, I think). If N is not what this formula would give you then I think there must be something wrong - it's a standard experiment in School and kids tend to get it right with very simple equipment,
I guess we should make sure we're talking about the same thing here. I assume you are talking about a row of pendulums that you view end-on and get a wave like pattern when they are started off at the same time.
I am just wondering whether the n you are using, starts from the value 1. I think it should start with value 0. n would stand for the additional pendulums and not the total number, I think. Think it over and see if that helps.
## 1. What is a Large Pendulum Wave?
A Large Pendulum Wave is a physical phenomenon that occurs when a row of equally spaced pendulums of increasing length are released simultaneously. The result is a wave-like motion where the pendulums swing back and forth in a synchronized pattern.
## 2. How does a Large Pendulum Wave work?
The Large Pendulum Wave works due to the conservation of energy and the law of gravity. When the pendulums are released, the shorter ones have a higher frequency and swing back and forth more quickly. As the pendulums get longer, their frequency decreases, causing them to swing slower. This results in a wave-like pattern that can continue for several minutes.
## 3. What factors affect the Large Pendulum Wave?
The main factors that affect the Large Pendulum Wave are the length and weight of the pendulums, the distance between them, and the initial angle at which they are released. These factors can be adjusted to create different patterns and speeds in the wave.
## 4. What are the real-life applications of the Large Pendulum Wave?
The Large Pendulum Wave is mainly used for educational and entertainment purposes. It can help students understand concepts of energy, frequency, and gravity in a hands-on and visual way. It is also a popular attraction at science museums and fairs.
## 5. Can a Large Pendulum Wave be created with any type of pendulum?
Yes, a Large Pendulum Wave can be created with any type of pendulum, as long as they are of equal length and weight and have a free range of motion. However, the wave may not be as pronounced or long-lasting if the pendulums are not specifically designed for this purpose.
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As of May 31, 2023, we have updated our Code of Conduct.
# Questions tagged [tpi]
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### TPI calculation by PDAL [closed]
Is there a convenient way to calculate the topographic position index by PDAL? I have LAS file and what I do is iteratively for every point calculate the TPI. The coordinate information I take from ...
79 views
### Calculate TPI for irregular DEM
When I use SAGA's Tool Topographic Position Index (TPI) or Tool TPI Based Landform Classification (in QGIS or SAGA itself) on a rectangular DEM the results are as expected but when I try them on a DEM ...
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### quite different results in spatialEco::tpi with circular and rectangular window
In R, I have an elevation raster. I want to calculate the topogographic position index using the spatialEco::tpi function. The CRS of the elevation raster is "+init=epsg:5677"; the CRS unit ...
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### Stripes in TPI from SRTM 30 DEM
The TPI is calculated with a 150 to 300m annulus in Google Earth Engine. The underlying Elevation-Data from SRTM does not show such striped. I did use raw SRTM from GEE for elevation, no reprojecting ...
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### Using SAGA GIS for TPI classification?
I processed one country with TPI, and I got a certain Landforms classification of terrain. (Topographic Position Index (TPI) calculation as proposed by Guisan et al. (1999)). When I processed only one ...
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### How to merge pixel values to the nearest maximum value based on area in a classified raster?
We have a raster image with 9 discrete values. Generated the image using SAGA's Terrain Analysis -> Morphometry -> TPI classification technique. Now, like to merge pixels which occupy certain area (...
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### How to conduct logistic regression between two rasters [closed]
I have two rasters. The first raster is a region showing sinkholes(depressions) of the region(cell values are 1 for sinkholes and 0 for non-sinkhole cells). The second raster is the TPI(Topographic ...
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1 vote
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### Will Topographic Position Index (Tpi) not work in scattered raster layers?
I have a non-continuous elevation raster layer and want to use tpi in the spatialEco package in R. The elevation raster is in EPSG:5677 and looks like this: I always get this (to me) rather cryptic ...
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## Saturday, August 23, 2008
### CONTRIBUTIONS OF ANCIENT INDIA: MATHEMATICS
The study of mathematics had reached a lofty height in ancient India. Here is a small account of the contributions of ancient Indian mathematicians to the world.
The concept of 0 (zero) was introduced in India as early as the 3rd century B.C. It was passed on to the Arabian scholars and also to the Chinese in the 9th century A.D. European scholars grasped the inmportance of zero in the 10th century A.D.
The value of “pi” was first calculated by the Indian Mathematician Budhayana, and he explained the concept of what is known as the Pythagorean Theorem. He discovered this in the 6th century, which was long before the European mathematicians.
Algebra, trigonometry and calculus also orignated from India. Quadratic equations were introduced by Aryabhatta-I. Quadratic equations were also used by Sridhar Acharya in the 11th century. The largest numbers the Greeks and the Romans used were 106 whereas Hindus used numbers as big as 10*53 ( i.e 10 to the power of 53 ) with specific names as early as 5000 B.C. during the Vedic period. Even today, the largest used number is Tera: 10*12( 10 to the power of 12 ).
The earliest available writing on astrology is Jatakatilaka by Sridhar Acharya, a Jain who also authored a work on general science called Sastrakavita and was patronised by Western Chalukya King Somesvara I.
Aryabhatta (AD 476 – 550) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. Aryabhata is the father of the Hindu-Arabic number system which has become universal today. His most famous works are the Aryabhatiya (AD 499 at age of 23 years) and Arya-Siddhanta.
Brahmagupta was an Indian mathematician and astronomer. Brahmagupta’s most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. Brahmagupta had very important contributions in varied branches of mathematics as linear equations, quadratic equations, indeterminate equations, series theory, the number zero, Diophantine analysis, Pythagorean triples, Pell’s equation, geometry and trigonometry.
Unknown said...
i felt greatby knowing about them thank you
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## Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)
Published by Pearson
# Chapter 3 - Introduction to Graphing - 3.3 Graphing and Intercepts - 3.3 Exercise Set - Page 179: 62
#### Answer
Refer to the graph below.
#### Work Step by Step
The graph of an equation of the form $x=h$ is a vertical line whose x-intercept is $(h, 0)$. Each point on the line has an x-coordinate of $h$. Thus, the graph of $x=6$ is a vertical line whose x-intercept is $(6, 0)$. Each point on the line has an x-coordinate of $6$. This means that the following points are all on the line: $(6, -2)$, $(6, 0)$, and $(6, 2)$ Plot the three points listed above then connect them using a line to complete the graph. (refer to the attached image in the answer part above for the graph)
After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.
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# 500 in Words
500 in words is written as Five Hundred. In both the International System of Numerals and the Indian System of Numerals, 500 is written as Five Hundred. The number 500 is a Cardinal Number as it represents some quantity. For example, “Rita went to the market and bought 500 kg of potatoes”. The number 500 kg represents the weight of potatoes.
500 in Words Five Hundred Five Hundred in Number 500
## 500 in English Words
We write 500 in English Words using the letters of the English alphabet. Therefore, we read 500 in English as “Five Hundred.”
## How to Write 500 in Words?
To write 500 in words, we shall use the place value chart. In the place value chart, write 0 in the ones place, 0 in the tens place, and 5 in the hundreds place. Now let us make a place value chart to write the number 500 in words.
Hundreds Tens Ones 5 0 0
Thus, we can write the expanded form as
5 × Hundred + 0 × Ten + 0 × One
= 5 × 100 + 0 × 10 + 0 × 1
= 500 + 0 + 0
= 500
= Five Hundred
500 is a natural number which is the successor of 499 and the predecessor of 501.
500 in words – Five Hundred
• Is 500 an odd number? – No
• Is 500 an even number? – Yes
• Is 500 a perfect square number? – No
• Is 500 a perfect cube number? – No
• Is 500 a prime number? – No
• Is 500 a composite number? – Yes
## Frequently Asked Questions on 500 in Words
### How to write 500 in words?
500 in words is written as Five Hundred.
### How to write 500 in the International and Indian System of Numerals?
In both the system of numerals, 500 is written as Five Hundred.
### How to write 500 in a place value chart?
In the place value chart, write 0 in both ones and tens place respectively and then 5 in the hundreds place.
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# station 1 electro static box
STATION 1
ELECTRO STATIC BOX
Notice the 2 Gold Foil leaves attached to the
end of the electrode.
Charge your plastic rod with fur by rubbing it
vigorously. Then move the rod towards the
electrode watching what the leaves do.
After touching the rod to the electrode
repeat the experiment using the silk
PRODUCT- DIAGRAM THE FLOW OF
CHARGES FROM THE ROD TO THE
LEAVES OF THE BOX
11. In an experiment, three microscopic latex spheres are sprayed into a chamber and become
charged with !3e, !5e, and "3e, respectively. Later, all three spheres collide simultaneously and
then separate. Which of the following are possible values for the final charges on the spheres?
X
Y
Z
(A)+4e
-4e
+5e
(B) -4e
+4.5e
+4.5e
(C) +5e
-8e
-7e
(D) +6e
+6e
-7e
(E) -6
+3
+3
STATION 3
PITH BALLS
In this set up you have 1 pith ball hanging by
itself, then two with balls hanging together.
Charge the plastic rod by rubbing it with the
fur pad. Then move it towards the individual
pith ball and note what happens.
Discharge the pith ball by touching it with
Now experiment with the double pith balls
by charging one ball, then both balls.
PRODUCT DRAW THE CHARGE TRANSFER
BETWEEN THE ROD AND BALLS
Students charge two balloons exactly the same way to approximately the same charge on each
balloon. Balloon #1 is held in place while the force on balloon #2 is measured at location A.
Keeping balloon #1 held in place, balloon #2 is moved to location B and the force is measured
again. How does the force on balloon #2 change after being moved from location A to B?
A.
B.
C.
D.
The force increases and remains in the same direction
The force increases and changes direction
The force decreases and remains in the same direction
The force decreases and changes direction
STATION 2
ELECTRIC FIELDS
Measure the distance between the 2 metal
pins on the conductive paper in meters.
Voltmeter to measure the amount of volts at
different places on the conductive paper.
Create a diagram by connecting areas of
similar voltage (equi potential lines)
Examine how the field is shaped.
Two small spheres have equal charges (q) and are separated by a distance (d).
PRODUCT DIAGRAM OF THE VARYING FIELDS The force exerted on each sphere by the other has a magnitude F. If the
CREATED BY THE TWO CHARGES
charges on each sphere are doubled and the distance is halved the force on
each sphere has a magnitude of:
a. F
b. 2F
c. 4F
d. 8F
e. 16F
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# Lesson 1
Human Frequency Table
These materials, when encountered before Algebra 1, Unit 3, Lesson 1 support success in that lesson.
### Lesson Narrative
The mathematical purpose of this lesson is to create and interpret data from a two-way table. Previously, students used tables to organize and interpret data. Students also encountered two-way tables in grade 8 when they used them to summarize data and find associations between variables. This lesson prepares students to use two-way tables to interpret and analyze data from a two-way table in the associated Algebra 1 lesson. Students look for and make use of structure (MP7) when they practice changing perspectives to focus on various aspects of two-way tables.
### Learning Goals
Teacher Facing
• Create two-way tables
• Use a two-way table to interpret data
### Student Facing
• Let’s use tables to organize data.
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# How Long Will It Take to Pay Off Your Loan?
If you’re wondering how long it will take to pay off your loan , you’re not alone. Many borrowers struggle to estimate their repayment timeline, especially if they have a high interest rate .
Luckily, there are some simple calculations you can do to get a better idea of how long it will take to pay off your loan. In this blog post, we’ll walk you through the steps so you can start planning your repayment strategy.
Checkout this video:
## Introduction
When you’re trying to pay off debt, it’s helpful to have a plan. This calculator can show you how long it will take to pay off your credit card debt with different monthly payment amounts and interest rates. Enter your loan information and click “Calculate” to see your results.
## How long will it take to pay off your loan?
Are you wondering how long it will take to pay off your loan? The answer depends on a few factors, including the size of your loan , the interest rate, and the length of your repayment term. Use this calculator to estimate how long it will take to pay off your loan .
### The length of your loan
The length of your loan plays a big role in how much you’ll ultimately spend on interest. That’s because the longer you borrow money, the more time interest has to accrue. That said, shorter-term loans often come with higher interest rates, so it’s important to strike a balance that works for your unique financial situation.
As a general rule of thumb, it will take you about five years to pay off a \$10,000 loan if you make \$200 monthly payments at an 8% interest rate. But if you extend your loan payoff timeline to 10 years, your monthly payments will go down to \$100— even though you’ll be paying more in interest overall.
Of course, these are just ballpark estimates— the actual amount of time it takes to pay off your loan will depend on the size of your monthly payments and the total amount you borrowed. Use our Loan Payoff Calculator to see how different variables affect your loan’s repayment timeline.
### The interest rate on your loan
The interest rate on your loan can have a big impact on how long it takes to pay it off. A higher interest rate means you’ll have to pay more interest over the life of the loan, and it will take longer to pay off the principal. Conversely, a lower interest rate means you’ll have to pay less interest and you can pay off the loan faster.
### The monthly payment on your loan
The monthly payment on your loan will depend on the total amount you borrowed, the interest rate, and the length of time you have to repay the loan. You can use our online calculator to estimate your monthly payment.
To calculate your monthly payment, enter the following information into the calculator:
-The total amount you borrowed
-The annual percentage rate (APR) of the loan
-The length of time you have to repay the loan (in years)
Click “Calculate” to see your monthly payment.
## Conclusion
To sum it up, how long it will take to pay off your loan depends on a variety of factors, including the type of loan, the interest rate, the repayment term, and your personal financial situation. While there is no one-size-fits-all answer, you can use this information to estimate how long it will take you to pay off your loan.
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LEARNATHON
III
Competition for grade 6 to 10 students! Learn, solve tests and earn prizes!
### Theory
1 Introduction to electricity 2 Electric current 3 Flow of charges 4 Conventional current 5 Electric potential difference 6 Electric circuit 7 Components in an electric circuit 8 Ohm's law 9 Experimental verification of Ohm's law 10 Conductors 11 Insulators 12 Resistance of a material 13 Electrical resistivity 14 Factors affecting electrical resistivity 15 Conductance of a material 16 System of resistors 17 Resistances in parallel 18 Combination of series and parallel connections 19 Heating effect of current 20 Joule's law of heating 21 Practical applications of heating effect 22 Other applications of heating effect 23 Electric power 24 Electrical energy 25 Mind map
### Exercises
1 Basics of electricity Difficulty: easy 2 2 Electrical appliances Difficulty: easy 2 3 Circuit connections Difficulty: easy 3 4 Electrical components Difficulty: easy 3 5 Symbols of electrical devices Difficulty: easy 2 6 Electrical instruments Difficulty: easy 2 7 Calculation of current Difficulty: medium 3 8 Effective resistance Difficulty: medium 4 9 Potential difference across the resistors Difficulty: medium 6 10 Circuit current problem Difficulty: medium 4 11 Potential difference Difficulty: medium 4 12 Resistance of a wire Difficulty: medium 4 13 Heating effect in appliances Difficulty: medium 6 14 Resistance calculation Difficulty: medium 5 15 Energy consumption Difficulty: medium 6 16 Resistance analysis Difficulty: medium 6 17 Consumption of current Difficulty: hard 6 18 Differences between electrical terms Difficulty: hard 6 19 Total resistance Difficulty: hard 7 20 Conductivity Difficulty: hard 9 21 Electrical connections Difficulty: hard 6 22 Electric bulb Difficulty: hard 7 23 Electrical components in a circuit Difficulty: hard 6
### Tests
1 Training test 1 Difficulty: medium 7 2 Training test 2 Difficulty: medium 10
### Teacher manual
1 Methodical recommendation
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Q:
# How do you convert 107 degrees Fahrenheit to Celsius?
A:
Converting degrees from Fahrenheit to Celsius is accomplished by subtracting 32 from the original number in Fahrenheit. Next, divide the answer by 1.8 to get the final answer in Celsius, which 107 degrees Fahrenheit equals 41.6 degrees Celsius. To convert from Celsius to Fahrenheit, reverse the equation.
## Keep Learning
1. Subtract 32 from the Fahrenheit temperature
To convert Fahrenheit to Celsius, the first step is to subtract 32 from the degrees Fahrenheit. For this equation, 32 subtracted from 107 equals 75.
2. Divide the answer by 1.80
Next, divide the answer from step one by 1.80 to get the final answer. For this equation, 75 divided by 1.80 equals 41.6.
To complete the conversion from Fahrenheit to Celsius, it's necessary to label the final answer. The final answer for this problem is 41.6 degrees Celsius.
Sources:
## Related Questions
• A: The formula to convert 350 degrees Fahrenheit to degrees Celsius is (5/9)(350 degrees Fahrenheit - 32) = degrees Celsius. The solution to this equation is ... Full Answer >
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• A: Converting degrees from Fahrenheit into Celsius involves just a few simple calculations. The degrees in Celsius is determined by subtracting 32 from the Fa... Full Answer >
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• A: The complete equation is C = 5/9 (F - 32), where F is the temperature in degrees Fahrenheit and C is the temperature in degrees Celsius. Subtract 32 from t... Full Answer >
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# hw12bsol-dist - Solutions CBE 320 Transport Phenomena...
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CBE 320 November 16, 2009 Transport Phenomena Problem Session XII Part B: Macroscopic Balances for Nonisothermal Systems 1. Work problem 15A.1 in BSL 2. Work problem 15A.2 in BSL 3. Work problem 15A.4 in BSL 4. Chilling Oil with an Ice Bath An incompressible oil is to be chilled using an ice bath as indicated in the figure below. Your job is to calculate the required tube length . The oil flows through the tube (I.D. 0.003175 m), entering the bath at T b 1 = 25 C. The oil is to leave the bath at T b 2 = 10 C. The oil mass flow rate is 0.01 kg/s. Since the bath is well-stirred, it is
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# Which is better: Simple or compound interest?
In document NELSON SENIOR MATHS ESSENTIALS 12 (Page 131-135)
simple or compound
interesl?
### simple interest? Is compound interest always better than
It's a common misconception that compound interest always gives a better return than simple interest. In this investigation, you are going to make some calculations to help you choose the investment with the better return.
What you have to do
• Copy the following table or print a copy from NelsonNet.
• Use the technology of your choice to complete the following 3 summaries. To make it easier, make the principal \$1000 in every calculation.
• Complete the class discussion questions after you have finished the calculations.
### Which is better: Simple or compound interest?
Part A
The interest rates and the term are the same. Simple interest investment
Term: 4 years Interest rate: 5% p.a. Term: 6 years Interest rate: 3% p.a. Term: 20 years Interest rate: 7.5% p.a.
PartB
Annually compound interest investment
Term: 4 years Interest rate: 5% p.a. Term: 6 years Interest rate: 3% p.a.
Term: 20 years Interest rate: 7.5% p.a.
Summary
Simple interest =
Compound interest =
Which investment is better?
Simple interest =
Compound interest =
Which investment is better? Simple interest =
Compound interest =
Which investment is better?
The term is the same but the compound rate is higher than the simple interest rate. Simple interest investment
Term: 4 years Interest rate: 5% p.a. Term: 6 years
Interest rate: 3.75% p.a. Term: 20 years Interest rate: 7.5% p.a.
Annually compounded interest investment
Term: 4 years Interest rate: 5.1 % p.a. Term: 6 years Interest rate: 4% p.a.
Term: 20 years Interest rate: 8.1 % p.a.
Summary
Simple interest =
Compound interest =
Which investment is better? Simple interest =
Compound interest = Which investment is better?
Simple interest =
Compound interest =
Which investment is better?
PartC
The terms are the same but the simple rate is higher than the compound interest rate.
Simple interest investment
Term: 4 years Interest rate: 5% p.a.
Term: 6 years Interest rate: 3% p.a. Term: 20 years Interest rate: 12% p.a.
Term: 3 years Interest rate: 12% p.a. Class discussion questions
Annually compounded interest investment
Term: 4 years
Interest rate: 4.8% p.a.
Term: 6 years
Interest rate: 2.75% p.a.
Term: 20 years Interest rate: 7.5% p.a.
Term: 3 years Interest rate: 7.5% p.a.
Summary Simple interest = Compound interest = Which investment is better? Simple interest =
Compound interest = Which investment is better? Simple interest =
Compound interest= Which investment is better? Simple interest =
Compound interest = Which investment is better?
• When the interest rates and the term are the same, does simple or compound interest give the better return?
• When the compound rate is higher than the simple interest rate, and the terms are the same, which type of investment produces the better return?
• When the simple rate is bigger than the compound rate, and the terms are the same, will one or the other type of interest always give the better return?
### IIJI INTEREST RATES AND SAVINGS
Compound interest is amazing. You might be surprised how quickly it helps regular savings accumulate. Banks and building societies have savings and investment accounts that pay compound interest. In addition, some accounts offer bonus interest to encourage you to deposit money without making any withdrawals.
Most bank websites and the MoneySmart website contain savings calculators. To complete the following example and exercises, you will need access to an online calculator, the spreadsheet on NelsonNet or a programmable calculator with a TVM menu.
I
### �
Monlhly saving� calculator
Monthly savings calculator
### 0 Example 6
Jessica has \$2400 in an account that pays 3.5% p.a. interest. She needs to have \$80 000 in 5 years from now. How much will she need to save per month to reach her goal?
### Solution
There are a variety of technologies we can use to solve this problem. This solution uses the Savings Term calculator on the
Commonwealth Bank website. It is located in the 'tools' section.
Enter 3.5 for the Interest Rate, 2400 for Jessica's Current Balance, 80 000 for the target Deposit Amount and 5 years for the Savings Term.
Enter:
Interest Rate
Current Balance of Savings Target Deposit Amount Savings Term Calculate
### I
Result:
Amount to Save Per Month
I 3.s %p.a. \$12400 \$180 000
years
Months Clear
\$11174.92
### I
Jessica will need to save \$1174.92 every month to reach her target.
,.
### ,,jp
Use the Commonwealth Bank's 'Monthly savings calculator' to determine how much each person needs to save every month to reach their target.
Home Interest Current Target
b Isaac 2% \$1750 \$8000
C Liam 5.1% Nil \$25 000
d Ellen 3.75% \$5000 \$45 000
e Aidan 9% Nil
Savings Monthly savings term required 4 years 3 years 5 years 6 years 5 years 6months
2 Amanda has \$4000 in her account and she needs \$20 000 in 5 years time.
a How much will she need to invest each month to reach her goal when interest rates are
3% p.a.?
b How much will she need to contribute per month if interest rates are 8% p.a.?
c If interest rates increase to 10% p.a., how much will Amanda need to contribute monthly? d Write a sentence to describe the relationship between interest rates and the amount you
need to save per month to reach a target.
Use the Commonwealth Bank's 'Savings term calculator' to answer questions 3 to 5. 3 Chelsea has \$1300 in a savings account that pays 4.2% p.a. interest. She deposits
\$300 per month into the account. How long will it take before there is \$6000 in her account? 4 Thomas has a special savings account that pays 5% p.a. interest. At the moment he only has
\$100 in the account but he is going to deposit \$500 into the account each month. When will there be more than \$18 000 in the account?
5 Chloe is going to start to save the deposit for her first home. She is going to deposit \$600 per month into an account. How much quicker will she be able to save \$50 000 when interest rates are 7% p.a. compared to 3% p.a.?
Use the Commonwealth Bank's 'Target deposit amount calculator' to answer questions 6 and 7. 6 Jonathan is saving \$320 each month in an account that pays 5% p.a. The current balance of the
account is \$4200. How much will be in the account in one and a half years time?
7 The balance in Maddison's acc_ount is \$2150. The account pays 4.7% p.a. interest and each month Maddison deposits \$420 into the account. How much will be in her account in 4 years time? 8 Group discussion and investigation question.
Imagine that you need to save \$30 000 for the deposit on your first house. a What savings interest rates are banks offering at the moment?
b How could you increase your income to make it possible to save more? c How much can you afford to save each month?
d How long will it take you to save the deposit?
e Imagine that you have a partner who can save the same amount per month that you can. How long will it take the two of you to save the deposit?
Suppose that interest rates fall when you are saving. What effect will this have on the time it will take to save the deposit?
In document NELSON SENIOR MATHS ESSENTIALS 12 (Page 131-135)
Outline
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https://www.storyboardthat.com/storyboards/e2887041/unknown-story
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# Day in the life of probability
More Options: Make a Folding Card
#### Storyboard Description
This storyboard does not have a description.
#### Storyboard Text
• Thank you
• Hey, bobby did you remember to study for the finals today.
• Sure
• Probability.
• What do you need to study anyway?
• Oh! All you have to remember is that 0 represents impossible and 1 represents certain.
• Umm? Lets go study in the library.
• Heres a question, what is the probability of you rolling a dice and getting 3?
• Good! But why is that answer correct?
• Because on a dice, there are 6 sides, and one 3, so it will be 1/6.
• Ummm, 1/6?
• To Be Continued......
• Does Bobby fail, the probability final? Find out next time on The day in the life of probability!
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### Teacher Resources
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https://www.riddlesandanswers.com/v/230990/while-building-a-medieval-cathedral-it-cost-37-guilders-to-hire-4-artists-and-3-stonemasons-or-33-g/
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# GUILDERS EXPENSE RIDDLE
#### Trending Tags
Terms · Privacy · Contact
## Guilders Expense Riddle
While building a medieval cathedral, it cost 37 guilders to hire 4 artists and 3 stonemasons or 33 guilders for 3 artists and 4 stonemasons. What would the expense of just one of each?
Hint:
Ten guilders. An artist cost 7 guilders, and a stonemason cost 3 guilders.
Did you answer this riddle correctly?
YES NO
Solved: 67%
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https://cs.stackexchange.com/questions/91202/why-is-the-complete-state-of-a-turing-machine-representable-by-a-finite-word
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# Why is the complete state of a turing machine representable by a finite word?
At the moment I'm just reading through this article on the word problem for groups (https://projecteuclid.org/euclid.bams/1183548590), and I'm wondering about a certain snippet.
On page 40 the article describes a word that can encode the state of a turing machine, namely $$S_{k_{u}}...S_{k_{1}}q_{i}S_{j_{1}}...S_{j_{v}}$$ where $$S_{k_{u}}...S_{k_{1}}S_{j_{1}}...S_{j_{v}}$$ is the tape expression, $$q_{i}$$ is the current internal state of the turing machine, and $$S_{j_{1}}$$ is the currently scanned symbol.
I'm just wondering why this word is necessarily finite. Why can't the input of the turing machine have an infinite amount of non-zero characters?
• Welcome on CS.SE. Note that we prefer plain text to text-as-an-image. When asking, one is expected to cut & paste the text (and possibly format the formulas using MathJax).
– chi
Commented Apr 27, 2018 at 10:34
• Ah, no problem, I'll quickly fix that up now Commented Apr 27, 2018 at 10:38
• Awesome, I've just changed it now, be sure to tell me if there's anything else I need to do to make this question better. Commented Apr 27, 2018 at 10:45
• I think it is OK now. It is self-contained: the link is a bonus -- useful, but the question can be answered even without accessing the link. It is short and to the point, which is always a good thing.
– chi
Commented Apr 27, 2018 at 10:48
I'm just wondering why this word is necessarily finite. Why can't the input of the turing machine have an infinite amount of non-zero characters?
No, it can not.
When the Turing machine starts, the tape contains the input (a finite-length word) followed by an infinite amount of blank symbols (often written as #). It is common to simply omit the trailing infinitely many blanks, and just represent a finite prefix instead, adding blanks as needed when the head moves right past the end of the tape.
When the machine runs, in each step it can at most (over-)write over one tape symbol, and move right by at most one position. This is an essential feature of any Turing machine.
Hence, after finitely many steps, we still have a finite amount of non-blank symbols. The tape again contains infinitely many trailing blanks (usually not represented), so it can be represented by a suitable finite prefix.
• "The tape starts with the input (a finite-length word) followed by an infinite amount of blank symbols (often written as #)." Isn't the tape infinitely long on both the left and right though? Commented Apr 27, 2018 at 10:24
• @user3201708 It depends on the actual definition, above I assume it's only unbounded on the right. Anyway, the whole argument can be adapted: after finite time, moving one position at a time, writing one cell at a time, the tape will still contain infinitely many blanks on the left and the right.
– chi
Commented Apr 27, 2018 at 10:26
• Sorry to be a pain, but is there any reason you only assumed that the turing machine is unbounded on the right? I can see that the two definitions should have equivilant results, but is having a bounded left standard or something? Commented Apr 27, 2018 at 10:30
• @user3201708 Both are somehow standard. Their expressive power is the same: there's no function which can be computed by one the other can not compute. Actually, you'll probably find other equivalent variants when reading about TM theory: multi-tape TMs (e.g. with a read-only input tape, a write-only output tape, and several working tapes) are a common variant, for instance.
– chi
Commented Apr 27, 2018 at 10:37
• Ah, awesome. Thanks for all your help! I wish I could upvote you, but apparently I don't have enough reputation as of yet, hopefully someone else will come along and do the job for me! Commented Apr 27, 2018 at 10:40
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https://www.doityourself.com/stry/crown-molding-cost-estimation
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# Crown Molding Cost Estimation
Estimating crown molding cost is important in order to know how much you should expect to pay for crown molding that you are looking to install in your home as a do-it-yourself project. In order to estimate the cost of crown molding you need to know a few things about the room in which the crown molding is being installed.
#### Measure the Room
It is necessary to measure the room where the crown molding installation will be installed. This does not necessarily mean that you need to climb up on a ladder to measure the top of the wall near the ceiling where the crown molding will be installed. For most homes the measurement can be taken at the baseboards. This will make it easier for you to obtain an accurate measurement necessary to translate it into the cost of crown molding needed for the room. When taking measurements of a room where the crown molding will be installed, make sure to measure all of the walls and opening in order to obtain the most accurate measurement. Be sure to also measure the room twice.
#### Calculate the Total Linear Feet
Once you have obtained the measurement of the room or rooms where the crown molding will be installed, you need to take those measurements and add them together. This will help you calculate the total number of linear feet the room is (linear feet is not the same as square feet, which takes the length of the room times the width). Calculating the linear size of the room only requires you to sum the total measurements. For example, if you have a room with 4 walls and measure them as follows; 10 feet, 10 feet, 8 feet, 12 feet, the total linear feet of the room is 40 feet (10 + 10 + 12 + 8 = 40) You will also need to round your numbers up to the nearest whole foot; for example a measurement of 8 feet, 7 inches would come out to 9 feet.
#### Using an Online Cost Estimator
If you go online you can find numerous websites that are available to provide you with the cost estimation for crown molding. The cost estimation calculators that are available online provide you with an estimation of cost for the crown molding including the cost of labor and materials. It should be noted that with any estimate, particularly those generated online, it can not truly reflect the costs for your given region of the country where the costs of materials and labor may be higher than a national average in which these costs are based on.
Sample Cost Estimates
Using the information provided by 1 website that calculates crown molding estimates, here are the costs for a room of 30, 40, 50, 75, 100, 125, 150 and 200 linear feet:
- 30 linear feet, \$142.50
- 40 linear feet, \$190.00
- 50 linear feet, \$237.50
- 75 linear feet, \$356.25
- 100 linear feet, \$475.00
- 125 linear feet, \$593.75
- 150 linear feet, \$712.50
- 200 linear feet, \$950.00
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https://www.tutorialsrack.com/articles/268/python-program-to-multiply-two-matrices
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;
# Python Program to Multiply Two Matrices
###### Tutorialsrack 23/04/2020 Python
In this Python program, we will learn how to multiply two matrices.
Here is the code of the program to multiply two matrices.
### Program 1: Program to multiply two matrices using nested For loops
##### Program to multiply two matrices using nested For loops
``````# Program to multiply two matrices using nested For loops
# 3x3 matrix
X = [[6,7,3],
[4 ,11,6],
[7 ,8,9]]
# 3x4 matrix
Y = [[5,8,1,2],
[6,7,3,0],
[4,5,9,1]]
# Output is 3x4
result = [[0,0,0,0],
[0,0,0,0],
[0,0,0,0]]
# iterate through rows of X
for i in range(len(X)):
# iterate through columns of Y
for j in range(len(Y[0])):
# iterate through rows of Y
for k in range(len(Y)):
result[i][j] += X[i][k] * Y[k][j]
print("\nOutput of Multiplication of Two Matrices:")
for r in result:
print(r)
``````
##### Output
Output of Multiplication of Two Matrices:
[84, 112, 54, 15]
[110, 139, 91, 14]
[119, 157, 112, 23]
### Program 2: Program to multiply two matrices using list comprehension
##### Program to multiply two matrices using list comprehension
``````# Program to multiply two matrices using list comprehension
# 3x3 matrix
X = [[6,7,3],
[4 ,11,6],
[7 ,8,9]]
# 3x4 matrix
Y = [[5,8,1,2],
[6,7,3,0],
[4,5,9,1]]
# Output is 3x4
result = [[sum(a*b for a,b in zip(X_row,Y_col)) for Y_col in zip(*Y)] for X_row in X]
print("\nOutput of Multiplication of Two Matrices:")
for r in result:
print(r)
``````
##### Output
Output of Multiplication of Two Matrices:
[84, 112, 54, 15]
[110, 139, 91, 14]
[119, 157, 112, 23]
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# 1. Doing practical work in groups
Estimation is an important skill in both mathematics and science and a useful skill to cultivate for all pupils. Simple balances can be made with very modest resources that allow pupils to approach measuring and estimating weight through practical investigation. You may like to make simple balances and plan and carry out these activities jointly with a science teacher in your school. This can be done by helping pupils to compare and contrast weights in different ways.
## Case Study 1: Estimating weight
Mrs Nkumu in Nigeria was on a teachers’ course at her local district offices and as part of the day’s lesson on numeracy the facilitator told the following story to them. Then she asked them what they thought the girls knew and what would they do next with these pupils if they were in their class.
‘Two girls, Ranke and Ade, were discussing the quantity of popcorn in two packets, A and B which looked the same shape and size. Ronke picked up the packets one after the other and was surprised that B felt heavier than A. She told Ade that B appeared to be heavier than A. Ade decided to put the two packets in the two pans of a simple balance. (See Resource 1: Simple balance). She observed that on the scale, packet B went down and so B is heavier than A. Ronke was right.’
The teachers worked in pairs and devised activities that encouraged estimation of heavier/lighter, then using a balance or scales to test their ideas. Each pair tried their lesson out with their class and reported back next session.
Mrs Nkumu found that her class enjoyed the lesson but that she did not have enough different objects for the pupils to use. Next time she said she would spend more time collecting objects and she would use smaller groups of 4-6 rather than over 10 each.
## Activity 1: Comparing weight
You will need 5 simple balances (see Resource 1) to carry out this activity and 5 sets of common objects e.g. stones, balls, tins, bottle tops etc that could be used with the balances.
Write instructions for your pupils on the board (see Resource 2: Worksheets) and show the class what you want them to do using any two objects.
Ask them to estimate which is heavier by giving the objects to 2 pupils to answer.
Now ask a pupil to test their idea out by putting them on the pan and let them decide.
Ask them which is the heaviest object and why they think so.
Organise your pupils into 5 groups, giving each group a set of objects and a balance. Ask pupils to find which object is heavier by estimating its weight and then by using the balance. (See Key Resource: Using group work in the classroom [Tip: hold Ctrl and click a link to open it in a new tab. (Hide tip)] )
Ask them to fill in a table of their results to share with the class to see if everyone agrees.
You could challenge older or more able pupils to see if they can order their objects from heaviest to lightest before testing. How could they test their answers using the simple balance?
Section 4: Working with weight
2. Introducing units to compare weights
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In the first part of this series, we looked at the basics of color — how we see color, and how we can represent it numerically in the various systems that deal with color in a quantitative manner. Now let’s look at what it takes for a display to be color-accurate, and why this can be a particular challenge in mobile devices.
Looking ahead, in the third and final part of the series, we’ll wrap up with some consideration of how the whole video chain contributes to the ability to deliver the right color.
So what do we mean by 'accurate color' in these terms, and what does the display have to do — and be — to produce it?
Remember this diagram from the first article? It’s a chart of the color coordinate system we introduced then, technically referred to as the CIE xy chromaticity diagram, or a representation of the Yxy color space.We’re taking a look at just what it takes for a display to be color-accurate, and why this can be a particular challenge in mobile devices. (Remember the “Y” here is luminancewhat we usually think of as “brightness,” and in this diagram would be the third dimension — an axis sticking “up out of the screen,” at right angles to the plane of the diagram shown.)
The triangle shown in this diagram is the color gamut that you get from the three primary colors at the corners of the triangle; in other words, the range of colors you can produce through various combinations of these three colors. So what do we mean by “accurate color” in these terms, and what does the display have to do — and be — to produce it?
This “space” (the total possible range of all Y, x, and y values) was derived from the curves that describe how the eye sees color in the first place, and so it covers the full range of color and brightness values that the eye can see. The full Yxy space is actually a three-dimensional volume, which turns out to be rather oddly shaped, as shown below.
The full three-dimensional Yxy color space
The important thing here, though, is that any color you can see is somewhere within that space.
We don’t often see the full 3D volume used in this sort of discussion, because of the obvious difficulties of accurately showing what’s going on in a 3D space through a 2D medium. So from here on out, I’ll also be using the simpler 2D xy diagram; just keep in mind that we’re actually talking about things that really need three numbers to correctly describe.
Since any particular display only has three primary colors to play with, we’re always going to see display gamuts as triangles within this space as we saw above. No display with any reasonable number of practical primary colors can ever hope to cover all the possible colors the eye can see. Their color gamuts will always be less than the full color space.
This doesn’t necessarily mean the best color possible comes from widest/largest color gamut we can get. Image capture devices (cameras) also have limits of their own, like any other delivery medium such as print or film. So the people who create the various sorts of image content, like movies and photographs, pretty much always work within an established standard color space. The term “color space”refers to both the total range of possible colors, as in the Yxy space we’ve been talking about, as well as the specific regions within that space that these various standards define. The most common standard space currently for digital photography is still the sRGB space, originally defined by HP and Microsoft back in 1996. It also just so happens the standard color space for digital television, a spec commonly known as “Rec. 709,” uses the same primaries as sRGB. The gamut for both of these is shown in the xy diagram above.
Neither standard is what you’d call a “wide gamut” spec, but both are larger than what’s provided by a lot of smartphone and tablet displays, especially LCDs. One of the advantages provided by OLED technology may be a wider color gamut. If you’re dealing with material, whether video or still images, created with the sRGB/Rec. 709 primaries in mind, you ideally want the display to use those same primaries. You clearly don’t want a smaller gamut, since then some colors in the image data simply wouldn’t be possible produced by the display. However, smaller-than-standard gamuts have long been the norm in mobile devices.
Smaller-than-standard gamuts have long been the norm in mobile devices
Using less saturated primaries (with more “white” in its makeup) makes for a brighter display, all else being equal, and more brightness for a given backlight level makes for longer battery life, always a key selling point for these products.
A wider-gamut display (and remember that a lot of displays are being marketed on the strength of having a really wide gamut) can be just as bad, too. Let’s say you’re dealing with a given image created assuming the sRGB standard is to be used. If some pixels in that image have RGB values of (255,0,0) — which just means “this pixel is supposed to be pure red” — what happens when the display uses the primaries shown in the diagram below?
The display will still give you a “pure red,” but it’s very different to the one whoever created the image (and was assuming the sRGB primaries) intended. It’s a purer, more saturated, more intense red. So even though the display’s gamut exceeded what’s required for sRGB, it still isn’t necessarily accurate.
Show a “pure red” – meaning RGB values of 255,0,0 – on a display with a different gamut than intended, and you get the wrong color. And that sort of error happens for any color in the space.
A few other major concerns determine whether or not a display is color accurate. Even if all of the primaries are spot-on, the display can still have problems with accuracy. If those pixels we were looking at earlier had RGB codes of (255,255,255) — all three colors set to their maximum level — generally we could assume it would mean “white,” but which white is intended?
Different color standards specify different “white points,” so the brightness of the three primaries at their maximums have to be set in the right relationship. The sRGB and Rec. 709 standards, both specify what’s known as the “D65” white (also often referred to as a “6500K color temperature”). Using the primaries specified for these, the relative brightness of each primary in terms of how much they contribute to the white is roughly 60 percent green, 30 percent red, and only 10 percent blue. If the maximum brightness of each primary isn’t controlled to hit these relative values, every color other than the pure primaries will be off to some degree, even though the primaries are dead on.
One last major source of color error has to do with the tone response, more commonly known as the 'gamma curve.'
One last major source of color error has to do with the tone response, commonly known as the “gamma curve,” of each of the primary channels. As covered in my article last November, you don’t want a display to give a straight linear response to the input signal — it’s supposed to respond along a specific curve. These color standards also describe the expected display response. It’s usually roughly equivalent to a “gamma” value somewhere in the range of 2.2 – 2.5. All three primary channels should provide the same response curve. If any of the three is a little high or a little low at any point in the response, that will result in color error whenever it’s called for. In the monitor and TV markets, where having the primaries match the sRGB/Rec. 709 set pretty closely is actually the norm, response curve errors across the primaries are often the biggest single cause of color error.
Speaking of color error, let’s talk about how the pros express just how much error you’re getting in a given situation. For any color a display is asked to make, there’s both the color it was supposed to be, and the color it actually displayed. Both of those, can be specified in terms of their color coordinates in a given space. So the most obvious way to express color error is simply to calculate how far apart these two points are in a given space.
A ΔE* value of 1.0 is supposed to represent a 'just noticeable difference,' or JND. It's just enough error for the human eye to see the difference in the two colors if you put areas of each color side-by-side.
This number is expressed as a value called “ΔE*“, commonly read as “delta E star.” The coordinate system and calculations used to get this value are intended to make it perceptually correlated, which just means the relative size of the ΔE* value corresponds to how far off you perceive the color to be. A ΔE* value of 1.0 is supposed to represent a “just noticeable difference,” or JND. It’s just enough error for the human eye to see the difference in the two colors if you put them side by side. A value of 5-10 represents a color error that’s fairly easy to detect, and anything that gets into the 10-20 range is pretty obviously wrong if compared with the intended or reference color.
Having looked at what’s needed (just not always achieved) for a display to be accurate, we’re ready to tie this all together. Stay tuned for Part 3, where we’ll be covering how color accuracy is — finally! — coming to the mobile device markets, and how Android now includes the features to enable this.
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CC-MAIN-2019-51
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longest
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https://www.numbersaplenty.com/1434739218179
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Search a number
1434739218179 is a prime number
BaseRepresentation
bin10100111000001101000…
…111110001011100000011
312002011022102221221000012
4110320031013301130003
5142001321214440204
63015035440415135
7205441114064351
oct24701507613403
95064272857005
101434739218179
11503518981063
121b208b3174ab
13a53abab1787
144d6282307d1
15274c2d7b76e
hex14e0d1f1703
1434739218179 has 2 divisors, whose sum is σ = 1434739218180. Its totient is φ = 1434739218178.
The previous prime is 1434739218169. The next prime is 1434739218203. The reversal of 1434739218179 is 9718129374341.
It is a weak prime.
It is a cyclic number.
It is not a de Polignac number, because 1434739218179 - 24 = 1434739218163 is a prime.
It is not a weakly prime, because it can be changed into another prime (1434739218109) by changing a digit.
It is a pernicious number, because its binary representation contains a prime number (19) of ones.
It is a polite number, since it can be written as a sum of consecutive naturals, namely, 717369609089 + 717369609090.
It is an arithmetic number, because the mean of its divisors is an integer number (717369609090).
Almost surely, 21434739218179 is an apocalyptic number.
1434739218179 is a deficient number, since it is larger than the sum of its proper divisors (1).
1434739218179 is an equidigital number, since it uses as much as digits as its factorization.
1434739218179 is an odious number, because the sum of its binary digits is odd.
The product of its digits is 9144576, while the sum is 59.
The spelling of 1434739218179 in words is "one trillion, four hundred thirty-four billion, seven hundred thirty-nine million, two hundred eighteen thousand, one hundred seventy-nine".
| 511
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| 2.984375
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https://labs.tib.eu/arxiv/?author=Ben%20Adcock
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• Frames and numerical approximation(1612.04464)
Nov. 5, 2018 math.NA, cs.NA
Functions of one or more variables are usually approximated with a basis: a complete, linearly-independent system of functions that spans a suitable function space. The topic of this paper is the numerical approximation of functions using the more general notion of frames: that is, complete systems that are generally redundant but provide infinite representations with bounded coefficients. While frames are well-known in image and signal processing, coding theory and other areas of applied mathematics, their use in numerical analysis is far less widespread. Yet, as we show via a series of examples, frames are more flexible than bases, and can be constructed easily in a range of problems where finding orthonormal bases with desirable properties (rapid convergence, high resolution power, etc.) is difficult or impossible. A key concern when using frames is that computing a best approximation requires solving an ill-conditioned linear system. Nonetheless, we construct a frame approximation via regularization with bounded condition number (with respect to perturbations in the data), and which approximates any function up to an error of order $\sqrt{\epsilon}$, or even of order $\epsilon$ with suitable modifications. Here $\epsilon$ is a threshold value that can be chosen by the user. Crucially, rate of decay of the error down to this level is determined by the existence of approximate representations of $f$ in the frame possessing small-norm coefficients. We demonstrate the existence of such representations in all of our examples. Overall, our analysis suggests that frames are a natural generalization of bases in which to develop numerical approximation. In particular, even in the presence of severely ill-conditioned linear systems, the frame condition imposes sufficient mathematical structure in order to give rise to accurate, well-conditioned approximations.
• Compressed sensing with sparse corruptions: Fault-tolerant sparse collocation approximations(1703.00135)
Aug. 30, 2018 math.NA
The recovery of approximately sparse or compressible coefficients in a Polynomial Chaos Expansion is a common goal in modern parametric uncertainty quantification (UQ). However, relatively little effort in UQ has been directed toward theoretical and computational strategies for addressing the sparse corruptions problem, where a small number of measurements are highly corrupted. Such a situation has become pertinent today since modern computational frameworks are sufficiently complex with many interdependent components that may introduce hardware and software failures, some of which can be difficult to detect and result in a highly polluted simulation result. In this paper we present a novel compressive sampling-based theoretical analysis for a regularized $\ell^1$ minimization algorithm that aims to recover sparse expansion coefficients in the presence of measurement corruptions. Our recovery results are uniform, and prescribe algorithmic regularization parameters in terms of a user-defined a priori estimate on the ratio of measurements that are believed to be corrupted. We also propose an iteratively reweighted optimization algorithm that automatically refines the value of the regularization parameter, and empirically produces superior results. Our numerical results test our framework on several medium-to-high dimensional examples of solutions to parameterized differential equations, and demonstrate the effectiveness of our approach.
• Optimal sampling rates for approximating analytic functions from pointwise samples(1610.04769)
April 5, 2018 math.NA
We consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in $M$, and that any method with faster exponential convergence must also be exponentially ill-conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith & Rivlin concerning the maximal behaviour of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points, and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results -- based on computing the maximal polynomial via a variant of the classical Remez algorithm -- confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree $N < M$ is possible only once $M$ is sufficiently large for there to be a subset of $N$ of the nodes that mimic the behaviour of the $N$th set of Chebyshev nodes.
• Uniform Recovery from Subgaussian Multi-Sensor Measurements(1610.05758)
Feb. 15, 2018 cs.IT, math.IT, math.FA
Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.
• Frames and numerical approximation II: generalized sampling(1802.01950)
Feb. 6, 2018 math.NA
In a previous paper [Adcock & Huybrechs, 2016] we described the numerical properties of function approximation using frames, i.e. complete systems that are generally redundant but provide infinite representations with coefficients of bounded norm. Frames offer enormous flexibility compared to bases. We showed that, in spite of extreme ill-conditioning, a regularized projection onto a finite truncated frame can provide accuracy up to order $\sqrt{\epsilon}$, where $\epsilon$ is an arbitrarily small threshold. Here, we generalize the setting in two ways. First, we assume information or samples from $f$ from a wide class of linear operators acting on $f$, rather than inner products with the frame elements. Second, we allow oversampling, leading to least-squares approximations. The first property enables the analysis of fully discrete approximations based, for instance, on function values only. We show that the second property, oversampling, crucially leads to much improved accuracy on the order of $\epsilon$ rather than $\sqrt{\epsilon}$. Overall, we show that numerical function approximation using truncated frames leads to highly accurate approximations in spite of having to solve an ill-conditioned system of equations. Once the approximations start to converge, i.e. once sufficiently many degrees of freedom are used, any function $f$ can be approximated to within order $\epsilon$ with coefficients of small norm.
• Approximating smooth, multivariate functions on irregular domains(1802.00602)
Feb. 2, 2018 math.NA
In this paper, we analyze a method known as polynomial frame approximation for approximating smooth, multivariate functions defined on irregular domains in $d$ dimensions, where $d$ can be arbitrary. This method is simple, and relies only on orthogonal polynomials on a bounding tensor-product domain. In particular, the domain of the function need not be known in advance. When restricted to a subdomain, an orthonormal basis is no longer a basis, but a frame. Numerical computations with frames present potential difficulties, due to the near-linear dependence of the finite approximation system. Nevertheless, well-conditioned approximations can be obtained via regularization; for instance, truncated singular value decompositions. We comprehensively analyze such approximations in this paper, providing error estimates for functions with both classical and mixed Sobolev regularity, with the latter being particularly suitable for higher-dimensional problems. We also analyze the sample complexity of the approximation for sample points chosen randomly according to a probability measure, providing estimates in terms of the Nikolskii-type inequality for the domain. For a large class of nontrivial domains, we show that the sample complexity for points drawn from the uniform measure is quadratic in the dimension of the polynomial space, independently of $d$.
• Compressed sensing approaches for polynomial approximation of high-dimensional functions(1703.06987)
June 9, 2017 math.NA
In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on $d$-dimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted $\ell^1$ minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension $d$. Hence the curse of dimensionality - the bane of high-dimensional approximation - is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.
• Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class(1601.01988)
June 2, 2017 math.NA, cs.IT, math.IT
In compressed sensing, it is often desirable to consider signals possessing additional structure beyond sparsity. One such structured signal model - which forms the focus of this paper - is the local sparsity in levels class. This class has recently found applications in problems such as compressive imaging, multi-sensor acquisition systems and sparse regularization in inverse problems. In this paper we present uniform recovery guarantees for this class when the measurement matrix corresponds to a subsampled isometry. We do this by establishing a variant of the standard restricted isometry property for sparse in levels vectors, known as the restricted isometry property in levels. Interestingly, besides the usual log factors, our uniform recovery guarantees are simpler and less stringent than existing nonuniform recovery guarantees. For the particular case of discrete Fourier sampling with Haar wavelet sparsity, a corollary of our main theorem yields a new recovery guarantee which improves over the current state-of-the-art.
• Robustness to unknown error in sparse regularization(1705.10299)
May 29, 2017 math.NA, cs.IT, math.IT
Quadratically-constrained basis pursuit has become a popular device in sparse regularization; in particular, in the context of compressed sensing. However, the majority of theoretical error estimates for this regularizer assume an a priori bound on the noise level, which is usually lacking in practice. In this paper, we develop stability and robustness estimates which remove this assumption. First, we introduce an abstract framework and show that robust instance optimality of any decoder in the noise-aware setting implies stability and robustness in the noise-blind setting. This is based on certain sup-inf constants referred to as quotients, strictly related to the quotient property of compressed sensing. We then apply this theory to prove the robustness of quadratically-constrained basis pursuit under unknown error in the cases of random Gaussian matrices and of random matrices with heavy-tailed rows, such as random sampling matrices from bounded orthonormal systems. We illustrate our results in several cases of practical importance, including subsampled Fourier measurements and recovery of sparse polynomial expansions.
• Recovery guarantees for compressed sensing with unknown errors(1702.04424)
May 9, 2017 math.NA
From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori estimates of the noise, which can be very hard to obtain in practice, especially when the noise term also includes unknown discrepancies between the finite model and data. In this work, we study the performance of l1-minimization when these estimates are not available, providing robust recovery guarantees for quadratically constrained basis pursuit and random sampling in bounded orthonormal systems. Several applications of this work are approximation of high-dimensional functions, infinite-dimensional sparse regularization for inverse problems, and fast algorithms for non-Cartesian Magnetic Resonance Imaging.
• Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates(1606.07698)
May 8, 2017 math.NA
We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example. Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries. The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum. As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data.
• Infinite-dimensional compressed sensing and function interpolation(1509.06073)
Jan. 21, 2017 math.NA, cs.IT, math.IT
We introduce and analyze a framework for function interpolation using compressed sensing. This framework - which is based on weighted $\ell^1$ minimization - does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees, and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted $\ell^1$ minimization based on this framework. This guarantee has several key benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches.
• Compressed Sensing and Parallel Acquisition(1601.06214)
Dec. 17, 2016 cs.IT, math.IT, math.FA
Parallel acquisition systems arise in various applications in order to moderate problems caused by insufficient measurements in single-sensor systems. These systems allow simultaneous data acquisition in multiple sensors, thus alleviating such problems by providing more overall measurements. In this work we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing recovery guarantees for which, subject to appropriate conditions, the number of measurements required per sensor decreases linearly with the total number of sensors. Throughout, we consider two different sampling scenarios -- distinct (corresponding to independent sampling in each sensor) and identical (corresponding to dependent sampling between sensors) -- and a general mathematical framework that allows for a wide range of sensing matrices (e.g., subgaussian random matrices, subsampled isometries, random convolutions and random Toeplitz matrices). We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. This model includes both sparse and distributed signals and clustered sparse signals. As our results show, optimal recovery guarantees for both distinct and identical sampling are possible under much broader conditions on the so-called sensor profile matrices (which characterize environmental conditions between a source and the sensors) for the sparse in levels model than for the sparse model. To verify our recovery guarantees we provide numerical results showing phase transitions for a number of different multi-sensor environments.
• Infinite-dimensional $\ell^1$ minimization and function approximation from pointwise data(1503.02352)
Dec. 15, 2016 math.NA, math.FA
We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require \textit{a priori} bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization; namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $\ell^1$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.
• Analyzing the structure of multidimensional compressed sensing problems through coherence(1610.07497)
Oct. 24, 2016 cs.IT, math.IT
Recently it has been established that asymptotic incoherence can be used to facilitate subsampling, in order to optimize reconstruction quality, in a variety of continuous compressed sensing problems, and the coherence structure of certain one-dimensional Fourier sampling problems was determined. This paper extends the analysis of asymptotic incoherence to cover multidimensional reconstruction problems. It is shown that Fourier sampling and separable wavelet sparsity in any dimension can yield the same optimal asymptotic incoherence as in one dimensional case. Moreover in two dimensions the coherence structure is compatible with many standard two dimensional sampling schemes that are currently in use. However, in higher dimensional problems with poor wavelet smoothness we demonstrate that there are considerable restrictions on how one can subsample from the Fourier basis with optimal incoherence. This can be remedied by using a sufficiently smooth generating wavelet. It is also shown that using tensor bases will always provide suboptimal decay marred by problems associated with dimensionality. The impact of asymptotic incoherence on the ability to subsample is demonstrated with some simple two dimensional numerical experiments.
• Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements(1411.0300)
Sept. 9, 2016 math.NA
We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$ derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as $k$ increases, it grows linearly in $k+1$ with the constant of proportionality $1/\mathrm{e}$. Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at $k+1$ times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval $s+1$ bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in $s+1$ for large $s$, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large $s$, we obtain the same result as in the case of univariate sampling with $s$ derivatives.
• Resolution-optimal exponential and double-exponential transform methods for functions with endpoint singularities(1510.07027)
Sept. 3, 2016 math.NA
We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and double-exponential mappings, in combination with Fourier-like approximation in a truncated domain. We show theoretically that this method is superior to variable transform techniques based on the standard exponential and double-exponential mappings. In particular, it can resolve oscillatory behaviour using near-optimal degrees of freedom, whereas the standard mappings require degrees of freedom that grow superlinearly with the frequency of oscillation. We highlight these results with several numerical experiments. Therein it is observed that near-machine epsilon accuracy is achieved using a number of degrees of freedom that is between four and ten times smaller than those of existing techniques.
• Sparsity and Parallel Acquisition: Optimal Uniform and Nonuniform Recovery Guarantees(1603.08050)
March 25, 2016 cs.IT, math.IT, math.FA
The problem of multiple sensors simultaneously acquiring measurements of a single object can be found in many applications. In this paper, we present the optimal recovery guarantees for the recovery of compressible signals from multi-sensor measurements using compressed sensing. In the first half of the paper, we present both uniform and nonuniform recovery guarantees for the conventional sparse signal model in a so-called distinct sensing scenario. In the second half, using the so-called sparse and distributed signal model, we present nonuniform recovery guarantees which effectively broaden the class of sensing scenarios for which optimal recovery is possible, including to the so-called identical sampling scenario. To verify our recovery guarantees we provide several numerical results including phase transition curves and numerically-computed bounds.
• Optimal Sparse Recovery for Multi-Sensor Measurements(1603.06934)
March 22, 2016 cs.IT, math.IT, math.FA
Many practical sensing applications involve multiple sensors simultaneously acquiring measurements of a single object. Conversely, most existing sparse recovery guarantees in compressed sensing concern only single-sensor acquisition scenarios. In this paper, we address the optimal recovery of compressible signals from multi-sensor measurements using compressed sensing techniques, thereby confirming the benefits of multi- over single-sensor environments. Throughout the paper, we consider a broad class of sensing matrices, and two fundamentally different sampling scenarios (distinct and identical respectively), both of which are relevant to applications. For the case of diagonal sensor profile matrices (which characterize environmental conditions between a source and the sensors), this paper presents two key improvements over existing results. First, a simpler optimal recovery guarantee for distinct sampling, and second, an improved recovery guarantee for identical sampling, based on the so-called sparsity in levels signal model.
• Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples(1405.3111)
Sept. 6, 2015 math.NA
In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes.
• On Asymptotic Incoherence and its Implications for Compressed Sensing of Inverse Problems(1402.5324)
July 7, 2015 math.NA, cs.IT, math.IT
Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to many inverse problems. Instead, the key property that permits efficient recovery in such problems is so-called local incoherence. Similarly, the standard notion of sparsity is also inadequate for many real world problems. In particular, in many applications, the optimal sampling strategy depends on asymptotic incoherence and the signal sparsity structure. The purpose of this paper is to study asymptotic incoherence and its implications towards the design of optimal sampling strategies and efficient sparsity bases. It is determined how fast asymptotic incoherence can decay in general for isometries. Furthermore it is shown that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence as a power law up to a constant factor. Sharp bounds on the asymptotic incoherence for Fourier sampling with polynomial bases are also provided. A numerical experiment is also presented to demonstrate the role of asymptotic incoherence in finding good subsampling strategies.
• Recovering piecewise smooth functions from nonuniform Fourier measurements(1410.0088)
Oct. 1, 2014 math.NA
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.
• On asymptotic structure in compressed sensing(1406.4178)
July 5, 2014 cs.IT, math.IT, math.FA
This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, asymptotic sparsity and multilevel sampling, can be utilised to better understand underlying phenomena in practical compressed sensing and improve results in real-world applications. The contribution of the paper is fourfold: First, it explains how the sampling strategy depends not only on the signal sparsity but also on its structure, and shows how to design effective sampling strategies utilising this. Second, it demonstrates that the optimal sampling strategy and the efficiency of compressed sensing also depends on the resolution of the problem, and shows how this phenomenon markedly affects compressed sensing results and how to exploit it. Third, as the new framework also fits analog (infinite dimensional) models that govern many inverse problems in practice, the paper describes how it can be used to yield substantial improvements. Fourth, by using multilevel sampling, which exploits the structure of the signal, the paper explains how one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passaging. This final observation raises the question whether universality is desirable even when such matrices are applicable. Examples of practical applications investigated in this paper include Magnetic Resonance Imaging (MRI), Electron Microscopy (EM), Compressive Imaging (CI) and Fluorescence Microscopy (FM). For the latter, a new compressed sensing approach is also presented.
• Breaking the coherence barrier: A new theory for compressed sensing(1302.0561)
June 23, 2014 math.NA, cs.IT, math.IT
This paper provides an extension of compressed sensing which bridges a substantial gap between existing theory and its current use in real-world applications. It introduces a mathematical framework that generalizes the three standard pillars of compressed sensing - namely, sparsity, incoherence and uniform random subsampling - to three new concepts: asymptotic sparsity, asymptotic incoherence and multilevel random sampling. The new theorems show that compressed sensing is also possible, and reveals several advantages, under these substantially relaxed conditions. The importance of this is threefold. First, inverse problems to which compressed sensing is currently applied are typically coherent. The new theory provides the first comprehensive mathematical explanation for a range of empirical usages of compressed sensing in real-world applications, such as medical imaging, microscopy, spectroscopy and others. Second, in showing that compressed sensing does not require incoherence, but instead that asymptotic incoherence is sufficient, the new theory offers markedly greater flexibility in the design of sensing mechanisms. Third, by using asymptotic incoherence and multi-level sampling to exploit not just sparsity, but also structure, i.e. asymptotic sparsity, the new theory shows that substantially improved reconstructions can be obtained from fewer measurements.
• A note on compressed sensing of structured sparse wavelet coefficients from subsampled Fourier measurements(1403.6541)
June 15, 2014 math.FA
This note complements the paper "The quest for optimal sampling: Computationally efficient, structure-exploiting measurements for compressed sensing" [2]. Its purpose is to present a proof of a result stated therein concerning the recovery via compressed sensing of a signal that has structured sparsity in a Haar wavelet basis when sampled using a multilevel-subsampled discrete Fourier transform. In doing so, it provides a simple exposition of the proof in the case of Haar wavelets and discrete Fourier samples of more general result recently provided in the paper "Breaking the coherence barrier: A new theory for compressed sensing" [1].
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## Narrow Search
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# The Satellites' Flight Configuration
In this lesson, learners will first watch a video about the orbit and formation of the MMS satellites to learn about their flight configuration. After, they will research similar facts about other types of satellites. Next, learners will compute the... (View More)
# Powering the Satellite
In this lesson, learners will first use computers to research and learn how solar panels convert sunlight into electricity. Next, they will calculate the surface area of solar panels board a satellite and their total power generated in various... (View More)
# Launch of the Satellites
In this lesson, learners will research facts about Atlas V rockets, which launched the MMS satellites. After, they will compute the speed of the launch rocket, given a data chart of time vs. distance from lift-off. Then, they will write a report... (View More)
# Graphs and Functions
Students will learn about NASA's Radiation Belt Storm Probes (RBSP), Earth's van Allen Radiation Belts, and space weather through reading a NASA press release and viewing a NASA eClips™ video segment. Then students will use simple linear functions... (View More)
Audience: Middle school
Materials Cost: Free
# Rational Number Operations
Students will learn about the twin STEREO spacecraft and how they are being used to track solar storms through reading a NASA press release and viewing a NASA eClips™ video segment. Then students will examine data to learn more about the frequency... (View More)
Audience: Middle school
Materials Cost: Free
# Angular Measure
Students will learn about the Transit of Venus through reading a NASA press release and viewing a NASA eClips™ video that describes several ways to observe transits. Then students will study angular measurement by learning about parallax and how... (View More)
Audience: Middle school
Materials Cost: Free
# Mean, Median and Mode
During the last sunspot cycle between 1996-2008, over 21,000 flares and 13,000 clouds of plasma exploded from the Sun's magnetically active surface. Students will learn more about space weather through reading a NASA press release and viewing a NASA... (View More)
Audience: Middle school
Materials Cost: Free
# Data Collection and Analysis
During the last sunspot cycle between 1996-2008, over 21,000 flares and 13,000 clouds of plasma exploded from the Sun's magnetically active surface. These events create space weather. Students will learn more about space weather and how it affects... (View More)
Audience: Middle school
Materials Cost: Free
# Translating Between Tables and Expressions
Students will learn about black holes through reading a NASA press release and viewing a NASA eClips™ video segment. Then students will use tables and mathematical expressions to compare black hole sizes and temperatures. Common Core State... (View More)
Keywords: Black holes; Supernova
Audience: Middle school
Materials Cost: Free
# Family Science Night Facilitators Guide
The 9-session NASA Family Science Night program emables middle school children and their families to discover the wide variety of science, technology, engineering, and mathematics being performed at NASA and in everyday life. Family Science Night... (View More)
Audience: Informal education
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https://www.folkstalk.com/tech/find-missing-number-array-javascript-with-code-examples/
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# Find Missing Number Array Javascript With Code Examples
Find Missing Number Array Javascript With Code Examples
This article will show you, via a series of examples, how to fix the Find Missing Number Array Javascript problem that occurs in code.
```let a = [5, 7],
count = 10,
missing = []
for (let i = 1; i <= count; i++) {
if (a.indexOf(i) === -1) {
missing.push(i)
}
}
console.log(missing)
```
The issue with Find Missing Number Array Javascript can be solved in a variety of ways, all of which are outlined in the list that follows.
```var a = [5],
count = 5;
var missing = new Array();
for (var i = 1; i <= count; i++) {
if (a.indexOf(i) == -1) {
missing.push(i);
}
}
console.log(missing); // to check the result.
```
Using a variety of different examples, we have learned how to solve the Find Missing Number Array Javascript.
## How do you find the missing number in an array?
Follow the steps mentioned below to implement the idea:
• Calculate the sum of the first N natural numbers as sumtotal= N*(N+1)/2.
• Traverse the array from start to end. Find the sum of all the array elements.
• Print the missing number as SumTotal – sum of array.
## How do you find the missing number in a integer array of 1 to 100 Javascript?
Example
• function findMissedNum(arrayOfNumbers, n = 100) {
• if(arrayOfNumbers. length === n) {
• console. log("no number is missed");
• }
• if(arrayOfNumbers. length < (n - 1) ) {
• console. log("more than one number is missed")
• }
• let totalSum = (n * (n+1)) / 2;
## How do you find a number in an array?
• If you need the index of the found element in the array, use findIndex() .
• If you need to find the index of a value, use indexOf() .
• If you need to find if a value exists in an array, use includes() .
• If you need to find if any element satisfies the provided testing function, use some() .
## How do you find multiple missing numbers in an array?
To find the multiple missing elements run a loop inside it and see if the diff is less than arr[i] – i then print the missing element i.e., i + diff. Now increment the diff as the difference is increased now. Repeat from step 2 until all the missing numbers are not found.08-Feb-2022
## How do you find the missing number in an array without sorting it?
Calculate the sum of number using (n+1) * (n+2)/2. Loop through all the elements from the array and subtract all the numbers form the sum. Then you will get the missed number.29-Feb-2020
## How do you find the missing number in a series?
In missing number series, a series is given with one missing number and you are asked to find the missing term. To find the missing number, we first identify the rule or formula which is applied in the given missing number series.
## How do you find the missing element in an unsorted array?
Method-2:
• Insert all the elements in an unordered_set.
• Find the minimum and maximum element of the array.
• Traverse the elements from minimum to maximum. Check if current element is present in the set or not. If not then check if this is kth missing by counting the missing elements.
## Can you find missing numbers 30 45 75?
This is Expert Verified Answer 30, 45, 75, 105, 165, ? On observing carefully, we find that the given numbers in the series are multiples of 15 and prime numbers starting from 2. Hence, 195 will come in place of question mark.14-Oct-2019
## How do you find two missing numbers in a sequence?
Find two Missing Numbers in a Sequence of Consecutive Numbers
• Approach is very simple, Add all the given numbers say S.
• Calculate sum of N numbers by formula n(n+1)/2 , say N.
• Find sum of two missing numbers a+b = N-S.
## How do you find missing elements in an array Java?
If a single number is missing in an integer array that contains a sequence of numbers values, you can find it basing of the sum of numbers or, basing on the xor of the numbers. The sum of n sequential numbers will be [n*(n+1)]/2. Using this get the sum of the numbers the n numbers.02-Aug-2019
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https://www.theatlantic.com/technology/archive/2015/07/how-math-can-defeat-bullies/397061/
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Do kids have an unheralded incentive to master math?
This week, I’m sharing responses to the question, “What insight or idea has thrilled or excited you?” This installment comes courtesy of John Allen Paulos, a Professor of Mathematics at Temple University who is here this week at The Aspen Ideas Festival. He adapted his answer from his forthcoming book, A Numerate Life: A Mathematician Explores the Vagaries of Life, His and Probably Yours.
He writes:
I could mention my first introduction to Godel’s theorem about the essential incompleteness of mathematics; or my first encounter with the Banach-Tarski theorem in topology showing that a sphere the size of a pea can be decomposed into a finite number of pieces and put back together to get a sphere the size of a basketball; or Russell’s paradox about the set of all sets that do not belong to themselves; or any number of counterintuitive results in probability theory. All of these mathematical ideas excited me in high school and college, but I will concentrate instead on the thrill I felt in elementary school when I saw that the power of simple arithmetic was sufficient to vanquish a bully, my fifth-grade teacher. It still evokes the same emotions in me that it did decades ago.
I was about 10 years old and enthralled with baseball. I loved playing the game and aspired to be a major league shortstop. (My father played in college and professionally in the minor leagues.) I also became obsessed with baseball statistics and noted that a relief pitcher for the then Milwaukee Braves had an earned run average (ERA) of 135. (The arithmetic details are less important than the psychology of the story, but as I dimly recall, the pitcher had allowed the opposing team to score five runs and had got only one batter out. Getting one batter out is equivalent to pitching one-third of an inning, one-27th of a complete nine-inning game––and allowing five runs in one-27th of an inning translates into an ERA of 5/(1/27) or 135.)
Impressed by this extraordinarily bad ERA, I mentioned it diffidently to my teacher during a class discussion of sports. He looked pained and annoyed and sarcastically asked me to explain the fact to my class. Being quite shy, I did so with a quavering voice, a shaking hand, and a reddened face. (A strikeout in self-confidence.) When I finished, he almost bellowed that I was confused and wrong and that I should sit down.
An overweight coach and gym teacher with a bulbous nose, he asserted that ERA’s could never be higher than 27, the number of outs in a complete game.
For good measure he cackled derisively.
Later that season, The Milwaukee Journal published the averages of all the Braves players. Since this pitcher hadn't pitched again, his ERA was 135, as I had calculated. I remember thinking then of mathematics as a kind of omnipotent protector. I was small and quiet and he was large and loud. But I was right and I could show him. This thought and the sense of power it instilled in me was exciting. So, still smarting from my earlier humiliation, I brought in the newspaper and showed it to him. He gave me a threatening look and again told me to sit down. His idea of good education apparently was to make sure everyone remained seated. I did sit down but this time with a slight smile on my face.
We both knew I was right and he was wrong.
Oddly, this particular teacher did give me a potent reason to study mathematics that I think is underrated: show kids that with it and logic, a few facts, and a bit of psychology, you can prevail over blowhards no matter your age or size. Not only that, but you can sometimes expose nonsensical claims as well. For many students, this may be a much better selling point than being able to solve mixture problems or using trigonometry to estimate the height of a flagpole from across a river. (Incidentally, this mindset is not unrelated to some of my adult writings.)
As with mathematical ideas, so with scientific ones: There were many standard ones from plate tectonics to the double helix of DNA that gave me cerebral whiplash when I first heard them. Instead, however, let me focus briefly on a philosophically flavored idea, that of atomic materialism, which thrilled me as a 10-year-old. (My 11th year was a thrilling one.)
I'd read and had also been told by my grandfather, who seemed proud of the ancient Greek lineage of the idea, that everything was composed of atoms. It seemed obvious to me that an atom couldn't think, and so I "thought" that this proved that humans couldn't think either. I was so pleased with my ground-breaking idea about our essentially zombie natures that I explained it at length (all of two paragraphs) on a piece of paper, folded the paper carefully, put it inside a small metal box, taped it very securely, and buried it near the swing in our backyard where future generations of unthinking humans could appreciate my deep thoughts on this matter.
My attraction to the idea of atomic materialism wasn’t just intellectual, if that’s not too heavy a term to apply to a 10-year-old, but also visceral. Lying on the floor watching television or wrestling with my brother, I often had the inchoate idea that, in an important sense, there was no essential difference between me and not-me, that everything was composed of the same stuff and that the air above my forehead and the brain inside it were just patterned differently. The notion of emergent qualities, properties, and abilities didn't complicate my youthful certainty about these matters, and the dreary conclusion I came to that we couldn’t really think was one that I oddly found quite cheering.
Again, the excitement provided by my absurd interpretation of this fundamental idea preceded and in a way laid the groundwork for my appreciation of many other scientific ideas. Had I been told of emergent properties immediately, however, I doubt I would have been nearly as excited.
Email conor@theatlantic.com to share an idea or insight that has thrilled or excited you.
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1. ## Proving an identity
Hi,
I hope someone can help. I'm trying to understand the process of proving an identity.
Let's say that you were trying to prove the following equation: sin2x/(1+cos2x) = tanx
I know that the equation is an identity since when I simplify the left-hand side, it is equivalent to the right-hand side.
But let's say that I replaced x with pi/4, and still showed that both sides were equivalent - would using only one value be enough to prove that the equation is an identity? I would think that by proving the equation using only one value (e.g. pi/4), would not be enough to prove that the equation is an identity. I thought that x takes into account all values, so it is more accurate than merely subbing in pi/4.
The definition of a trigonometric identity is as follows:
I want to place emphasis on the fact that the definition says that the solution set for trigonometric identities is all real numbers (and not including the solutions that are undefined). So with that being said, if x was only equal to pi/4, then how could the equation possibly be an identity?
I would appreciate some guidance.
Sincerely,
Olivia
2. ## Re: Proving an identity
Originally Posted by otownsend
Let's say that you were trying to prove the following equation: sin2x/(1+cos2x) = tanx
I know that the equation is an identity since when I simplify the left-hand side, it is equivalent to the right-hand side.
But let's say that I replaced x with pi/4, and still showed that both sides were equivalent - would using only one value be enough to prove that the equation is an identity? I would think that by proving the equation using only one value (e.g. pi/4), would not be enough to prove that the equation is an identity. I thought that x takes into account all values, so it is more accurate than merely subbing in pi/4.
The definition of a trigonometric identity is as follows:
I want to place emphasis on the fact that the definition says that the solution set for trigonometric identities is all real numbers (and not including the solutions that are undefined). So with that being said, if x was only equal to pi/4, then how could the equation possibly be an identity?
Note that $2\left( {\dfrac{\pi }{4}} \right) = \left( {\dfrac{\pi }{2}} \right)$ so
\begin{align*}\dfrac{\sin \left( {\frac{\pi }{2}} \right)}{1-\cos \left( {\frac{\pi }{2}} \right)}&=\dfrac{1}{1-0}\\&=1\\&=\tan \left( {\frac{\pi }{4}} \right) \end{align*}
3. ## Re: Proving an identity
They gave an example of it working. It may be left up to the reader to prove the identity. Such as this:
$\dfrac{\sin(2x)}{1+\cos(2x)} = \dfrac{2\sin x\cos x}{1+2\cos^2 x-1} = \dfrac{\sin x \cancel{2\cos x}}{\cancel{2\cos x} \cos x} = \dfrac{\sin x}{\cos x} = \tan x$
This proves the identity.
4. ## Re: Proving an identity
Thank you for responding... however, I'm more wondering whether it is incorrect style to prove an identity by solving for a specific variable as opposed to just x?
5. ## Re: Proving an identity
Originally Posted by otownsend
Thank you for responding... however, I'm more wondering whether it is incorrect style to prove an identity by solving for a specific variable as opposed to just x?
A statement, an identity, is never proved by example.
An example can disprove a statement called a counter-example; but it can never ever prove a statement.
6. ## Re: Proving an identity
What if the identity held true to an example such as when x = pi/4? What would that be communicating? Based on what you have told me, I would think that this would not be enough to claim the whole equation for all cases as an identity. Please let me know if my thinking about this is right...
7. ## Re: Proving an identity
Originally Posted by otownsend
What if the identity held true to an example such as when x = pi/4? What would that be communicating? Based on what you have told me, I would think that this would not be enough to claim the whole equation for all cases as an identity. Please let me know if my thinking about this is right...
No, it just proves that the world is flat.
8. ## Re: Proving an identity
Right, and it also proves that global warming isn't real.
9. ## Re: Proving an identity
Originally Posted by otownsend
Right, and it also proves that global warming isn't real.
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# Poisson - Finding and Removing outlier
Show 40 post(s) from this thread on one page
Page 1 of 2 12 Last
• Oct 1st 2013, 10:58 AM
laban1
Poisson - Finding and Removing outlier
Hi !
Sometimes I have lot of "zeroes" sometimes not.
It's for practical sampling, with a lot of figures, so I'd
rather have an easy enough approach then a perfect but complicated solution.
Instruments sometimes give unexpected values quite off.
Do I have to solve
a) upper bound: P(x>cutoff) = 0,003 (setting same level as Std3 in a Norm)
Is there a faster/better rule of thumb here?
It's for a practical point of view so it doesn't have to be perfect.
b) And how do I set lower bound? seems even more tricky!
c) When I found an outlier in a rule I set up,
How do I go about it? Do I remove it from my sample-set completely
directly when I discover it, or do I keep it if it
is within expected level of probability?
Hope you can help!
Thanks! (Hi)
• Oct 1st 2013, 09:47 PM
chiro
Re: Poisson - Finding and Removing outlier
Hey laban1.
Can you outline your sample properties? What is your distribution (assumed)? What are you using for outlier detection? (Cooks distance, something else maybe)?
• Oct 2nd 2013, 03:01 AM
laban1
Re: Poisson - Finding and Removing outlier
- Poisson (as in title), Lambda = np is fairly stable in each of the studied Areas.
but Lambda can very from very small to very large in different Areas.
What are you using for outlier detection? (Cooks distance, something else maybe)?
Cooks distance? Don't know. Suppose it could be a multiple of sdt?? Hence my question.
• Oct 2nd 2013, 06:43 PM
chiro
Re: Poisson - Finding and Removing outlier
There is an attribute in Poisson modelling called over-dispersion.
I think you should check it out and use your favourite software package like R or SAS to estimate the over-dispersion co-efficient and incorporate it into your analysis.
You could throw away the 0's if you have good enough justification to throw them out, but if that is not the case then take a look at over-dispersion and consider looking at other similar methods to account for this skewed behaviour in the Poisson.
Overdispersion - Wikipedia, the free encyclopedia
I can't tell you what to do with the data in terms of throwing out outliers or censoring data, but I do know that for your kind of problem, over-dispersion analyses is a good first start.
• Oct 4th 2013, 01:14 PM
laban1
Re: Poisson - Finding and Removing outlier
Hi!
I have basic knowledge of over-dispersion. I was more into outlier.
• Oct 6th 2013, 02:10 AM
laban1
Re: Poisson - Finding and Removing outlier
Any input?
• Oct 6th 2013, 05:48 PM
chiro
Re: Poisson - Finding and Removing outlier
You will have to decide whether the outliers are justified to be thrown out or whether you have to use something like over-dispersion.
We don't know enough about the context of the data to answer that for you.
• Oct 7th 2013, 08:18 AM
laban1
Re: Poisson - Finding and Removing outlier
Hi! What do you need to know?
• Oct 7th 2013, 05:51 PM
chiro
Re: Poisson - Finding and Removing outlier
You should decide first of all whether the outliers should stay or be thrown out.
To do this, you need to figure out whether the outliers are representative of the data and what you are trying to answer or if they are not.
• Oct 7th 2013, 06:39 PM
SlipEternal
Re: Poisson - Finding and Removing outlier
Quote:
Originally Posted by chiro
We don't know enough about the context of the data to answer that for you.
Quote:
Originally Posted by laban1
Hi! What do you need to know?
From the post just before yours, it appears chiro is asking for the context of the data. The more information you provide about the data, the better our understanding of it will become. The question "what do you need to know?" is difficult to answer from this end. We have no access to the data. We have no feel for what the data looks like, where it came from, how it is being evaluated, the circumstances of how it is obtained, potential expected causes for outliers (and possible methods for detecting them), potential dependencies that could be examined to rule out outliers, the level of accuracy you want in your findings, etc.
Did you read up on Cook's Distance as chiro suggested? You could also look at Identifying outliers. Check and see if any of those methods seem suitable for your model. The section below titled "Working with Outliers" might also be of interest. You asked if you should delete outlier data. That practice is frowned upon.
• Oct 13th 2013, 02:37 AM
laban1
Re: Poisson - Finding and Removing outlier
Sorry for the late feedback
It seems pretty advanced, for me, Cooks distance and all !
Just to choose method as you point out, is not even given.
I'm playing with an option to expand my measuring time so that instead of getting Lambda = 3
for one hour, I could sum 4 hour and get Lambda = 12 and thus I would get approx Norm
and could apply "3std-rule".
I have quite a lot of data, and I expect the process to be fairly stable (Lambda stable) over time.
Would that be an option to consider?
Pros and cons?
Thanks!
I'm measuring waterflow during time of least expected flow, 1hour btw 03-04 at night.
I have different flow-meaters and I expect there are calibration-problems as well.
• Oct 13th 2013, 05:26 PM
chiro
Re: Poisson - Finding and Removing outlier
Again you have to decide whether its justified to take out the outliers or not.
If you can't justify it, then you will probably have to resort to something like over-dispersion or a more general form of model.
• Oct 15th 2013, 10:29 AM
laban1
Re: Poisson - Finding and Removing outlier
Quote:
Originally Posted by laban1
Sorry for the late feedback
It seems pretty advanced, for me, Cooks distance and all !
Just to choose method as you point out, is not even given.
I'm playing with an option to expand my measuring time so that instead of getting Lambda = 3
for one hour, I could sum 4 hour and get Lambda = 12 and thus I would get approx Norm
and could apply "3std-rule".
I have quite a lot of data, and I expect the process to be fairly stable (Lambda stable) over time.
Would that be an option to consider?
Pros and cons?
Thanks!
I'm measuring waterflow during time of least expected flow, 1hour btw 03-04 at night.
I have different flow-meaters and I expect there are calibration-problems as well.
Any input on this?
• Oct 15th 2013, 04:55 PM
chiro
Re: Poisson - Finding and Removing outlier
You may not necessarily get normality by using a higher rate.
You should look at asymptotic results, particular with regard to the deviance statistic (which is a chi-square statistic) if you have a big enough sample.
• Oct 17th 2013, 08:31 AM
laban1
Re: Poisson - Finding and Removing outlier
Quote:
Originally Posted by chiro
You may not necessarily get normality by using a higher rate.
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# Number 788
### Properties of number 788
Cross Sum:
Factorization:
2 * 2 * 197
Divisors:
1, 2, 4, 197, 394, 788
Count of divisors:
Sum of divisors:
Prime number?
No
Fibonacci number?
No
Bell Number?
No
Catalan Number?
No
Base 2 (Binary):
Base 3 (Ternary):
Base 4 (Quaternary):
Base 5 (Quintal):
Base 8 (Octal):
Base 32:
ok
sin(788)
0.51392673921436
cos(788)
-0.85783407878243
tan(788)
-0.59909806794317
ln(788)
6.6694980898579
lg(788)
2.8965262174896
sqrt(788)
28.071337695236
Square(788)
### Number Look Up
Look Up
788 which is pronounced (seven hundred eighty-eight) is a special figure. The cross sum of 788 is 23. If you factorisate 788 you will get these result 2 * 2 * 197. 788 has 6 divisors ( 1, 2, 4, 197, 394, 788 ) whith a sum of 1386. The number 788 is not a prime number. The number 788 is not a fibonacci number. The number 788 is not a Bell Number. The number 788 is not a Catalan Number. The convertion of 788 to base 2 (Binary) is 1100010100. The convertion of 788 to base 3 (Ternary) is 1002012. The convertion of 788 to base 4 (Quaternary) is 30110. The convertion of 788 to base 5 (Quintal) is 11123. The convertion of 788 to base 8 (Octal) is 1424. The convertion of 788 to base 16 (Hexadecimal) is 314. The convertion of 788 to base 32 is ok. The sine of 788 is 0.51392673921436. The cosine of the number 788 is -0.85783407878243. The tangent of the figure 788 is -0.59909806794317. The root of 788 is 28.071337695236.
If you square 788 you will get the following result 620944. The natural logarithm of 788 is 6.6694980898579 and the decimal logarithm is 2.8965262174896. You should now know that 788 is great number!
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## 10. Stereo Rectification
### What is Stereo Rectification
Stereo Rectification is called stereo parallelization processing in Japanese. Translate the right camera (not the left camera) to the left (angle) from the right image taken by the right camera and the left camera and the right image taken from the left image by the right camera (this is the same as the original right image) This is a process to obtain a virtual left image taken without changing. This virtual left image is obtained by transforming the original left image. It is interesting that stereo parallelization is possible because there is no way to obtain an image taken from an arbitrary position and angle unless all three-dimensional information of the target is known. Stereo parallel processing is an indispensable process for stereo vision. On the other hand, the purpose of stereo vision is to obtain three-dimensional information of the target, so it is very important that stereo parallel processing is possible. That's it. As will be seen later, the right image also needs to be converted in order to move it to the left and take the left image.
### Coordinate system transformation
Consider a transformation between two 3D coordinate systems $$OXYZ$$ and $$oxyz$$. The two coordinate systems are essentially equal, but here $$OXYZ$$ is the world coordinate system or global coordinate system, and $$oxyz$$ is the camera coordinate system attached to the camera. $$O$$ and $$o$$ represent the origin of each. The vector is expressed as being independent of the coordinate system. I will write $$\vec{Oo}=\vec{t}$$. Then $$P$$ can be expressed as \begin{align*} \vec{OP}=\vec{Oo}+\vec{oP}=\vec{t}+\vec{oP}=\vec{t}+x\vec{e_1}+y\vec{e_2}+z\vec{e_3} \end{align*} as a point in space. If $$\vec{e_1}$$, $$\vec{e_2}$$, and $$\vec{e_3}$$ are the basis vectors of the coordinate system $$oxyz$$, the coordinates of the $$P$$ point in the coordinate system $$oxyz$$ are $$(x, y, z)$$. On the other hand, the coordinates of the $$P$$ point in the coordinate system $$OXYZ$$ are assumed to be $$(X, Y, Z)$$. Here, if all the vectors are displayed as components using the coordinate system $$OXYZ$$, the result is \begin{align*} \vec{OP}=\begin{pmatrix} X \\ Y \\ Z \end{pmatrix} .\end{align*} Also assume that it is \begin{align*} \vec{t}=\begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix}, \vec{e_1}=\begin{pmatrix} e_{11} \\ e_{12} \\ e_{13} \end{pmatrix}, \vec{e_2}=\begin{pmatrix} e_{21} \\ e_{22} \\ e_{23} \end{pmatrix}, \vec{e_3}=\begin{pmatrix} e_{31} \\ e_{32} \\ e_{33} \end{pmatrix} .\end{align*} Then \begin{align*} \vec{OP}=\vec{t}+x\vec{e_1}+y\vec{e_2}+z\vec{e_3} \end{align*} becomes \begin{align*} \begin{pmatrix} X \\ Y \\ Z \end{pmatrix}= \begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix}+ \begin{pmatrix} e_{11} \\ e_{12} \\ e_{13} \end{pmatrix}x+ \begin{pmatrix} e_{21} \\ e_{22} \\ e_{23} \end{pmatrix}y+ \begin{pmatrix} e_{31} \\ e_{32} \\ e_{33} \end{pmatrix}z= \begin{pmatrix} e_{11} & e_{21} & e_{31} \\ e_{12} & e_{22} & e_{32} \\ e_{13} & e_{23} & e_{33} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}+ \begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix} \end{align*} in the component display. \begin{align*} \begin{pmatrix} X \\ Y \\ Z \end{pmatrix}= \begin{pmatrix} e_{11} & e_{21} & e_{31} \\ e_{12} & e_{22} & e_{32} \\ e_{13} & e_{23} & e_{33} \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix}+ \begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix} \end{align*} is expressed as \begin{align*} X=R^{-1}x+t \end{align*} by newly writing \begin{align*} X=\begin{pmatrix} X \\ Y \\ Z \end{pmatrix}, x=\begin{pmatrix} x \\ y \\ z \end{pmatrix}, R^{-1}=\begin{pmatrix} e_{11} & e_{21} & e_{31} \\ e_{12} & e_{22} & e_{32} \\ e_{13} & e_{23} & e_{33} \end{pmatrix}, t=\begin{pmatrix} t_1 \\ t_2 \\ t_3 \end{pmatrix} .\end{align*} When $$R$$ is an inverse matrix of $$R^{-1}$$ and multiplied on both sides from the left, it becomes \begin{align*} RX=x+Rt ,\end{align*} but $$T=-Rt$$ is represented as \begin{align} x=RX+T \label{eq:1} .\end{align} Note that $$X$$ is the coordinate of the $$P$$ point in the coordinate system $$OXYZ$$, and $$x$$ is the coordinate of the $$P$$ point in the coordinate system $$oxyz$$. The expression \eqref{eq:1} is "$$X$$ is rotated by $$R$$ in the coordinate system $$OXYZ$$ and only $$T$$ "The coordinate in the coordinate system $$OXYZ$$ of the translated position is the coordinate of $$P$$ point in the coordinate system $$oxyz$$." This is the coordinate transformation formula. \begin{align*} \begin{pmatrix} x \\ y \\ z \end{pmatrix}= \begin{pmatrix} r_{11} & r_{12} & r_{13} \\ r_{21} & r_{22} & r_{23} \\ r_{31} & r_{32} & r_{33} \end{pmatrix} \begin{pmatrix} X \\ Y \\ Z \end{pmatrix}+ \begin{pmatrix} T_1 \\ T_2 \\ T_3 \end{pmatrix} \end{align*} which wrote this in the component display can be expressed as \begin{align*} \begin{pmatrix} x \\ y \\ z \end{pmatrix}= \begin{pmatrix} r_{11} & r_{12} & r_{13} & T_1 \\ r_{21} & r_{22} & r_{23} & T_2 \\ r_{31} & r_{32} & r_{33} & T_3 \end{pmatrix} \begin{pmatrix} X \\ Y \\ Z \\ 1 \end{pmatrix} .\end{align*} Write this as \begin{align} x=(R|T)\tilde{X} \label{eq:2} .\end{align} $$\tilde{X}$$ means $$X$$ with $$1$$ added, and is called homogeneous coordinate representation. It is said that homogeneous coordinates (also called homogeneous coordinates) are for handling points at infinity. Since the infinity point does not appear in this story, we will use only $$\tilde{X}$$. Using homogeneous coordinates, the transformation of the coordinate system can be expressed as a matrix product.
I want to think about how the coordinate system itself rotates and translates, but it is quite troublesome because the rotation matrix and the movement vector are displayed in the first place. For the time being, we will not consider the rotation or translation of the coordinate system itself. Since \begin{align*} \vec{e_1}=\begin{pmatrix} e_{11} \\ e_{12} \\ e_{13} \end{pmatrix}, \vec{e_2}=\begin{pmatrix} e_{21} \\ e_{22} \\ e_{23} \end{pmatrix}, \vec{e_3}=\begin{pmatrix} e_{31} \\ e_{32} \\ e_{33} \end{pmatrix} \end{align*} is an orthonormal vector, it is \begin{align*} \begin{pmatrix} (\vec{e_1})^T \\ (\vec{e_2})^T \\ (\vec{e_3})^T \end{pmatrix} \begin{pmatrix} \vec{e_1} & \vec{e_2} & \vec{e_3} \end{pmatrix} =\begin{pmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \end{pmatrix} \begin{pmatrix} e_{11} & e_{21} & e_{31} \\ e_{12} & e_{22} & e_{32} \\ e_{13} & e_{23} & e_{33} \end{pmatrix} =\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} .\end{align*} That is, \begin{align*} R=\begin{pmatrix} e_{11} & e_{12} & e_{13} \\ e_{21} & e_{22} & e_{23} \\ e_{31} & e_{32} & e_{33} \end{pmatrix} =(R^{-1})^T \end{align*}\begin{align*} R^{-1}=\begin{pmatrix} e_{11} & e_{21} & e_{31} \\ e_{12} & e_{22} & e_{32} \\ e_{13} & e_{23} & e_{33} \end{pmatrix} =R^T .\end{align*}
### Camera and perspective transformation
Consider a pinhole camera in which light from an object always passes through the pinhole. Align the origin $$o$$ of the 3D coordinate $$oxyz$$ with this pinhole. Next, consider the plane that intersects with the $$oz$$ axis at $$f$$ point $$(0, 0, f)$$ and calls it the imaging surface. However, $$f$$ is positive and the imaging surface is between the pinhole and the object. $$f$$ is also called the focus. In an actual pinhole camera, the imaging surface is on the opposite side of the object across the pinhole, so this is a virtual arrangement to improve the visibility of the equation. Place the two-dimensional pixel coordinates $$fuv$$ with the $$f$$ point as the origin on the imaging surface so that the $$fv$$ axis and the $$oy$$ axis are parallel, so that the $$fu$$ axis and the $$ox$$ axis are parallel. The 3D coordinates $$oxyz$$ where the 2D pixel coordinates $$fuv$$ are arranged in this way are called camera coordinates.
When the light emitted from the point $$(x, y, z)$$ on the object passes through the point $$(u, v)$$ on the imaging surface, it is $$u=fx/z$$ because of the relationship of $$x:z=u:f$$. If the pixel coordinates are in pixels, the pixel size is $$p$$ and $$u=fx/{pz}$$. Similarly, $$v=fy/{pz}$$. These are written as \begin{align*} s \begin{pmatrix} u \\ v \\ 1 \end{pmatrix}= \begin{pmatrix} f/p & 0 & 0 \\ 0 & f/p & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} .\end{align*} Because this is $$su=fx/p, sv=fy/p, s=z$$, if you use the third formula $$s=v$$, the first formula will be $$u=fx/{pz}$$ and the second formula will be $$v=fy/{pz}$$. Since \begin{align*} s \begin{pmatrix} u \\ v \\ 1 \end{pmatrix}= \begin{pmatrix} f_x/p & 0 & c_x \\ 0 & f_y/p & c_y \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \end{align*} is $$su=f_xx/p+c_xz, sv=f_yy/p+c_yz, s=z$$, using the third formula $$s=v$$, the first formula represents $$u=f_xx/{pz}+c_x$$ and the second formula represents $$v=f_yy/{pz}+c_y$$. $$(c_x, c_y)$$ is the pixel coordinate of the center (intersection of the $$oz$$ axis and the imaging surface). This is for the case where the origin of the pixel coordinates is shifted from the $$oz$$ axis. When $$f_x/p$$ and $$f_y/p$$ are newly expressed as \begin{align} s \begin{pmatrix} u \\ v \\ 1 \end{pmatrix}= \begin{pmatrix} f_x & 0 & c_x \\ 0 & f_y & c_y \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} \label{eq:3} \end{align} as $$f_x$$ and $$f_y$$, $$f_x$$ and $$f_y$$ are focal lengths in units of pixel size. There are two focal lengths, $$f_x$$ and $$f_y$$, because the $$x$$ and $$y$$ directions may have different values. Originally, $$c_x$$ and $$c_y$$ are the center positions in pixel size. This transformation from the camera coordinate system to the pixel coordinate system of the imaging surface is called perspective transformation. \begin{align*} \tilde{m}= \begin{pmatrix} u \\ v \\ 1 \end{pmatrix} \end{align*} has a homogeneous coordinate representation. If homogeneous coordinates are used, the perspective transformation can also be expressed as a matrix product. The $$s$$ from the above equations is needed when using homogeneous coordinates with any real number. The expression \eqref{eq:3} can be written as \begin{align} s\tilde{m}=Ax \label{eq:4} .\end{align} $$A$$ is called camera matrix. Combined with \eqref{eq:1} in the previous paragraph, \begin{align} s\tilde{m}=A(RX+T) \label{eq:5} ,\end{align} combined with \eqref{eq:2} in the previous paragraph, becomes \begin{align*} s\tilde{m}=A(R|T)\tilde{X} .\end{align*} These formulas are called camera basic formulas. The basic camera equation represents the transformation from the global coordinate system to the pixel coordinate system, that is, how the real world is photographed.
### Relationship between cameras
\begin{align} x_l=R_lX+T_l \label{eq:6} \end{align}\begin{align} x_r=R_rX+T_r \label{eq:7} \end{align} is obtained by preparing two formulas \eqref{eq:1} for the transformation from the world coordinate system to the camera coordinate system for the left camera and the right camera. Multiply $$R$$ from the left on both sides of the expression \eqref{eq:6} to get \begin{align*} Rx_l=RR_lX+RT_l .\end{align*} If this $$R$$ is $$R$$ which becomes $$R_r=RR_l$$, it can be transformed to \begin{align*} Rx_l=R_rX+RT_l=x_r-T_r+RT_l \end{align*}\begin{align*} x_r=Rx_l+T_r-RT_l \end{align*} using this and \eqref{eq:7}. When $$T_r-RT_l$$ is newly placed as $$-T$$, it becomes \begin{align} x_r=Rx_l-T \label{eq:8} .\end{align} This means that if $$x_l$$ is rotated by $$R$$ and translated by $$-T$$, it becomes $$x_r$$, but in order to say that, the relationship of \begin{align*} R_r=RR_l \end{align*}\begin{align*} T_r=RT_l-T \end{align*} is necessary. It should be noted here that the $$R$$ and $$T$$ are not reflected in the camera itself by rotating or translating, although they are reflected in the camera. Now, these and \eqref{eq:8} represent the relative positions of the two cameras. I will change \eqref{eq:8} to \begin{align} x_l=R^{-1}(x_r+T) \label{eq:9} \end{align} for later use.
### Stereo parallel processing
The \eqref{eq:4} and \eqref{eq:1}, which are written separately for the basic camera \eqref{eq:5}, are prepared for the \begin{align} s_l\tilde{m_l}=A_lx_l \label{eq:10} \end{align}\begin{align*} x_l=R_lX+T_l \end{align*} for the left camera and the \begin{align*} s_r\tilde{m_r}=A_rx_r \end{align*}\begin{align} x_r=R_rX+T_r \label{eq:11} \end{align} for the right camera. Multiply $$A_l^{-1}$$ on both sides of \eqref{eq:10} from the left, transform it to \begin{align*} s_lA_l^{-1}\tilde{m_l}=x_l ,\end{align*} and use \eqref{eq:9} to become \begin{align*} s_lA_l^{-1}\tilde{m_l}=R^{-1}(x_r+T) .\end{align*} If you use \eqref{eq:11}, you can transform it with \begin{align*} s_lA_l^{-1}\tilde{m_l}=R^{-1}(R_rX+T_r+T) .\end{align*} Multiplying $$R$$ from the left on both sides results in \begin{align*} s_lRA_l^{-1}\tilde{m_l}=R_rX+T_r+T .\end{align*} Furthermore, if both sides are multiplied by $$A_r$$ from the left, it will become \begin{align} s_lA_rRA_l^{-1}\tilde{m_l}=A_r(R_rX+T_r+T) \label{eq:12} .\end{align} \begin{align*} s_l\tilde{\dot{m_l}}=A_r(R_rX+T_r+T) \end{align*} can be obtained by using $$\tilde{\dot{m_l}}$$ converted from $$\tilde{m_l}$$ by \begin{align} \tilde{\dot{m_l}}=A_rRA_l^{-1}\tilde{m_l} \label{eq:13} .\end{align} Comparing this with the basic formula \begin{align} s_r\tilde{m_r}=A_r(R_rX+T_r) \label{eq:14} \end{align} of the right camera, it can be seen that the left pixel image converted by the formula \eqref{eq:13} can obtain a projection image when the object is translated by $$T$$ in the same right camera. You Furthermore, in order to make the $$T$$ move in the direction of the $$ox$$ axis of the right camera, considering the rotation matrix $$L$$ that becomes the \begin{align*} LT=c\begin{pmatrix} e_{r11} \\ e_{r12} \\ e_{r13} \end{pmatrix}=b \end{align*} and multiplying both sides of the \eqref{eq:12} and \eqref{eq:14} from the left by the $$A_rLA_r^{-1}$$, \begin{align*} s_lA_rLRA_l^{-1}\tilde{m_l}=A_r(LR_rX+LT_r+b) \end{align*}\begin{align*} s_rA_rLA_r^{-1}\tilde{m_r}=A_r(LR_rX+LT_r) \end{align*} is completed and the stereo parallelization is completed. Pixel conversion needs to be performed on both images, but the \begin{align} \tilde{\ddot{m_l}}=A_rLRA_l^{-1}\tilde{m_l} \label{eq:15} \end{align}\begin{align} \tilde{\ddot{m_r}}=A_rLA_r^{-1}\tilde{m_r} \label{eq:16} \end{align} and the converted right image $$\tilde{\ddot{m_r}}$$ and left image $$\tilde{\ddot{m_l}}$$ are taken by a camera (right image) and the camera is accurately translated by $$-c$$ in the direction of the $$ox$$ axis. This is the left image. Here, for the first time, we considered the parallel movement of the camera itself, but it is thought that the situation in which the image was translated by $$c$$ in the direction of the $$ox$$ axis occurred because the camera was translated by $$-c$$ in the direction of the $$ox$$ axis. Because it is. It should also be noted that the $$L$$ has an arbitrary degree of freedom with the $$ox$$ axis as the rotation axis. Next, the OpenCV initUndistortRectifyMap function that performs this image conversion is explained.
### initUndistortRectifyMap function
The argument of OpenCV's initUndistortRectifyMap function is
• InputArray cameraMatrix
• InputArray distCoeffs
• InputArray R
• InputArray newCameraMatrix
• Size size
• int m1type
• OutputArray map1
• OutputArray map2
. In the conversion of the expression \eqref{eq:15}, which is the conversion of the left image, cameraMatrix = $$A_l$$, distCoeffs = distortion coefficient of the left camera, R = $$LR$$, newCameraMatrix = $$A_r$$. For the conversion of the right image, the expression \eqref{eq:16}, cameraMatrix = $$A_r$$, distCoeffs = Right camera distortion coefficient, R = $$L$$, newCameraMatrix = $$A_r$$.
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A181772 Kissing numbers for the laminated lattices Lambda(1), Lambda(2), Lambda(8), Lambda(24). 0
2, 6, 240, 196560 (list; graph; refs; listen; history; text; internal format)
OFFSET 1,1 COMMENTS Given on p. 8 of Dixon, with "coincidence" involving Fibonacci numbers. Since there is no indication of how the sequence 1,2,8,24 might be extended, I have marked this as "fini" and "full". - N. J. A. Sloane, Nov 12 2010 Let x = {1, 2, 8, 24}. Then (Lambda_x/x + 1)^2 - 1 = {8, 15, 960, 67092480} and is either a cake number (A000125) or the product of consecutive cake numbers. For instance, 960 = 1 * 2 * 4 * 8 * 15 = (Lambda_8/8 + 1)^2 - 1 and 67092480 = 1 * 2 * 4 * 8 * 15 * 26 * 42 * 64 = (Lambda_24/24 + 1)^2 - 1. This is interesting, at least in part, since x^2 = {1, 4, 64, 576} is also a cake number. - Raphie Frank, Dec 06 2012 REFERENCES J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Chap. 6. LINKS Geoffrey Dixon, Integral Octonions, Octonion XY-Product, and the Leech Lattice, Nov 11, 2010. CROSSREFS Cf. A002336. Sequence in context: A332692 A100359 A206033 * A183398 A052342 A199125 Adjacent sequences: A181769 A181770 A181771 * A181773 A181774 A181775 KEYWORD nonn,fini,full AUTHOR EXTENSIONS Definition rewritten by N. J. A. Sloane, Nov 12 2010 STATUS approved
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# Tutor profile: Tomas C.
Tomas C.
Tutoring since 2015
## Questions
### Subject:R Programming
TutorMe
Question:
What is the difference between & and && in R?
Tomas C.
The functional & is a vectorized operation, meaning it can return a vector. For example: x <- 1:10 x %% 2 == 0 & x > 5 will return a vector [1] FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE TRUE On the other hand, && only evaluates the first element of each vector x <- 1:10 x %% 2 == 0 && x > 5 [1] FALSE
### Subject:Statistics
TutorMe
Question:
What is a general method to estimate the variance of an estimator when it is not possible to derive its formula?
Tomas C.
Bootstrap is an excellent option. It takes samples with replacement from the sample obtained, calculates the estimate with each subsample as the original sample, and estimates the variability through the variability in the estimates of the subsamples. This method also allows to obtain the whole sampling distribution of an estimator, meaning we could estimate much more interesting things than just the variance.
### Subject:Data Science
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Can you include explanatory variables in a linear model when it is not related linearly with the response variable?
Tomas C.
Yes. The term linear model does not mean that each explanatory variable is linearly related to the outcome. It just means that each of the terms of the model is related to the outcome linearly. Then, the model $$y = \beta_0 + \beta_1x + \beta_2x^2$$ is linear because each predictor, $$x$$ and $$x^2$$ is linearly related to $$y$$ through each $$\beta$$ coefficient. However, it does not mean that $$y$$ and $$x$$ are linearly related. The relationship shown there is quadratic, indeed.
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Created by dvannice90
# 1st Math
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How Long?
Try measuring with paper clips to find out how long the pictured objects are. This worksheet is great for measurement practice and building counting skills too.
Which is Longer?
Which is longer: a hockey puck or a pair of scissors? In this worksheet, kids get a math boost as they put groups of things in order from shortest to longest.
Measurements of Length
Is a flamingo taller than a duck? In this worksheet, kids put each group of things in order from shortest to tallest.
Measurement: Length in Centimeters
First graders practice measurement in centimeters as they measure ribbons long and short.
Which is Heavier?
Give your child's logical reasoning skills a boost with this worksheet that asks him to put each group in order from lightest to heaviest.
Estimating Volume
Help your child practice his skills with volume with this printable worksheet, which asks him to estimate how much liquid things can hold.
Estimating Length
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### Take the Test 2
Online Math Practice Test 2 with answers
(Numbers - Prealgebra - Addition, Subtraction, Multiplication, Division)
1. 7654 x 132
2. 2736 - 1789
3. 5152 ÷ 14
4. 5678 x 39
5. 2733 + 11231
6. 5678 - 2345
7. 98999 - 23110
8. 789 x 99
9. 12364 ÷ 22
10. 8098 + 7875
11. 7894 - 2367
12. 8098 + 7875
13. 30807÷ 63
14. 7678 + 1234
15. 7890 - 999
16. 1278 x 44
17. 2323 + 1119
18. 43218 + 91267
19. 913456 - 23119
20. 7789 x 99
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Categories
Recent Files
# Decimals Fractions Percents Tenths 3blank Converting Tos 3rd Grade 7th Changing Repeating
By Rachel R. Mueller on May 15 2018 22:18:57
When teaching decimal numbers, first review the basics of thousands, hundreds, tens and ones and then introduce (or review) tenths, hundredths and thousandths.
Children may find the easiest way to work this out is by using the column method, where all the numbers are placed in a column, one on top of the other. It is vital that they remember that the decimal points of each number must line up. Once they’ve worked out using pencil and paper, let them check on a calculator as using decimals accurately on a calculator is an important real life skill too.
Understanding decimals is vital for real-life, everyday maths. Phoebe Doyle offers some teacher tips to help your Key Stage 2 child understand what decimals are and how to use this knowledge in calculations.
These worksheets can help your students review decimals number concepts. Worksheets include place value, naming decimals to the nearest tenth and hundredth place, adding decimals, subtracting decimals, multiplying, dividing, and rounding decimals.
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# ML Aggarwal Solutions Class 9 Mathematics Solutions for Trigonometric Ratios Exercise 17 in Chapter 17 - Trigonometric Ratios
Question 12 Trigonometric Ratios Exercise 17
Given A is an acute angle and cosec A = √2, find the value of
2 \sin ^{2} A+3 \cot ^{2} A /
\left(\tan ^{2} A-\cos ^{2} A\right)
The values of all trigonometric functions dependent on the value of the ratio of sides in a right-angled triangle are known as trigonometric ratios. The trigonometric ratios of a right-angled triangle's sides with regard to any of its acute angles are known as that angle's trigonometric ratios.
Let triangle ABC be a right-angled at B and A is an acute angle.
Given that cosec A = √2
Which implies,
AC/BC = √2/1
Let AC = √2x
Then BC = x
In right angled triangle ABC
By using Pythagoras theorem,
We get
\begin{array}{l} A C^{2}=A B^{2}+B C^{2} \\ (\sqrt{2} x)^{2}=A B^{2}+x^{2} \\ A B^{2}=2 x^{2}-x^{2} \\ A B=x \end{array}
Sin A = perpendicular/ hypotenuse
= BC/AC
= 1/ √2
Cot A = base/ perpendicular
= x/x
= 1
Tan A = perpendicular/ base
= BC/AB
= x/x
= 1
Cos A = base/ hypotenuse
= AB/AC
= x/ √2x
= 1/√2
Substituting these values we get
2 \sin ^{2} A+3 \cot ^{2} A /\left(\tan ^{2} A-\cos ^{2} A\right)=8
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Learn about the Life and Accomplishments of Galileo...
Interactive Excitement in a Feature Packed Learning Tool
Check out our new Augmented Reality Apps! Check out our Giants in Math and Science Series: With this app students learn about the life and accomplishments of Galileo. It discusses Galileo's work and shows why he deserves the title 'Father of Modern Science'. Galileo Galilei was an Italian mathematician, physicist and astronomer. He was born in Pisa, in the Tuscany region of Italy. Initially, following his father's wishes, Galileo studied medicine, but soon he became more intrigued by mathematics and the physical sciences. One of Galileo's legendary experiments involved measuring the speed at which cannonballs fall. Galileo conducted this experiment by dropping canonballs and objects of different weights from the famous Leaning Tower of Pisa. With this app students learn about the life and accomplishments of Galileo. One of his many challenges was teaching about his scientific discoveries when they conflicted with the teachings of the Church. For example, Galileo studied the planets using a telescope that he designed. He discovered four moons orbiting Jupiter and he proved mathematically that Venus is between the Earth and the Sun and that it orbits the Sun not the Earth. During the time of the Inquisition these ideas were heresy and Galileo was punished for writing about them. Students will enjoy using this graphically illustrated, interactive learning tool. The timeline feature presents a sequential view of Galileo's scientific and mathematical journey. A quiz function helps students demonstrate their comprehension of the reading material. Galileo's work showed how the scientific method could be used to make monumental discoveries about the natural world and we hope this app will inspire students to study math and science. The app shows that Galileo certainly deserves the title 'Father of Modern Science'.
Special features make using Galileo Galilei • The Father of Modern Science fun: 1. Tap buttons to learn about the major events in his life. 2. interesting facts are presented using colorful charts. 3. Tap the quiz or game buttons to practice what you've learned. 4. Read about some of the significant discoveries in science, mathematics and astronomy that Galileo made during his lifetime. 5. Sound effects make the interaction even more fun. 6. Available for iPad and Apple TV devices. Available now for Apple TV Bring the excitement of exploring the basic physics, mathematics and astronomy to your classroom or living room with the Apple TV version of the Galileo Galilei - The Father of Modern Science.
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# Ruthless Diffraction Physics Strategies Exploited
In reality, the total amount of diffraction that occurs in any wave depends upon the wavelength of that wave. When interference is constructive, the high level of the wave increases. Interfering waves don’t will need to be spherical or originate from the very same source.
Last, you might have a diffraction pattern around an edge, which maynot be explained by the HUP. A ray of sunlight consists of many wavelengths that in combination seem to be colourless. Diffraction of light is the simplest to recognize, as it takes just a glance.
Waves can travel immense distances though the oscillation at the same point is extremely small. The consequent beam has a bigger aperture, and hence a decrease divergence.
We’ll understand that the degree of diffraction by means of a doorway can be large. If you are quite careful and hold the diffraction grating so that the area of view is close to, but doesn’t incorporate the sun, you can observe an extremely fine spectra. This phenomena is a result of the refraction of sound waves.
Likewise the source just under the surface of the slit will interfere destructively with the source located just beneath the center of the slit at the exact angle. Moreover, due to that, even massive objects and slits lead them to diffract. Every one of the slits function as a source for circular expanding waves.
Let’s look at what it says about absorbance. Since the wave speed is the exact same everywhere, there is not any refraction, and the wave doesn’t change direction as it propagates. It is known as interference.
## What You Should Do to Find Out About Diffraction Physics Before You’re Left Behind
The solution of a single physics problem is frequently the solution of another one apparently quite unrelated. Unfortunately, you are going to be liable for any expenses incurred in return to sender parcels in the event the info you provided was inaccurate. Our clients can expect to receive much better products every time a buy order is placed with us.
It would be recommended to assess the topic of optical gratings in an optics book of your selection. This is called the law of reflection.
Despite the fact that the above diagrams help give a sense of the notion of diffraction, only real-world photography can present its visual effects. The prelab is due by the start of the lab period. Nowadays, the technology necessary to test this notion is in its infancy.
The quantity of reflection depends upon the dissimilarity of both spaces. One of the absolute most powerful parts of evidence for light being some kind of wave motion is it also shows diffraction. Rather, a wave will undergo certain behaviors as soon as it encounters the close of the medium.
Thus, portions of the surface will be stationary. A batter has the capability to transport energy from her to the softball with a bat. On a warm day, the air close to the ground will become warmer than the remaining part of the air and the speed of sound close to the ground increases.
A number of the speakers are often quite small and utilize sound reflection off walls to help disperse the sounds instead of dispersion. Decibels aren’t a measure of just how much sound is created, but how much sound is perceived. Basically, the reflection of sound is truly much like the reflection of light.
Every time a speaker cone pushes out, it’s increasing pressure. You might find yourself wondering how to have a huge depth of field without using a rather small aperture. In Pattern 1, gauge the separation between the centers of the very first dark regions on both sides of the central bright region.
In the event the body isn’t resisted by any form of friction, the motion remains. This is necessary for landscape photography in which you need both foreground and background in focus. Simply because, if light doesn’t diffract, you wouldn’t see anything that’s not directly hit by light.
The different harmonics are those which will occur, with a variety of amplitudes, in stringed instruments. Physics’s been around for a long, long moment. Waves display several standard phenomena.
Such oscillation is known as free oscillation. This is known as the SUPERPOSITION PRINCIPLE. Diffraction is an easy wave phenomenon.
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Home >> AIMS >> Class11 >> Math >> Sets
# If A = {x: x is a natural number}, B ={x: x is an even natural number} C = {x: x is an odd natural number} and D = {x: x is a prime number}, find A $\cap$ D
A $\cap$ D = {x: x is a prime number} = D
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# Super-D Numbers
• Last Updated : 17 Feb, 2023
Given an integer N, the task is to check N is a super-d Number.
Super-D Number is a number N such that D*ND contains a substring made of D digits containing D only, where D is more than 1 and less than 10
Examples:
Input: N = 261
Output: Yes
Explanation:
It will be true for D = 3
D*ND = 3*2613 = 53338743
which contains a substring made of 3 digits 333 containing 3 only.
Input: N = 10
Output: No
Approach: The idea is to create every possible string concatenating digit D, D number of times and then will check if the concatenation is present as a substring in D*ND or not, where D will be in the range [2, 9].
Below is the implementation of the above approach:
## C++
`#include ``#include // include string library for string manipulation``#include // include cmath library for pow function` `using` `namespace` `std;` `// Function to check if N is a super-d number``bool` `isSuperdNum(``int` `n)``{`` ``for` `(``int` `d = 2; d < 10; d++)`` ``{`` ``string subString = ``""``; ``// create an empty string`` ``for` `(``int` `i = 0; i < d; i++)`` ``{`` ``subString += to_string(d); ``// add d as string to the subString`` ``}`` ``if` `(to_string(d * ``pow``(n, d)).find(subString) != string::npos)`` ``// check if subString is present in d * n^d as a substring`` ``return` `true``;`` ``}`` ``return` `false``;``}` `// Driver Code``int` `main()``{`` ``int` `n = 261;`` ``if` `(isSuperdNum(n) == ``true``)`` ``cout << ``"Yes"` `<< endl;`` ``else`` ``cout << ``"No"` `<< endl;`` ``return` `0;``}`
## Java
`// Java implementation to``// check if N is a super-d number``class` `GFG{`` ` `// Function to check if N``// is a super-d number``static` `boolean` `isSuperdNum(``int` `n)``{`` ``for` `(``int` `d = ``2``; d < ``10``; d++)`` ``{`` ``String subString = newString(d);`` ``if` `(String.valueOf(`` ``(d * Math.pow(n, d))).contains(subString))`` ``return` `true``;`` ``}`` ``return` `false``;``}` `// Driver Code``private` `static` `String newString(``int` `d)``{`` ``String ans = ``""``;`` ``for` `(``int` `i = ``0``; i < d; i++)`` ``{`` ``ans += String.valueOf(d);`` ``}`` ``return` `ans;``}` `// Driver Code``public` `static` `void` `main(String[] args)``{`` ``int` `n = ``261``;`` ``if` `(isSuperdNum(n) == ``true``)`` ``System.out.println(``"Yes"``);`` ``else`` ``System.out.println(``"No"``);``}``}` `// This code is contributed by Rajput-Ji`
## Python3
`# Python3 implementation to``# check if N is a super-d number` `# Function to check if N``# is a super-d number``def` `isSuperdNum(n):`` ``for` `d ``in` `range` `(``2``, ``10``):`` ``substring ``=` `str``(d) ``*` `d;`` ``if` `substring ``in` `str``(d ``*` `pow``(n, d)):`` ``return` `True`` ``return` `False` `# Driver Code``n ``=` `261``if` `isSuperdNum(n) ``=``=` `True``:`` ``print``(``"Yes"``)``else` `:`` ``print``(``"No"``)`
## C#
`// C# implementation to``// check if N is a super-d number``using` `System;` `class` `GFG{`` ` `// Function to check if N``// is a super-d number``static` `bool` `isSuperdNum(``int` `n)``{`` ``for``(``int` `d = 2; d < 10; d++)`` ``{`` ``String subString = newString(d);`` ``if` `(String.Join(``""``,`` ``(d * Math.Pow(n, d))).Contains(subString))`` ``return` `true``;`` ``}`` ``return` `false``;``}` `private` `static` `String newString(``int` `d)``{`` ``String ans = ``""``;`` ` ` ``for``(``int` `i = 0; i < d; i++)`` ``{`` ``ans += String.Join(``""``, d);`` ``}`` ``return` `ans;``}` `// Driver Code``public` `static` `void` `Main(String[] args)``{`` ``int` `n = 261;`` ` ` ``if` `(isSuperdNum(n) == ``true``)`` ``Console.WriteLine(``"Yes"``);`` ``else`` ``Console.WriteLine(``"No"``);``}``}` `// This code is contributed by Rajput-Ji`
## Javascript
`// Function to check if N is a super-d number``function` `isSuperdNum(n) {`` ``for` `(let d = 2; d < 10; d++) {`` ``let substring = String(d).repeat(d);`` ``if` `(String(d * Math.pow(n, d)).includes(substring)) {`` ``return` `true``;`` ``}`` ``}`` ``return` `false``;``}` `// Driver code``let n = 261;``if` `(isSuperdNum(n)) {`` ``console.log(``"Yes"``);``} ``else` `{`` ``console.log(``"No"``);``}`
Output:
`Yes`
Time Complexity: O(1)
Auxiliary Space: O(1) as it is using constant space
References: OEIS
My Personal Notes arrow_drop_up
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https://app.seesaw.me/activities/mdavbu/properties-of-addition-and-multiplication-exit-ticket
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Match the expressions to the correct property. ____ 22 x (8 x 12) = (22 x 8) x 12 ____ 15 + 28 + 18 = 28 + 15 + 18 ____ 16 + (11 + 10) = (16 + 11) + 10 ____ 33 x 4 = 4 x 33 ____ 8 X 1 = 8 ____ 82 + 0 = 82 A. Commutative Property of Multiplication B. Associative Property of Addition C. Identity Property of Addition D. Commutative Property of Addition E. Associative Property of Multiplication F. Identity Property of Multiplication Evaluate (solve) the following expression: 25 + (34-8) / 2 Use the Distributive Property of Multiplication to rewrite this expression: 8 X 25
Student Instructions
### Properties of Addition and Multiplication - Exit Ticket
Please match the expressions to the correct property, and answer the questions at the end.
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# Tip Calculator
dollar
percentage
Count
Dollar
Dollar
Dollar
Dollar
Tip Calculator: The Tip usually defined as money given with gratitude by the customer to the service worker in addition to the actual price. It is commonly given to service workers for voluntary service performed to the customers.
This Tip culture is not often seen in all countries and it is not mandatory to follow this tipping process. Generally, Tips are offered to service workers for the service they performed to customers in Restaurants or bars, Saloons, Drivers and so on.
All countries do not follow this tipping culture, in some countries it is considered as social etiquette and in some countries, it is considered as an insult to the service worker.
So, because of these differences, Tip culture is mainly followed in some countries only. In some areas, the tips are gradually included in the bill and the customers do not need to leave tips for servers.
But in some situations, one can see that the service workers do not have any extra compensation for their good service. So, in those cases customers can offer tips to the workers are gratitude.
Generally, 15% of meal price is offered as a tip before tax meal price. If the sales tax percentages range in between 6% to 9% then about 14% of after-meal tax price is offered as the tip.
In countries like the US and Canada tip service is widely in practice and also employees may have some compensation if they receive good tips representing their good service.
### How to calculate Tip using Tip Calculator
This free tip calculator provides user to solve issues with Tip offering in the way of providing an Online Tip Calculator.
• Enter the total bill amount you need to pay.
• Enter the Tip percentage followed in your region or country.
• Enter the number of service workers offered you a Good service.
• Click on option saying “Calculate”.
• Instantly the Tip calculator performs its duty and displays the result.
Another way to calculate the Tip is just to follow your state sales tax. For suppose, if your state sales tax is 15% then just double or triple on your choice of interest and just leave tip on the table.
The Tip Calculator works at free of cost and users can change the amount or tip percentage as many times as possible and determine the tip amount.
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https://artofproblemsolving.com/wiki/index.php?title=2016_AMC_8_Problems/Problem_1&oldid=127511
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2016 AMC 8 Problems/Problem 1
The longest professional tennis match ever played lasted a total of $11$ hours and $5$ minutes. How many minutes was this?
$\textbf{(A) }605\qquad\textbf{(B) }655\qquad\textbf{(C) }665\qquad\textbf{(D) }1005\qquad \textbf{(E) }1105$
Solution
It is best to split 11 hours and 5 minutes into 2 parts, one of 11 hours and another of 5 minutes. We know that there is $60$ minutes in a hour. Therefore, there are $11 \cdot 60 = 660$ minutes in 11 hours. Adding the second part(the 5 minutes) we get $660 + 5 = \boxed{\textbf{(C)}\ 665}$.
Solution 2
The best method comes when you remember your multiplication tables. Thus trivial, we get our answer of $\boxed{{(C) } 665}$.
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# Red blood cell indices: Wikis
Note: Many of our articles have direct quotes from sources you can cite, within the Wikipedia article! This article doesn't yet, but we're working on it! See more info or our list of citable articles.
# Encyclopedia
Red blood cell indices are blood tests that provide information about the hemoglobin content and size of red blood cells. Abnormal values indicate the presence of anemia and which type of anemia it is.[1]
## Mean corpuscular volume
Mean corpuscular volume (MCV) is the average size of a red blood cell and is calculated by dividing the hematocrit (Hct) by the red blood cell count.
• $MCV = \frac{Hct}{RBC}$
• Normal range: 80-100 fL
## Mean corpuscular hemoglobin
Mean corpuscular hemoglobin (MCH) is the average amount of hemoglobin (Hb) per red blood cell and is calculated by dividing the hemoglobin by the red blood cell count.
• $MCH = \frac{Hb}{RBC}$
• Normal range: 27-31 pg/cell
## Mean corpuscular hemoglobin concentration
Mean corpuscular hemoglobin concentration (MCHC) is the average concentration of hemoglobin per red blood cell and is calculated by dividing the hemoglobin by the hematocrit.
• $MCHC = \frac{Hb}{Hct}$
• Normal range: 32-36 g/dL
## Worked example
Measure Units Conventional units Conversion
Hct 40%
Hb 100 grams/liter 10 grams/deciliter (deci- is 10-1)
RBC 5E+12 cells/liter 5E+12 cells/liter
MCV = Hct / RBC 8E-14 liters/cell 80 femtoliters/cell (femto- is 10-15)
MCH = Hb / RBC 2E-11 grams/cell 20 picograms/cell (pico- is 10-12)
MCHC = Hb / HCT 250 grams/liter 25 grams/deciliter (deci is 10-1)
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Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel
Abstract: In this paper, the existence and uniqueness of the solution of Fredholm-Volterra integral equation is considered (NF-VIE) with continuous kernel; then we used a numerical method to reduce this type of equations to a system of nonlinear Volterra integral equations. Runge-Kutta method (RKM) and Bolck by block method (BBM) are used to solve the system of nonlinear Volterra integral equations of the second kind (SNVIEs) with continuous kernel. The error in each case is calculated.
1. Introduction
Integral equations of various types and kinds play an important role in many branches of linear and nonlinear function analysis and their applications in the theory of elasticity, engineering, mathematical physical and contact mixed problems. Therefor, many different methods are used to obtain the solution of the Volterra integral equation. In [1] Linz, studied analytical and numerical methods of Volterra equation. In [2], Mirzaee and Rafei used the BBM for the numerical solution of the nonlinear two-dimensional Volterra integral equations. In the references [3] - [8] the authors considered many different methods to solve linear and nonlinear system of Volterra integral equations numerically with continuous and singular kernels. In [9], Al-waqdani studied linear F-VIE with continuous kernel and solved the linear SVIEs numerically with continuous kernel.
$\mu \varphi \left(x,t\right)=f\left(x,t\right)+\lambda {\int }_{a}^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y+\lambda {\int }_{0}^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau$ (1)
Equation (1) is called the NF-VIE in the space ${L}_{2}\left[a,b\right]×C\left[0,T\right],T<\infty$. Here the Fredholm integral term is considered in position with a positive continuous kernel $K\left(x,y\right)$ for all $x,y\in \left[a,b\right]$, while the Volterra integral term is considered in time with a positive continuous kernel $F\left(t,\tau \right)$ for all $t,\tau \in \left[0,T\right],T<\infty$. The free term $f\left(x,t\right)$ is known continuous function in the space ${L}_{2}\left[a,b\right]×C\left[0,T\right]$, while $\varphi \left(x,t\right)$ is unknown function representing the solution of the nonlinear integral Equation (1). The numerical coefficient $\lambda$ is called the parameter of the integral equation, may be complex, and has physical meaning, while the constant parameter $\mu$ defines the kind of the integral Equation (1).
2. Existence of Solution of NF-VIE
To prove the existence of a unique solution of Equation (1) using fixed point theorem.
We write it in the integral operator form:
$\stackrel{¯}{W}\varphi =f+W\varphi$ (2)
where
$W\varphi =K\varphi +F\varphi$ (3)
$K\varphi =\lambda {\int }_{a}^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y,\text{\hspace{0.17em}}\text{\hspace{0.17em}}F\varphi =\lambda {\int }_{0}^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau$ (4)
Then, we assume the following conditions:
i) The kernel of Fredholm integral term satisfies:
$|K\left(x,y\right)|\le {A}_{1}$ ( ${A}_{1}$ is a constant).
ii) The kernel of Volterra integral term satisfies:
$|F\left(t,\tau \right)|\le {A}_{2}$ ( ${A}_{2}$ is a constant).
iii) The given function $f\left(x,t\right)$ with its partial derivatives is continuous in ${L}_{2}\left[a,b\right]×C\left[0,T\right]$ where:
$‖f\left(x,t\right)‖=\mathrm{max}{\int }_{0}^{t}{\left({\int }_{a}^{b}{|f\left(x,\tau \right)|}^{2}\text{d}x\right)}^{\frac{1}{2}}\text{d}\tau ={A}_{3}$ ( ${A}_{3}$ is a constant).
iv) The known continuous function $\gamma \left(x,t,\varphi \left(x,t\right)\right)$, for the constant $B>{B}_{1},B>p$, the following conditions:
a) ${\left({\int }_{0}^{t}{\int }_{a}^{b}{|\gamma \left(x,t,\varphi \left(x,t\right)\right)|}^{2}\text{d}x\text{d}t\right)}^{\frac{1}{2}}\le {B}_{1}{‖\varphi \left(x,t\right)‖}_{{L}_{2}\left[a,b\right]×C\left[0,T\right]}$
b) $‖\gamma \left(x,t,{\varphi }_{1}\left(x,t\right)\right)-\gamma \left(x,t,{\varphi }_{2}\left(x,t\right)\right)‖\le N\left(x,t\right)|{\varphi }_{1}\left(x,t\right)-{\varphi }_{2}\left(x,t\right)|$
where ${‖N\left(x,t\right)‖}_{{L}_{2}\left[a,b\right]×C\left[0,T\right]}=p$.
Theorem 1:
If the condition i)-iv) are verified, then Equation (1) has unique solution in the Banach space ${L}_{2}\left[a,b\right]×C\left[0,T\right],T<\infty$.
The provement of this theorem depends on the following two lemmas:
Lemma 1:
Under the conditions i)-iv-a), the operator $\stackrel{¯}{W}$ defined by (2), maps the space ${L}_{2}\left[a,b\right]×C\left[0,T\right],T<\infty$ into itself.
Proof:
In view of Formula (2) and (3) we get:
$\begin{array}{c}‖\stackrel{¯}{W}\varphi \left(x,t\right)‖\le ‖f\left(x,t\right)‖+|\lambda |‖{\int }_{a}^{b}|K\left(x,y\right)||\gamma \left(y,t,\varphi \left(y,t\right)\right)|\text{d}y‖\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\lambda |‖{\int }_{0}^{t}|F\left(t,\tau \right)||\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)|\text{d}\tau ‖\end{array}$
Using the conditions (i)-(iii), then applying Cauchy-Schwarz inequality, we have: $\begin{array}{c}‖\stackrel{¯}{W}\varphi \left(x,t\right)‖\le {A}_{3}+|\lambda |{A}_{1}{\left({\int }_{a}^{b}{|\gamma \left(y,t,\varphi \left(y,t\right)\right)|}^{2}\text{d}y\right)}^{\frac{1}{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\lambda |{A}_{2}{\left({\int }_{0}^{t}{|\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)|}^{2}\text{d}\tau \right)}^{\frac{1}{2}}\end{array}$
In the light of the condition (iv-a), the above inequality take the form:
$‖\stackrel{¯}{W}\varphi \left(x,t\right)‖\le {A}_{3}+\alpha ‖\varphi \left(x,t\right)‖,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\alpha =|\lambda |{B}_{1}\left({A}_{1}+{A}_{2}+L\right)$ (5)
The lost inequality (5) shows that, the operator $\stackrel{¯}{W}$ maps the ball ${S}_{r}$ into itself, where
$r=\frac{{A}_{3}}{1-|\lambda |{B}_{1}\left({A}_{1}+{A}_{2}L\right)}$ (6)
Since $r>0,{A}_{3}>0$, therefore we have $\alpha <1$. Moreover, the inequality (5) involves the boundedness of the operator W of Equation (2) where:
$‖W\varphi \left(x,t\right)‖\le \alpha ‖\varphi \left(x,t\right)‖$ (7)
Also, the inequalities (5) and (7) define the boundedness of the operator $\stackrel{¯}{W}$.
Lemma 2:
If the conditions (i),(ii) and (iv-b) are satisfied, then the operator $\stackrel{¯}{W}$ is contractive in space ${L}_{2}\left[a,b\right]×C\left[0,T\right],T<\infty$.
Proof:
For two functions ${\varphi }_{1}\left(x,t\right)$ and ${\varphi }_{2}\left(x,t\right)$ in the space ${L}_{2}\left[a,b\right]×C\left[0,T\right]$ Formula (2), (3) leads to:
$\begin{array}{c}‖\stackrel{¯}{W}{\varphi }_{1}\left(x,t\right)-\stackrel{¯}{W}{\varphi }_{2}\left(x,t\right)‖\le |\lambda |‖{\int }_{a}^{b}|K\left(x,y\right)||\gamma \left(y,t,{\varphi }_{1}\left(y,t\right)\right)-\gamma \left(y,t,{\varphi }_{2}\left(y,t\right)\right)|\text{d}y‖\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\lambda |‖{\int }_{0}^{t}|F\left(t,\tau \right)||\gamma \left(x,\tau ,{\varphi }_{1}\left(x,\tau \right)\right)-\gamma \left(x,\tau ,{\varphi }_{2}\left(x,\tau \right)\right)|\text{d}\tau ‖\end{array}$
Using the condition (iv-b), then apply Cauchy-Schwarz inequality we have:
$\begin{array}{c}‖\stackrel{¯}{W}{\varphi }_{1}\left(x,t\right)-\stackrel{¯}{W}{\varphi }_{2}\left(x,t\right)‖\le |\lambda ||K\left(x,y\right)|{\left({\int }_{a}^{b}{N}^{2}\left(x,t\right){|{\varphi }_{1}\left(y,t\right)-{\varphi }_{2}\left(y,t\right)|}^{2}\text{d}y\right)}^{\frac{1}{2}}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+|\lambda ||F\left(t,\tau \right)|{\left({\int }_{a}^{b}{N}^{2}\left(x,\tau \right){|{\varphi }_{1}\left(x,\tau \right)-{\varphi }_{2}\left(x,\tau \right)|}^{2}\text{d}\tau \right)}^{\frac{1}{2}}\end{array}$
Finally, with the aid of conditions (i), (ii), and (iv-b) we obtain:
$‖\stackrel{¯}{W}{\varphi }_{1}\left(x,t\right)-\stackrel{¯}{W}{\varphi }_{2}\left(x,t\right)‖\le \alpha ‖{\varphi }_{1}\left(x,t\right)-{\varphi }_{2}\left(x,t\right)‖$ (8)
In equality (8) shows that, the operator $\stackrel{¯}{W}$ is continuous in the space ${L}_{2}\left[a,b\right]×C\left[0,T\right],T<\infty$, then $\stackrel{¯}{W}$ is a contraction operator under the condition $\alpha <1$.
3. The SVIEs
Consider:
$\begin{array}{l}\varphi \left(x,t\right)=f\left(x,t\right)+\lambda \underset{a}{\overset{b}{\int }}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left(\mu =1\right)\end{array}$ (9)
when $a=b=0$, Equation (9) becomes:
$\varphi \left(0,t\right)=f\left(0,t\right)+\lambda {\int }_{0}^{t}F\left(t,\tau \right)\gamma \left(0,\tau ,\varphi \left(0,\tau \right)\right)\text{d}\tau$
Then,
${\varphi }_{0}\left(t\right)={f}_{0}\left(t\right)+\lambda \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,{\varphi }_{0}\left(\tau \right)\right)\text{d}\tau$ (10)
where ${u}_{0}\left(t\right)=u\left(0,t\right),{f}_{0}\left(t\right)=f\left(0,t\right)$.
Formula (10) represents Volterra integral equation of the second kind at $a=b=0$. For representing (9) as a VIEs, we use the numerical method. Divide the interval $\left[a,b\right]$ as $a={x}_{0}\le {x}_{1}\le \cdots \le {x}_{N}=b$. Using the quadrature formula, Equation (9) becomes:
$\begin{array}{l}{\int }_{a}^{b}K\left(x,y\right)\gamma \left(y,t,\varphi \left(y,t\right)\right)\text{d}y\\ =\underset{m=0}{\overset{n}{\sum }}{u}_{m}K\left({x}_{n},{x}_{m}\right)\gamma \left({x}_{n},t,\varphi \left({x}_{n},t\right)\right)\end{array}$ (11)
where $n=1,2,\cdots ,N$, and ${u}_{0}=\frac{1}{2}{h}_{0},{u}_{n}=\frac{1}{2}{h}_{n},{u}_{i}={h}_{i},\left(i\ne 0,n\right)$.
Using (11) in (9), we have:
$\begin{array}{c}\varphi \left(x,t\right)=f\left(x,t\right)+\lambda {\int }_{0}^{t}F\left(t,\tau \right)\gamma \left(x,\tau ,\varphi \left(x,\tau \right)\right)\text{d}\tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\underset{m=0}{\overset{n}{\sum }}{u}_{m}K\left({x}_{n},{x}_{m}\right)\gamma \left({x}_{n},t,\varphi \left({x}_{n},t\right)\right)\end{array}$ (12)
Then:
${\varphi }_{n}={W}_{n}\left(t\right)+\lambda \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left({x}_{n},\tau ,{\varphi }_{n}\left(\tau \right)\right)\text{d}\tau$ (13)
where ${W}_{n}\left(t\right)={f}_{n}\left(t\right)+\lambda \underset{m=0}{\overset{n-1}{\sum }}{u}_{m}K\left({x}_{n},{x}_{m}\right)\gamma \left({x}_{m},t,{\varphi }_{m}\left(t\right)\right)$, $n=0,1,\cdots ,N$. Formula (13) represents a NSVIEs of the second kind, and we have N unknown functions ${\varphi }_{n}\left(t\right)$ corresponding to time interval $\left[0,T\right]$.
4. Some Numerical Methods for Solving SVIEs
4.1. RKM
In this section, the RKM is used to solve NF-VIE of the second kind. By divide the interval $\left[a,b\right]$ as $a={x}_{0}\le {x}_{1}\le \cdots \le {x}_{i}\le \cdots \le {x}_{N}=b$, $i=0,1,\cdots ,N$ and using the quadrature formula, the integral Equation (1) represent a NSVIEs as:
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,{\varphi }_{n}\left(\tau \right)\right)\text{d}\tau$
where ${W}_{n}\left(t\right)={f}_{i}\left(t\right)+\underset{i=0}{\overset{n}{\sum }}{u}_{i}{k}_{ni}\gamma \left(t,{\varphi }_{i}\left(t\right)\right)$.
To solve the NSVIEs:
$\phi \left(t\right)=\underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau +F\left(t\right)+\underset{i=0}{\overset{n}{\sum }}{u}_{i}{k}_{ni}\gamma \left(t,\phi \left(t\right)\right)\text{ }\text{ },\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=0,1,\cdots ,N$ (14)
where
$\begin{array}{l}\phi \left(t\right)={\left({\varphi }_{1}\left(t\right),{\varphi }_{2}\left(t\right),\cdots ,{\varphi }_{l}\left(t\right)\right)}^{\text{T}},\\ \gamma \left(\tau ,\phi \left(\tau \right)\right)=\left(\gamma \left(\tau ,{\phi }_{1}\left(\tau \right)\right),\gamma \left(\tau ,{\phi }_{2}\left(\tau \right)\right),\cdots ,\gamma \left(\tau ,{\phi }_{l}\left(\tau \right)\right)\right),\\ F\left(t\right)={\left({f}_{1}\left(t\right),{f}_{2}\left(t\right),\cdots ,{f}_{l}\left(t\right)\right)}^{\text{T}}\end{array}$
and
$\begin{array}{l}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ =\left[\begin{array}{cccc}{F}_{1,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{1,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{1,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ {F}_{2,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{2,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{2,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ ⋮& ⋮& \ddots & ⋮\\ {F}_{s,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{s,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{s,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\end{array}\right]\end{array}$
Then, we get
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=0,1,\cdots ,N$ (15)
Now, applying the RKM for solve (15):
Suppose that:
$F\left(t,\tau \right)=\underset{s}{\sum }{\psi }_{s}\left(t\right){\chi }_{s}\left(\tau \right)$ (16)
Substituting from (16) into (15),
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\underset{0}{\overset{t}{\int }}\underset{s}{\overset{}{\sum }}{\psi }_{s}\left(t\right){\chi }_{s}\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau$
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\sum {\psi }_{s}\left(t\right){\int }_{0}^{t}{\chi }_{s}\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau$
Then, we have,
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\underset{s}{\sum }{\psi }_{s}\left(t\right){z}_{s}\left(t\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}t>0$ (17)
where
${z}_{s}\left(t\right)=\underset{0}{\overset{t}{\int }}{\chi }_{s}\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)d\tau$ (18)
By derivative (18), we have,
${{z}^{\prime }}_{s}\left(t\right)={\chi }_{s}\left(\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right),\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{z}_{s}\left(0\right)=0$
Now, apply the RKM to this system of equations to give,
$\stackrel{˜}{z}\left({v}_{p}h\right)=h\underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}{\chi }_{s}\left({v}_{q}h\right)\gamma \left({v}_{q}h,\stackrel{˜}{\phi }\left({v}_{q}h\right)\right)\text{ }\text{ },\text{\hspace{0.17em}}\text{\hspace{0.17em}}p=1,2,\cdots ,m$
$\begin{array}{c}{\stackrel{˜}{\varphi }}_{n}\left({v}_{p}h\right)={W}_{n}\left({v}_{p}h\right)+\underset{s}{\sum }{\stackrel{˜}{z}}_{s}\left({v}_{p}h\right){\psi }_{s}\left({v}_{p}h\right)\\ ={W}_{n}\left({v}_{p}h\right)+\underset{s}{\sum }{\psi }_{s}\left({v}_{p}h\right)h\underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}{\chi }_{s}\left({v}_{q}h\right)\gamma \left({v}_{q}h,\stackrel{˜}{\phi }\left({v}_{q}h\right)\right)\\ ={W}_{n}\left({v}_{p}h\right)+h\underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}\gamma \left({v}_{q}h,\stackrel{˜}{\phi }\left({v}_{q}h\right)\right)\underset{s}{\sum }{\psi }_{s}\left({v}_{p}h\right){\chi }_{s}\left({v}_{q}h\right)\end{array}$
By using Equation (16) to give,
${\stackrel{˜}{\varphi }}_{n}\left({v}_{p}h\right)={W}_{n}\left({v}_{p}h\right)+h\underset{q=0}{\overset{p-1}{\sum }}{A}_{pq}F\left({v}_{p}h,{v}_{q}h\right)\gamma \left({v}_{q}h,\stackrel{˜}{\phi }\left({v}_{q}h\right)\right)\text{ }\text{ },\text{\hspace{0.17em}}\text{\hspace{0.17em}}p=1,2,\cdots ,m$ (19)
which is approximate solution for Equation (15).
Now, if $m=4$ consider the Pouzet’s derivation, we define:
$\begin{array}{l}{p}_{i,j}\left(t\right)={\varphi }_{i,j-1}\left(t\right)\\ {q}_{i,j}\left(t\right)={G}_{i,j}\left({t}_{j+\frac{1}{2}}\right)+\frac{h}{2}F\left({t}_{j+\frac{1}{2}},{t}_{j}\right){p}_{i,j}\\ {r}_{i,j}\left(t\right)={G}_{i,j}\left({t}_{j+\frac{1}{2}}\right)+\frac{h}{2}F\left({t}_{j+\frac{1}{2}},{t}_{j+\frac{1}{2}}\right){q}_{i,j}\\ {s}_{i,j}\left(t\right)={G}_{i,j}\left({t}_{j+1}\right)+hF\left({t}_{j+1},{t}_{j+1}\right){r}_{i,j}\end{array}$ (20)
${\varphi }_{i,j}\left(t\right)={G}_{i,j}\left({t}_{j+1}\right)+\frac{h}{6}\left[F\left({t}_{j+1},{t}_{j}\right){p}_{i,j}+2F\left({t}_{j+1},{t}_{j+\frac{1}{2}}\right)\left[{q}_{i,j}+{r}_{i,j}\right]+F\left({t}_{j+1},{t}_{j+1}\right){s}_{i,j}\right]$
The function ${\varphi }_{i,j}\left(t\right)$ is unknown function, such that
${G}_{i,j}\left(t\right)={W}_{i}\left(t\right)+\frac{h}{6}\underset{n=0}{\overset{j-1}{\sum }}\left[F\left(t,{t}_{n}\right){p}_{i,n}+2F\left(t,{t}_{n+\frac{1}{2}}\right)\left[{q}_{i,n}+{r}_{i,n}\right]+F\left(t,{t}_{n+1}\right){s}_{i,n}\right]$ (21)
where ${G}_{i,0}\left(t\right)={W}_{i}\left(t\right)$, where $j=2,3,\cdots ,N-1$.
when $j=1$, Equation (10) becomes:
${\varphi }_{i,1}\left(t\right)={G}_{i,1}\left({t}_{2}\right)+\frac{h}{6}\left[F\left({t}_{2},{t}_{1}\right){p}_{i,1}+2F\left({t}_{2},{t}_{1+\frac{1}{2}}\right)\left[{q}_{i,1}+{r}_{i,1}\right]+F\left({t}_{2},{t}_{2}\right){s}_{i,1}\right]$
such that ${G}_{i,1}=W\left({t}_{2}\right)$ and
${p}_{i,1}\left(t\right)={W}_{i}\left( t 1 \right)$
$\begin{array}{l}{q}_{i,1}\left(t\right)={G}_{i,1}\left({t}_{1+\frac{1}{2}}\right)+\frac{h}{2}F\left({t}_{1+\frac{1}{2}},{t}_{1}\right){p}_{i,1}\\ {r}_{i,1}\left(t\right)={G}_{i,1}\left({t}_{1+\frac{1}{2}}\right)+\frac{h}{2}F\left({t}_{1+\frac{1}{2}},{t}_{1+\frac{1}{2}}\right){q}_{i,1}\\ {s}_{i,1}\left(t\right)={G}_{i,1}\left({t}_{2}\right)+hF\left({t}_{2},{t}_{2}\right){r}_{i,1}\end{array}$
Since the function ${\varphi }_{i,j}\left(t\right)={\varphi }_{i,j}$ is the approximate solution at $\left({x}_{i},{t}_{j}\right)$ for Equation (1).
4.2. BBM
In this section, we use the BBM for solving the NF-VIE of the second kind. The interval $\left[a,b\right]$ is divided into steps of width h, ${t}_{j}=jh,j=0,1,\cdots ,n$ and $h=\left(b-a\right)/n$. the approximate solution of ${\varphi }_{i}\left(t\right)$ will be define at mesh-points ${t}_{j}$ and denoted by ${\varphi }_{ij},j=0,1,\cdots ,n$ such as ${\varphi }_{ij}$ is an approximation to ${\varphi }_{i}\left({t}_{j}\right)$.
To solve the NSVIEs:
$\phi \left(t\right)=F\left(t\right)+\lambda \underset{j=0}{\overset{i}{\sum }}{u}_{j}{K}_{nj}\gamma \left(t,\varphi \left(t\right)\right)+\lambda \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=0,1,\cdots ,N$ (22)
where
$\begin{array}{l}\phi \left(t\right)={\left({\varphi }_{1}\left(t\right),{\varphi }_{2}\left(t\right),\cdots ,{\varphi }_{l}\left(t\right)\right)}^{\text{T}},\\ \gamma \left(\tau ,\phi \left(\tau \right)\right)=\left(\gamma \left(\tau ,{\phi }_{1}\left(\tau \right)\right),\gamma \left(\tau ,{\phi }_{2}\left(\tau \right)\right),\cdots ,\gamma \left(\tau ,{\phi }_{l}\left(\tau \right)\right)\right),\\ F\left(t\right)={\left({f}_{1}\left(t\right),{f}_{2}\left(t\right),\cdots ,{f}_{l}\left(t\right)\right)}^{\text{T}}\end{array}$
and
$\begin{array}{l}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ =\left[\begin{array}{cccc}{F}_{1,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{1,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{1,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ {F}_{2,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{2,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{2,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\\ ⋮& ⋮& \ddots & ⋮\\ {F}_{s,1}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& {F}_{s,2}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)& \cdots & {F}_{s,s}\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\end{array}\right]\end{array}$
Then, we get
${\varphi }_{n}\left(t\right)={W}_{n}\left(t\right)+\lambda \underset{0}{\overset{t}{\int }}F\left(t,\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}n=0,1,\cdots ,N$ (23)
Rewrite Equation (23) as follows:
${\varphi }_{i}\left({t}_{k}\right)={W}_{i}\left({t}_{k}\right)+\lambda \underset{0}{\overset{{t}_{pm}}{\int }}{F}_{i,s}\left({t}_{k},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau +\lambda \underset{{t}_{pm}}{\overset{{t}_{n}}{\int }}{F}_{i,s}\left({t}_{k},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau$
$\begin{array}{c}{\varphi }_{i}\left({t}_{k}\right)={f}_{i}\left({t}_{k}\right)+\lambda \underset{n=0}{\overset{i}{\sum }}{u}_{n}{k}_{in}\gamma \left({t}_{k},{\varphi }_{n}\left({t}_{k}\right)\right)+\lambda \underset{0}{\overset{{t}_{pm}}{\int }}{F}_{i,s}\left({t}_{k},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau \\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \underset{{t}_{pm}}{\overset{{t}_{n}}{\int }}{F}_{i,s}\left({t}_{k},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau \end{array}$ (24)
where $s=1,2,\cdots$.
If the values ${\varphi }_{i0},{\varphi }_{i1},\cdots ,{\varphi }_{i,pm}$ are known, then the first integral can be approximated by standard quadrature methods, and the second integral is obtain by a quadrature rule using values of the integrand at $\tau ={t}_{pm},{t}_{pm+1},\cdots ,{t}_{p\left(m+1\right)}$.
Since the values of ${\varphi }_{i}$ at these points are unknown, we have a system of ${l}_{p}$ nonlinear equations by applied the BBM:
$\begin{array}{c}{\varphi }_{ik}={f}_{i}\left({t}_{k}\right)+\lambda \underset{n=0}{\overset{i}{\sum }}{u}_{n}{K}_{in}\gamma \left({t}_{k},{\varphi }_{n}\left({t}_{k}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left[h\underset{j=0}{\overset{mp}{\sum }}{v}_{kj}{F}_{i,s}\left({t}_{k},{\tau }_{j}\right)\left(\gamma \left({\tau }_{j},{\varphi }_{ij}\right),\cdots ,\gamma \left({\tau }_{j},{\varphi }_{lj}\right)\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left[h\underset{j=mp}{\overset{\left(m+1\right)p}{\sum }}{v}_{kj}^{*}{F}_{i,s}\left({t}_{k},{\tau }_{mp+j}\right)\left(\gamma \left({\tau }_{mp+j},{\varphi }_{1,mp+j}\right),\cdots ,\gamma \left({\tau }_{mp+j},{\varphi }_{l,mp+j}\right)\right)\right]\end{array}$ (25)
For $n=mp+1,mp+2,\cdots ,\left(m+1\right)p$, $m=0,1,\cdots ,\left(N-1\right)$, where ${v}_{kj},{v}_{kj}^{*}$ depend on the quadrature rule used.
Now, for the Modified method of two Blocks we take $p=2$, this integration over $\left[0,{t}_{2m}\right]$ can be accomplished by Simpson’s rule, and the integral over $\left[{t}_{2m},{t}_{n}\right]$ by using a quadratic interpolation of the integrand at the point ${t}_{2m},{t}_{2m+1},{t}_{2m+2}$, then Equation (23) becomes:
${\varphi }_{i,2m+1}={W}_{i}\left({t}_{2m+1}\right)+\lambda \underset{0}{\overset{\left(2m+1\right)h}{\int }}{F}_{i,s}\left({t}_{2m+1},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau$ (26)
and
$\begin{array}{l}{\varphi }_{i,2m+2}\\ ={W}_{i}\left({t}_{2m+2}\right)+\lambda \underset{0}{\overset{\left(2m+2\right)h}{\int }}{F}_{i,s}\left({t}_{2m+2},\tau \right)\gamma \left(\tau ,\phi \left(\tau \right)\right)\text{d}\tau \end{array}$ (27)
where $i=1,2,\cdots ,l;m=0,1,\cdots$.
Therefore, by Equation (25) the approximate solution is computed by:
$\begin{array}{c}{\varphi }_{i,2m+1}={W}_{i}\left({t}_{2m+1}\right)+\lambda \left[\frac{h}{3}\underset{j=0}{\overset{2m}{\sum }}{v}_{j}{F}_{i,s}\left({t}_{2m+1},{\tau }_{j}\right)\left(\gamma \left({\tau }_{j},{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_{j},{\varphi }_{lj}\right)\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left[\frac{h}{12}\left[5{F}_{i,s}\left({t}_{2m+1},{\tau }_{2m}\right)\left(\gamma \left({\tau }_{2m},{\varphi }_{1,2m}\right),\cdots ,\gamma \left({\tau }_{2m},{\varphi }_{l,2m}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+8{F}_{i,s}\left({t}_{2m+1},{\tau }_{2m+1}\right)\left(\gamma \left({\tau }_{2m+1},{\varphi }_{1,2m+1}\right),\cdots ,\gamma \left({\tau }_{2m+1},{\varphi }_{l,2m+1}\right)\right)\\ \text{ }\text{ }\begin{array}{c}\text{ }\\ \text{ }\end{array}-F\left({t}_{2m+1},{\tau }_{2m+2}\right)\left(\gamma \left({\tau }_{2m+2},{\varphi }_{1,2m+2}\right),\cdots ,\gamma \left({\tau }_{2m+2},{\varphi }_{l,2m+2}\right)\right)\right]\right]\end{array}$ (28)
$\begin{array}{l}{\varphi }_{i,2m+2}\\ ={W}_{i}\left({t}_{2m+2}\right)+\lambda \frac{h}{3}\underset{j=0}{\overset{2m+2}{\sum }}{{v}^{\prime }}_{j}{F}_{i,s}\left({t}_{2m+2},{\tau }_{j}\right)\left(\gamma \left({\tau }_{j},{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_{j},{\varphi }_{lj}\right)\right)\end{array}$ (29)
where ${u}_{i,0}={g}_{i}\left({t}_{0}\right)$.
Thus, replace the second term in Equation (28) by using integration formulas, then we get:
$\begin{array}{c}{\varphi }_{i,2m+1}={W}_{i}\left({t}_{2m+1}\right)+\lambda \left[\frac{h}{3}\underset{j=0}{\overset{2m}{\sum }}{v}_{j}{F}_{i,s}\left({t}_{2m+1},{\tau }_{j}\right)\left(\gamma \left({\tau }_{j},{\varphi }_{1j}\right),\cdots ,\gamma \left({\tau }_{j},{\varphi }_{lj}\right)\right)\right]\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+\lambda \left[\frac{h}{6}\left[{F}_{i,s}\left({t}_{2m+1},{\tau }_{2m}\right)\left(\gamma \left({\tau }_{2m},{\varphi }_{1,2m}\right),\cdots ,\gamma \left({\tau }_{2m},{\varphi }_{l,2m}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{F}_{i,s}\left({t}_{2m+1},{\tau }_{2m+\frac{1}{2}}\right)\left(\left(\frac{3}{8}{\varphi }_{1,2m}+\frac{3}{4}{\varphi }_{1,2m+1}-\frac{1}{8}{\varphi }_{1,2m+2}\right),\cdots ,\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{ }\left(\frac{3}{8}{\varphi }_{l,2m}+\frac{3}{4}{\varphi }_{l,2m+1}-\frac{1}{8}{\varphi }_{l,2m+2}\right)\right)\\ \text{ }\text{ }\begin{array}{c}\text{ }\\ \text{ }\end{array}+{F}_{l,s}\left({t}_{2m+1},{\tau }_{2m+1}\right)\left(\gamma \left({\tau }_{2m+1},{\varphi }_{1,2m+1}\right),\cdots ,\gamma \left({\tau }_{2m+1},{\varphi }_{l,2m+1}\right)\right)\right]\right]\end{array}$ (30)
$\begin{array}{c}{\varphi }_{i,2m+2}={W}_{i}\left({t}_{2m+2}\right)+\lambda \left[\frac{h}{3}{F}_{i,s}\left({t}_{2m+2},{\tau }_{0}\right)\left(\gamma \left({\tau }_{0},{\varphi }_{10}\right),\cdots ,\gamma \left({\tau }_{0},{\varphi }_{l0}\right)\right)\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}+4{F}_{i,s}\left({t}_{2m+2},{\tau }_{1}\right)\left(\gamma \left({\tau }_{1},{\varphi }_{11}\right),\cdots ,\gamma \left({\tau }_{1},{\varphi }_{l1}\right)\right)+\cdots \\ \text{ }\text{ }\begin{array}{c}\text{ }\\ \text{ }\end{array}+{F}_{i,s}\left({t}_{2m+2},{\tau }_{2m+2}\right)\left({u}_{1,2m+2},\cdots ,{u}_{l,2m+2}\right)\right]\end{array}$ (31)
where
$\begin{array}{l}{v}_{0}={v}_{2m}=1,\text{\hspace{0.17em}}\text{\hspace{0.17em}}{v}_{j}=3-{\left(-1\right)}^{j},\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,2,\cdots ,2m-1\\ {{v}^{\prime }}_{0}={{v}^{\prime }}_{2m+2},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{{v}^{\prime }}_{j}=3-{\left(-1\right)}^{j},\text{\hspace{0.17em}}\text{\hspace{0.17em}}j=1,2,\cdots ,2m+1\end{array}$
Finally, we construct $2l$ nonlinear equations from (30) and (31) to find the unknown functions ${\varphi }_{i,2m+1},{\varphi }_{i,2m+2}$. The resulting system is solved by using modified Newton-Raphson method.
5. Numerical Examples
We solve two examples by RKM and BBMat $N=20,50$, $T=0.01,0.1,0.3$, $\lambda =1$ and $\mu =1$.
In Tables 1-6: fExact®Exact solution, fR.K.®approximate solution of RKM, ER.K.®the absolute error of RKM, fB.B.®approximate solution of BBM, EB.B.®the absolute error of BBM.
Example 1
Consider: $\varphi \left(x,t\right)=xt-\frac{{x}^{2}{t}^{2}}{5}-\frac{{x}^{2}{t}^{6}}{5}+{\int }_{0}^{1}{x}^{2}{y}^{2}{\varphi }^{2}\left(y,t\right)\text{d}y+{\int }_{0}^{t}t\tau {\varphi }^{2}\left(x,\tau \right)\text{d}\tau$
Exact solution $\varphi \left(x,t\right)=xt$
Case 1: $N=20$
Table 1. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.01, N = 20.
Table 2. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.1, N = 20.
Table 3. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.3, N = 20.
Case 2: $N=50$
Table 4. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.01, N = 50.
Table 5. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.1, N = 50.
Table 6. The values of exact, approximate solutions, and errors by using RKM and BBM at T = 0.3, N = 50.
6. Conclusions
From the previous discussions we conclude the following:
1) As N is increasing the errors are decreasing.
2) As x and t are increasing in $\left[0,1\right]×\left[0,1\right]$, the errors due to RKM and BBM are also increasing.
3) The errors due to the BBM are less than the errors due to RKM (i.e. BBM the better than RKM to solve NF-VIE with continuous kernel).
Cite this paper: Al-Bugami, A. and Al-Juaid, J. (2020) Runge-Kutta Method and Bolck by Block Method to Solve Nonlinear Fredholm-Volterra Integral Equation with Continuous Kernel. Journal of Applied Mathematics and Physics, 8, 2043-2054. doi: 10.4236/jamp.2020.89152.
References
[1] Linz, P. (1985) Analytic and Numerical Methods for Volterra Equations. SIAM, Philadelphia.
https://doi.org/10.1137/1.9781611970852
[2] Mirzaee, F. and Rafei, Z. (2011) The Block by Block Method for the Numerical Solution of the Nonlinear Two-Dimensional Volterra Integral Equations. Journal of king Saud University-Science, 23, 191-195.
https://doi.org/10.1016/j.jksus.2010.07.008
[3] Maleknejad, K. and Shahrezaee, M. (2004) Using Runge-Kutta Method for Numerical Solution of the System of Volterra Integral Equation. Applied Mathematics and Computation, 149, 399-410.
https://doi.org/10.1016/S0096-3003(03)00148-6
[4] Maleknejad, K. and Ostadi, A. (2017) Numerical Solution of System of Volterra Integral Equations with Weakly Singular Kernels and Its Convergence Analysis. Applied Numerical Mathematics, 115, 82-98.
https://doi.org/10.1016/j.apnum.2016.12.005
[5] Armand, A. and Gouyandch, Z. (2014) Numerical Solution of the System of Volterra Integral Equations of the First Kind. IJIM, 6, 9.
[6] Biazar, J. and Babolian, E. (2003) Solution of a System of Volterra Integral Equations of the First Kind by Adomian Method. Applied Mathematics and Computation, 139, 249-258.
https://doi.org/10.1016/S0096-3003(02)00173-X
[7] Yaghouti, M.R. (2012) A Numerical Method for Solving a System of Volterra Integral Equations. WAP, 2, 18-33.
[8] Katani, R. and Shahmorad, S. (2010) Block by Block Method for the Systems of Nonlinear Volterra Integral Equations. Applied Mathematical Modelling, 34, 400-406.
https://doi.org/10.1016/j.apm.2009.04.013
[9] Al-Wagdani, M.M. (2016) On Some Numerical Treatments for Solving Volterra Integral Equation and System of Volterra Integral Equations. Taif University, Taif.
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# Algebra
posted by on .
State whether each equation represents a linear, quadratic, or exponential function. Tell how you decided.
1. y = 9^3x + 4
2. 4x = 5 - y
3. y = 2x^2 + 23 – 14x
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FREE Mcq On Residues Complex Analysis.PDF. You can download and read online PDF file Book Mcq On Residues Complex Analysis only if you are registered here.Download and read online Mcq On Residues Complex Analysis PDF Book file easily for everyone or every device. And also You can download or readonline all file PDF Book that related with Mcq On Residues Complex Analysis book. Happy reading Mcq On Residues Complex Analysis Book everyone. It's free to register here toget Mcq On Residues Complex Analysis Book file PDF. file Mcq On Residues Complex Analysis Book Free Download PDF at Our eBook Library. This Book have some digitalformats such us : kindle, epub, ebook, paperbook, and another formats. Here is The Complete PDF Library
Mcq On Residues Complex Analysis
COMPLEX ANALYSIS: SOLUTIONS 5 - Page 13/27. Read PDF Mcq On Residues Complex Analysis NTNU Complex Variable Solvedproblems Pavel Pyrih 11:03 May 29, 2012 ( Public Domain ) Contents 1 Residue Theorem Problems 2 2 Zero Sum Theorem For Residues Problems 76 3 Power Series 14th, 2024
Mcq On Residues Complex Analysis Free Pdf Books
2021Complex Analysis, Probability And Statistical Methods ...,” Chandrika Prasad And Reena Garg Advanced Engineering Mathematics”,Khanna Publishing,2018. Author: MSRIT_MATHS Created Date: 1/14/2020 11:21:43 PM ... May 3th, 2021Analysis Of Disc Brake Squeal Using The Complex Eigenvalue 4th, 2024
MCQ ASSOCIATION MCQ 15.1 (c) Attribute MCQ 15.2 (c ...
MCQ 15.27 If All Frequencies Of Classes Are Same, The Value Of Chi-square Is: (a) Zero (b) One (c) Infinite (d) All Of The Above MCQ 15.28 In Order To Carry Out A χ2-test On Data In A Contingency Table, The Observed Values In The Table Should Be: (a) Close 19th, 2024
Ms. Excel MCQ Bank - MCQ Questions Collection » MCQ Sets
A. The Edit>Copy Format And Edit>Paste Format Commands Form The Menu. B. The Copy And Apply Formatting Dialog Box, Located Under The Format>Copy And Apply Menu. C. There Is No Way To Copy And Apply Formatting In Excel – You Have To Do It Manually D. The Format … 19th, 2024
MCQ SAMPLING AND SAMPLING DISTRIBUTIONS MCQ 11.1 …
MCQ 11.74 When Sampling Is Done With Or Without Replacement, Is Equal To: MCQ 11.75 If X Represent The Number Of Units Having The Specified Characteristic And N Is The Size Of The Sample, Then Popula 8th, 2024
MCQ 6.1 (d) Classical Probability MCQ 6.2 (d) Probability ...
MCQ 6.1 When The Possible Outcomes Of An Experiment Are Equally Likely To Occur, This We Apply: (a) Relative Probability (b) Subjective Probability (c) Conditional Probability (d) Classical Probability MCQ 6.2 A Number Between 0 And 2th, 2024
Contents Complex Analysis Practice Problems 2.0 Complex 2 ...
PRELIM PROBLEM SOLUTIONS THE GRAD STUDENTS + KEN Contents 1. Complex Analysis Practice Problems 2.0 1 2. Real Analysis Practice Problems 2.0 4 3. Algebra Practice Problems 2.0 8 1. Complex Analysis Practice Problems 2.0 Complex 2.0 #9.2 Let Dbe A Domain Which Contains In Its Interior The Closed Unit Disk Jzj 1. Let F(z) Be Analytic In 3th, 2024
1 Mcq Math Question Chapter Complex Number
Check The Below Multiple Choice Questions For 9th Class Maths Chapter 1-Number System. MCQs On Class 9 Maths Chapter 1 Number System Check The Below NCERT MCQ Questions For Class 9 Maths Chapter 1 Number Systems With Answers Pdf Free Download. MCQ Questions For Class 9 Maths With Answers Were Prepared Based On The Latest Exam Pattern. 8th, 2024
AOAC Official Method 2007.01 Pesticide Residues In Foods ...
ª 2007 AOAC IN TER NA TIONAL Table 2007.01A. Interlaboratory Study Results For Incurred Pesticides (and Chlorpyrifos-methyl) Analyte Matrix Avg. Concn S R A RSD R B, % S R C, Ng/g Rec., % RSD R D, % HorRat No. Of Labs Outlier Labs E Chlorpyrifos-methyl Grapes 165 14 8.5 35 83 21 1.00 11 6-C, 4-C Lettuces 178 20 11 30 89 17 0.81 10 11-SG 13th, 2024
C-terminal Region (residues 287 To 339) Interacts ...
A Key Step In This Process Is The Recognition Of The Histone H3 Variant CENP-A In The Centromeric Nucleosome By The Kinetochore Protein CENP-N. We Report Cryo–electron ... Centromere-specific H3 Variant CENP-A, Nucleosomal DNA, And CENP-N. H2A H2B CENP-N Y51 T60 Y58 L64 F66 R73 F71 L81 CENP-A Q87 F106 H104 L91 E107 H4 Y52 E54 R56 V61 L63 DNA ... 18th, 2024
An Improved Prediction Of Catalytic Residues In Enzyme ...
An Improved Prediction Of Catalytic Residues In Enzyme Structures Yu-Rong Tang1, Zhi-Ya Sheng1,2, Yong-Zi Chen And Ziding Zhang3 Bioinformatics Center, College Of Biological Sciences, China Agricultural University, Beijing 100094, China 1Both Authors Contributed Equally To This Work. 5th, 2024
Management Of Agrochemical Residues In Foods
Various Food Industries, And Consumers Alike. This Results In The Imposition Of Much Stricter Sanitary And Phytosanitary (SPS) Standards By Major Importers On Food Products Especially From Developing Countries. ... (APO) And The Council Of Agriculture Management Of Agrochemical ... Regulatory And Non-regulatory Measures At Appropriate Points In ... 7th, 2024
Evaluation Of Certain Veterinary Drug Residues In Food
The Risk Assessment Of Chemicals In Food (EHC 240) 15 2.6 Feedback From JMPR On Ongoing Work On General Criteria For Interpretation Of Toxicological Data 15 2.7 Extrapolation Of MRLs To Minor Species 16 2.8 MRLs For Veterinary Drug Residues In Honey 25 2.9 Scope Of MRLs Established By JE 13th, 2024
Biomass Residues For Power Generation: A Simulation Study ...
Using Biomass Residues From Liberia’s Major Rubber And Oil Palm Plantations. The Model Is Constructed Using The NetLogo Software (v 5.2.0) [1]. We Evaluate The Use Of Gasifiers Powered By Residues From Replanting In Major Rubber And Oil Palm 20th, 2024
Volumes Of Individual Amino Acid Residues In Gas-Phase ...
Duction,15,16 And Proton-transfer Reactivity Studies;14d,17 Kinetic Energy Release Measurements;18 Microscopy Studies Of The High- Energy Impacts Of Proteins On Surfaces;19 And Measurements Of Collision Cross Sections By Ion Scattering20 And Ion Mobility Methods.21-23 In This Study, We Use Ion Mobility Techniques To Determi 11th, 2024
Residues Arg568 And Phe592 Contribute To An Antigenic ...
W Il D- Typ E.S Q Un Cs Of H A G Pr Im Es Ud F Oc Nth V A Lb R Equst . Alv Co W If Dby Ng D MTS13 Va R Int Sw Elyx Pdc B - Ously.14 Epitope Mapping Of Anti-ADAMTS13 Antibodies R Ec Omb I Na Td Sr Pl F H W E Rb Ou Nd Tp IG Sha( EH L C ) 50mMT PH 7.6 (I Nvit R Og E),5 0 M MNaCl Ck 1% W/ B Ser 3th, 2024
Using Mutagenesis To Explore Conserved Residues In The RNA ...
67.3–72.8 °C), Indicating That Y148 Contributes Significantly To The Structural Stability Of The NP. Compared To WT And Other Mutant NPs, Y1 17th, 2024
Chapter 10 Quadratic Residues - Trinity College Dublin
Chapter 10 Quadratic Residues 10.1 Introduction De Nition 10.1. We Say That A2Z Is A Quadratic Residue Mod Nif There Exists B2Z Such 21th, 2024
COMMUNITY REFERENCE LABORATORIES RESIDUES …
2.11. Screening Methods Screening Methods Are Defined In Commission Decision 2002/657 [1] As "methods Used To Detect The Presence Of An Analyte Or Class Of Analytes At The Level Of Interest. These Methods Have The Capability For A High Sample Throughput And Are Used To Sift Larg 21th, 2024
Pharmaceutical Residues In Freshwater - OECD
Pharmaceutical Production And Formulation, Patient Use, Use In Food Production And Improper Disposal (Figure 1). The Presence Of Pharmaceuticals In Freshwater And Terrestrial Ecosystems Can Result In The Uptake Of Pharmaceuticals Into Wildlife, And Have The Poten 7th, 2024
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Packaging, Labeling Or Holding Operations For Dietary Supplements 111.70: What Specifications Must You Establish? (b) For Each Component That You Use In The Manufacture Of A Dietary Supplement, ... Known Or Reasonably Foreseeable Hazards That May Be Present In The Food For Any Of The Follow 9th, 2024
The Calculus Of Residues - The University Of Oklahoma ...
(z −z 0)m, (7.1) Where φ(z) Is Analytic In The Neighborhood Of Z = Z 0. Now We Have Seen That If C Encircles Z 0 Once In A Positive Sense, I C Dz 1 (z −z 0)n = 2πiδn,1, (7.2) Where The Kronecker δ-symbol Is Defined By δm,n = ˆ 0, M 6=n, 1, M = N.. (7.3) Proof: By Cauchy’s Theorem We May Take C To 23th, 2024
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Yadava, Chandrika Prasad Singh, "An Investigation Of The Toxicology And Residues Of Rabon In Poultry." ( 2th, 2024
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M I The Ith Modulus Of An RNS (residue Number System) M Dynamic Range Of An RNS = ~i~t0.k-~l Mi M Dynamic Range, With M I Exluded; I.e., M/m~ N Smallest Integer Such That 2" = 1 Mod M P Position Sum Of Residues Or Ps 10th, 2024
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AFM1 In Milk (Creppy, 2002). AFM1 Is Relatively Stable During Pasteurization, Sterilization And Storage Of Milk And Milk-based Products. Even Low-concentration Intake Of AFM1 Is A Real Threat To Human Health, Particularly To Children As The Main Consumers Of Dairy Products. The Toxicity Of A 3th, 2024
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# Search by Topic
#### Resources tagged with 2D shapes and their properties similar to Track Design:
Filter by: Content type:
Age range:
Challenge level:
### There are 67 results
Broad Topics > Angles, Polygons, and Geometrical Proof > 2D shapes and their properties
### Track Design
##### Age 14 to 16 Challenge Level:
Where should runners start the 200m race so that they have all run the same distance by the finish?
### A Rational Search
##### Age 14 to 18 Challenge Level:
Investigate constructible images which contain rational areas.
### Pentagonal
##### Age 14 to 16 Challenge Level:
Can you prove that the sum of the distances of any point inside a square from its sides is always equal (half the perimeter)? Can you prove it to be true for a rectangle or a hexagon?
##### Age 14 to 16 Challenge Level:
The sides of a triangle are 25, 39 and 40 units of length. Find the diameter of the circumscribed circle.
### Playground Snapshot
##### Age 7 to 14 Challenge Level:
The image in this problem is part of a piece of equipment found in the playground of a school. How would you describe it to someone over the phone?
### Efficient Packing
##### Age 14 to 16 Challenge Level:
How efficiently can you pack together disks?
### Lawnmower
##### Age 14 to 16 Challenge Level:
A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of. . . .
##### Age 14 to 16 Challenge Level:
Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .
### Roaming Rhombus
##### Age 14 to 16 Challenge Level:
We have four rods of equal lengths hinged at their endpoints to form a rhombus ABCD. Keeping AB fixed we allow CD to take all possible positions in the plane. What is the locus (or path) of the point. . . .
### Squaring the Circle
##### Age 11 to 14 Challenge Level:
Bluey-green, white and transparent squares with a few odd bits of shapes around the perimeter. But, how many squares are there of each type in the complete circle? Study the picture and make. . . .
### Crescents and Triangles
##### Age 14 to 16 Challenge Level:
Can you find a relationship between the area of the crescents and the area of the triangle?
### Efficient Cutting
##### Age 11 to 14 Challenge Level:
Use a single sheet of A4 paper and make a cylinder having the greatest possible volume. The cylinder must be closed off by a circle at each end.
### Rolling Coins
##### Age 14 to 16 Challenge Level:
A blue coin rolls round two yellow coins which touch. The coins are the same size. How many revolutions does the blue coin make when it rolls all the way round the yellow coins? Investigate for a. . . .
##### Age 14 to 16 Challenge Level:
Investigate the properties of quadrilaterals which can be drawn with a circle just touching each side and another circle just touching each vertex.
### Arclets Explained
##### Age 11 to 16
This article gives an wonderful insight into students working on the Arclets problem that first appeared in the Sept 2002 edition of the NRICH website.
### Holly
##### Age 14 to 16 Challenge Level:
The ten arcs forming the edges of the "holly leaf" are all arcs of circles of radius 1 cm. Find the length of the perimeter of the holly leaf and the area of its surface.
### Circumspection
##### Age 14 to 16 Challenge Level:
M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.
### Witch's Hat
##### Age 11 to 16 Challenge Level:
What shapes should Elly cut out to make a witch's hat? How can she make a taller hat?
### Approximating Pi
##### Age 14 to 18 Challenge Level:
By inscribing a circle in a square and then a square in a circle find an approximation to pi. By using a hexagon, can you improve on the approximation?
### Gym Bag
##### Age 11 to 16 Challenge Level:
Can Jo make a gym bag for her trainers from the piece of fabric she has?
### Dividing the Field
##### Age 14 to 16 Challenge Level:
A farmer has a field which is the shape of a trapezium as illustrated below. To increase his profits he wishes to grow two different crops. To do this he would like to divide the field into two. . . .
### First Forward Into Logo 4: Circles
##### Age 7 to 16 Challenge Level:
Learn how to draw circles using Logo. Wait a minute! Are they really circles? If not what are they?
### Rolling Around
##### Age 11 to 14 Challenge Level:
A circle rolls around the outside edge of a square so that its circumference always touches the edge of the square. Can you describe the locus of the centre of the circle?
### LOGO Challenge - Circles as Animals
##### Age 11 to 16 Challenge Level:
See if you can anticipate successive 'generations' of the two animals shown here.
### Hex
##### Age 11 to 14 Challenge Level:
Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.
### Salinon
##### Age 14 to 16 Challenge Level:
This shape comprises four semi-circles. What is the relationship between the area of the shaded region and the area of the circle on AB as diameter?
### From One Shape to Another
##### Age 7 to 14
Read about David Hilbert who proved that any polygon could be cut up into a certain number of pieces that could be put back together to form any other polygon of equal area.
### LOGO Challenge 12 - Concentric Circles
##### Age 11 to 16 Challenge Level:
Can you reproduce the design comprising a series of concentric circles? Test your understanding of the realtionship betwwn the circumference and diameter of a circle.
### LOGO Challenge 11 - More on Circles
##### Age 11 to 16 Challenge Level:
Thinking of circles as polygons with an infinite number of sides - but how does this help us with our understanding of the circumference of circle as pi x d? This challenge investigates. . . .
### LOGO Challenge 6 - Triangles and Stars
##### Age 11 to 16 Challenge Level:
Recreating the designs in this challenge requires you to break a problem down into manageable chunks and use the relationships between triangles and hexagons. An exercise in detail and elegance.
### From All Corners
##### Age 14 to 16 Challenge Level:
Straight lines are drawn from each corner of a square to the mid points of the opposite sides. Express the area of the octagon that is formed at the centre as a fraction of the area of the square.
### Trapezium Four
##### Age 14 to 16 Challenge Level:
The diagonals of a trapezium divide it into four parts. Can you create a trapezium where three of those parts are equal in area?
### Poly Plug Rectangles
##### Age 5 to 14 Challenge Level:
The computer has made a rectangle and will tell you the number of spots it uses in total. Can you find out where the rectangle is?
### What's Inside/outside/under the Box?
##### Age 7 to 14
This article describes investigations that offer opportunities for children to think differently, and pose their own questions, about shapes.
### What Shape?
##### Age 7 to 14 Challenge Level:
This task develops spatial reasoning skills. By framing and asking questions a member of the team has to find out what mathematical object they have chosen.
### Curvy Areas
##### Age 14 to 16 Challenge Level:
Have a go at creating these images based on circles. What do you notice about the areas of the different sections?
### Towering Trapeziums
##### Age 14 to 16 Challenge Level:
Can you find the areas of the trapezia in this sequence?
### What Shape for Two
##### Age 7 to 14 Challenge Level:
'What Shape?' activity for adult and child. Can you ask good questions so you can work out which shape your partner has chosen?
### Semi-detached
##### Age 14 to 16 Challenge Level:
A square of area 40 square cms is inscribed in a semicircle. Find the area of the square that could be inscribed in a circle of the same radius.
### Circle Packing
##### Age 14 to 16 Challenge Level:
Equal circles can be arranged so that each circle touches four or six others. What percentage of the plane is covered by circles in each packing pattern? ...
### Bow Tie
##### Age 11 to 14 Challenge Level:
Show how this pentagonal tile can be used to tile the plane and describe the transformations which map this pentagon to its images in the tiling.
### Blue and White
##### Age 11 to 14 Challenge Level:
Identical squares of side one unit contain some circles shaded blue. In which of the four examples is the shaded area greatest?
### Some(?) of the Parts
##### Age 14 to 16 Challenge Level:
A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle
### Square Areas
##### Age 11 to 14 Challenge Level:
Can you work out the area of the inner square and give an explanation of how you did it?
### Not So Little X
##### Age 11 to 14 Challenge Level:
Two circles are enclosed by a rectangle 12 units by x units. The distance between the centres of the two circles is x/3 units. How big is x?
### Floored
##### Age 11 to 14 Challenge Level:
A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?
### Tied Up
##### Age 14 to 16 Short Challenge Level:
How much of the field can the animals graze?
### Three Four Five
##### Age 14 to 16 Challenge Level:
Two semi-circles (each of radius 1/2) touch each other, and a semi-circle of radius 1 touches both of them. Find the radius of the circle which touches all three semi-circles.
### Lying and Cheating
##### Age 11 to 14 Challenge Level:
Follow the instructions and you can take a rectangle, cut it into 4 pieces, discard two small triangles, put together the remaining two pieces and end up with a rectangle the same size. Try it!
### 2001 Spatial Oddity
##### Age 11 to 14 Challenge Level:
With one cut a piece of card 16 cm by 9 cm can be made into two pieces which can be rearranged to form a square 12 cm by 12 cm. Explain how this can be done.
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## Saturday, December 9, 2023
Circuit FLIP FLOP LED Flasher Police Using Transistor
As a beginner in electronic projects, I loved making this simple LED Flip-Flop circuit! It's fun to build and also teaches you about basics of electronics. Now a days with Arduino and similar development boards, it has become rare to see such elegant basic electronic projects without any microcontroller. So let's use basic electronic components to build this magical circuit and learn how magically it works as well!
Let's keep the transistor as our starting point and start rigging up the circuit as shown in the figure.
Connect the emitter of transistor 1 (Let's call it Q1) to the emitter of
the other transistor (Q2).
Connect Q1's base to the negative of the capacitor (C2).
Connect Q2's base to the negative of the capacitor (C1).
Connect the collector of Q1 to positive of the capacitor (C1).
Connect the collector of Q2 to positive of the capacitor (C2).
Let's go ahead with the LED part. It's nothing new, by now you would know that a LED is almost always connected in series with a resistor. You might also know that this LED is called as 'current limiting resistor'.
Connect 470 ohm resistor to positive of LED 1
Connect the negative lead of the LED 1 to the positive of the capacitor C1 (to the same node where collector of Q1 is connected).
Connect 470 ohm resistor to positive of LED 2
Connect the negative lead of the LED 2 to the positive of the capacitor C2 (to the same node where collector of Q2 is connected).
By now you would have observed that the LED part of the circuit is exactly symmetrical with respect to right and left plane.
Now connect a lead of 10K resistor to the negative of capacitor C1 and similarly another resistor to negative of C2.
To drive an LED you'll need a 3.3 forward voltage minimum. So any DC power supply between 4.5V to 9V would do great!
Here, I've used a 9V battery for my convenience.
Connect Positive of the power source to all the other ends of the resistors that we've not connected yet.
Connect Negative of the power source to the emitter of both the transistor that we've shorted.
Although the circuit seems quite simple to build, it has quite complex explanation for the way it works. To understand it in the most simple terms, as you can see in the circuit, transistor Q1 is controlled by the resistor-capacitor pair (10K resistor) which acts as oscillator (with the help of transistor which works as stable multivibrator).
For simplicity, let's consider the transistors as a switch between the collector and the emitter. For the base of the transistor, let's represent it as a diode (because it has the same behavior).
Initially, when we power up the circuit, both the transistors act like open switch. The base of both transistors has a threshold voltage of 0.7 and below that voltage, the transistors will remain open.
Now, at the collector of the transistors, the voltage will be supply voltage. Now, since the middle resistors (10k) are of much greater value than the outer resistors (470 ohm) the capacitors will take some time to fully charge up.
As the capacitor starts to charge, because of component tolerances, one side of the circuit will be slightly faster in charging up the capacitor than the other. After a while, this capacitor reaches the threshold value of 0.7V and this causes the transistor to power up and closes the transistor. The LED of that side will light up now.
By this time, the other side of the circuit would have undergone the same process so before that part of the LED starts glowing, this first transistor closes and causes the LED to light up, the voltage at this end of the capacitor drops to zero And the voltage built up on the capacitor becomes negative and causes the other side of the transistor to open. The same process repeats on the other side as well and it keeps on repeating.
What is the very interesting thing that happens here is how when the left side capacitor (Q1) starts to charge and reaches the threshold voltage, the circuit closes the right transistor (Q2) and lights up the right side LED (LED 2). As soon as the LED is turned ON, the voltage at the capacitor becomes negative (because when the charging of the capacitor reaches the threshold, it starts to conduct and hence it drops to zero volts). This negative voltage at the base of the other transistor (Q1) and hence this transistor acts as open. And this cycle keeps repeating
Calculation of Frequency and Applications of the Circuit.
It is important that the internal resistors must be of the same value. Also both the capacitors must be of the same value.
The oscillation frequency is set by the internal Resistor-Capacitor network and it is given by f=1/(1.38RC)
Since I've chosen R=10k and C=100uf, I'll get frequency around 0.7 Hertz per second.
If you want to blink a single LED, simply remove the other LED and connect the resistor directly.
This circuit has many applications,
They can be used to get a square wave.
They are used as indicators.
They are used in toys to produce flashing lights.
They are also used as turning indicators.
Circuit FLIP FLOP LED Flasher Police Using Transistor
As a beginner in electronic projects, I loved making this simple LED Flip-Flop circuit! It's fun to build and also teaches you about basics of electronics. Now a days with Arduino and similar development boards, it has become rare to see such elegant basic electronic projects without any microcontroller. So let's use basic electronic components to build this magical circuit and learn how magically it works as well!
Let's keep the transistor as our starting point and start rigging up the circuit as shown in the figure.
Connect the emitter of transistor 1 (Let's call it Q1) to the emitter of
the other transistor (Q2).
Connect Q1's base to the negative of the capacitor (C2).
Connect Q2's base to the negative of the capacitor (C1).
Connect the collector of Q1 to positive of the capacitor (C1).
Connect the collector of Q2 to positive of the capacitor (C2).
Let's go ahead with the LED part. It's nothing new, by now you would know that a LED is almost always connected in series with a resistor. You might also know that this LED is called as 'current limiting resistor'.
Connect 470 ohm resistor to positive of LED 1
Connect the negative lead of the LED 1 to the positive of the capacitor C1 (to the same node where collector of Q1 is connected).
Connect 470 ohm resistor to positive of LED 2
Connect the negative lead of the LED 2 to the positive of the capacitor C2 (to the same node where collector of Q2 is connected).
By now you would have observed that the LED part of the circuit is exactly symmetrical with respect to right and left plane.
Now connect a lead of 10K resistor to the negative of capacitor C1 and similarly another resistor to negative of C2.
To drive an LED you'll need a 3.3 forward voltage minimum. So any DC power supply between 4.5V to 9V would do great!
Here, I've used a 9V battery for my convenience.
Connect Positive of the power source to all the other ends of the resistors that we've not connected yet.
Connect Negative of the power source to the emitter of both the transistor that we've shorted.
Although the circuit seems quite simple to build, it has quite complex explanation for the way it works. To understand it in the most simple terms, as you can see in the circuit, transistor Q1 is controlled by the resistor-capacitor pair (10K resistor) which acts as oscillator (with the help of transistor which works as stable multivibrator).
For simplicity, let's consider the transistors as a switch between the collector and the emitter. For the base of the transistor, let's represent it as a diode (because it has the same behavior).
Initially, when we power up the circuit, both the transistors act like open switch. The base of both transistors has a threshold voltage of 0.7 and below that voltage, the transistors will remain open.
Now, at the collector of the transistors, the voltage will be supply voltage. Now, since the middle resistors (10k) are of much greater value than the outer resistors (470 ohm) the capacitors will take some time to fully charge up.
As the capacitor starts to charge, because of component tolerances, one side of the circuit will be slightly faster in charging up the capacitor than the other. After a while, this capacitor reaches the threshold value of 0.7V and this causes the transistor to power up and closes the transistor. The LED of that side will light up now.
By this time, the other side of the circuit would have undergone the same process so before that part of the LED starts glowing, this first transistor closes and causes the LED to light up, the voltage at this end of the capacitor drops to zero And the voltage built up on the capacitor becomes negative and causes the other side of the transistor to open. The same process repeats on the other side as well and it keeps on repeating.
What is the very interesting thing that happens here is how when the left side capacitor (Q1) starts to charge and reaches the threshold voltage, the circuit closes the right transistor (Q2) and lights up the right side LED (LED 2). As soon as the LED is turned ON, the voltage at the capacitor becomes negative (because when the charging of the capacitor reaches the threshold, it starts to conduct and hence it drops to zero volts). This negative voltage at the base of the other transistor (Q1) and hence this transistor acts as open. And this cycle keeps repeating
Calculation of Frequency and Applications of the Circuit.
It is important that the internal resistors must be of the same value. Also both the capacitors must be of the same value.
The oscillation frequency is set by the internal Resistor-Capacitor network and it is given by f=1/(1.38RC)
Since I've chosen R=10k and C=100uf, I'll get frequency around 0.7 Hertz per second.
If you want to blink a single LED, simply remove the other LED and connect the resistor directly.
This circuit has many applications,
They can be used to get a square wave.
They are used as indicators.
They are used in toys to produce flashing lights.
They are also used as turning indicators.
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Tuesday, April 16, 2024
# How Much Power Does 20 Solar Panels Produce
## What Are The Labour Costs
Solar Basics Pt 2: How Much Energy Does One Solar Panel Produce?
When you are given a quote for your solar panel system, labour costs are typically included. There are two common ways of calculating the labour costs for solar panel installations in the UK.
One way is to calculate approximately £300-£500 per person, per day. As it usually requires two people to install the system, a final installation fee is normally £600-£1,000. If you are installing a larger system, it may take more than one day and the costs will naturally rise.
Another way to calculate the installation fee is to charge 20 pence per watt. So, for a 4kW system, you would be paying 20 pence for 4000 watts, resulting in £800.
## What Is Solar Panel Output
The power rating of your system is a measure of how big your generation system is, not how much energy it will produce. This is analogous to a car engine, where the size of the engine gives you an indication of how powerful it is, but does not itself tell you how much petrol it will use, although the two are related.
In addition to knowing the output rating of your solar power system, you should also understand how many your solar system can be expected to produce. Knowing this number will help you calculate the revenues and savings you can expect to receive from your solar panels.
## Solar Power In Crisis: ‘my Panels Generate Enough Power For Two Loads Of Washing’
Endless energy from the sun looked like a long-term solution for running our homes. But now the state has pulled the plug on the subsidies that made panels affordable for many. What happens now?
Sit back, relax, and read this story with an untroubled conscience: it has been created on a laptop and mobile phone powered entirely by the rays of the sun. This feat would surely astound the most idealistic Greek philosopher or Victorian entrepreneur. It would confirm their wildest hopes for humanitys progress. Perhaps they would be even more amazed that it was possible via a coalition of Chinese companies, British roofers and local councils. Oh, and government support, which is set to be abruptly withdrawn.
When I call Leo Murray, hes standing by a fake sun a 10ft helium balloon filled with LED lights in Ravenscourt Park, London. The campaigns director for 10:10, a charity encouraging positive action on climate change, Murrays lightbulb-like brightness is dimmed by the governments desire to slash solar support. Who wouldnt want to exceed our renewables targets, he wonders, when surveys show that solar is the most popular form of energy, with 80% support: We explain to the public how we all contribute towards solar it adds a couple of quid on our bills each year and we cant find anyone who is anti-renewable energy.
Recommended Reading: Where Is Solar Energy Stored
## The Electricity Produced By One Solar Panel
The energy output of a solar panel varies based on the size of the panel you have and its efficiency.
On average a modern solar panel produces between 250 to 270 watts under ideal sunlight and temperature conditions. This is the power equivalent to lighting 84 compact fluorescent bulbs for 1 hour a day . Such panel consists of 60 solar cells. One solar cell generates 5 watts, with the efficiency between 15 to 18 percent. The size of the panel in this case is around 65 inches in length and 40 inches in width .
But you can have also panels generating as much as 325 watts. You should know, though, that panels with higher output are more expensive and are usually installed when free space is a limiting factor.
## How To Use A Calculator To Calculate How Many Solar Panels You Need
Our solar calculator can estimate the number of panels needed for your home. The tool works by taking your average monthly energy usage in kilowatt hours .
• Calculators that use kilowatt hours are more accurate because they consider your exact energy needs
• Those that ask for home area must assume the electricity consumption
Each electric company has a different power bill format, but they all display your electricity consumption for the billing period. The exact description will vary, but you should look for a term such as kWh used or kWh consumed.
Solar calculators also ask for your home location to determine how much sunshine is available. Based on those two values, they can estimate the system size in kilowatts. Some solar calculators assume a wattage for panels, while others ask you directly. Based on the total kilowatts and the rated watts per panel, the calculator can determine how many are needed.
It is important to understand the difference between kWh savings and dollar savings. kWh savings refer to the amount of energy that your panels produce. On the other hand, dollar savings are obtained when the solar electricity production is multiplied by the kWh price.
Read Also: How Much Electricity Does A 4kw Solar System Produce
## Solar Panel Power Output
Lets start off with the basics a solar panel’s output is expressed in watts. On average, a domestic solar panel has a power output of around 265 watts, although it can range anywhere from as little as 225 watts to more than 350 watts.
The higher the wattage of a solar panel, the more electricity it can produce under the same conditions.
To calculate how much electricity a solar panel will produce in a day, you simply have to multiply its wattage by the number of sunlight hours.
So, for example, a home in Cambridge typically receives four hours of sunshine a day. If this home has a 280-watt solar panel, it will generate 1,120 watt-hours or 1.1 kilowatt-hours of electricity that day .
If that same home had a 320-watt solar panel, it would be able to produce 1,280Wh or 1.2kWh of power on that same day .
Take a look at the table below to compare the most powerful panels on the market although you should be aware that high-power panels aren’t necessarily for everyone. There are some excellent all-rounders out there, and our guide on the best solar panels will give you a better idea of which panel is best for you.
Company
340W
## How Long Do Solar Panels Take To Pay For Themselves
As previously stated, the repayment period is determined by a variety of circumstances. Its impossible to determine how long your solar panel will take to pay for itself, but we can give you an estimate.
The payback period can range from 5 to 16 years in general.
The payback period is uncertain because it is influenced by personal characteristics like consumption rates, which we may not be aware of. You can quickly figure out how long it will take you to pay off your loan. How? Continue reading to find out more.
Also Check: How Much Does Solar Energy Cost Per Kwh
## Can I Power My Whole Home With Solar Panels
Statistics from the U.S. Energy Information Administration show that in 2019 the average energy usage per U.S. household was 10,649 KWh, thats around 29 KWh a day. Remember that this is a ballpark figure, and your energy usage may be higher or lower. If you look at your electricity bills for a full year and then you can calculate precisely how much energy your household uses.
The big-ticket items in terms of electrical usage are to do with heating and cooling. More than 50% of domestic electricity usage comes from aircon, home heating, water heaters, and refrigerators. If you use a lot of these, then youre likely to have a higher energy requirement than the national average.
If you were to install solar panels with a combined power of 5 kW in an area that receives, on average, four hours of direct sunlight a day, then your annual solar energy production would be 7,300 kWh. Thats 69% of an average households annual energy needs very handy indeed.
So it is possible to meet most or even all of your power needs from solar power, provided you have enough roof space and the funds for an extensive enough system.
Lets take a look at how achievable those solar power energy returns are for you.
## The Time To Go Solar Is Now
How many kWh can 200watt solar panel produce in a day?
Sunrun will ensure that you have the best number and style of solar panels to optimize your rooftops solar power production. You can rest easy with a customized solar solution from Sunrun. Our systems are designed for your house structure, lifestyle, energy and financial goals.
We have the resources and experience to maximize your solar systems performance. Well guide you every step of the way from contract through installation and maintenance. And, well be there to support and guide you for many years to come.
#### Sunrun
Also Check: Do You Have To Register As A Sole Proprietor
## Is Geographical Location A Factor
Location, location, location. The effectiveness of solar panels is strongly influenced by where you live. The US states with the most sunshine and the clearest skies are the best places to install solar panels. But that doesnt mean that you cant get good returns in other states.
Studies show that the most productive states in terms of the energy collected by solar panels are Arizona and New Mexico. In these states, a 1 kW solar setup will generate nearly 5 kWh of power per day on average. But if you install the same panels in Washington State, you can expect just 3 kWh per day and only 2.8 in Minnesota.
Wherever you live, you can use solar panels to make a real difference in your energy costs. If youre in a cloudier, more northern state, you will need to install more panels to push up the energy output.
This is because in places with less direct sunlight, you need to spend more money on a larger solar panel set up to get the same energy youd get from a small setup in a sunnier place.
Solar power is more economical in places with a lot of direct sunlight.
## How Big Is A 5kw Solar System
Alright, lets get down to business. First, lets take a look at exactly how big a 5kW installation really is.
#### What does 5kW actually mean?
A 5kW solar installation produces 5 kilowatts of electricity under perfect conditions. With LED light bulbs using about 9 watts , a 5kW installation could power 555 LEDs indefinitely as long as perfect conditions remained 24/7 .
Over the course of an hour, one 9 watt LED uses 9 watt-hours of electricity. A 5kW solar installation, under perfect conditions, produces 5 kilowatt-hours over the course of an hour, under perfect conditions. Over 10 hours, it produces 50 kWh.
In reality, we arent lucky enough to have perfect conditions. The sun goes down each night. Storms come in. Rain pours down. Some places, like Washington and New York, just dont enjoy the same ultra-strong sunlight than Arizona and Nevada do.
In reality, your solar installation will produce less than its nameplate capacity. Exactly how much will depend on your location. The National Renewable Energy Labs PVWatts solar calculator gives us a glimpse into what we can expect a 5kW installation to produce over a year in different areas of the US. Lets take a look:
• Anchorage, AK: 4,487 kWh/year
• Reno, NV: 8,371 kWh/year
#### How many solar panels is that?
Speaking of roof space, just how much does a 5kW solar installation take up? Lets take a look!
#### How much space does that take on my roof?
Also Check: How Much Does A 20kw Solar System Cost
Also Check: How Much Does A Solar System Cost
## Factors That Influence How Many Solar Panels You Need: Size And Output
To determine how many solar panels are needed to power a house, several factors must be considered. For example, if there are two identical homes powered by solar energy in California and New York, with exactly the same energy usage, the California home will need fewer solar panels because the state gets more sunshine.
The following are some of the most important factors to consider when figuring out many solar panels you need:
## What Are The Factors That Affect The Solar Panel Output
It is best to know the solar panel output beforehand because it will help you in the future.
There are several solar panels available on the market due to the boom of the solar industry. Hence, it has become challenging to choose the right one. However, if you know about the factors that influence the solar panels output, you will surely make the right move.
Multiple factors determine how much energy do solar panels produce. We have listed down the primary ones here for your understanding. Have a look-
Don’t Miss: How Many Solar Panels For 3000 Sq Ft Home
## First Things First A 20 Kw Solar Installation Is Big
The average home solar installation in the United States is 5.6 kW, so a 20 kW system is almost 4 times bigger!
If youre interested in installing a 20 kW solar system, chances are this is a commercial installation or your electricity use is really high compared to the national average of about 900 kilowatt-hours per month. Maybe you live in an area that regularly sees summer temperatures over 100 degrees F and so you have to crank that air conditioning all summer to stay comfortable . Or maybe you live in a large house , so you have a lot of lighting and space to air condition. If this is the case, great! You are a prime candidate to go solar. The higher your electricity bill, generally the more you can save by installing a solar system.
You can’t correctly size your solar PV system unless you know how much electricity your home uses now . The easiest way to figure this out is to look at past electricity bills, which should tell you how much power you’ve used in the previous month or quarter. From this you can figure out the average daily usage. This is even easier if you have a smart meter installed you should be able to see your daily usage either on the bill or by checking your account online.
Your power consumption is measured and billed in kilowatt-hours .
A typical Australian home uses 1520kWh per day. But households can vary considerably in their electricity consumption, depending on the number of people living there, the area they live in, whether they use gas for cooking or hot water, and many other factors. For example, a single-person home will typically use about 812kWh per day on average, while a household of five people with a pool could use 3040kWh per day.
#### Time of day and seasonal usage
It’s important to consider when you use electricity. Is your home generally empty during weekdays, with everyone at work or school, so that your main power consumption comes in the evening? If so, your solar panels might not be used most effectively, as it’s better to use the generated power during the day than export it to the grid.
Also consider whether some days are more power-hungry that others the weekend for instance, when everyone is at home. And do you use more power in summer or in winter ?
You May Like: Are Dirty Solar Panels Less Efficient
## How Much Power Do Solar Panels Produce
4.3/5Solar panelsolar paneliscan produceenergy
16 panels
One may also ask, how many solar panels do I need for 1000 kWh per month? A home that consumes 1,000 kWh per month will normally need between 20 and 30 solar panels. The exact number changes depending on the specifications of the chosen panel model, as well as the sunshine available at the project site.
Additionally, how much power does a 300 watt solar panel produce?
For example, if a 300watt solar panel in full sunshine actively generates power for one hour, it will have generated 300 watt-hours of electricity. That same 300watt panel produces 240 volts, which equals 1.25 Amps.
How much does a 250 watt solar panel cost?
Solar companies can typically get a single solar panel at a price of \$0.75 per watt. Therefore, if the solar panel output is 250 watts, that single panel might cost you \$187.50. However, if a homeowner is trying to buy one or two panels on their own for a small DIY project, they will likely pay closer to \$1 per watt.
## Calculating The Size Of Your Solar Pv System
Calculating the KWH Your Solar Panels Produce
Now you know how much power you typically use and the times of day you use it. What capacity will your solar PV system need to be to cover your power usage?
First, were assuming youll have a grid-connected system. This is by far the most common type and it simply means you have solar panels generating electricity during the day, and a grid connection to supply electricity when the solar panels arent generating enough . See grid-connected vs off-grid for more.
#### How much electricity can you expect per kW of solar panels?
Solar PV systems are rated in watts or kilowatts . Youll see systems described as 4kW, 5kW, 10kW and so on.
1kW of solar panels = 4kWh of electricity produced per day .
For each kW of solar panels, you can expect about 4kWh per day of electricity generation. So a 6.6kW solar system will generate about 26.4kWh on a good day .
Its just a general rule the actual amount of electricity generated per kW of solar panels depends on your location, the time of year and the amount of sunlight youre getting, the quality of the system, the orientation of the panels, how old they are, and so on. In southern regions such as Hobart it could be as low as 3.5kWh per day, while the same 1kW of panels in Darwin could generate 5kWh.
Recommended Reading: How Much Do The Tesla Solar Panels Cost
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# Hotel Nights Calculator
Calculate the number of hotel nights and estimate the total cost of the stay based on the entered arrival date, departure date, and optionally, the price per night.
\$
Result:
\$
## How to use this calculator?
3. If desired, input the price for 1 night for the estimated total cost of the stay.
4. Tick the "Add 1 day" box if you anticipate late check-out, and the hotel counts it as an additional night.
## How do hotels calculate nights?
Hotel nights usually represent the nights you spend in a hotel.
If you plan to leave the hotel after the check-out time on a given day, even if you won't spend that night there, you should include this night in your calculation (use the "Add 1 day" option in the calculator).
For example, if you check in at 11 AM on Monday:
• Check out before 11 AM on Tuesday, and many hotels would count it as one night.
• Check out at 6 PM on Tuesday (after the check-out time) or 8 PM on Wednesday, and it would be considered two nights.
Be aware of specific check-in and check-out times some hotels enforce. It's crucial to understand their policies, how they calculate nights, and any potential charges for early check-ins or late check-outs.
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Week 7 Ch 8 Checkpoint
# Week 7 Ch 8 Checkpoint - Section 8.1 Page 600 Problem 14...
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Section Page # Problem # Please put your work and/or answer in this column Corrections by Instructor Credit Earned 8.1 600 14 Find the median of each set of numbers. 1, 4, 9, 15, 25, 36 15 + 9 = 24 24 2 = 12 600 18 Find the mode of each set of numbers. 41, 43, 56, 67, 69, 72 There is no mode 600 20 Find the mode of each set of numbers. 9, 8, 10, 9, 9, 10, 8 9, it appears the most
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Section Page # Problem # Please put your work and/or answer in this column Corrections by Instructor Credit Earned 600 24 A salesperson drove 238, 159, 87, 163, and 198 miles (mi) on a 5- day trip. What was the mean number of miles driven per day? 238 + 159 + 87 + 163 + 198 = 845 8455 = 169 602 36a The following scores were recorded on a 200-point final examination: 193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. (a) Find the mean final examination score. 193 + 185 + 163 + 186 + 192 + 135 + 158 + 174 + 188 + 172 + 168 + 183 + 195 + 165 + 183 = 2,640 2,460 ÷ 15 = 176 602 36b The following scores were
Section Page # Problem # Please put your work and/or answer in this column Corrections by Instructor Credit Earned recorded on a 200-point final examination: 193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183. (b) Find the median final examination score. 135, 158, 163, 165, 168, 172, 174, 183, 183, 185, 186, 188, 192, 193, 195 183 602 36c The following scores were recorded on a 200-point final examination: 193, 185, 163, 186, 192, 135, 158, 174, 188, 172, 168, 183, 195, 165, 183.
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Published:
# The Machine Learning Interview Checklist
Generally speaking, interviews for Ph.D. positions, internships in machine learning are generally structured into three parts:
• mathematics,
• programming, and
• machine learning. This is sometimes accompanied by a presentation of a paper or your work. In the following, I collected the most frequent topics you will potentially encounter.
## Mathematics
It is not all you need for a machine learning interview, but reading and comprehending each concept in the freely available Mathematics for Machine Learning book is what can lay the foundations on the math and fundamental machine learning side. Regarding mathematical concepts, you need to be fluent at least in:
• Linear algebra
• Probability
• Analysis
• Optimization
### Linear algebra
Since neural networks consist of matrices and tensors, it is essential to be aware of the corresponding operations—knowledge of matrices is a must, tensors are a nice-to-have, but you need to have an intuition that tensor can be thought of “generalized matrices into higher dimensions”. Matrix knowledge includes how linear equation systems and matrices correspond, what are the computational aspects of matrix algebra (cost) and how matrices can express linear transformations—unfortunately, bonus points cannot be collected for knowing the eponymous film series.
Linear equations can be thought as (intersecting) hyperplanes in a vector space, so characterization of a vector space is essential. When we connect matrices—what we do since they can describe the equation system—to vector spaces, we also want to measure angles and distances, so being aware of norms, inner products is also a must.
When we talk about matrices, decompositions such as Singular Value Decomposition (SVD) and Principal Component Analysis (PCA) cannot be neglected. Here the key is to understand what these decompositions mean, and also what we can do with them (low-rank approximations and even unsupervised representation learning in simple cases).
This figure from the MML book can be very helpful to comprehend all things matrix philogeny:
#### Checklist
• matrix operations
• multiplication (non-commutative, exceptions exist)
• inverse
• Moore-Penrose pseudoinverse
• transpose
• conjugate
• trace
• determinant
• connection to invertibility
• geometric interpretation
• matrix properties and their connections
• definiteness
• (anti-)symmetry
• square matrices
• similar matrices
• constructing (anti-)symmetric matrices from a general matrix
• orthogonal matrices (rotations)
• matrices and linear transformations
• matrix decompositions
• low-rank approximations
• spectral decomposition
• diagonalization
• Principal Component Analysis (PCA)
• Singular Value Decomposition (SVD)
• Cholesky
• QR
• LU
• Eigendecomposition
• characteristic polynomial
• equation systems
• relation to matrices
• classification based on number of solutions
• geometric interpretation (intersection of hyperplanes)
• Gaussian elimination
• underdetermined problems
• overdetermined problems
• approximate/least-squares solution (when no exact solution exists)
• geometric and algebraic multiplicity (connection to eigenvalue problem)
• inner products
• meaning (“orthogonality”)
• $p$-norms
• metrics
• triangle inequality
• Cauchy-Schwarz inequality
• vector spaces
• inner product spaces, normed spaces
• subspaces
• linear (in)dependence
• orthogonal bases
• span
• orthogonal projection
• orthogonal complement
• orthogonalization (Gram-Schmidt)
• image space
• kernel/null space
### Probability
Understanding Bayes’s theorem is the bread and butter for several machine learning algorithms. Besides helping with the Monty Hall Problem, it is a fundament of a broad range of methods aiming to infer unmeasured quantities. However, not everything is Bayesian estimation: Maximum Likelihood and Maximum A Posterior methods are also frequently used. Factorization, independence and the latter’s relation to (un)correlatedness are further essential concepts, since assumptions on distributions generally are about them. To juggle with distributions, we need to distinguish them by names such as marginal, conditional, or joint; to manipulate them (read: to use Bayes’s theorem), we need the Sum and teh Product rules. Be also aware of the special role of the Gaussian, and exponential families (to get conjugate priors for Bayes estimation, leading to closed-form solutions) can also become handy. Staying with the Gaussian, its prominence in the Central Limit Theorem is the basis for using the Gaussian assumption on distributions. When we want to transform probability densities, then the change of variables formula will be our tool. There are properties (sufficient statistics) that can describe distributions concisely, the mean and variance are such (they are sufficient to describe a Gaussian) and they have interesting properties. It is also worth knowing that they are instantiations of (central) moments, which can be thought as a general family of descriptors for probability distributions. A pinch of information theory, i.e., quantities related to the information content of a random variable, are used in several fields of machine learning. (Differential) Entropy is the most important concept, but cross entropy and mutual information are also essential.
#### Checklist
• Probability mass function (pmf)
• Probability density function (pdf)
• Cumulative density function (cdf)
• Sufficient statistics
• Mean
• (Co)variance
• Correlation
• (Conditional) Independence
• Uncorrelatedness vs Independence
• Factorization of a joint distribution
• Conditional distribution
• Marginal distribution, marginalization
• Sum rule
• Product rule
• Bayes’s Theorem
• likelihood
• prior
• posterior
• evidence (marginal likelihood)
• Relation of probabilistic estimation methods
• Bayes estimation (full Bayes)
• prior predictive distribution
• posterior predictive distribution
• Maximum A Posteriori (MAP) estimation
• Maximum Likelihood (MLE) estimation
• Markov factorization, Markov property
• Gaussian distribution with special properties
• Central Limit Theorem (CLT)
• Exponential families (conjugacy)
• Change of variables formula
• Information theory
• (Differential) entropy
• Cross entropy
• Mutual information
I would also recommend some online courses, particularly the Bayesian Statistics Specialization on Coursera - with flashcards for the first two courses available on my blog; here for the first, and here for the second course. Additionally, there are related aspects in the Probabilistic Graphical Methods Specialization on the same website, with flashcards by yours truly for PGM1, PGM2, and PGM3.
### Analysis
Analysis is mostly a tool for optimization in the context of machine learning - if you happen to be a theorist, it can be much more, but let’s stick to the fundamentals. Knowing what the derivative is, what it means (also for vector valued functions) is essential to understand optimization methods and the Taylor approximation. So the Jacobian and the Hessian should be a trusted acquaintance. The former is also required for the change of variables formula mentioned above.
#### Checklist
• Differentiation
• Total differential
• Partial derivative
• Directional derivative
• Jacobian matrix
• Hessian matrix
• Taylor series
• Taylor approximation
• Differentiation identities
• Chain rule
• Quotient rule
• Composite function
### Optimization
Since machine learning evolves around mostly gradient-based optimizers, Stochastic Gradient Descent (SGD) makes the top of the list. Taking a step back, you should be aware about the family of first-order methods, and why they are popular (computationally low cost). But beware of the caveats: local optima, setting the step size and co. So as a follow-up, ponder why second-order methods should (they are aware of the curvature) and should not (calculating the Hessian is costly) be used. When we need to incorporate prior knowledge/constraints, then a Lagrange-multiplier will come handy. It is also good to know why we love convex optimization (i.e., to wonder about the good old days when problems were convex, as ML problems will not be convex in most cases). It might also be useful to know about the Karush-Kuhn-Tucker (KKT) conditions, which collect necessary conditions for the solutions of constrained optimization problems, including problems with inequality constraints in nonlinear programs (a synonym for optimization problem). To transition towards ML-related topics, here you should know optimizers such as ADAM, Nesterov momentum, or RMSProp - the key is they use an averaging procederure (‘momentum’) to incorporate previous update(s). As a final twist, knowing conceptually how modern ML frameworks implement gradient calculation (automatic differentiation) also belongs to the good-to-know facts.
• First-order methods
• Full-batch vs Mini-batch
• Second-order methods
• Why they are useful for convex problems
• Geometric interpretation (using curvature)
• Drawbacks (expensive)
• Gradient conditions for (unconstrained) minima
• First-order (necessary)
• Second-order (sufficient)
• Karush-Kuhn-Tucker (KKT) conditions
• Momentum
• RMSProp
• Automatic differentiation (concept)
• forward pass
• backward pass
The hilariously-titled All the Math You Missed (But Need to Know for Graduate School) also looks promising, but as far as I can tell from skimming it, it is for the next level.
## Machine learning
### Fundamentals
The categories of machine learnin (supervised, unsupervised, and reinforcement learning are the three main categories, but self-supervised and semi-supervised learning also belongs) are a must.
For supervised methods, classification and regression are the categories you need to be aware of, including what loss functions (cross entropy vs mean squared error) are used. Additionally, the Support Vector Machine (SVM) with its hinge loss also often comes up. Flavors of regression (linear, polynomial) are also prevalent.
For unsupervised, PCA from linear algebra is a trusted friend, but it needs to be accompanied by k-means and Gaussian Mixture Models (GMMs), including how the latter two relate to each other (GMM is “soft” k-means).
Reinforcement learning: model-free and model-based, offline and online RL are useful categories to keep in mind. To my best knowledge, these will mostly come up during an interview if you want to work in the field of reinforcement learning.
Nitty-gritty details of what coulf (and will) go wrong during training and how to fight them also comprises the essential toolbox of a machine learning engineer/researcher, including data preparation, architecture design. This is to find a trade-off between underfitting and overfitting.
#### Checklist
• Categorization
• Supervised
• Classification
• Support Vector Machines (SVMs)
• margin
• support vector
• loss functions
• Cross Entropy
• Hinge Loss (SVM)
• Regression
• Mean Squared Error
• Unsupervised
• Principal Component Analysis (PCA)
• k-Means
• Gaussian Mixture Models (GMMs) + Expectation-Maximization (EM)
• (Variational) Autoencoders
• Reinforcement Learning
• Semi-supervised Learning
• Self-supervised Learning
• Overfitting
• Reasons
• Solutions
• Underfitting
• Reasons
• Solutions
• Exploration vs exploitation dilemma
• Cross-validation
• Latent variable models
• Batch Normalization
• Residual Networks (ResNets)
• LSTM
• Data preprocessing (e.g., whitening)
• Data augmentations
### Specifics
Of course, they are the field-specific knowledge like ResNets, Transformers, CNNs for computer vision. Here generally what is expected to provide a high-level, intuitive understanding of modern/state-of-the-art methods, but it is generally not required to comb through arXiv digest daily.
## Programming
Know machine learning frameworks (PyTorch and TensorFlow are the big two, but JAX is on the rise as far as I can tell), and you can shine if you can compare them. If you use any of those, then you should prepare to state why you use it.
• documentation,
• unit testing, and
• Continuous Integration As an extra, you can add in knowledge about container-based frameworks such as Singularity or Docker - most cluster infrastructures are based on those.
For Python, realpython.com is my go-to resource, you can learn about all these concepts there.
#### Checklist
• Frameworks
• PyTorch
• TensorFlow
• Keras
• JAX
• Best practices
• Documentation
• Unit testing
• Continuous Integration
• Container-based tools
• Singularity
• Docker
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Data Sizes And Speeds
Educational level: this is a secondary education resource.
Completion status: this resource is ~75% complete.
Resource type: this resource is a lesson.
Completion status: this resource is considered to be ready for use.
## Names for different sizes of data
When choosing a new computer we come across terms such as "300GB hard drive" and "500MB download", and to the uninitiated, this can be somewhat disconcerting. Data in a computer is represented in a series of bits. Since the birth of computers, bits have been the language that control the processes that take place inside that mysterious box called your computer. In this article, we look at the very language that your computer uses to do its work.
### Bit
A bit is a binary unit, simply a 1 or a 0. A true or a false. It is the most basic unit of data in a computer. It's like the dots and dashes in Morse code for a computer. Bits are machine readable.
### Nibble
A nibble is 4 bits, or half of a byte. One hexadecimal digit is one nibble in size.
### Byte
In computer science a byte is a unit of measurement of information storage, that equals '8 bits', can be used to represent letters and numbers.
For example, the number 01000001 is 8 bits long, and represents the letter A in 8-bit encoding.
### Word
Unfortunately the term "word" has two definitions. 1) The word size for a computer is the number of bits that the central processing unit (CPU) of a particular computer can handle at one time. These word sizes range from a nibble to more than 128 bits. 2) Word size = 16 bits (or two bytes). This second definition was pretty much driven by all of the people that were writing software to be used to program computers.
### KB
A kilobyte, or KB, is a unit of data that equals 1024 bytes, or 210. This is not to be confused with the decimal kilo which means 1000 or 103. The difference is because the term was coined by computer scientists. Powers of 2 do not fit into 1000 neatly, therefore using the decimal system in binary computing would be computationally wasteful.
### MB
A megabyte, or MB, is a unit of data that equals 1,048,576 bytes, or 220. This is equal to a kilobyte squared, 10242.
### GB
1 GB
A gigabyte, or GB, is a unit of data that equals 1,073,741,824 bytes, or 230. This is equal to a kilobyte cubed, 10243. Because of the difference between computer metrics and decimal metrics, storage devices are usually advertised with gigabytes presented as 1 billion bytes rather than 1.07 billion bytes, thereby understating their true capacity. This explains why there are inconsistencies when comparing the actual size of a hard drive to the presented size.
### TB
A terabyte, or TB, is a unit of data that equals 1,099,511,627,776 bytes, or 240. This is equal to a kilobyte to the fourth power, 10244, being approximately one trillion bytes, or 1024 gigabytes. Consumer storage devices are often measured in terabytes.
### PB
14 petabytes stored within.
A petabyte, or PB, is a unit of data that equals 1,125,899,906,842,624 bytes, or 250. This is equal to a kilobyte to the fifth power, 10245, or roughly one quadrillion bytes. Large data centers, such as those operated by Google, can handle petabytes of data every day. Microsoft stores on 900 servers a total of approximately 14 petabytes. Cisco Systems predicts "the global Internet networks will deliver 12.5 petabytes every 5 minutes in 2016."
## Measurements of Data Speed
Data transfer speeds can be measured in bits per second, or in bytes per second. A byte is (generally) 8 bits long. Network engineers describe network speeds in bits per second, while web browsers usually measure a file download rate in bytes per second. A lowercase "b" usually means a bit, while an uppercase "B" represents a byte. In networking, metric prefixes (e.g. kilo, mega, and giga) refer to their decimal, not binary meaning.
### bps
Known as bits per second, bps was the main way of describing data transfer speeds several decades ago.
### Kbps
Kilobits per second, or 1000 bits per second. The quality of compressed audio files (e.g. MP3s) are typically measured in Kbps.
### Mbps
Megabits per second, or 1,000,000 bits per second. Internet service providers usually measure their Internet connectivity in Mbps.
### Gbps
Gigabits per second, or 1,000,000,000 bits per second. Modern local area networks, Internet infrastructure, and consumer Internet connections in some countries can operate at these speeds.
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An often cited example of the importance of getting units correct in design calculations is the [[[[https://en.wikipedia.org/wiki/Mars_Climate_Orbiter | Mars Climate Orbiter]]. The orbiter got too close to the planet, causing it to pass through the upper atmosphere and disintegrate. The problem was that computer software produced an output in non-SI units of pound-seconds (lbf×s) instead of the metric units of newton-seconds (N×s) specified in the contract between NASA and Lockheed.
to:
An often cited example of the importance of getting units correct in design calculations is the [[https://en.wikipedia.org/wiki/Mars_Climate_Orbiter | Mars Climate Orbiter]]. The orbiter got too close to the planet, causing it to pass through the upper atmosphere and disintegrate. The problem was that computer software produced an output in non-SI units of pound-seconds (lbf×s) instead of the metric units of newton-seconds (N×s) specified in the contract between NASA and Lockheed.
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(:keywords unit conversion, Mathcad, university course:)
(:description Unit Conversions in Mathcad - Problem-Solving Techniques for Chemical Engineers at Brigham Young University:)
An often cited example of the importance of getting units correct in design calculations is the [[[[https://en.wikipedia.org/wiki/Mars_Climate_Orbiter | Mars Climate Orbiter]]. The orbiter got too close to the planet, causing it to pass through the upper atmosphere and disintegrate. The problem was that computer software produced an output in non-SI units of pound-seconds (lbf×s) instead of the metric units of newton-seconds (N×s) specified in the contract between NASA and Lockheed.
In future missions problems with units have been addressed. Imagine attempting to perform a fully automated landing on Mars when the software is using miles instead of kilometers. How would this impact the final result?
(:html:)
<iframe width="560" height="315" src="//www.youtube.com/embed/Ki_Af_o9Q9s?rel=0" frameborder="0" allowfullscreen></iframe>
(:htmlend:)
Mathcad is a tool to arrange, calculate, and visualize engineering calculations and units can be a part of each number or equation. Use the following worksheet to step through example problems related to using units effectively in a Mathcad worksheet.
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4 m/s as our estimated windspeed, we must scale our estimate down, mul-
tiplying it by (4/6)3 0.3. (Remember, wind power scales as wind-speed
cubed.)
On the other hand, to estimate the typical power, we shouldn’t take the
mean wind speed and cube it; rather, we should find the mean cube of the
windspeed. The average of the cube is bigger than the cube of the average.
But if we start getting into these details, things get even more complicated,
because real wind turbines don’t actually deliver a power proportional to
wind-speed cubed. Rather, they typically have just a range of wind-speeds
within which they deliver the ideal power; at higher or lower speeds real
wind turbines deliver less than the ideal power.
### Variation of wind speed with height
Taller windmills see higher wind speeds. The way that wind speed increases
with height is complicated and depends on the roughness of the
surrounding terrain and on the time of day. As a ballpark figure, doubling
the height typically increases wind-speed by 10% and thus increases the
power of the wind by 30%.
Some standard formulae for speed v as a function of height z are:
1. According to the wind shear formula from NREL [ydt7uk], the speed
varies as a power of the height:
where v10 is the speed at 10 m, and a typical value of the exponent α
is 0.143 or 1/7. The one-seventh law (v(z) is proportional to z1/7) is
used by Elliott et al. (1991), for example.
2. The wind shear formula from the Danish Wind Industry Association
[yaoonz] is
where z0 is a parameter called the roughness length, and vref is the
speed at a reference height zref such as 10 m. The roughness length
for typical countryside (agricultural land with some houses and shel-
tering hedgerows with some 500-m intervals – “roughness class 2”)
is z0 = 0.1 m.
In practice, these two wind shear formulae give similar numerical answers.
That’s not to say that they are accurate at all times however. Van den Berg
(2004) suggests that different wind profiles often hold at night.
Figure B.7. Top: Two models of wind speed and wind power as a function of height. DWIA = Danish Wind Industry Association; NREL = National Renewable Energy Laboratory. For each model the speed at 10 m has been fixed to 6 m/s. For the Danish Wind model, the roughness length is set to z0 = 0.1 m. Bottom: The power density (the power per unit of upright area) according to each of these models.
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Purchase Solution
# Turbomachinery: Impulse & Reaction Turbines
Not what you're looking for?
Question 4: An impulse air turbine has a total (on static) inlet pressure and temperature of 11,25 bar, 600 degrees K and exhausts to the atmosphere at 0.91 bar. The nozzle efficiency is 87% and its exit angle is 71 degrees. The rotor efficiency is 65%. If the blade speed if 190 m/s at the mean radius of 12cm, calculate the power output for full admission to the 2cm long rotor blades.
Question 5: It has been decided to use the following velocity diagrams for a STEAM reaction turbine stage which is expected to have a nozzle and rotor efficiency of 80% when the stage inlet total ((not static) pressure and temperature are 17 bar, 400 degrees C. Using the small segment of the enthalpy-entropy chart for steam that is suppled, plot the hs diagram for the nozzle and rotor and hence determine the outlet pressures of each.
Question 6: It has been decided to use the following velocities for a reaction turbine stage which is expected to have a nozzle efficiency of 90% and a rotor efficiency of 85% when the stage inlet total (not static) pressure and temperature are 5 bar, 1050 K. Calculate the as yet known velocities and then determine the specific work output and the outlet pressure from the rotor.
##### Solution Summary
THis solution includes complete step-by-step calculations and answers. The explanation is 683 words. Illustrations are used for the geometry of the velocities for (1) through (3) and the thermodynamic analysis for (1) and (2).
##### Solution Preview
Notations and assumptions:
[ ]* = total features (like p*, T*)
cp = specific heat at constant pressure (for air at low temperatures we will take cp ...
##### Variables in Science Experiments
How well do you understand variables? Test your knowledge of independent (manipulated), dependent (responding), and controlled variables with this 10 question quiz.
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# Races – Solutions
Have given below solutions to the races questions that can be found here.
1. Two friends A and B simultaneously start running around a circular track . They run in the same direction. A travels at 6m/s and B runs at b m/s. If they cross each other at exactly two points on the circular track and b is a natural number less than 30, how many values can b take?
Let track length be equal to T. Time taken to meet for the first time = T / relative speed
= T/(6-b) or T/(b-6)
Time taken for a lap for A = T/6
Time taken for a lap for A = T/b
So, time taken to meet for the first time at the starting point = LCM (T/6, T/b) = T / HCF (6,b)
Number of meeting points on the track = Time taken to meet at starting point/Time taken for first meeting = Relative speed / HCF (6,b). For a more detailed discussion on this have a look at the last few slides in this presentation.
So, in essence we have to find values for b such that 6-b/ HCF(6,b) = 2 or b-6/ HCF(6,b) = 2
b = 2, 10, 18 satisfy this equation. So, there are three different values that b can take.
2. Three friends A, B and C decide to run around a circular track. They start at the same time and run in the same direction. A is the quickest and when A finishes a lap, it is seen that C is as much behind B as B is behind A. When A completes 3 laps, C is the exact same position on the circular track as B was when A finished 1 lap. Find the ratio of the speeds of A,B and C?
Let track length be equal to T. When a completes a lap, let us assume B has run a distance of (t-d). At this time, C should have run a distance of (t-2d)
After three laps C would have traveled a distance of 3 * (t-2d) = 3t – 6d.
After 3 laps C is in the same position as B was at the end one lap. So, the position after 3t-6d should be the same as t-d. Or, C should be at a distance of d from the end of the lap. C will have completed less than 3 laps (as he is slower than A), so he could have traveled a distance of either t-d or 2t-d.
=> 3t-6d = t-d => 2t = 5d => d = 0.4t => The distances covered by A,B and C when A completes a lap will be t, 0.6t and 0.2t respectively. Or, the ratio of their speeds is 5:3:1
In the second scenario, 3t-6d = 2t-d => t = 5d=> d = 0.2t => The distances covered by A,B and C when A completes a lap will be t, 0.8t and 0.6t respectively. Or, the ratio of their speeds is 5:4:3
The ratio of the speeds of A, B and C is either 5:3:1 or 5:4:3
### Related posts
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• November 24, 2010 Races – Basic Questions Have given below a few basic questions in Races 1. In a 100m race, A can give B a start of 12 meters or 3 seconds. What is the speed of B in m/sec and how long will A take to run […] Posted in Speed Time Distance
• November 25, 2010 Races Basic Questions Solutions Have given below the solutions to the basic questions on Races. Races Races from rajeshb1980 View more presentations from rajeshb1980. Posted in Speed Time Distance
• November 18, 2010 Speed Time Distance – Two Questions Have given below two questions on Speed Time Distance. These are slightly difficult questions. For a recap of the basics, have a look at questions here and solutions here . 1. City A to […] Posted in Speed Time Distance
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# Number Field
### Some articles on number field, numbers, number, field, fields:
Lattice Sieving
... It is almost exclusively used in conjunction with the number field sieve ... The algorithm implicitly involves the ideal structure of the number field of the polynomial it takes advantage of the theorem that any prime ideal above some rational prime p can be written as ... One then picks many prime numbers q of an appropriate size, usually just above the factor base limit, and proceeds by For each q, list the prime ideals above q by factorising the ...
Emmy Noether - Contributions To Mathematics and Physics - Third Epoch (1927–35) - Noncommutative Algebra
... Noether also was responsible for a number of other advancements in the field of algebra ... dimensional central division algebra over a number field splits locally everywhere then it splits globally (so is trivial), and from this, deduced their Hauptsatz ("main theorem") every finite ... that all maximal subfields of a division algebra D are splitting fields ...
Wieferich@Home - Properties - Connection With Fermat's Last Theorem
... K = Q(ξ) is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers (such an extension is known as a number field or in this ...
Modulus (algebraic Number Theory) - Definition
... Let K be a global field with ring of integers R ... If K is a number field, ν(p) = 0 or 1 for real places and ν(p) = 0 for complex places ... If K is a function field, ν(p) = 0 for all infinite places ...
### Famous quotes containing the words field and/or number:
Frankly, I’d like to see the government get out of war altogether and leave the whole field to private industry.
Joseph Heller (b. 1923)
As equality increases, so does the number of people struggling for predominance.
Mason Cooley (b. 1927)
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Difference Between Similar Terms and Objects
# Difference Between Vector and Matrix
Vector vs Matrix
Mathematics is used by man in the different fields that interest him. It is used in engineering, natural and social science, medicine, and other disciplines. It has been used ever since man discovered numbers and learned how to count.
It was first used by man to record time, for land measurement, in making patterns for painting and weaving, and in trading. The Egyptians and Babylonians were the first to use mathematics in taxation, construction, and astronomy, and the Greeks were the first to study mathematics as a science.
Mathematics has many fields which include geometry and algebra. Linear algebra in particular is a branch of mathematics that deals with the study of vector spaces and linear operations which are represented by a matrix or matrices.
A vector is defined as a mathematical quantity that has magnitude and direction, such as velocity. It is represented by a letter which is also what is used to represent a real number or a scalar quantity. To distinguish it from a real number, it is typed in boldface with an arrow above it. A unit vector is a vector with a magnitude of 1 and is denoted with a carat (^) above the variable.
Vectors are used in geometry to simplify three-dimensional problems, and many quantities in physics are vector quantities. A vector has the ability to simultaneously represent magnitude and direction. An example is the wind which has both speed and direction and so are other moving objects.
A matrix, on the other hand, is a rectangular array of numbers which is a key tool in linear algebra. It is used to represent linear transformations and keep track of coefficients in linear equations. Matrices are also used in physics, graph theory, computer graphics, calculus, and serialism.
An item in a matrix is called an element or an entry, and it is represented by a lower-case letter with two subscript indices. The matrix is represented by an upper-case letter and notated by brackets or parentheses.
It can have a row (row vector) or a column (column vector) which defines the components of vectors. Higher dimensional arrays of numbers or matrices define components of a generalization of a vector which is called a tensor.
Summary:
1.A matrix is a rectangular array of numbers while a vector is a mathematical quantity that has magnitude and direction.
2.A vector and a matrix are both represented by a letter with a vector typed in boldface with an arrow above it to distinguish it from real numbers while a matrix is typed in an upper-case letter.
3.Vectors are used in geometry to simplify certain 3D problems while matrices are key tools used in linear algebra.
4.A vector is an array of numbers with a single index while a matrix is an array of numbers with two indices.
5.While a vector is used to represent magnitude and direction, a matrix is used to represent linear transformations and keep track of coefficients in linear equations.
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Help us improve. Rate this post! (8 votes, average: 3.25 out of 5)
1. can we also say that a matrix is a placeholder of a vector?
2. Overall the entry was useful but the language was very gendered (e.g. “Mathematics is used by man in the different fields that interest him.”) How about using the word (hu)mankind or people?
• Anna: I agree that using the term “mankind” is sexist, and should be abolished. How about: “mathematics is used by humanity in the different fields that interest people”..? We are all one human race, and shouldn’t divide ourselves by sex either, because we are all spiritual entities having a human experience. Men can sometimes still be chauvinistic – thinking of themselves as superior to women, sort of like how Jewish people can think of themselves as superior to other ‘races’ (my mother is Jewish) – both are wrong minded!!
• How about we stop feeling hurt for the sake of feeling hurt and start using our time productively? Having this type of mindset will bring you no joy in life. You can do better.
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2018 NAIPC Practice Contest 08
#### Start
2018-02-24 18:00 UTC
## 2018 NAIPC Practice Contest 08
#### End
2018-02-24 23:00 UTC
The end is near!
Contest is over.
Not yet started.
Contest is starting in -390 days 18:43:37
5:00:00
0:00:00
# Problem JWater
A water company is trying to provide water from its pumping station to a mansion. The company owns $n$ water stations, numbered $1 \ldots n$, which are connected by a variety of pipes. Water can flow through both directions of a pipe, but the total amount of water that can flow through the pipe is bounded by the capacity of the pipe.
The water company is constantly improving the pipes, increasing the capacity of various pipes. The water company is conducting $k$ improvements (each of which is permanent after it is executed). An improvement consists of taking a pipe between two locations and increasing its capacity by a fixed amount, or installing a pipe between two locations which are not directly connected by a pipe.
After each improvement, the water company wants to know the maximum amount of water the mansion could receive.
## Input
Each input will consist of a single test case. Note that your program may be run multiple times on different inputs. The first line of input contains three integers, $n$ ($2 \le n \le 100$), $p$ ($0 \le p \le \frac{n (n-1)}{2}$), and $k$ ($1 \le k \le 10\, 000$), where $n$ is the number of stations, $p$ is the number of initial pipes, and $k$ is the number of improvements. The first station in the list is always the pumping station, and the second is always the mansion.
The next $p$ lines will describe the pipes in the initial setup. The lines will each contain three integers, $a$, $b$ ($1 \le a < b \le n$) and $c$ ($1 \le c \le 1\, 000$), which indicates that stations $a$ and $b$ are connected by a pipe with capacity $c$. No $(a, b)$ pair will appear more than once in this section.
The next $k$ lines will describe the improvements. The lines will each contain three integers, $a$, $b$ ($1 \le a < b \le n$) and $c$ ($1 \le c \le 1\, 000$), which indicates that the pipe connecting stations $a$ and $b$ has its capacity increased by $c$ (if there is currently no pipe between $a$ and $b$, then one is created with capacity $c$). Note that it is possible for an $(a,b)$ pair to be repeated in this section.
## Output
Output $k+1$ integers, each on its own line, describing the maximum amount of water that can reach the mansion. The first number is the amount of water reaching the mansion in the initial configuration. The next $k$ numbers are the amounts of water reaching the mansion after each improvement.
Sample Input 1 Sample Output 1
3 2 1
1 3 10
2 3 1
2 3 15
1
10
Sample Input 2 Sample Output 2
6 10 2
1 3 2
1 4 6
1 5 1
3 5 8
4 5 7
2 4 3
2 5 4
2 6 1
5 6 9
3 6 5
2 6 9
1 6 3
8
9
12
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Order Homework Help from Pro Experts
# Phase Sequence and Numbering of Phases
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When 3 phases ona certain order are able to achieve their maximum point or their peak; that sequence can be expressed as phase sequence.
In case of 3 phase e.m.f. generation in consideration offield system the assumption is mainly dependent on clockwise rotation. This presumption is based on e.m.f. of phases.
Let us consider these 3 phases as‘l,’‘m’ and ‘n.’As per this, the phase lag between ‘l’and ‘m’ is observed to have an angular difference of 120°. Similarly, thephase lag between ‘m’ and ‘n’ do also have similar variation by 120 electrical degrees. Sofinally, it can be clearly seen that angular variation between ‘n’ and ‘l’ in clockwise movement is 240 electrical degrees.
So, the sequential pattern in which phases of these e.m.f. (m, n, and l) are placed to acquire their highest value will be denoted as lmn. This positioning and value are known as phase sequence or phase order.
This sequence follows as clockwise motion with the order as ‘l,’‘m’ and finally to ‘n.’In case this rotational movement of this field structure is forced to move in counter clockwise movement or reversed direction, its resultant will also follow the same path. Those 3 phases in that order would be able to achieve their maximum voltage ina reversed order, thereby relaying the new phase order as ‘l,’‘n’ and then ‘m.’
As per this counter rotation, the new e.m.f. of phase showcases the fact that in this new phase, ‘n’ will be lagging behind ‘l.’But in place of 120 electrical degrees, the angular difference would be of 240 electrical degrees.
In generalizedsense, this phase orderof voltages which on application to a load can help in determiningthis sequence. The purpose of this sequencing is based on the connected 3 phase lines. The reversal of this phase sequence is possible when any of the line pairs are interchanged with each other.
The outcome for reversal of sequence can be seen in an induction motor, where its motor rotation takes place in counter clockwise direction.
Explaining via examples
Let us consider these 3 phases to be either numbered or be taken as in the form of colors. For this purpose, let us stick to colors (primary colors). So, we will be using the initials for blue, red and yellow, taking it as BRY. So, in this sequence regarding any of the 3 phase system, 2 of the potential orders having phase voltages or 3 coils can pass via their peak value. It can be RBY (red, blue and then yellow) or it can be RYB (red, yellow and then blue).
So, in accordance with this convention,
• RBY can be taken to be negative
• RYE can be taken to be positive
Links of Next Electrical Engineering Topics:-
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# Why are discontinuities of monotonic $f : (a, b) \to \mathbb R$ countable? [duplicate]
I'm struggling to find an elegant proof of the following problem
Let $f : (a, b) \to \mathbb R$ be non-decreasing, $a, b \in \mathbb R$, then $f$ only has countably many discontinuities.
My intuition was to show by contradiction that the set of discontinuities $N \subseteq (a,b)$ is discrete, i.e. all discontinuities are isolated. From there on it's easy to prove that there is an injective function $N \to \mathbb Q$.
But does my first step make sense? Say we had non-isolated discontinuities like $\epsilon > 0, x_0 \in N$ such that $B_\epsilon(x_0) \subseteq N$ - how could one derive a contradiction?
I've already shown that $$\lim_{x \nearrow x_0} f(x) \text{ and } \lim_{x \searrow x_0} f(x)$$ exist for all points $x_0 \in (a, b)$ and that $f$ is continuous at $x_0$ iff both limits equal. I just somehow fail to do the final step properly. Any thoughts, please?
-
## marked as duplicate by David Mitra, Asaf Karagila, t.b., Davide Giraudo, Zev ChonolesDec 29 '11 at 0:08
Discontinuities need not be isolated. If the left and right limits are not equal, then they define a non-empty open interval. In that interval is a rational number. So...? – Qiaochu Yuan Dec 28 '11 at 20:23
@Alex: the Cantor set doesn't have the discrete topology as a subset of the reals. – Qiaochu Yuan Dec 28 '11 at 20:24
It's not true that $N$ needs to be discrete. It can even be dense: Chose an injection $h:\mathbb Q\to \mathbb N$, and let $f(x)=x+\sum_{y\in \mathbb Q \cap (a,x)} 2^{-h(y)}$ which is strictly increasing but discontinuous at every rational number. – Henning Makholm Dec 28 '11 at 20:27
I realized that and deleted my comment. Sorry to leave yours hanging. – Alex Becker Dec 28 '11 at 20:28
In addition to 84870 which David Mitra links to, there are: math.stackexchange.com/questions/56831/… and math.stackexchange.com/questions/14458/… – Jonas Meyer Dec 28 '11 at 22:17
No, the set of discontinuities need not be discrete. For example, it's quite possible to have a discontinuity at $0$ and also at $1/n$ for each positive integer $n$.
Hint: for each discontinuity, there are rational numbers that are not in $f((a,b))$.
Certainly $\lim_{x \nearrow x_0} f(x) \leq \lim_{x \searrow x_0} f(x)$. Also, $\sum\limits_{x_0\in (a,b)}\lim_{x \searrow x_0} f(x) - \lim_{x \nearrow x_0} f(x)$ is finite. It is not hard to show that since this is finite, all but countably many terms are $0$.
EDIT: When I shoot from the hip I forget that if $f(x)$ is unbounded you need to normalize it as $f(x)/\max\{|f(y)| : y\in (\frac{x+a}{2},\frac{x+b}{2})\}$ which is defined because $f(x)$ is monotonic and has all the discontinuities of $f(x)$.
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## 181693
181,693 (one hundred eighty-one thousand six hundred ninety-three) is an odd six-digits prime number following 181692 and preceding 181694. In scientific notation, it is written as 1.81693 × 105. The sum of its digits is 28. It has a total of 1 prime factor and 2 positive divisors. There are 181,692 positive integers (up to 181693) that are relatively prime to 181693.
## Basic properties
• Is Prime? Yes
• Number parity Odd
• Number length 6
• Sum of Digits 28
• Digital Root 1
## Name
Short name 181 thousand 693 one hundred eighty-one thousand six hundred ninety-three
## Notation
Scientific notation 1.81693 × 105 181.693 × 103
## Prime Factorization of 181693
Prime Factorization 181693
Prime number
Distinct Factors Total Factors Radical ω(n) 1 Total number of distinct prime factors Ω(n) 1 Total number of prime factors rad(n) 181693 Product of the distinct prime numbers λ(n) -1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) -1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 12.1101 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0
The prime factorization of 181,693 is 181693. Since it has a total of 1 prime factor, 181,693 is a prime number.
## Divisors of 181693
2 divisors
Even divisors 0 2 2 0
Total Divisors Sum of Divisors Aliquot Sum τ(n) 2 Total number of the positive divisors of n σ(n) 181694 Sum of all the positive divisors of n s(n) 1 Sum of the proper positive divisors of n A(n) 90847 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 426.255 Returns the nth root of the product of n divisors H(n) 1.99999 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors
The number 181,693 can be divided by 2 positive divisors (out of which 0 are even, and 2 are odd). The sum of these divisors (counting 181,693) is 181,694, the average is 90,847.
## Other Arithmetic Functions (n = 181693)
1 φ(n) n
Euler Totient Carmichael Lambda Prime Pi φ(n) 181692 Total number of positive integers not greater than n that are coprime to n λ(n) 181692 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 16436 Total number of primes less than or equal to n r2(n) 8 The number of ways n can be represented as the sum of 2 squares
There are 181,692 positive integers (less than 181,693) that are coprime with 181,693. And there are approximately 16,436 prime numbers less than or equal to 181,693.
## Divisibility of 181693
m n mod m 2 3 4 5 6 7 8 9 1 1 1 3 1 1 5 1
181,693 is not divisible by any number less than or equal to 9.
• Arithmetic
• Prime
• Deficient
• Polite
• Prime Power
• Square Free
## Base conversion (181693)
Base System Value
2 Binary 101100010110111101
3 Ternary 100020020101
4 Quaternary 230112331
5 Quinary 21303233
6 Senary 3521101
8 Octal 542675
10 Decimal 181693
12 Duodecimal 89191
20 Vigesimal 12e4d
36 Base36 3w71
## Basic calculations (n = 181693)
### Multiplication
n×i
n×2 363386 545079 726772 908465
### Division
ni
n⁄2 90846.5 60564.3 45423.2 36338.6
### Exponentiation
ni
n2 33012346249 5998112227019557 1089815004863864370001 198011757678730108978591693
### Nth Root
i√n
2√n 426.255 56.6386 20.6459 11.2685
## 181693 as geometric shapes
### Circle
Diameter 363386 1.14161e+06 1.03711e+11
### Sphere
Volume 2.51248e+16 4.14845e+11 1.14161e+06
### Square
Length = n
Perimeter 726772 3.30123e+10 256953
### Cube
Length = n
Surface area 1.98074e+11 5.99811e+15 314702
### Equilateral Triangle
Length = n
Perimeter 545079 1.42948e+10 157351
### Triangular Pyramid
Length = n
Surface area 5.71791e+10 7.06884e+14 148352
## Cryptographic Hash Functions
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| 915,116,322
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# Mass using triple integrals
1. Mar 18, 2012
### 1MileCrash
1. The problem statement, all variables and given/known data
Find the mass m of the pyramid with base in the plane z = 9 and sides formed by the three planes y = 0 and y - x = 5 and 6x + y + z = 28, if the density of the solid is given by δ(x,y,z) = y.
2. Relevant equations
3. The attempt at a solution
This problem is driving me insane. It takes me about 45 minutes of algebra to evaluate this incorrectly set up integral..
I integrated y in the order dz dy dx, limits, respectively:
9 to 28-6x-y
0 to 5+x
0 to 2
Is that correct? I don't really know how to get the limits for x.. this is so hard to picture in my mind!
To get 2, I solved the system 6x + y + z = 28 with z = 9 and y = 5 + x, and for 0, I just guessed.
2. Mar 18, 2012
### OldEngr63
May I suggest that you draw a sketch and get a clear grip on the limits before you try to evaluate anything.
3. Mar 18, 2012
### 1MileCrash
Well, that's what I'm trying to do..
4. Mar 18, 2012
### 1MileCrash
I drew it again, the only thing different is that my order is dzdxdy.
I don't understand how to picture these things, it's very difficult.. I just drew an xy plane noting that z = 9 for the base, and looked at that. What would you do?
5. Mar 18, 2012
### 1MileCrash
I think this is far beyond me.
6. Mar 18, 2012
### 1MileCrash
Nevermind, found a good explanation online of something called the "shadow method."
7. Mar 18, 2012
### LCKurtz
Have you figured out it gets tricky if you use any order of integration other than dzdxdy?
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0
# How many cm is 205 ft?
Updated: 9/20/2023
Wiki User
11y ago
205 ft = 6,248.4 cm
Wiki User
11y ago
Earn +20 pts
Q: How many cm is 205 ft?
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Related questions
### How many feet in 205 meters?
672.57 ft Algebraic Steps / Dimensional Analysis Formula 205 m*100 cm 1 m*1 in 2.54 cm*1 ft 12 in=672.5721785 ft Direct Conversion Formula 205 m*1 ft 0.3048 m=672.5721785 ft
### How many feet are in 205 cm?
205 cm = 6.73 feet.
### How many centimeters are in 205 inches?
520.7 cm Algebraic Steps / Dimensional Analysis Formula 205 in* 2.54 cm 1 in = 520.7 cm
2,460 yd .615ft
### What is 205 cm converted to feet and inches?
205 cm is approximately 6 feet 8.7 inches when converted to the imperial system.
### How many ft are in 205 yd?
One yard is equal to 3 feet, therefore 205 x 3 = 615 ft
### How many cm in 2.9 ft?
1 ft = 30.48 cm So... 2.9 ft × 30.48 cm/ft = 88.392 cm
20.5 cm 20.5 cm
### How many ft in an cm?
in 1 cm there is 0.0328084 ft
### How many cm is 3.6 ft.?
3.6 ft is 109.72800 cm
### How many cm are in 6.3 ft?
6.3 ft = 192.024 cm
### How many cm in 883 ft?
883 ft = 26,913.84 cm
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# How do projective representations act on the QFT vacuum?
Let $$U:\mathcal{G}\to \mathcal{U}(\mathcal{H})$$ be a unitary projective representation of a symmetry group $$\mathcal{G}$$ on a Hilbert space $$\mathcal{H}$$. It satisfies the composition rule:
$$U(g_1)U(g_2)=e^{i\phi(g_1,g_2)}U(g_1g_2).\tag{1}$$
Now suppose there is a state $$|\Omega\rangle$$ which only changes by a phase under the symmetry:
$$\tag{2} U(g)|\Omega\rangle = e^{if(g)}|\Omega\rangle,$$
for some real phases $$f(g)$$. Then acting on $$|\Omega\rangle$$ with each side of (1) we find:
$$\phi(g_1,g_2)=-f(g_1g_2)+f(g_1)+f(g_2) \mod 2\pi \tag{3}$$
So $$\phi(g_1,g_2)$$ is a trivial cocycle. By defining the "improved" symmetry operators $$\tilde{U}(g)=e^{-i f(g)}U(g)\tag{4}$$
we get a true, non-projective representation of $$\mathcal{G}$$.
To summarise, if a projective representation contains a vector which is left invariant up to phases, then it is not an "intrinsically projective" representation: rescaling the unitaries by phases makes it into a true representation.
But in any QFT without symmetry-breaking there is such an invariant state: the vacuum! So the above argument shows that unbroken symmetries can't be represented projectively in QFT.
My question: What about spinor representations of the Lorentz group - they're projective, aren't they?
• Indeed, you just proved that if $\phi$ is a non-trivial cocycle (i.e., the rep is truly projective) then neither the vacuum -- nor any other particular state -- can be invariant, not even up to a phase. This is a trivial statement, it is obvious that there are no non-trivial one-dimensional projective representations (given that a projective rep is a map to $PU(n)$ and $PU(1)\equiv\emptyset$). In conclusion, any anomalous symmetry in quantum mechanics is spontaneously broken, which is a rather well known fact. Congrats on rediscovering it! Mar 2, 2023 at 0:32
• I'll choose to take your last sentence at face value. So, thank you! Mar 2, 2023 at 20:54
• @AccidentalFourierTransform Do you know a grad-level (i.e. not too advanced, no nlab links pls!) reference which discusses anomalies in terms of projective representations? Mar 3, 2023 at 2:58
• possibly useful? physics.stackexchange.com/q/687500/84967 Mar 9, 2023 at 20:51
1. Yes, OP is right. More generally, one could consider a projective (not necessarily unitary) representation $$\rho:G\to {\rm End }(V)$$ where $$\rho(g)\rho(h)~=~c(g,h)\rho(gh),\tag{1}$$ and where $$c(g,h)\in \mathbb{C}^{\times}$$ is a 2-cocycle. Assume that there exists a 1-dimensional invariant subspace of $$V$$, $$\rho(g)|\Omega\rangle~=~b(g)|\Omega\rangle,\qquad |\Omega\rangle~\in~V\backslash\{0\}, \qquad b(g)~\in~ \mathbb{C}^{\times}~:=~\mathbb{C}\backslash\{0\}, \tag{2}$$ i.e. a 1-dimensional projective subrepresentation. So $$c(g,h)~=~b(gh)^{-1}b(g)b(h)\tag{3}$$ is a 2-coboundary, i.e $$\rho$$ can be lifted to an ordinary linear representation $$\tilde{\rho}(g)~:=~b(g)^{-1}\rho(g),\qquad \tilde{\rho}(g)\tilde{\rho}(h)~=~\tilde{\rho}(gh),\tag{4}$$ such that $$\tilde{\rho}$$ acts trivially on the vacuum $$\left.\tilde{\rho}\right|_{{\rm span}|\Omega\rangle}~=~ \left.{\bf 1}\right|_{{\rm span}|\Omega\rangle}, \tag{5}$$ cf. OP's title question.
2. OP's last subquestion about spinor representations for the restricted Lorentz group $$SO^+(1,3)$$ seems already answered in OP's other post here, i.e. one should consider its double cover $$Spin^+(1,3)~\cong~ SL(2,\mathbb{C})\tag{6}$$ instead.
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https://www.mathworks.com/matlabcentral/cody/problems/546-is-a-the-inverse-of-b/solutions/2756812
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Cody
# Problem 546. Is A the inverse of B?
Solution 2756812
Submitted on 29 Jul 2020 by Jerry Wright
This solution is locked. To view this solution, you need to provide a solution of the same size or smaller.
### Test Suite
Test Status Code Input and Output
1 Pass
A=[2,4;3,5]; B=[-2.5,2;1.5,-1]; y_correct = 1; assert(isequal(isInverse(A,B),y_correct))
y = logical 1
2 Pass
A=[1,2;3,5]; B=[-5,2;3,-1]; y_correct = 1; assert(isequal(isInverse(A,B),y_correct))
y = logical 1
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http://encyclopedia.kids.net.au/page/co/Convergence_in_distribution
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## Encyclopedia > Convergence in distribution
Article Content
# Convergence of random variables
Redirected from Convergence in distribution
In probability theory, several different notions of convergence of random variables are investigated. These will be presented here. Throughout, we assume that (Xn) is a sequence of random variables, and X is a random variable, and all of them are defined on the same probability space (Ω, P).
### Convergence in distribution
We say that the sequence Xn converges towards X in distribution, if
$\lim_{n\rightarrow\infty}P(X_n\leq a)=P(X\leq a)$
for every real number a at which the cumulative distribution function of the limiting random variable X is continuous.
Essentially, this means that the probability that the value of X is in a given range is very similar to the probability that the value of Xn is in that range, if only n is large enough. This notion of convergence is used in the central limit theorems.
Convergence in distribution is also called convergence in law, since the word "law" is sometimes used as a synonym of "probability distribution." Another name is weak convergence.
### Convergence in probability
We say that the sequence Xn converges towards X in probability if
$\lim_{n\rightarrow\infty}P\left(\left|X_n-X\right|\geq\varepsilon\right)=0$
for every ε > 0.
This means that if you pick a tolerance ε and choose n large enough, then the value of Xn will be almost guaranteed to be within that tolerance of the value of X. This notion of convergence is used in the weak law of large numbers.
Convergence in probability implies convergence in distribution.
### Almost sure convergence
We say that the sequence Xn converges almost surely or almost everywhere or with probability 1 or strongly towards X if
$P\left(\lim_{n\rightarrow\infty}X_n=X\right)=1.$
This means that you are virtually guaranteed that the values of Xn approach the value of X. This notion of convergence is used in the strong law of large numbers.
Almost sure convergence implies convergence in probability.
### Convergence in mean
We say that the sequence Xn converges towards X in mean or in the L1 norm if
$\lim_{n\rightarrow\infty}E\left(\left|X_n-X\right|\right)=0$
where E denotes the expected value.
This means that the expected difference between Xn and X gets as small as desired if n is chosen big enough. This convergence is considered in Lp spaces (where p = 1).
Convergence in the mean implies convergence in probability. There is no general relation between convergence in mean and almost sure convergence however.
### Convergence in mean square
We say that the sequence Xn converges towards X in mean square or in the L2 norm if
$\lim_{n\rightarrow\infty}E\left(\left|X_n-X\right|^2\right)=0.$
This means that the expected squared difference between Xn and X gets as small as desired if n is chosen big enough. This convergence is considered in Lp spaces (where p = 2).
Convergence in mean square implies convergence in mean.
All Wikipedia text is available under the terms of the GNU Free Documentation License
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# Leetcode Multiply Strings problem solution
In this Leetcode Multiply Strings problem solution, we have given two non-negative integers num1 and num2 represented as strings, return the product of num1 and num2, also represented as a string.
## Problem solution in Python.
```class Solution:
def multiply(self, num1, num2):
num1_num = 0
for each in num1:
asc = ord(each)
num = asc - 48
num1_num = num1_num*10+num
num2_num = 0
for each in num2:
asc = ord(each)
num = asc - 48
num2_num = num2_num*10+num
mul = num1_num * num2_num
if mul == 0:
return '0'
strlist = []
while mul != 0:
digit = mul%10
strlist.append(digit)
mul = (mul- digit)//10
result = ''
for each in strlist:
newstr = chr(each+48)
result = newstr + result
return result
```
## Problem solution in Java.
```public String multiply(String num1, String num2) {
if ("0".equals(num1) || "0".equals(num2)) {
return "0";
}
int num1Len = num1.length();
int num2Len = num2.length();
int t;
int[] res = new int[num1Len + num2Len];
for (int i = num1Len - 1; i >= 0; i--) {
for (int j = num2Len - 1, r = num1Len - 1 - i; j >= 0; j--, r++) {
res[r] += (num1.charAt(i) - '0') * (num2.charAt(j) - '0');
if (res[r] > 9) {
t = res[r];
res[r] = t % 10;
res[r + 1] += t / 10 % 10;
}
}
}
StringBuilder sb = new StringBuilder();
for (int i = 0; i < res.length; i++) {
if (sb.length() == 0 && res[res.length - 1 - i] == 0) {
continue;
}
sb.append(res[res.length - 1 - i]);
}
return sb.toString();
}
```
## Problem solution in C++.
```class Solution {
public:
string multiply(string num1, string num2) {
int n1 = num1.length(), n2 = num2.length();
if(n1 == 0 || n2 == 0) return "";
string res(n1+n2, '0');
int c = 0;
for(int i = n1-1; i >= 0; i--){
for(int j = n2-1; j >= 0; j--){
int tmp = (num1[i]-'0')*(num2[j]-'0')+c+res[i+j+1]-'0';
res[i+j+1] = tmp%10+'0';
c = tmp/10;
}
res[i] += c;
c = 0;
}
int start = 0;
while(start < res.length() && res[start] == '0')
start++;
if(start == res.length())
return "0";
return res.substr(start);
}
};
```
## Problem solution in C.
```char* multiply(char* arg1, char* arg2) {
int len = strlen(arg1) + strlen(arg2);
char *result = malloc(sizeof(char) * (len + 1));
int acc;
int k = 0;
for (int i = 0; i < len+1; i++)
result[i] = '0';
for (int i = strlen(arg1)-1; i >= 0; i--){
for (int j = strlen(arg2)-1; j >= 0; j--){
acc = (arg1[i] - '0') * (arg2[j] - '0') + (result[i + j + 1] - '0');
result[i+j+1] = (acc % 10) + '0';
result[i+j] = ((acc /10) + (result[i + j] - '0')) + '0';
}
}
result[len] = 0;
while (result[k] == '0')
k++;
return strlen(result+k) > 0 ? result+k : "0";
}
```
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Question DetailsNormal
\$ 85.00
STAT 200 OL1/US1 Sections Final Exam Spring 2018
Question posted by
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 1 of 7
STAT 200
OL1/US1 Sections
Final Exam
Spring 2018
The final exam will be posted at 12:01 am on March 2, and it is
due at 11:59 pm on March 4, 2018. Eastern Time is our reference
time.
This is an open-book exam. You may refer to your text and other course materials
for the current course as you work on the exam, and you may use a calculator.
You must complete the exam individually. Neither collaboration nor consultation
with others is allowed. It is a violation of the UMUC Academic Dishonesty and
Plagiarism policy to use unauthorized materials or work from others.
Show all of your supporting work and reasoning. Answers that come straight
from calculators, programs or software packages without any explanation will not
be accepted. If you need to use technology (for example, Excel, online or handheld calculators, statistical packages) to aid in your calculation, you must cite the
sources and explain how you get the results.
This exam has 200 total points; 10 points for each question.
You must include the Honor Pledge on the title page of your submitted final exam.
Exams submitted without the Honor Pledge will not be accepted.
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 2 of 7
1. True or False. Justify for full credit.
(a) If P(A) = 0.4, P(B) = 0.5, and A and B are independent, then P(A AND B) = 0.9.
(b) If the variance of a data set is 0, then all the observations in this data set must be zero.
(c) The mean is always equal to the median for a normal distribution.
(d) A 95% confidence interval is wider than a 90% confidence interval of the same parameter.
(e) In a two-tailed test, the value of one test statistic is 2. The test statistic follows a
distribution with the distribution curve shown below. If we know the shaded area is 0.03,
then we have sufficient evidence to reject the null hypothesis at 0.05 level of significance.
2. Choose the best answer. Justify for full credit.
(a) Among the Senators in the current Congress, 54% are Republicans. The value 54% is a
(i) statistic
(ii)
(iii) parameter
cannot be determined
(b) The hotel ratings are usually on a scale from 0 star to 5 stars. The level of this measurement is
(i)
(ii)
(iii)
(iv) interval
nominal
ordinal
ratio
(c) In a career readiness research, 100 students were randomly selected from the psychology
program, 150 students were randomly selected from the communications program, and 120
students were randomly selected from cyber security program. This type of sampling is called:
(i)
(ii)
(iii)
(iv) cluster
convenience
systematic
stratified
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 3 of 7
3. Choose the best answer. Justify for full credit.
(a) A study of 10 different weight loss programs involved 500 subjects. Each of the 10 programs
had 50 subjects in it. The subjects were followed for 12 months. Weight change for each
subject was recorded. You want to test the claim that the mean weight loss is the same for the
10 programs. What statistical approach should be used?
(i)
(ii)
(iii)
(iv) t-test
linear regression
ANOVA
confidence interval
(b) A STAT 200 instructor teaches two classes. She wants to test if the variances of the score
distribution for the two classes are different. What type of hypothesis test should she use?
(i)
(ii)
(iii)
(iv) t-test for two independent samples
t-test for matched samples
z-test for two samples
F- test
4. The frequency distribution below shows the distribution for IQ scores for a random sample of
IQ Scores Frequency Relative Frequency
50 - 69 24
70 - 89
90 -109 453
110 - 129 0.25
130 - 149 25
Total 1000
(a) Complete the frequency table with frequency and relative frequency. Express the relative
frequency to three decimal places.
(b) What percentage of the adults in this sample has an IQ score of at least 110?
(c) Does this distribution have positive skew or negative skew? Why or why not?
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 4 of 7
5. The five-number summary below shows the grade distribution of two STAT 200 quizzes for a
sample of 500 students.
Minimum Q1 Median Q3 Maximum
Quiz 1 15 35 55 85 100
Quiz 2 20 35 50 90 100
For each question, give your answer as one of the following: (i) Quiz 1; (ii) Quiz 2; (iii) Both quizzes
have the same value requested; (iv) It is impossible to tell based on the given information. Then
(a)
(b)
(c) Which quiz has less range in grade distribution?
Which quiz has the greater percentage of students with grades 85 and over?
Which quiz has a greater percentage of students with grades less than 60?
6. A sample of 10 LED light bulbs consists of 1 defective and 9 good light bulbs. A quality
control technician wants to randomly select two of the light bulbs for inspection. Find the
probability that the first selected light bulb is good and the second light bulb is also good.
(Show all work. Just the answer, without supporting work, will receive no credit.)
(a) Assuming the two random selections are made with replacement.
(b) Assuming the two random selections are made without replacement.
7. There are 1000 students in a high school. Among the 1000 students, 250 students take AP
Statistics, and 300 students take AP French. 100 students take both AP courses. Let S be the
event that a randomly selected student takes AP Statistics, and F be the event that a randomly
selected student takes AP French. Show all work. Just the answer, without supporting work,
(a) Provide a written description of the complement event of (S OR F).
(b) What is the probability of complement event of (S OR F)?
8. Consider rolling two fair dice. Let A be the event that the sum of the two dice is 8, and B be
the event that the first one is a multiple of 3.
(a) What is the probability that the sum of the two dice is 8 given that the first one is a multiple of
3? Show all work. Just the answer, without supporting work, will receive no credit.
(b) Are event A and event B independent? Explain.
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 5 of 7
9. Answer the following two questions. (Show all work. Just the answer, without supporting
(a) UMUC Stat Club is sending a delegate of 2 members to attend the 2018 Joint Statistical
Meeting in Vancouver, Canada. There are 10 qualified candidates. How many different ways
can the delegate be selected?
(b) A bike courier needs to make deliveries at 6 different locations. How many different routes can
he take?
10. Assume random variable x follows a probability distribution shown in the table below.
Determine the mean and standard deviation of x. Show all work. Just the answer, without
supporting work, will receive no credit.
x -2 0 1 2 3
P(x) 0.1 0.1 0.4 0.2 0.2
11. Mimi is into figure skating these days due to the Winter Olympics. She started learning toe loop
jumps a couple weeks ago. Her probability of landing a successful single toe loop jump is 0.30.
She plans to attempt 10 single toe loop jumps in her practice session tomorrow.
(a) Let X be the number of successful single toe loop jumps that Mimi lands. As we know, the
distribution of X is a binomial probability distribution. What is the number of trials (n),
probability of successes (p) and probability of failures (q), respectively?
(b) Find the probability that Mimi lands at least one successful single toe loop jump in her 10
attempts. (round the answer to 3 decimal places) Show all work. Just the answer, without
supporting work, will receive no credit.
12. Assume the weights of men are normally distributed with a mean of 170 lbs and a standard
deviation of 30 lbs. Show all work. Just the answer, without supporting work, will receive no
credit.
(a) Find the 75th percentile for the distribution of men’s weights.
(b) What is the probability that a randomly selected man weighs more than 200 lbs?
13. Assume the SAT Mathematics Level 2 test scores are normally distributed with a mean of 500
and a standard deviation of 100. Show all work. Just the answer, without supporting work, will
(a) If a random sample of 64 test scores is selected, what is the standard deviation of the sample
mean?
(b) What is the probability that 64 randomly selected test scores will have a mean test score that is
between 475 and 525?
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 6 of 7
14. A survey showed that 80% of the 1600 adult respondents believe in global warming. Construct a
90% confidence interval estimate of the proportion of adults believing in global warming. Show
all work. Just the answer, without supporting work, will receive no credit.
15. In a study designed to test the effectiveness of garlic for lowering cholesterol, 49 adults were
treated with garlic tablets. Cholesterol levels were measured before and after the treatment. The
changes in their LDL cholesterol (in mg/dL) have a mean of 3 and standard deviation of 14.
Construct a 95% confidence interval estimate of the mean change in LDL cholesterol after the
garlic tablet treatment. Show all work. Just the answer, without supporting work, will receive
no credit.
16. Mimi is interested in testing the claim that banana is the favorite fruit for more than 80% of the
adults. She conducted a survey on a random sample of 100 adults. 85 adults in the sample
chose banana as his / her favorite fruit.
Assume Mimi wants to use a 0.05 significance level to test the claim.
(a) Identify the null hypothesis and the alternative hypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without
supporting work, will receive no credit.
(d) Is there sufficient evidence to support the claim that banana is the favorite fruit for more than
17. In a study of freshman weight gain, the measured weights of 5 randomly selected college
students in September and April of their freshman year are shown in the following table.
Weight (lb)
Student September April
1 147 150
2 127 125
3 140 150
4 156 160
5 154 156
Is there evidence to suggest that the mean weight of the freshmen in April is greater than the
mean weight in September?
Assume we want to use a 0.05 significance level to test the claim.
(a) Identify the null hypothesis and the alternative hypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without supporting
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without
supporting work, will receive no credit.
STAT 200: Introduction to Statistics Final Examination, Spring 2018 OL1 Page 7 of 7
(d) Is there sufficient evidence to support the claim that the mean weight of the freshmen in April
is greater than the mean weight in September? Justify your conclusion.
18. In a pulse rate research, a simple random sample of 40 men results in a mean of 80 beats per
minute, and a standard deviation of 11.3 beats per minute. Based on the sample results, the
researcher concludes that the pulse rates of men have a standard deviation greater than 10 beats
per minutes. Use a 0.05 significance level to test the researcher’s claim.
(a) Identify the null hypothesis and alternative hypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without
supporting work, will receive no credit.
(c) Determine the P-value for this test. Show all work; writing the correct P-value, without
supporting work, will receive no credit.
(d) Is there sufficient evidence to support the researcher’s claim? Explain.
19. The UMUC Daily News reported that the color distribution for plain M&M’s was: 40%
brown, 20% yellow, 20% orange, 10% green, and 10% tan. Each piece of candy in a random
sample of 100 plain M&M’s was classified according to color, and the results are listed below.
Use a 0.05 significance level to test the claim that the published color distribution is correct.
Color Brown Yellow Orange Green Tan
Number 35 25 16 10 14
(a) Identify the null hypothesis and the alternative hypothesis.
(b) Determine the test statistic. Show all work; writing the correct test statistic, without
supporting work, will receive no credit.
(c) Determine the P-value. Show all work; writing the correct P-value, without supporting
(d) Is there sufficient evidence to support the claim that the published color distribution is
20. A STAT 200 instructor believes that the average quiz score is a good predictor of final exam
score. A random sample of 8 students produced the following data where x is the average quiz
score and y is the final exam score.
x 85 50 72 65 78 60 100 70
y 85 75 75 64 80 65 95 60
(a) Find an equation of the least squares regression line. Show all work; writing the correct
equation, without supporting work, will receive no credit.
(b) Based on the equation from part (a), what is the predicted final exam score if the average quiz
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https://forum.xojo.com/t/port-source-from-m-to-xojo/14150
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# Port source from .m to Xojo
Hello,
I’m trying to port a .M file (Objective C?) to Xojo.
The function prints prime numbers in a spiral fashion, that is if a number is prime it gets painted else it is omitted. The code is only a few lines but I’m struggling.
The source:
[code]% A first attemp to draw Ulam spiral.
% see: http://en.wikipedia.org/wiki/Ulam_spiral
% S. E. Mousavi
% June 17, 2011
m = 1001; % size of A
m2 = m*m;
length = 1;
direction = 1; % 1:R, 2:U, 3:L, 4:D
A = zeros(m, m);
x = ceil(m/2);
y = x;
ind = 1;
A(y, x) = ind;
while (ind < m2)
for i = 1:2
for j = 1:length
ind = ind + 1;
if (ind > m2), break; end
switch direction
case 1
x = x + 1;
case 2
y = y - 1;
case 3
x = x - 1;
case 4
y = y + 1;
end
A(y, x) = ind;
end
if (ind > m2), break; end
direction = mod(direction, 4) + 1;
end
length = length + 1;
end
B = isprime(A);
imagesc(B);
axis equal;
axis off;[/code]
My Xojo code sofar:
[code] dim X as Integer
dim n as Integer = 1000
dim Y as Integer
dim i as integer
dim direction as integer = 1
dim ind integer = 1
x = Round(Canvas1.Width/2)
y = Round(Canvas1.Height/2)
for i = 1 to n
``````ind = ind+1
direction = (direction mod 4 )+ 1
select case direction
case 1
x = x + 1
case 2
y = y - 1
case 3
x = x - 1
case 4
y = y + 1
end select
if IsPrime(i) then
g.Pixel(x,y) = rgb(0,0,0)
end if
``````
next[/code]
IsPrime is a separate standard function and returns true when a number is Prime
The problem I have is the Ind value. It controls the direction of the drawing. I don’t get that running.
The spiral fashion is like
It is called the Ulam spiral when you only show the primes as dots
In the paint event of a canvas
Cocoa?
You’re moving around in a square of 2 x 2 pixels.
I know, but how to solve it?
Something like this?
[code] Dim start As Integer = 10
Dim maximum As Integer = 1000
Self.Maximize()
Dim x As Integer = Me.Width \ 2
Dim y As Integer = Me.Height \ 2
Dim distance As Integer = 40
Dim counter As Integer = 1
Dim direction As Integer = 0
g.DrawString(“stepper”, 10, 20)
g.DrawString(“counter”, 80, 20)
g.DrawString(“direction”, 150, 20)
For stepper As Integer = start To maximum
``````If counter = 0 Then
counter = Sqrt(stepper - start - 1)
direction = (direction + 1) Mod 4
Else
counter = counter - 1
End
g.DrawString(Str(stepper), 10, (stepper - start + 2) * 20)
g.DrawString(Str(counter), 80, (stepper - start + 2) * 20)
g.DrawString(Str(direction), 150, (stepper - start + 2) * 20)
Select Case direction
Case 0
x = x + distance
Case 1
y = y - distance
Case 2
x = x - distance
Case 3
y = y + distance
End
If x > 220 Then
g.DrawString(Str(stepper), x, y)
End
``````
Next[/code]
Very good! Only the self.maximize throws a stack overflow. Never mind.
I modified the code so I can paint pixels instead of numbers:
[code] Dim start As Integer = 1
Dim maximum As Integer = 10000
’ Self.Maximize()
Dim x As Integer = Me.Width \ 2
Dim y As Integer = Me.Height \ 2
Dim distance As Integer = 1
Dim counter As Integer = 1
Dim direction As Integer = 0
For stepper As Integer = start To maximum
``````If counter = 0 Then
counter = Sqrt(stepper - start - 1)
direction = (direction + 1) Mod 4
Else
counter = counter - 1
End
Select Case direction
Case 0
x = x + distance
Case 1
y = y - distance
Case 2
x = x - distance
Case 3
y = y + distance
End
If IsPrime(Stepper) then
g.Pixel(x,y) = rgb(0,0,0)
end if
``````
Next[/code]
The thing is, when I enlarge the maximum integer, It becomes slow because of the IsPrime function. I will try to optimize that.
Thanks very much, you have really put me on track!
[quote=63629:@Alexander van der Linden]Ulam spiral.
S. E. Mousavi[/quote]
AHHHHH Matlab
Yeah this is not quite a straightforward port
Yes!
It draws and calculates the primes up to 200,000 in about 2 secs.
some optimization… first create a Picture object and draw into that cuts time in half…
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CC-MAIN-2024-22
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http://www.research.stlouisfed.org/fred2/series/WTFSRFA?cid=32215
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| 805,410,571
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Factors Supplying Reserve Balances - Total Factors Supplying Reserve Funds
2014-11-19: 4,546,462 Millions of Dollars (+ see more)
Weekly, Ending Wednesday, Not Seasonally Adjusted, WTFSRFA, Updated: 2014-11-20 3:48 PM CST
Click and drag in the plot area or select dates: Select date: 1yr | 5yr | 10yr | Max to
Total factors supplying reserve funds are the sum of "Reserve Bank credit," "gold stock," the "special drawing right certificate account," and "Treasury currency outstanding.
Restore defaults | Save settings | Apply saved settings
w h
Graph Background: Plot Background: Text:
Color:
(a) Factors Supplying Reserve Balances - Total Factors Supplying Reserve Funds, Millions of Dollars, Not Seasonally Adjusted (WTFSRFA)
Total factors supplying reserve funds are the sum of "Reserve Bank credit," "gold stock," the "special drawing right certificate account," and "Treasury currency outstanding.
Factors Supplying Reserve Balances - Total Factors Supplying Reserve Funds
Integer Period Range: to copy to all
Create your own data transformation: [+]
Need help? [+]
Use a formula to modify and combine data series into a single line. For example, invert an exchange rate a by using formula 1/a, or calculate the spread between 2 interest rates a and b by using formula a - b.
Use the assigned data series variables above (e.g. a, b, ...) together with operators {+, -, *, /, ^}, braces {(,)}, and constants {e.g. 2, 1.5} to create your own formula {e.g. 1/a, a-b, (a+b)/2, (a/(a+b+c))*100}. The default formula 'a' displays only the first data series added to this line. You may also add data series to this line before entering a formula.
will be applied to formula result
Create segments for min, max, and average values: [+]
Graph Data
Graph Image
Suggested Citation
``` Board of Governors of the Federal Reserve System (US), Factors Supplying Reserve Balances - Total Factors Supplying Reserve Funds [WTFSRFA], retrieved from FRED, Federal Reserve Bank of St. Louis https://research.stlouisfed.org/fred2/series/WTFSRFA/, November 28, 2014. ```
Retrieving data.
Graph updated.
Recently Viewed Series
Subscribe to our newsletter for updates on published research, data news, and latest econ information.
Name: Email:
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| 872,575,949
| 3,777
|
# Math
posted by .
Place >, <, or = between each of the following pairs to make true sentences
4/5 and 120/150
-6/ and -7/6
-4/20 and 4/-20
• Math -
Find common denominators. -7/6 is an improper fraction for -1 1/6. It doesn't matter whether the negative sign is in the numerator or the denominator.
### Answer This Question
First Name School Subject Your Answer
### Related Questions
More Related Questions
Post a New Question
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CC-MAIN-2017-26
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https://www.automateexcel.com/how-to/sort-by-numbers/
| 1,723,646,473,000,000,000
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| 517,167,258
| 45,185
|
How to Sort by Number in Excel & Google Sheets
Written by
Editorial Team
Reviewed by
Laura Tsitlidze
Last updated on October 29, 2023
This tutorial demonstrates how to sort by number in Excel and Google Sheets.
Sort by Number
Say you have the data shown below, an unsorted list of numbers in Column B. Let’s use this example to show how to sort by number.
To sort the numbers in ascending order, use the sort command in Excel. Select the column to sort, and in the Ribbon, go to Home > Sort & Filter > Sort Smallest to Largest.
Now Column B is sorted, with the numbers in ascending order.
Filter Then Sort by Number
You can also sort numbers by first applying a filter to the column.
1. Click on any cell in the column you want to sort. In the Ribbon, go to Home > Sort & Filter > Filter.
1. Click on the filter button, which has appeared in cell B1, and choose Sort Smallest to Largest.
The result is the same as using the sort command.
Possible Issues While Sorting
Excel’s sorting command works with both alphabetical and numerical data, but sometimes cell content can be misread and not accurately sorted. For example, if any number in your list is stored as text (because of bad formatting, leading spaces, or something else), that value will not be sorted correctly. Say the value in cell B6 has a leading space and you sort by Column B.
As you can see, this cell has a warning that the number is stored as text. Apart from this value, all other cells are sorted correctly.
To resolve this problem, click on the warning icon next to the cell, and choose Convert to Number.
Now, if you sort the data again, the value is sorted with the others.
Sort by Number in Google Sheets
Let’s use the example data from above to demonstrate sorting in Google Sheets. Select the range to sort (B2:B9) and in the Menu, go to Data > Sort range by column B, A → Z.
As in Excel, this sorts Column B from the smallest value to the largest.
Column Options
Google Sheets has various column options. For this example specifically, column options in Google Sheets function like an Excel filter.
Position the mouse on a column letter and click on the arrow when it appears next to the letter. Then click on Sort sheet A → Z.
Again, as in the sections above, you get numbers in Column B sorted in ascending order.
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# Easy Statistics: Linear and Non-Linear Regression
## An easy introduction to Ordinary Least Squares, Logit and Probit regression and tips for regression modelling.
An easy introduction to Ordinary Least Squares, Logit and Probit regression and tips for regression modelling.
### What you'll learn:
• The theory behind linear and non-linear regression analysis.
• To be at ease with regression terminology.
• The assumptions and requirements of Ordinary Least Squares (OLS) regression.
• To comfortably interpret and analyse regression output from Ordinary Least Squares.
• To learn and understand how Logit and Probit models work.
• To learn tips and tricks around Non-Linear Regression analysis.
• Practical examples in Stata
• None
### Description:
Make sure to check out my twitter feed for promo codes and other updates.
Three courses combined. Linear and Non-Linear Regression and Regression Modelling.
Learning and applying new statistical techniques can often be a daunting experience.
“Easy Statistics” is designed to provide you with a compact, and easy to understand, course that focuses on the basic principles of statistical methodology.
This course will focus on the concept of linear regression, non-linear regression and regression modelling. Specifically Ordinary Least Squares, Logit and Probit Regression.
The first two parts will explain what regression is and how linear and non-liner regression works. It will examine how Ordinary Least Squares (OLS) works and how Logit and Probit models work. It will do this without any complicated equations or mathematics. The focus of this course is on application and interpretation of regression. The learning on this course is underpinned by animated graphics that demonstrate particular statistical concepts.
No prior knowledge is necessary and this course is for anyone who needs to engage with quantitative analysis.
The main learning outcomes are:
1. To learn and understand the basic statistical intuition behind Ordinary Least Squares
2. To be at ease with general regression terminology and the assumptions behind Ordinary Least Squares
3. To be able to comfortably interpret and analyze complicated linear regression output from Ordinary Least Squares
4. To learn tips and tricks around linear regression analysis
5. To learn and understand the basic statistical intuition behind non-linear regression
6. To learn and understand how Logit and Probit models work
7. To be able to comfortably interpret and analyze complicated regression output from Logit and Probit regression
8. To learn tips and tricks around non-linear Regression analysis
Specific topics that will be covered are:
• What kinds of regression analysis exist
• Correlation versus causation
• Parametric and non-parametric lines of best fit
• The least squares method
• R-squared
• Beta’s, standard errors
• T-statistics, p-values and confidence intervals
• Best Linear Unbiased Estimator
• The Gauss-Markov assumptions
• Bias versus efficiency
• Homoskedasticity
• Collinearity
• Functional form
• Zero conditional mean
• Regression in logs
• Practical model building
• Understanding regression output
• Presenting regression output
• What kinds of non-linear regression analysis exist
• How does non-linear regression work?
• Why is non-linear regression useful?
• What is Maximum Likelihood?
• The Linear Probability Model
• Logit and Probit regression
• Latent variables
• Marginal effects
• Dummy variables in Logit and Probit regression
• Goodness-of-fit statistics
• Odd-ratios for Logit models
• Practical Logit and Probit model building in Stata
The computer software Stata will be used to demonstrate practical examples.
Regression Modelling
The third part provides useful practical tips for regression modelling.
Understanding how regression analysis works is only half the battle. There are many pitfalls to avoid and tricks to learn when modelling data in a regression setting. Often, it takes years of experience to accumulate these. In these sessions, we will examine some of the most common modelling issues. What is the theory behind them, what do they do and how can we deal with them? Each topic has a practical demonstration in Stata. Themes include:
• Fundamental of Regression Modelling – What is the Philosophy?
• Functional Form – How to Model Non-Linear Relationships in a Linear Regression
• Interaction Effects – How to Use and Interpret Interaction Effects
• Using Time – Exploring Dynamics Relationships with Time Information
• Categorical Explanatory Variables – How to Code, Use and Interpret them
• Dealing with Multicollinearity – Excluding and Transforming Collinear Variables
• Dealing with Missing Data – How to See the Unseen
### Who this course is for:
• Academic students of any level.
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• Business users and managers who engage with quantitative reports.
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• Anyone who has an interest in, or needs to engage, with statistical regression.
### Course Details:
• 5 hours on-demand video
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` Find the direction in which plane should be headed to reach B`
11 months ago
Arun
22559 Points
``` Dear Abhishek Wind travels at a Speed =u=20m/sAeroplane travels at a Speed =v=150m/sTo reach the point B, The pilot should travel along AB.Let us assume that θ be the angle at which the plane heads towards east of line AB.so angle between two velocities is θ+30°tanθ°=usin(θ+30)/(v+ucos(θ+30))sinθ°/cosθ°=usin(θ+30)/(v+ucos(θ+30))By cross multiplying we getsinθ°(v+ucos(θ+30))=cosθusin(θ+30)sinθ° [v+ucosθcos30° - usinθsin30)]=ucosθ°sinθ°cos30°+ucosθ°cosθ°sin30Since Cos(A+B)=cosAcosB-SinASinB]Sin(A+B)=sinACosB+cosAsin BVsinθ°+ucosθ°sinθ°√3/2-usin²θ1/2=ucosθ°sinθ°√3/2+ucos²θ°1/2on solving the above equation,we getvsinθ°-u/2sin²θ=u/2cos²θVsinθ°=(u/2)[cos²θ+sin²θ]V sinθ°=u/2 [ ∵cos²θ+sin²θ=1]sinθ=u/2v=20/2x150=20/300=1/15θ=sin⁻¹[1/15]=sin⁻¹[0.066]∴ in order to reach point B, the plane should move with an angle of sin⁻¹[1/15] east of line AB.
```
11 months ago
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Search a number
33242032 = 242398693
BaseRepresentation
bin111111011001…
…1101110110000
32022112212111101
41332303232300
532002221112
63144254144
7552360355
oct176635660
968485441
1033242032
111784525a
12b171354
136b6b855
1445b462c
152db9757
hex1fb3bb0
33242032 has 20 divisors (see below), whose sum is σ = 64683360. Its totient is φ = 16549568.
The previous prime is 33242009. The next prime is 33242047. The reversal of 33242032 is 23024233.
It is a super-2 number, since 2×332420322 = 2210065382978048, which contains 22 as substring.
It is an unprimeable number.
It is a polite number, since it can be written in 3 ways as a sum of consecutive naturals, for example, 523 + ... + 8170.
It is an arithmetic number, because the mean of its divisors is an integer number (3234168).
Almost surely, 233242032 is an apocalyptic number.
It is an amenable number.
33242032 is a deficient number, since it is larger than the sum of its proper divisors (31441328).
33242032 is a wasteful number, since it uses less digits than its factorization.
33242032 is an evil number, because the sum of its binary digits is even.
The sum of its prime factors is 8940 (or 8934 counting only the distinct ones).
The product of its (nonzero) digits is 864, while the sum is 19.
The square root of 33242032 is about 5765.5903427143. The cubic root of 33242032 is about 321.5356912906.
Adding to 33242032 its reverse (23024233), we get a palindrome (56266265).
It can be divided in two parts, 3324 and 2032, that added together give a triangular number (5356 = T103).
The spelling of 33242032 in words is "thirty-three million, two hundred forty-two thousand, thirty-two".
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# ARTIFICIAL INTELLIGENCE 6 Fuzzy Logic
## Presentasi berjudul: "ARTIFICIAL INTELLIGENCE 6 Fuzzy Logic"— Transcript presentasi:
ARTIFICIAL INTELLIGENCE 6 Fuzzy Logic
Kecerdasan Buatan ARTIFICIAL INTELLIGENCE 6 Fuzzy Logic Sumber: Slide Kluliah Prof. Handayani Tjandrasa ITS Handayani Tjandrasa
DEFINISI "If it is sunny and warm today, I will drive fast"
Fuzzy logic: Cara untuk menyatakan variasi atau ketidaktepatan (tidak mutlak) dalam logika Misal dalam bahasa alami: "If it is sunny and warm today, I will drive fast" Linguistic variables: Temp: {freezing, cool, warm, hot} Cloud Cover: {overcast, partly cloudy, sunny} Speed: {slow, fast}
True/False Variables King(Richard) Greedy(Richard) Evil(Richard)
“Richard is greedy” is True or False: Greedy(Richard) {0,1}
Fuzzy Sets Variabelnya dapat mempunyai nilai dalam batasan [0,1]
Misal: Greedy(Richard) = 0.7 Fuzzy Linguistic Variables: digunakan untuk menyatakan nilai yang bervariasi dalam suatu spektrum Misal: Temp: {Freezing, Cool, Warm, Hot} Membership Function: Fungsi keanggotaan untuk menyatakan derajat kebenaran Contoh: “How warm is it?”
Membership Functions Temp: {Freezing, Cool, Warm, Hot}
Derajat kebenaran atau keanggotaan (membership)
Membership Functions Seberapa dingin 36 F° ? 30% Cool dan 70% Freezing
Fuzzy Disjunction Fuzzy Conjunction
AB max(A, B) (AB = C) (C = 0.75) Fuzzy Conjunction AB min(A, B) (AB = C) (C = 0.375)
Contoh A B A B A B A
Struktur Kontroler Fuzzification Inference Mechanism Defuzzification
Petakan variabel input dalam nilai fuzzy Inference Mechanism Pendekatan reasoning dan deduksi aksi kontrol Defuzzification Konversi nilai fuzzy output ke aksi kontrol
Contoh Input: Temp & Cover Output: Speed
Temp: {Freezing, Cool, Warm, Hot} Cover: {Sunny, Partly, Overcast} Speed: {Slow, Fast}
Rules If it's Sunny and Warm, drive Fast
Kecerdasan Buatan Rules If it's Sunny and Warm, drive Fast Sunny(Cover)Warm(Temp) Fast(Speed) If it's Cloudy and Cool, drive Slow Cloudy(Cover)Cool(Temp) Slow(Speed) Handayani Tjandrasa
Example Speed Calculation
How fast will I go if it is 65 F° 25 % Cloud Cover ?
Fuzzification: Berapa besar kecepatan bila temp
Fuzzification: Berapa besar kecepatan bila temp. 65 F° dan 25 % tertutup awan? 65 F° Cool = 0.25, Warm= 0.75 25% Cover Sunny = 0.8, Cloudy = 0.2
If it's Sunny and Warm, drive Fast Sunny(Cover)Warm(Temp)Fast(Speed)
0.8 0.7 = 0.7 Fast = 0.7 If it's Cloudy and Cool, drive Slow Cloudy(Cover)Cool(Temp)Slow(Speed) 0.2 0.4 = 0.2 Slow = 0.2 Speed = weighted mean = (2*25+7*75)/(9) = 63.8 mph
Fuzzy Linguistic Variables
Fan Speed Set stop {0, 0, 0} Set slow {50, 30, 10} Set medium {60, 50, 40} Set fast {90, 70, 50} Set blast {, 100, 80} Air Temperature Set cold {50, 0, 0} Set cool {65, 55, 45} Set just right {70, 65, 60} Set warm {85, 75, 65} Set hot {, 90, 80}
Rules Air Conditioning Controller: IF Cold then Stop If Cool then Slow
If OK then Medium If Warm then Fast IF Hot then Blast
Fuzzy Air Conditioner
Kecerdasan Buatan TERIMA KASIH Handayani Tjandrasa
Presentasi serupa
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This came to my attention on Chess Stack Exchange.
Geir, I'm sorry to say this, but I found out today that your 32 ply position is BUSTED: 10. Kf2 works as an alternative to 10. Kg2.
However, Stockfish analysis confirms that your 25 ply position is sound, and thus still the record.
(Read Only) pid=20430
White’s 9th move is Kxh1, so I am pretty sure 10.Kf2 is illegal. Otherwise I would agree that 10.Kf2 is a cook.
(Read Only) pid=20442
Oh, my oh my! You're right, so it isn't busted. I accidentally swapped the king and the knight. My bad Geir!
(Read Only) pid=20443
I have found a way to make 20, 21 or 25 plies without captures. The number of plies depends on how you treat threefold repetitions.
(= 9+7 )
1.h7 Rd8 2.Bf7 Rh8 3.Bg8 Kb8 4.Kb1 Kc7 5.Kc1 Kd6 6.Kd1 Ke5 7.Ke1 Kf4 8.Kf1 Kg3 9.Kg1 Kh3 10.Kh1 Kg3 11.Kg1 Kh3 12.Kh1 Kg3 13.Kg1 draw.
The reason this is all forced is that White wins if he gets to play Kg2, while Black wins if he gets to play Kh2. With Kg2 vs Kg4, White plays a5 to put Black in zugzwang, but with Kf1 vs Kh2, the pawn move only serves to delay the zugzwang by one move.
The question now is how long this line is to be considered. If you count the moves before any position is repeated, that is 20 plies (10...Kg3). If you include the move that makes the first repeated position, that is 21 plies (11.Kg1). If you count the moves to the threefold repetition, that is 25 plies (13.Kg1).
(Read Only) pid=20476
My suggestion is that due to the theme, you should count
to the first unforced move of one side. And that would be the
automatic draw, for fuzzy values of "move" and "forced" :-)
(Read Only) pid=20478
Hope I’m not making a dumb error here, but it seems to me that in the 3Rep study, White is safe as soon as he plays 9. Kg1. At that point he can freely advance Pa4 without risking loss. So the sequence of Nunn single moves ends at 18. That pawn still served a purpose in that it stopped Black dawdling on the 4th rank.
(Read Only) pid=20479
9. Kg1 Kh3 10. a5 Kg3 11. Kf1 Kh2 B+ or 11. Kh1 Kf2 B+.
(Read Only) pid=20480
Thanks, Joost. Sorry I was remembering the board wrong when I was cycling. I will have another look maybe later today when I'm freer.
(Read Only) pid=20487
(49) Posted by Geir Sune Tallaksen Østmoe [Wednesday, Jan 20, 2021 22:46]
I just realized I can add two plies like this:
(= 9+7 )
1.Bh5 Rd1 2.h7 Rd8 etc.
(Read Only) pid=20499
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## Intermediate Algebra for College Students (7th Edition)
$16 - 30i$
RECALL: (1) $(a-b)^2 = a^2 - 2ab + b^2$ (2) $i^2=-1$ Use rule (1) above with $a=5$ and $b=3i$ to obtain: $=5^2 - 2(5)(3i) + (3i)^2 \\=25 - 30i + 9i^2$ Use rule (2) above to obtain: $=25 - 30i + 9(-1) \\=25 - 30i+(-9) \\=16 - 30i$
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