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$1)$ Implies that $5(p-1)= 3q$, which gives $q= \frac{5(p-1)}{3}$. Then you can insert $q= \frac{5(p-1)}{3}$ in $(2)$. Then find what is $p$, and then you can find what $q$ is.
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$$\frac{p-1}{q} = \frac{q-1}{2p+1} = \frac {3}{5}$$ so $$\frac{p-1}{q}=\frac {3}{5}$$ and $$\frac {3}{5}=\frac{q-1}{2p+1}.$$ \begin{eqnarray} 5p-5&=3q&\\ 6p+3&=&5q-5 \end{eqnarray} $$5p-3q=5$$ $$6p-5q=-8$$ $$30p-18q=30$$ $$30p-25q=-40$$ $$-7q=-70$$ $$q=10$$ $$p=7.$$
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$p = 7$? :) ${}{}$ – Alex Nov 1 '12 at 19:37
is there a problem ? – Iuli Nov 1 '12 at 19:47
From $$\dfrac{p-1}{q} = \dfrac{q-1}{2p+1} = \dfrac {3}{5}$$ follow the system of linear equations with two unknowns
$$5(p-1)=3q$$ $$5(q-1)=3(2p+1)$$ or $$5p-3q=5$$ $$6p-5q=-8$$
wich can be solved using for example Cramer rule
$\Delta=-25+18=-7$
$\Delta_p=-25-24=-49$
$\Delta_q=-40-30=-70$
$$p=\frac{\Delta_p}{\Delta}=\frac{-49}{-7}=7$$,$$q=\frac{\Delta_q}{\Delta}=\frac{-70}{-7}=10$$
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# Getting an X for Chinese Remainder Theorem (CRT)
how do I get modulo equations to satisfy a given X in CRT.
For example say I have X = 1234. I choose mi as 5, 7, 11, 13. This satisfies the simple requirements of Mignotte's threshold secret sharing scheme. More precisely given in my example k = n = 4, and the product of any k - 1 is smaller then X how come simply computing the remainder of each won't give equations that solve to X = 1234.
In the case of the example,
x = 4 mod 5
x = 2 mod 7
x = 2 mod 11
x = 12 mod 13
Which resolves to 31264 (won't CRT produce the smallest?)
Any hints?
The final result of the CRT calculation must be reduced modulo 5 x 7 x 11 x 13 = 5005. This gives the correct answer.
Here is a much simpler way to immediately obtain the sought answer. Contrast the solution below to the much longer solution in your link, which involves calculations with much larger numbers and performs $4$ inversions vs. the single simple inversion below. Always search for hidden innate structure in a problem before diving head-first into brute-force mechanical calculations!
The key insight is: the congruences split into pairs with obvious constant solutions by CCRT, viz.
\begin{align}\rm\quad\quad\quad\quad\quad x\equiv \ \ \ 2\ \ \:(mod\ 7),\ \ x\equiv \ \ \ 2\ \ \:(mod\ 11)\ \iff\ x\equiv \ \ \ \color{#0a0}2\ \ (mod\ \color{#0a0}{77})\\[0.3em] \rm\quad\quad\quad\quad\quad x\equiv -1\ \ (mod\ 5),\,\ \ x\equiv\ {-}1\ \ (mod\ 13)\ \iff\ x\equiv \color{#c00}{-1}\ \ (mod\ \color{#c00}{65})\end{align}
So we reduced the above four original LHS equations to the above two RHS equations, which are easy to solve by CRT = Chinese Remainder Theorem. Indeed, applying Easy CRT below
$\rm\quad\quad\quad\quad\quad x\equiv\ \color{#0a0}{2 + 77}\ \bigg[\displaystyle\frac{\color{#c00}{-1}-\color{#0a0}2}{\color{#0a0}{77}}\ mod\,\, \color{#c00}{65}\bigg]\,\ \ (mod\ 77\cdot65)$ | {
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In the brackets $\,\rm\displaystyle\left[\, mod\ \ 65\!:\ \ \frac{-3}{77} \equiv \frac{-3}{12} \equiv \frac{-1}4 \equiv \frac{64}4 \equiv \color{#d0f}{16}\,\right]\quad$ (see Beware below)
This yields $\rm\ \ x\ \equiv\ \color{#0a0}{2 + 77}\,[\,\color{#d0f}{16}\,] \equiv 1234\,\ \ (mod\ 77\cdot 65)\quad$ QED
Theorem $\:$ (Easy CRT) $\rm\ \$ If $\rm\ m,\:n\:$ are coprime integers then $\rm\ m^{-1}\$ exists $\rm\ (mod\ n)\ \$ and
$\rm\displaystyle\qquad\quad\quad\quad \begin{eqnarray}\rm x&\equiv&\!\rm\ a\ \ (mod\ m) \\ \rm x&\equiv&\!\rm\ b\ \ (mod\ n)\end{eqnarray} \ \iff\ \ x \equiv\, a + m\ \bigg[\frac{b-a}{m}\ mod\ n\,\bigg]\,\ \ (mod\ m\:n)$
Proof $\rm\ (\Leftarrow)\ \ \ mod\ m:\,\ x \equiv a + m\ [\,\cdots\,] \equiv a,\$ and $\rm\ mod\ n\!\!:\,\ x \equiv a + (b\!-\!a)\ m/m \equiv b$
$\rm (\Rightarrow)\ \$ The solution is unique $\rm\ (mod\,\ mn)\$ since if $\rm\ x',\:x\$ are solutions then $\rm\ x'\equiv x\$ mod $\rm\:m,n\:$ therefore $\rm\ m,n\ |\ x'-x\ \Rightarrow\ mn\ |\ x'-x\ \$ since $\rm\ \:m,n\:$ coprime $\rm\:\Rightarrow\ lcm(m,n) = mn\ \ \$ QED
Note $\$ Easy CRT is not only easy to apply, but also very easy to remember. Namely note $\rm\ x\equiv a\pmod{\! m}\iff x = a + m\,k,\:$ for some integer $\rm\:k,\,$ This further satisfies the second congruence iff $\rm\ mod\ n\!:\ x = a + m\,k\equiv b$ $\iff$ $\rm k\:\equiv (b-a)/m,\$ hence the Easy CRT formula. This explains the $(\Leftarrow)$ proof: fill in the dots in $\rm\:x\equiv a + m\ [\,\cdots\,]\:$ to make $\rm\,x\equiv b\pmod{\! n}$
Beware $\$ Modular fraction arithmetic is well-defined only for fractions with denominator coprime to the modulus. See here for further discussion.
Below is the solution you linked to on "Math Celebrity" (cached to avoid link rot). | {
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Below is the solution you linked to on "Math Celebrity" (cached to avoid link rot).
• @Bill Dubuque: But the OP was just giving an example! Your solution is completely useless for any other, similar problem. Unless you can automate it, of course :-) – TonyK Feb 3 '11 at 17:27
• @TonyK: Most certainly not true. Most math problems do have interesting structure, esp. problems that are designed for tests, competitions etc. In fact this is frequently true even in research problems. Indeed, speaking as a number theorist, I can tell you that methods like the above are very useful in practice. In mathematics, intuition always trumps brute force. – Bill Dubuque Feb 3 '11 at 17:34
• @Bill Dubuque: Well, your methods might be much, much simpler for you. But if they require years of practice, they're not so useful for people like me (or, presumably, the OP). – TonyK Feb 3 '11 at 18:05
• @Tonyk: The above solution can be understood by a bright high-school student. Simple optimizations like the above arise frequently in elementary number theoretical problems, so it is well-worth knowing them. – Bill Dubuque Feb 3 '11 at 18:54
• @TonyK: The optimization I employed above is algorithmic and is frequently applicable in practice. But that was not my point in presenting the above. Rather, it was to emphasize conceptual vs. algorithmic thought - intuition vs. brute force. Mathematical problems are far from random. Typically they involve much structure and their solution requires insightful exploitation of such innate structure - something that cannot be algorithmic (indeed many math problems are algorithmically unsolvable - that's what makes them interesting). – Bill Dubuque Feb 4 '11 at 0:37 | {
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# Weighted Mean | {
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Variance is also based on the mean of your data. 0 (2016-2017), but may apply to other versions as well. This is a correct assumption if the same technique is used to measure the same parameter repeatedly. , Features, Capabilities, and Epics) to produce maximum economic benefit. Here is the mean of 1, 2, 3 and 4: Add up the numbers, divide by how many numbers: Mean = 1 + 2 + 3 + 44 = 104 = 2. weighted mean shift free download. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Abstract It is well known that the out-of-sample performance of Markowitz’s mean-variance portfolio criterion can be negatively affected by estima- tion errors in the mean and covariance. using excel to calculate the weighted standard deviation Does anyone know the formula for computing the weighted standard deviation? I was able to calculate the weighted average (16. You simply add up all the item values and divide by the total. This calculator will calculate the weighted average APR for all of your credit cards that have a current balance. Basically I calculate the most recent RSI value and it's exponentially weighted mean based on the window length selected. Water-Quality Characteristics. I am trying to calculate the mean profit percentage, but a free item is given when a certain amount of goods are bought. geometric mean concentration at which shellfish beds or swimming beaches must be closed. Weighted means are often used for frequency data. Weighted blankets, sometimes referred to as gravity blankets, were once a tool of therapists and psychiatry clinics. Purpose: To report design of a simplified external transmit-receive coil array for 7 Tesla (T) prostate MRI, including demonstration of the array for tumor localization using T2-weighted imaging (T2WI) at 7T before prostatectomy. Adding all of these totals up yields a weighted average of 5. | {
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(T2WI) at 7T before prostatectomy. Adding all of these totals up yields a weighted average of 5. Ranking, Matrix/Rating Scale, Multiple Choice, Multiple Textboxes, and Slider questions calculate an average or weighted average. How to calculate weighted mean in IBM SPSS? Using the results of the survey I need to create the rating of governmental organizations. A method of computing a kind of arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others. org Dictionary. Acronym/Abbreviation meaning of CAC The acronym CAC() means : Capital Access Corporation. test – here they are (the weighted variance is from Gavin Simpson, found on the R malining list): View Code RSPLUS. Compute the weighted mean of a variable. Suppose your teacher says, "The test counts twice as much as the quiz and the final exam counts three times as much as the quiz". Dec 10, 2010 · The IRR, also commonly referred to as the dollar weighted return, is the measurement of a portfolio’s actual performance between two dates, including the effects from all cash inflows and outflows. Recommended Articles. … If all the weights are equal, then the weighted mean is the same as the arithmetic mean. Finally, the query uses a GROUP BY clause to combine the data so that the calculation is performed for each gender. WMC is sampling-, scale-, and contrast-invariant, and is sparse on natural images. A weighted average, for example, takes into account the proportional. The AVERAGE function below calculates the normal average of three scores. Weighted Vest FAQs Weighted vests are often recommended for children with autism. 04 (df = 3, p <. It is based on kernel weighted sample statistics such as the mean (Nadaraya-Watson estimator) but also standard deviation, skewness, kurtosis, deciles, etc. Follow the example below to calculate the weighted average interest rate for a federal loan consolidation. Find the weighted average of class grades (with equal weight) | {
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rate for a federal loan consolidation. Find the weighted average of class grades (with equal weight) 70,70,80,80,80,90: Since the weight of all grades are equal, we can calculate these grades with simple average or we can cound how many times each grade apear and use weighted average. mean, or in other words searching the explicit expression of the weighted mean standard deviation distribution, under the safely motivated restriction of independent measures obeying Gaussian distributions. also i thought weighted mean measures the tendency of respondents answers to various questions. It is an average in which each quantity to be averaged is assigned a. Under Medicare Advantage, the federal government still foots the. weighted dataset, others simply recommend replacing the weighted base sizes with the unweighted base sizes in the test statistic formula, while still using the weighted summary statistics such as mean and standard deviation. Most companies weight the revenue based on the opportunity success %. The choice of Galloway et al. This is because the basic average of a group of numbers is the same calculation as a weighted average except that the weights of all the numbers are calculated as being the same. In calculating a weighted average, each number in the data set is. Weighted Mean (ratio statistics algorithms) A / S = n Σ i = 1 f i A i n Σ i = 1 f i S i = n Σ i = 1 f i S i R i n Σ i = 1 f i S i This is the weighted mean of the ratios weighted by the sales prices in addition to the usual case weights. Free Arithmetic Mean (Average) Calculator - find the average of a data set step-by-step. Another Tip: If number of boys and girls is same, i. The first or shorter echo (TE < 30msec) is proton density (PD) weighted or a mixture of T1 and T2. then the variance of the mean could be estimated with. The idea of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. Given N jobs where every | {
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also occurs in a more general form in several other areas of mathematics. Given N jobs where every job is represented by following three elements of it. weighted-mean definition: Noun (plural weighted means) 1. Define weighted. … So let's set up the ability to do a weighted mean … versus a mean. To calculate a weighted average in Excel, simply use the SUMPRODUCT and the SUM function. In this article, I'll go through a basic description of what a weighted GPA is, why it matters for you, and how you can calculate your own weighted GPA if your school uses this type of scale. a point lying far from the rest). The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from samples of the same population with different sample sizes: ¯ = ∑ = ∑ =. The WAM can be used to gain an average of an applicant results using their grades and unit credit value. Get Your Weighted And Unweighted GPA In Just A Few Minutes! This quick and easy online GPA calculator computes both weighted and unweighted high school grade point averages (GPA). However, Property Use Details can change over the course of the 12 months, so we time-weight them so they are appropriately attributed. It is challenging to achieve high weighted efficiency with low-power microinverters, typically because these devices are required to be low cost. Variance[wd] / n where n is sample size. Weighted arithmetic mean is used when the values are of different importance - weight p to be assigned to each value. More periods with zero is less scale sensitive than the MAPE and the MAD. Specify the plan goal weights for each of the three goal levels (the total must be 100 percent). POSIXct, colMeans for row and column means. Definition of weighted mean in the Definitions. The tutorial is mainly based on the weighted. Get Your Weighted And Unweighted GPA In Just A Few Minutes! This quick and easy online GPA calculator computes both weighted and unweighted high school grade point averages | {
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easy online GPA calculator computes both weighted and unweighted high school grade point averages (GPA). It often occurs, however, that one must combine two or more measurements of the same quantity with differing errors. Title: Variance of a Weighted Mean Created Date: 20160808202903Z. Use 'weighted average' in a Sentence. A combination of medication, lifestyle changes and other treatments can help limit the impact of anxiety and its symptoms. µ = P n Pi=1 w ix i n i=1 w i (45) It is equivalent to the simple mean when all the weights are equal, since µ = P n P i=1 wx i n i=1 w = w P n x i nw = 1 n X i=1 x i (46) If the samples are all different, then weights can be thought of as sample frequencies, or they can be used to calculate probabilities where p i = w i/ P w i. mean() while aggregating a data frame. …Let's consider an academic course as our example. Dec 30, 2016 · Weighted Average is a free application for students. 04 (df = 3, p <. As a result, practitioners usually rely on vendor specified parameters, such as mean-time-to-failure (MTTF), to model failure processes, although many are skeptical of the accuracy of those models [4,5,33]. 5 for Stock C, or 3. The first section which takes up columns A, B, and C is the weighted averages. Select a stem that prompts several possible responses, with one that is the most accurate. For example, say 239 students in Economics 201. Jul 24, 2014 · Functional diversity can be quantified using one single trait at a time or multiple traits (see the descriptions of multiple-indices in other page). The weighted mean is similar to an arithmetic mean (the most common type of average), where instead of each of the data points contributing equally to the final average, some data points contribute more than others. A weighted grade or score is average of a set of grades, where each grade (g) carries a different weight (w) of importance. an object containing the values whose weighted mean is to be computed. w: a numerical vector of | {
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an object containing the values whose weighted mean is to be computed. w: a numerical vector of weights the same length as x giving the weights to use for elements of x arguments to be passed to or from methods. Examples of how to use “weighted mean” in a sentence from the Cambridge Dictionary Labs. Instead of each data point contributing equally to the final mean, some data points contribute more "weight" than others. In simple terms, the category "total" will be equal to the sum of the scores in each grade item each multiplied by its grade weight, and that sum being finally divided by the sum of all weights. On the face of it, this does look as nice. RE: Geometric and weighted mean Perhaps it is worth pointing out that *an* average is a relatively arbitrary way of "summarising" a bunch of numbers into a single number. Sep 08, 2015 · How do I change my gradebook aggregation to ‘Weighted Mean of Grades’? By default, the Moodle gradebook aggregation (the way it adds your grade values) is ‘Simple Weighted Mean of Grades. When an assignment is “weighted”, it means there is a grade for that assignment. Catetan yén lamun kabéh beuratna sarua, weighted méan sarua jeung arithmetic mean. For example, if the chest is the target area, we might perform a set of barbell bench press, followed by a set of barbell incline press, and finished with a set of weighted parallel dips. A flow-weighted mean is the mean of a quantity after it is weighted proportional to a corresponding flow rate. The fact that x only gives one radiation field is the same as giving 0 radiation for the second field and so on, which means I do the following:. The mean of each area will be obtained using the formula: x = Σx/N (Downie and Heat, 1970) The numerical findings of the study will be statistically analysed and interpreted using the frequency count. rm: a logical value indicating whether na values in x should be stripped before the computation proceeds. For example, in the county data set, population | {
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should be stripped before the computation proceeds. For example, in the county data set, population is a natural weighting. Most content comes from the ECPR Winter School in Methods and Techniques R course, that I had the pleasure of teaching this February. A similar procedure is used for a 10% trimmed mean. Select Weighted Goals as the plan goals type. Weighted Mean calculator for calculating the weighted mean statistics for the given set of data. DESCRIPTION. The F1 score can be interpreted as a weighted average of the precision and recall, where an F1 score reaches its best value at 1 and worst score at 0. A weighted grade or score is average of a set of grades, where each grade (g) carries a different weight (w) of importance. Unlike most credit card interest calculators, this calculator will calculate the current finance charge for each card, and then compute the credit card average APR using a weighted formula. This paper proposes a novel on-line portfolio selection strategy named “Confidence Weighted Mean Reversion ” (CWMR). There's one more skill you'll need to calculate weighted scores: A simple average, which in "math speak" is more properly called the mean. 3% then you are essentially implying a growth rate of POSITIVE 3. Find more details on making DIY. • So it is better to use WAPE for volume weighted MAPE and WMAPE for dollar weighted or Cost-weighted measures. What is a weighted KPI calculations For weighted KPI calculations, the base weight of each KPI is factored against the weighted counts of each KPI by using the following formulas. For this reason, consider using Mean Absolute Deviation (MAD) alongside MAPE, or Consider that even fast moving consumer goods companies these days to average MAPEs over multiple time series. (the "Gold Book"). Featured Services & News. Weighted Mean Formula (Table of Contents). Wondering what a weighted blanket is and what is the purpose of having one? Learn how weighted blankets can benefit sleep issues, anxiety, | {
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what is the purpose of having one? Learn how weighted blankets can benefit sleep issues, anxiety, autism, ADHD, insomnia and more. These two measurements can be combined to give a weighted average. Follow this example use the following data set. The sample size weighted correlation may be used in correlating aggregated data Description. the weighted mean of is given by (4) Weighted means have many applications in physics, including finding the center of mass and moments of inertia of an object with a known density distribution and computing and electric and magnetic multipole moments of charge and current distributions, respectively. For instance if you have 3 objects, A,B and C. w: a numerical vector of weights the same length as x giving the weights to use for elements of x. Sincerely, Jacob. Weighted Shortest Job First (WSJF) is a prioritization model used to sequence jobs (eg. 1 Introduction The Least Mean Square (LMS) algorithm, introduced by Widrow and Hoff in 1959 [12] is an adaptive algorithm, which uses a gradient-based method of steepest decent [10]. The WACC is also the minimum average rate of return it must earn on its current assets to satisfy its shareholders, investors, or creditors. I have been working on a problem for the last year (in graph theory/algorithms). The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E), in which price is in the numerator. % % Example: % X = rand(3,5);. He has an entire dice set that almost always gets the best possible roll. Weighted average refers to the mathematical practice of adjusting the components of an average to reflect the importance of certain characteristics. Definition of weighted in the Definitions. What is the Weighted Average? The weighted average scoring model applies a "weight" to the matrix questions based on responses to the first item in a side-by-side matrix. wi(yi 0 1xi) 2. The weighted mean is similar to the arithmetic mean where | {
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in a side-by-side matrix. wi(yi 0 1xi) 2. The weighted mean is similar to the arithmetic mean where instead of each of the data points contributing equally to the final average, some data points contribute more than others. Example sentences with "weighted mean", translation memory add example en The conclusion was that the only items differentiating the SIDS infants from the control infants were a "lower mean weight and height at birth, the previous occurrence of cyanosis [bluish skin and mucous membranes caused by lack of oxygen in the blood] or pallor during. Activity recording is turned off. 1 Introduction The Least Mean Square (LMS) algorithm, introduced by Widrow and Hoff in 1959 [12] is an adaptive algorithm, which uses a gradient-based method of steepest decent [10]. Often used to account for area changes between meridians at varying latitudes by using the cosine of the latitude as the weights. Besides the stock market example given above, another situation were weighed average is used is in grading. mean fitness of the population W(bar) The sum of the fitnesses of the genotypes of a population weighted by their proportions; hence a weighted mean fitness. Specifically, overtime is normally calculated at 1. Each grade item can be given a weight to change its importance in the overall mean. Weighted Shortest Job First (WSJF) is a prioritization model used to sequence jobs (eg. and Wilks, A. Weighted Mean: A mean where some values contribute more than others. Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. The weight of the category should be entered as a decimal, make sure that the sum of the weights equal 1. In simple terms, the category "total" will be equal to the sum of the scores in each grade item each multiplied by its grade weight, and that sum being finally divided by the sum of all weights. This | {
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multiplied by its grade weight, and that sum being finally divided by the sum of all weights. This calculator will calculate the weighted average APR for all of your credit cards that have a current balance. What does weighted mean mean? Information and translations of weighted mean in the most comprehensive dictionary definitions resource on the web. All or only selected records: You can either use all the points in the point theme for the analysis or only a selected subset of points. The weights cannot be negative. So let's have a look at the basic R syntax and the definition of the weighted. It is calculated according to the following formula: M = mark received in a course U = units of credit for a course. This article will show you how to use Excel's SUMPRODUCT and SUM functions individually and how to combine the two to calculate a weighted average. Formula: Where x is the repeating value. 0015948968 4 2 5442. Field {Case_Field} Field used to group features for separate mean center calculations. 15 * (73 + 80 + 85 + 88) +. …But sometimes, some data points…are more important than others. Trump talks Medicare in a retirement enclave where doctors are a golf-cart ride away. and Wilks, A. weighted definition: 1. Jun 07, 2016 · 61. Mar 18, 2017 · 2. A method of computing a kind of arithmetic mean of a set of numbers in which some elements of the set carry more importance (weight) than others. Paper 3: Issues in calculating average effect sizes in meta-analyses Dr. Metascore is a weighted average in that we assign more importance, or weight, to some critics and publications than others, based on their quality and overall stature. Slugging average is sometimes referred to as slugging percentage. But an equally weighted mean would be the pooled mewn and it expected value would be [65+73]/2. Section 2 formally formulates the on-. Under Medicare Advantage, the federal government still foots the. A weighted GPA is calculated by awarding additional points to classes that | {
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still foots the. A weighted GPA is calculated by awarding additional points to classes that are considered more challenging than the basic curriculum. Free Arithmetic Mean (Average) Calculator - find the average of a data set step-by-step. The Benefits of a Weighted Sales Pipeline. 739 × 10-3 mm 2 /sec, SD = 0. Water-Quality Characteristics. Parameters: axis: {index (0), columns (1)} Axis for the function to be applied on. …If the professor had four exams,…and these were your four exam scores,…calculating your average would be easy, 80%. May 11, 2017 · The weighted mean is the weighted average. On the other hand, its weighted version is very useful for evaluating inequalities for definite integrals. The time-weighted formula is essentially a geometric mean of a number of holding-period returns that are linked together or compounded over time (thus, time-weighted). Among them, M55 showed a gene expression pattern consistent with behavioral changes after stress exposure, and the gene ontology analysis revealed that this was involved in nervous system development, gland. Data are classified using the Harmonized System of trade at the six- or eight-digit level. Sleep through the night and relieve anxiety with high-quality weighted blankets for adults from Weighting Comforts. Jul 24, 2014 · Functional diversity can be quantified using one single trait at a time or multiple traits (see the descriptions of multiple-indices in other page). 4) but have been unsuccessful in computing the weighted SD. Aug 29, 2013 · If such an effect is possible then the simplest and safest thing to do is compute the weighted sum for the m values in each of the n experiments. References. To calculate a weighted average in Excel, simply use the SUMPRODUCT and the SUM function. E(Z1)=65 and E(Z2)=73 The population mean is [2(65)+73]/3. It is challenging to achieve high weighted efficiency with low-power microinverters, typically because these devices are required to be low cost. How to Calculate | {
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microinverters, typically because these devices are required to be low cost. How to Calculate Weighted Average Price Per Share Calculating your weighted average price per share can help you assess the performance of an investment that was made in several transactions. …Let's consider an academic course as our example. What is the uncertainty of the weighted average? What's the correct procedure to find the uncertainty of the average?. Weighted average definition, a mean that is computed with extra weight given to one or more elements of the sample. Another Tip: If number of boys and girls is same, i. May 01, 2012 · PowerPivot Weighted Average Measure Compared to Non-Weighted Average. The weighted average (or weighted mean, as statisticians like to call it) is easy to compute in SAS by using either PROC MEANS or PROC UNIVARIATE. If trim is non-zero, a symmetrically trimmed mean is computed with a fraction of trim observations deleted from each end before the mean is computed. - joran Jun 12 '12 at 4:35 3 @Frank Hover over the down triangle beneath the vote count next to your Q. If I use the following data. Math, I am in the 9th grade, and our math teacher is explaining "weighted averaging. title = "Amplitude-weighted mean velocity: Clinical utilization for quantitation of mitral regurgitation", abstract = "Objectives. Weighted blankets, sometimes referred to as gravity blankets, were once a tool of therapists and psychiatry clinics. The weighted mean is used a lot by teachers. How to use weighted in a sentence. net dictionary. It is useful in many situations e. Flow-weighted mean concentration. If you're seeing this message, it means we're having trouble loading external resources on our website. Myometrial Invasion in Endometrial Cancer: Diagnostic Accuracy of Diffusion-weighted 3. Weighted arithmetic mean. when they tried on their own, funded by WW. He has an entire dice set that almost always gets the best possible roll. In this lesson from Alanis Business | {
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entire dice set that almost always gets the best possible roll. In this lesson from Alanis Business Academy, we describe the weighted mean, review the equation to solve for the weighted mean, and actually solve for a weighted mean. If all the weights are equal, then the weighted mean equals the arithmetic mean (the regular "average" you're used to). So, we use a weighted average (weighted mean or scaled average) to give greater value to some data over other data. If this keyword is not present or is zero, then the mean is computed across all dimensions of the input array. Michael Ho Zheng Sun
Jack Xin§. size }" if weights. 35 mg/L Flow Weighted C once tra i = 0. What is the acronym meaning/definition of CAC ?. We now see the mean as a weighted sum of the distinct values, where each value is weighted according to its proportion in the total list of numbers. Ranking, Matrix/Rating Scale, Multiple Choice, Multiple Textboxes, and Slider questions calculate an average or weighted average. Often used to account for area changes between meridians at varying latitudes by using the cosine of the latitude as the weights. Weighted average refers to the mathematical practice of adjusting the components of an average to reflect the importance of certain characteristics. (mean age, 57 years) who had. mean function in R, so far I couldn’t find a implementation of weighted. For weighted KPI calculations, the base weight of each KPI is factored against the weighted counts of each KPI by using the following formulas. n an average calculated by taking into account not only the frequencies of the values of a variable but also some other factor such as their variance. A weighted average is an average in which one element may contribute more heavily to the final result than another element. - joran Jun 12 '12 at 4:35 3 @Frank Hover over the down triangle beneath the vote count next to your Q. Among them, M55 showed a gene expression pattern consistent with behavioral changes after | {
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to your Q. Among them, M55 showed a gene expression pattern consistent with behavioral changes after stress exposure, and the gene ontology analysis revealed that this was involved in nervous system development, gland. Apr 21, 2017 · However, anxiety disorders can cause crippling feelings of fear, worry, or restlessness. This is a correct assumption if the same technique is used to measure the same parameter repeatedly. If using aggregated data, the correlation of the means does not reflect the sample size used for each mean. 2 mg/L Flow Weighted Concentration = 0. wi(yi 0 1xi) 2. Weighted average refers to the mathematical practice of adjusting the components of an average to reflect the importance of certain characteristics. WMC is sampling-, scale-, and contrast-invariant, and is sparse on natural images. The weighted arithmetic mean (or weighted average) is used if one wants to combine average values from samples of the same population with different sample sizes: ¯ = ∑ = ∑ =. The monthly returns are then compounded to arrive at the annual return. His head was pounding, his body seemed to ache from head to toe, and there was a tightness in his chest that pressed on his lungs. Weighted Average Item Price Report (WAIPR) and the Regional and Statewide Average Awarded Price Report (RSWAAPR) The Weighted Average Item Price Report (WAIPR) and the Regional and Statewide Average Awarded Price Report (RSWAAPR) are reports produced using information from NYSDOT's Trns•Port BAMS⁄DSS. Weighted gene co-expression network analysis identified 60 characteristic modules that correlated with stress or the FKBP5 genotype. 0026132544 5 1 209. 03 Progressive Less than 0. It is also called weighted average. Tseng Department of Biostatistics Department of Human Genetics. Thus, the weighted mean makes it possible to find the average student grade in the case where only the class means and the number of students in each class are available. In other words, each value to be averaged is | {
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and the number of students in each class are available. In other words, each value to be averaged is assigned a certain weight. Now, they have gone mainstream. Since we're all engineers, most students can figure out how this works at the end of the termbut how do you take it into account to calculate your current grade halfway through?. A weighted score or weighted grade is merely the average of a set of grades, where each set carries a different amount of importance. This is a correct assumption if the same technique is used to measure the same parameter repeatedly. Hi R experts, I need know how calculate a weighted mean by group in a data frame. Weighted voting. mean, part of R for Data Science: Lunchbreak Lessons. The experts in survey were estimating the organizations by. 03 Neutral 0. Section 2 formally formulates the on-. We will first create a Category which will use Weighted mean of grades. The Greedy Strategy for activity selection doesn’t work here as a schedule with more jobs may have smaller profit or value. Weighted Mean is a statistical method which calculates the average by multiplying the weights with its respective mean and taking its sum. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. The average is calculated by adding a range of numbers together and then dividing this total by the number of values in the range. The solid lines are the weighted mean while the dashed lines are the standard mean. Featured Services & News. A weighted mean (or weighted average) is like an ordinary mean, but the observations don't contribute equally - more emphasis is placed on some data values than others; they are weighted by a bigger or smaller amount than 1/n. Given a numeric vector of data values, x, and another numeric vector of weights, w, the weighted mean is the sum of the data value times the weights divided by the sum of the weights. 'Mikhail' instead of just 'Mihail')?. His head was pounding, his body seemed to ache | {
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of the weights. 'Mikhail' instead of just 'Mihail')?. His head was pounding, his body seemed to ache from head to toe, and there was a tightness in his chest that pressed on his lungs. This implies that we trim the lowest 5% of the data as well as the highest 5% of the data. Note that contrary to the weights w i in Eq. The weighted mean measures the average of the weighted data points. It is also called weighted average. Also called weighted mean. In a scenario where grades are weighted by category and the number of assignments in a category changes, no changes to the course points or syllabus would be necessary. What is a Weighted Mean? A weighted mean is a kind of average. Time-Weighted Return Formula. Meaning of weighted mean. A mean where some values contribute more than others. Input the above in your Ti-84 or TI-83 calculator Instructions on how to Create a List TI-84 | TI-83 tutorial. 631 × 10-3 mm 2 /sec, SD = 0. 03 Progressive Less than 0. If a professor decides to increase or decrease the workload based on the particular needs of any group of students, weighted grades make that easy. Here is the mean of 1, 2, 3 and 4: Add up the numbers, divide by how many numbers: Mean = 1 + 2 + 3 + 44 = 104 = 2. When totaling the individual values, each is multiplied by a weighting factor, and the total is then divided by the sum of all the weighting factors. Weighted means are often used for frequency data. These two measurements can be combined to give a weighted average. Weighted mean. Notice that the groups differ considerably in sample size. Mean: The arithmetic mean, also known as the simple mean or equal weighted mean. It is based on kernel weighted sample statistics such as the mean (Nadaraya-Watson estimator) but also standard deviation, skewness, kurtosis, deciles, etc. The above weighted average formula returns the value 849. Moving Average charts in JMP Each point on a Uniformly Weighted Moving Average (UWMA) chart, also called a Moving Average chart, is the | {
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on a Uniformly Weighted Moving Average (UWMA) chart, also called a Moving Average chart, is the average of the w most recent subgroup means (called “span”), including the present subgroup mean. Synonyms for weighted at Thesaurus. Generally the first item measures attributes like "Importance," "Need," or "Expectation". Input the above in your Ti-84 or TI-83 calculator Instructions on how to Create a List TI-84 | TI-83 tutorial. The area-weighted mean patch size, on the other hand, is a weighted mean, where the weights are based on the size of the patch. However, if you don’t want to spend big on Weighted belts, then you should absolutely go for Dark Iron Fitness which comes with all the basic features one could expect in Weighted belts. prepared and arranged in a way that is likely to produce a particular effect, usually an…. This method assumes that all of the trials have measured the outcome on the same scale. It is generally used to find average of variables that are expressed as a ratio of two different measuring units e. The T2-weighted sequence can be employed as a dual echo sequence. Weighted Mean. This has been a guide to Weighted Average in Excel. In calculating a weighted average, each number in the data set is. What does it mean republicans trying to phase out higher federal payments of medicare 3 Oct 2019. an object containing the values whose weighted mean is to be computed. a numerical vector of weights the same length as x giving the weights to use for elements of x. Next, obviously the weighted mean is $$\hat\mu=\sum_ip_ix_i,$$ and the variance:$$\hat\sigma^2=\sum_ip_i(x_i-\hat\mu)^2$$. Apr 21, 2017 · However, anxiety disorders can cause crippling feelings of fear, worry, or restlessness. By modifying sequence parameters such as repetition time (TR) and echo time (TE), for example, anatomical images can emphasize contrast between gray and white matter (e. Weighted mean The calculation of the mean discussed in the handout assumes that the standard | {
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(e. Weighted mean The calculation of the mean discussed in the handout assumes that the standard deviation of each individual measurement is the same. VWAP is typically used with intraday charts as a way to determine the general direction of intraday prices. This is a correct assumption if the same technique is used to measure the same parameter repeatedly. weighted average w/ matlab. Chapter 3 Weighted Mean study guide by john_lindsey8 includes 43 questions covering vocabulary, terms and more. Definition - What does Weighted Average Cost of Capital (WACC) mean? The weighted average cost of capital represents a weighted average of the after-tax cost of debt and the cost of equity where the weighting is based on a company’s target debt-equity ratio, measured at market. If this keyword is present, then the mean is only calculated across a single dimension. SUMPRODUCT should be the sum of the products of the respective values and weights. Nov 29, 2019 · 10 intermittent fasting side effects that might mean it’s not a great fit for you These are the 6 full-body home exercises the Springbok women’s vice captain swears by World Aids Day: Could. Weighted Averages Date: 11/02/98 at 21:09:00 From: Jacob Smith Subject: Help with "weighted Averaging" Dr. Weighted gene co-expression network analysis identified 60 characteristic modules that correlated with stress or the FKBP5 genotype. Source Score, x Weight,w Homework 60 5% Midterm 75 35% Project 80 20% Speech 70 15% Final Exam 62 25% 2. The weighted arithmetic mean of a set of numbers X 1, X 2, , X N with respective weights of w 1, w 2, , w N is defined as: Example : For a statistics course, the final exam grade is weighted four times as much as each quiz score. It includes the product id and invoice id along with some other information. Column A shows all the student’s percentage scores he/she received on tests, quizzes, homework assignments, and participation. Creates a classification table, from raw data in the spreadsheet, | {
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assignments, and participation. Creates a classification table, from raw data in the spreadsheet, for two observers and calculates an inter-rater agreement statistic (Kappa) to evaluate the agreement between two classifications on ordinal or nominal scales. Let us understand more with an example:. It is calculated according to the following formula: M = mark received in a course U = units of credit for a course. A mean where some values contribute more than others. 35 mg/L Flow Weighted C once tra i = 0. wRC+ takes the statistic Runs Created and adjusts that number to account for important external factors -- like ballpark or era. 739 × 10-3 mm 2 /sec, SD = 0. How to use weighted in a sentence. This method is sensitive to extreme observations (e. The arithmetic mean-geometric mean (AM-GM) inequality states that the arithmetic mean of non-negative real numbers is greater than or equal to the geometric mean of the same list. Here are the high-level steps for using weighted goals to manage a plan: Define goal IDs on the Goals page. | {
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# [ASK] Is Expected Frequency Rounded Up or Down?
#### Monoxdifly
A die is tossed 100 times. What's the expected frequency that the number appears will be 4?
The probability is 1/6, so the expected frequency is 1/6 × 100, but that results in a fraction (16 2/3). Do we need to round it up or down? Is the answer 16 or 17?
#### romsek
MHF Helper
You don't need to round it at all. The correct answer is $\dfrac{50}{3}$
True you won't see 2/3 of a die but that doesn't matter. An expectation is an average and as such there's no reason to expect it must be an integer.
#### Monoxdifly
Okay then, thanks.
#### TKHunny
If you are forced to round, for whatever reason, you will need to decide exactly how to do that.
Typically, a simple up or down methodology will prove unsatisfactory as you will violate the definition of a probability distribution:
Down 2 dp
1/3 = 0.33
1/3 = 0.33
1/3 = 0.33
0.33 * 3 = 0.99 and that is NOT 1.00
Likewise, fudging (forced footing) proves unsatisfactory:
1/3 = 0.33
1/3 = 0.33
1/3 = 0.34
They are not really equal, are they.
Unless you are in a purely theoretical pursuit, you just have to make up your mind what is and is not suitable.
#### Monoxdifly
Oh well, no rounding then, thanks. | {
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# How should I calculate $\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+…+n^n}{n^n}$ [duplicate]
How should I calculate the below limit
$$\lim_{n\rightarrow \infty} \frac{1^n+2^n+3^n+...+n^n}{n^n}$$ I have no idea where to start from.
• If this question does not come out from you, then where did you get this question? – Megadeth Oct 12 '15 at 5:37
• The limit should at least bigger than $1 + e^{-1} + e^{-2} + \cdots = \frac{e}{e-1}$ as for all $n$, the last term is $1$, the second last is $\left(\frac{n-1}{n}\right)^n = (1- \frac 1n)^n$ which tends to $e^{-1}$. So on so forth. – user99914 Oct 12 '15 at 6:05
• From $\frac{x_1+x_2+\cdots+x_n}{n}\geq\sqrt[n]{x_1x_2\cdots x_n}$ we write $$(\frac1n)^n+(\frac2n)^n+\cdots+(\frac1n)^n\geq n\frac{n!}{n^n}\to e$$ – Nosrati Oct 12 '15 at 6:29
• @Maryam $n^n\sim e^nn!$, so $n\dfrac{n!}{n^n}$ does not approach $e$. – Quang Hoang Oct 12 '15 at 6:35
• Have you tried Stolz-Cesaro ? – Lucian Oct 12 '15 at 6:49
First we use an observation by @Stan in the comment. Note that as $(1 +\frac{x}{n})^n$ is increasing in $n$ whenever $|x|<n$,
$$\left(\frac{k}{n}\right)^n = \left(1 + \frac{k-n}{n}\right)^n \le e^{k-n},$$
(here we assume that $x:= k-n$ is fixed and varies the remaining two $n$'s. This sequence is increasing and tends to $e^{k-n}$, as $|x| = |k-n| < n$. See here). Then we have
$$\begin{split} \frac{1^n + 2^n + \cdots + n^n}{n^n} &= \sum_{k=1} ^n \left(\frac{k}{n}\right)^n \\ &\le \sum_{k=1}^n e^{k-n} \\ &= 1 + e^{-1} + e^{-2} + \cdots e^{1-n} \\ &\le \frac{1}{1-e^{-1}} = \frac{e}{e-1}. \end{split}$$ This implies
$$\limsup_{n\to \infty} \frac{1^n + 2^n + \cdots + n^n}{n^n} \le \frac{e}{e-1}.$$
On the other hand, fix $k$. Then for all $n >k$, we have
$$\begin{split} \frac{1^n + 2^n + \cdots + n^n}{n^n} &\ge \frac{(n-k)^n + (n-k+1)^n + \cdots + n^n} {n^n}\\ &= \left( 1 - \frac kn\right)^n + \left( 1 - \frac {k-1}n\right)^n + \cdots +1 \end{split}$$
Then for all $\epsilon >0$, there is $N\in \mathbb N$ so that | {
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Then for all $\epsilon >0$, there is $N\in \mathbb N$ so that
$$\left| \left( 1 - \frac {j-1}n\right)^n - e^{-(j-1)} \right| < \epsilon$$
whenever $n \ge N$ and for all $j = 1, 2 , \cdots, k+1$ (Note $k$ is fixed, so this $N$ can be found)
In particular, this implies
$$\frac{1^n + 2^n + \cdots + n^n}{n^n} \ge e^{-k} + e^{-(k-1)} + \cdots + 1 - (k+1) \epsilon.$$
Thus
$$\liminf_{n\to \infty} \frac{1^n + 2^n + \cdots + n^n}{n^n} \ge e^{-k} + e^{-(k-1)} + \cdots + 1 - (k+1) \epsilon.$$
Now let $\epsilon \to 0$ and then $k \to \infty$, we have
$$\liminf_{n\to \infty} \frac{1^n + 2^n + \cdots + n^n}{n^n} \ge \frac{1}{1-e^{-1}} = \frac{e}{e-1}.$$
This implies
$$\lim_{n\to \infty} \frac{1^n + 2^n + \cdots + n^n}{n^n} = \frac{e}{e-1}.$$
• Another proof for $e^{k-n} \geq (k/n)^n = e^{n \ln (k/n)}$: $\Leftarrow k-n \geq n \ln (k/n) \Leftarrow \frac{k}{n}-1 \geq \ln \frac{k}{n}$ $\Leftarrow$ for $x>0$, $x-1 \geq \ln x$ $\Leftarrow$ Let $y(x) = x-1-\ln x$, then the stationary point should satisfy $y'=1-\frac{1}{x}=0$; meanwhile $y''=x^{-2}>0$. Thus $y_{\min} = y(1) = 0$, for all $x>0$, $y(x) \geq 0$. – Stan Oct 12 '15 at 8:01
• Another proof for the upper bound: $$1^n+2^n+\cdots+(n-k-1)^n < \int_1^{n-k} x^n\,dx < (n-k)^{n+1}/(n+1) < (n-k)^n.$$ Thus when we take only the terms from $(n-k)^n$ to $n^n$ in the numerator, the neglected terms are at most $(n-k)^n/n^n$, which tends to $e^{-k}$ as $n\to\infty$. Now we have the limit sandwiched between two explicit functions of $k$; let $k\to\infty$. – Greg Martin Oct 12 '15 at 8:10
• Nice proof! +1 :) – ZFR Nov 21 '15 at 7:50 | {
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# Associativity of a binary operator
I am reading the book "A first course in abstract algebra", and i am reading the chapter called subgroups. It says that, for general situations we do not use "*" to denote the binary operation on a group, but we simply use addition symbol "+" or multiplication symbol ".". Then it says, we denote the product a.a.a.a.a...a for n factors by an. But i am confused here: is that always associative? For example, for 3 factors, a.a.a is a3 according to what book says, but are we sure that a.(a.a)=(a.a).a=a3 is always true? Thank you.
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Do you know what "group" means? – Chris Eagle Mar 14 '13 at 20:53
Yes, i do. Why? – bigO Mar 14 '13 at 20:54
Then you know the answer to this question. – Chris Eagle Mar 14 '13 at 20:54
This is why i am asking this, i must be missing something – bigO Mar 14 '13 at 20:55
Note that not all binary operations need be associative. For example, take the integers under subtraction: $-4=(1-2)-3\neq 1-(2-3)=0$. – user1729 Mar 15 '13 at 14:54
Recall: One of the key properties satisfied by all groups is that its binary operation is associative on the set defined by the group.
If you are referring to Fraleigh's text, see
Definition 4.1: "A group $\langle G, *\rangle$ is a set $G$, closed under a binary operation $*$, such that the following axioms of satisfied:
$\mathcal G_1$: For all $a, b, c \in G,$ we have $$(a*b)*c = a*(b*c)\tag{associativity of *}$$
Of course, $\mathcal G_2,\;\text{and}\;\mathcal G_3\;$ are crucial, as well: the existence of an identity element $e\in G$, and for each $g\in G$, its inverse, $g^{-1}$ is also in $G$, including the definitions of the identity and how we define the inverse of a group element.
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Oh thanks, that's what i have been missing. – bigO Mar 14 '13 at 20:55 | {
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Oh thanks, that's what i have been missing. – bigO Mar 14 '13 at 20:55
Not only, As @amWhy noted, the binary operation is associative but also depend on how you show the binary operation, the figures of composition of elements varies: $$\cdot\longrightarrow a*b=a\cdot b$$ so $$\underbrace{a\cdot a\cdot a\cdot ...\cdot a}_{\large n\;times}=a^n~$$ And $$+\longrightarrow a*b=a+ b$$ so $$\underbrace{a+ a+ a+ ...+ a}_{\large n\;times}=na$$
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Don't know why the downvote? +1 – amWhy Mar 14 '13 at 21:09
Thank you! I did not downvote, just to let you know – bigO Mar 14 '13 at 21:28
@amWhy: Thanks Amy for the edit. :-) – Babak S. Mar 15 '13 at 5:32 | {
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# Win a coin game in 10th round
Two players are tossing a fair coin in one round. If it is heads, the first one gets a dollar from the second player. Otherwise, the first player gives a dollar to the second one. If both players have 6 dollars in the beginning of the game, what is the probability that the first player wins all the money exactly on 10th round.
• I am considering sample space to be 2^10; however, a friend of mine considers it to be 960. Whose idea is erroneous and why? Thanks. – Ulugbek Abdullaev Oct 26 '16 at 18:06
• The sample space is not as big as $2^{10}$, because the sequence $HHHHHHTTTT$ and $HHHHHHHHHH$ are considered exactly the same game (they quit playing after the sixth heads). Of course, if you think that they continue playing after one has lost just to "see what would happen", but without any money involved, and analyze those games, then the sample space has size $2^{10}$. It all depends on your interpretation. – Arthur Oct 26 '16 at 18:07
• @Arthur I got your point. Could you elaborate how to exclude those unnecessary? I'm just stuck.. – Ulugbek Abdullaev Oct 26 '16 at 18:41
• Hint: out of those ten rounds, how many Heads were there? How many Tails? We know the last one was $H$...what about the next to last? – lulu Oct 26 '16 at 19:04
• @BruceET It's very nearly the whole story. We see that we want $8$ Heads and $2$ Tails...there must be at least one $T$ in the first $6$ slots and and both $T$ must be in the first $8$. Easy to count. – lulu Oct 26 '16 at 21:58
If the coin is tossed 10 times, there could be wins at trials number 6, 8, and 10.
A win at the 6th requires six heads in a row (probability $.5^6 = 0.015625.$)
A first win at the 8th requires that exactly one of the first six tosses must be tails, followed by two heads (probability $6(.5)^8 = 0.0234375.$) | {
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The simulation below confirms these two results (within simulation error), and suggests that the probability of a first win at the 10th toss must have probability about 0.0265 (two or three place accuracy). I will leave it to you use similar logic to find a combinatorial formula and the exact probability.
m = 10^6; frst.win = numeric(m)
for(i in 1:m)
{ toss = sample(c(-1,1), 10, rep=T) # vector of 1's (Hs) and -1's (Ts)
cs = cumsum(toss) # 1st player's cumulative totals
frst.win[i] = match(6, cs) } # toss on which cum tot first reaches 6
table(frst.win)/m
frst.win
6 8 10
0.015633 0.023540 0.026513 | {
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# What is $\gcd(0,0)$?
What is the greatest common divisor of $0$ and $0$? On the one hand, Wolfram Alpha says that it is $0$; on the other hand, it also claims that $100$ divides $0$, so $100$ should be a greater common divisor of $0$ and $0$ than $0$.
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(o,o)? <- see the face scratching the head? – zerosofthezeta Sep 16 '13 at 4:57
It is typically not defined. – copper.hat Sep 16 '13 at 4:58
– anon Sep 16 '13 at 5:23
The word "greatest" in "Greatest Common Divisor" does not refer to being largest in the usual ordering of the natural numbers, but to being largest in the partial order of divisibility on the natural numbers, where we consider $a$ to be larger than $b$ only when $b$ divides evenly into $a$. Most of the time, these two orderings agree whenever the second is defined. However, while, under the usual order, $0$ is the smallest natural number, under the divisibility order, $0$ is the greatest natural number, because every number divides $0$.
Therefore, since every natural number is a common divisor of $0$ and $0$, and $0$ is the greatest (in divisibility) of the natural numbers, $\gcd(0,0)=0$. | {
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Similarly $\gcd(0,n)=n$ for all $n\in\Bbb N$ (including $n=0$). – anon Sep 16 '13 at 5:22
Thanks, great answer! How did you figure this out? Where do mathematicians store such commonly agreed on, but not completely obvious definitions? When did people start defining gcd this way, I assume they didn't when Euclidean algorithm was invented 300BC? – Konstantin Weitz Sep 16 '13 at 6:55
For the most part, the partial order of divisibility coincides with the usual ordering wherever it is defined. So if $\le$ represents the usual ordering and $\vert$ represents the partial order of divisibility, then $x\vert y\Rightarrow x\le y$, except if $y=0$. Moreover, if $x$ is a common divisor of two non-zero natural numbers $a$ and $b$, then $x\le\gcd(a,b)\Rightarrow x\vert\gcd(a,b)$. So in most cases, you can define $\gcd(a,b)$ to be the greatest common divisor of $a$ and $b$, where 'greatest' refers to the usual ordering. That's what Euler did (and left $\gcd(0,0)$ undefined). – Donkey_2009 Sep 16 '13 at 11:55
Euclid, I mean.... – Donkey_2009 Sep 16 '13 at 14:50
I answered this already in a comment at MO: "The best way to think about this is that the "gcd" of two natural numbers is the meet of them in the lattice of natural numbers ordered by divisibility. Note that $0$ is the top element in the divisibility order. The meet of the top element with itself is itself. So $0 = \gcd(0, 0)$ is the answer. 'Greatest' is an unfortunate misnomer in this case."
The book Mathematics Made Difficult has a nice little section on this. It should perhaps better be called "highest common factor" (hcf).
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You lost the race to William. – dfeuer Sep 16 '13 at 5:05
we all lost to him :) – DanielY Sep 16 '13 at 5:06
@dfeuer Well, I had answered the question already at MO. I wasn't racing. – user43208 Sep 16 '13 at 5:06
Win or lose, "the meet of [two numbers] in the lattice of natural numbers ordered by divisibility" is a definition of delightful pithiness. – Jordan Gray Sep 16 '13 at 11:00 | {
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Another way to think about this is ideals. The gcd of two natural numbers $a, b$ is the unique non-negative natural number that generates the ideal $\langle a, b \rangle$. So in this case, $\langle 0 ,0 \rangle$ is just the $0$ ideal so the gcd is $0$.
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Defining $\gcd(a,b)$ for $\def\Z{\mathbf Z}a,b\in\Z$, to be the non-negative generator of the ideal $a\Z+b\Z$ gives $\gcd(a,0)=|a|$ for all $a$, including for $a=0$. Similarly one can define $\def\lcm{\operatorname{lcm}}\lcm(a,b)$ to be the non-negative generator of the ideal $a\Z\cap b\Z$, which gives $\lcm(a,0)=0$ for all$~a$ (note that since this is the only common multiple in this case, it is unlikely to provoke much discussion).
Adding these cases to the usual definitions of the $\gcd$ and $\lcm$ for nonzero integers causes no problems; all usual formulas remain valid. In fact, if one wants the rule $\gcd(xy,xz)=|x|\gcd(y,z)$ to hold for all integers $x,y,z$, one is forced to put $\gcd(0,0)=0$.
On the other hand, I don't think it is absolutely vital for mathematics to have $\gcd(0,0)$ defined, in the same way as $0+0$, $0\times0$ and $0^0$ need to be defined (and $0/0$ needs to be undefined) in order for the usual rules of algebra that one uses all the time to be valid. I think it would suffice to qualify the rule I cited by "$x\neq0$" if one wants to leave $\gcd(0,0)$ undefined; other rules like $\gcd(a,b)=\gcd(a,a-b)$ do not seem to equate $\gcd(0,0)$ with something else. The reason that leaving it undefined is not so dramatic is that when considering divisibility, $0$ is often excluded anyway; for instance it has to be put aside in the theorem of Unique Factorisation.
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Simply said - this depends on your definition.
Clearly, if $d=\gcd(a,b)$, you require $d\mid a$, $d\mid b$, i.e., it is a common divisor.
But there are two possibilities how to express that it is greatest common divisor. | {
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But there are two possibilities how to express that it is greatest common divisor.
One of them is to require $$c\mid a \land c \mid b \Rightarrow c\le d$$ and the other one is $$c\mid a \land c \mid b \Rightarrow c\mid d.$$
Clearly, if you use the first definition, $\gcd(0,0)$ would be the largest integer, so it does not exists. If you use the second one, you get $\gcd(0,0)=0$. (Note that $0$ is the largest element of the partially ordered set $(\mathbb N,\mid)$.)
As far as I can say, the first definition appears in some text which are "for beginners"; for example here. (It was one of the first results from Google Books when searching for "gcd(0,0)".)
I would say that for students not knowing that $\mid$ is in fact a partial order, the first definition might feel more natural. But once you want to use this in collection with more advanced stuff (for example, g.c.d. as generator of an ideal generated by $a$ and $b$), then the second definition is better.
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This helped. I still feel a little icky about (ℕ,∣) as a partial order, because 0∣0 feels weird to me, and this whole argument requires that reflexivity. – rampion Apr 8 at 0:46
In the general framework of integral domains (commutative rings with unity, without nontrivial zero-divisors), we define a greatest common divisor of two elements $a,b$ of an integral domain $R$ as any element $c\in R$ satisfying:
• $c$ divides both $a$ and $b$, that is, there are $x,y\in R$ such that $a=cx$ and $b=cy$.
• If $d\in R$ divides both $a$ and $b$, then $d$ divides $c$.
We write $c=\gcd(a,b)$ in this case. The definition implies that if $c,c^\prime=\gcd(a,b)$, then $c$ and $c^\prime$ are associates, that is $c=uc^\prime$ for some invertible element $u$ in $R$. This is equivalent to say that the principal ideals generated by $c$ and $c^\prime$ are the same. Therefore a non-ambiguous definition is as follows: | {
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"$\gcd(a,b)$ is $\$ any minimal$\$ the minimum element in the family $\mathcal F=\{Rd: Rd\supseteq Ra+Rb\}$ of principal ideals containing $a$ and $b$, ordered by inclusion, wherever such minimum ideal exists."
Since $R0+R0=R0$, we see that $R0=0$, the trivial ideal of $R$, is the $\gcd$ of $0$ and $0$.
In general you cannot guarantee that $\gcd$s exist. A sufficient condition is your integral domain $R$ to be a UFD (Unique Factorization Domain).
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Use \gcd instead of $GCD$. Also, ^\prime can be abbreviated all the way down to '. – dfeuer Sep 16 '13 at 5:11
I've tried editing it myself, but you've interrupted me twice, so I'll let you do it. – dfeuer Sep 16 '13 at 5:11
@dfeuer Thanks for the suggestion. By the way, that ' shortcut works in ordinary $\LaTeX$? – Matemáticos Chibchas Sep 16 '13 at 5:12
Yep! It even works in Plain $\TeX$! $x'''$ just takes four keystrokes (six if you count dollar signs). – dfeuer Sep 16 '13 at 5:16
anon- Not on my keyboard. On my keyboard, backtick is the key to the left of 1, and tilde is obtained by holding shift and pressing the hash key (which is next to Enter). In general, the location of punctuation characters varies in keyboards used in different countries. – Hammerite Sep 16 '13 at 9:21
From Wikipedia:
The greatest common divisor of $a$ and $b$ is well-defined, except for the case $a=b=0$, when every natural number divides them.
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Well, this isn't really the right answer, is it? William got it right. – user43208 Sep 16 '13 at 5:22
All I'm saying is that Wikipedia got it wrong in this case (or at least it's misleading to suggest that $\gcd(0, 0)$ is not well-defined, because as has been clearly explained, it's perfectly well-defined and indeed it's $0$). – user43208 Sep 16 '13 at 5:32
The guy got his answer, that's the most important :) – DanielY Sep 16 '13 at 5:33
If we take Euclid's algorithm: | {
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If we take Euclid's algorithm:
$\text{gcd}(a, b) = \left\{ \begin{array}{l l} a & \quad \text{if$b = 0$}\\ \text{gcd($b$,$a$mod$b$)} & \quad \text{otherwise} \end{array} \right.$
as the definition of GCD, then $\text{gcd}(a, 0) = 0$ for any $a$, because the stopping case of $b = 0$ is reached immediately. If $a = 0$, then we get $\text{gcd}(0, 0) = 0$.
So, it's possible that Wolfram Alpha obtains its result as a side effect of using Euclid's algorithm rather than deliberate thought as to what gcd(0, 0) should return. But it does fortunately coincide with William's “partial order of divisibility” explanation.
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# Math Help - Stokes Theorem - double integral
1. ## Stokes Theorem - double integral
Use Stokes’ Theorem to evaluate $\int_c^.F \cdot dr$
C: is the boundary of the portion of $z = x^2 + y^2$ below z = 4, oriented downward,
$F = $
Can someone please check over my work thanks!
First I found N --> $N = \frac{<2x, 2y, -1>}{\sqrt{4x^2 + 4y^2 + 1^2}}$
Then i found $\bigtriangledown \times F = <0, 0, -1>$
from using the formula $\bigtriangledown \times F = $
Now $\bigtriangledown \times F \cdot N = \frac{1}{\sqrt{4x^2 + 4y^2 + 1^2}}$
Then $dS = {\sqrt{4x^2 + 4y^2 + 1}}dA$
So putting it all together....
$\int\int_R^. \frac{1}{\sqrt{4x^2 + 4y^2 + 1^2}} \cdot \sqrt{4x^2 + 4y^2 + 1}$
The square roots cancel and reduce to....
$\int\int_R^. 1 dA$
This double integral of 1 dA is equal to the Area of the Region
And using $z = x^2 + y^2$ and $z = 4$ we come up with $r^2 = 4 .... r = 2$
$A(R) = \pi r^2 = \pi(2^2) = 4\pi$
Does $4\pi$ look correct for the answer?
Thanks for looking
2. I've always disliked the notation $\vec{n}dS$ because used that unthinkingly results in doing two square roots which, just as you see here, cancel!
Instead think of $d\vec{S}$ as the "vector differential of surface area" and calculate it directly. We can write the surface $z= x^2+ y^2$ in a vector equation with x and y as parameter: $\vec{r}(x,y)= x\vec{i}+ y\vec{j}+ z\vec{k}= x\vec{i}+ y\vec{j}+ (x^2+ y^2)\vec{k}$. The derivatives with respect to x and y give two tangent vectors, $\vec{i}+ 2x\vec{k}$ and $\vec{j}+ 2y\vec{k}$. The cross product of those gives the "vector differential of surface area": $2x\vec{i}+ 2y\vec{j}- \vec{k}$ where I have chosen the order of multiplication to get a negative value for $\vec{k}$, "oriented downward".
You are correct that $\nabla\times \vec{F}= -\vec{k}$ so that $\nabla\vec{F}\cdot\vec{n} dS= \nabla\vec{F}\cdot d\vec{S}$[tex]= (0(2x)+ 0(2y)- 1(-1))dxdy= dxdy.
That gives $4\pi$, not $-4\pi$. | {
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That gives $4\pi$, not $-4\pi$.
Your normal vector, $\frac{-2x\vec{i}-2y\vec{j}+ \vec{k}}{\sqrt{4x^2+ 4y^2+ 1}}$ was oriented upward, not downward. | {
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## Palindromes
Problem 4 at Project Euler poses the following challenge:
A palindromic number reads the same both ways. The largest palindrome made from the product of two 2-digit numbers is 9009 = 91 × 99.
Find the largest palindrome made from the product of two 3-digit numbers.
### A Naïve Approach
A naïve solution would be to multiply all the three-digit numbers by one another, pull out all the palindromes, and find the greatest:
var isPalindrome = function(n){
n = n.toString();
return n === n.split("").reverse().join("");
};
var findBiggest = function(){
var allProducts = [];
for (var n = 100; n < 1000; n++){
for (var m = 100; m < 1000; m++){
allProducts.push(m * n);
}
}
var palindromes = allProducts.filter(isPalindrome);
return Math.max.apply(null, palindromes);
};
But the naïve appraoch means making a huge-ass array with 810,000 numbers in it, consuming 3.09MiB of memory and taking many milliseconds to build. We can do better with a little number theory.
#### Don’t include obvious duplicates
$m \times n = n \times m$
Instead of starting m at 100 for each value of n, start it at n. This avoids duplicates and cuts array size roughly in half, to only 405,450 entries.
#### Don’t include multiples of 10
Observe that $n \times 10$ always ends in $0$. We don’t ordinarily write positive non-zero integers in decimal with leading zeros, so we can safely exclude all multiples of 10 from $n$ and $m$, saving us another 76,995 entries in allProducts, almost 19%, for a new total of 328,445 entries.
#### Assume the greatest palindrome has six digits | {
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#### Assume the greatest palindrome has six digits
This is a pretty safe assumption in the instance the Project Euler problem describes, but may not be safe in other situations. Of our remaining 404,550 products of two three-digit decimal numbers, only 69,787 (17.25%) of them do not have six digits. It turns out these 17.25% include over 60% of the palindromes, but all we need is one six-digit palindrome in there to safely ignore the five-digit ones. $101,101 = 143\times707$ and has six digits; that’s enough to let us dismiss five-digit entries.
But how do we exclude five-digit products from our array without having to spend time calculating those products? We don’t. But limiting ourselves to condisering only six-digit products (actually, to products with an even number of digits) lets us do something nifty.
As it turns out, palindromic numbers with an even number of digits have a nifty property: they all share a common, easily identifiable factor.
Let $\mathbb{P}\subset\mathbb{N}$ be the set of palindromic numbers with $k+1$ digits $a_0...a_k$ in an arbitrary base $b:b\geq2$:
$\mathbb{P}=\left\{ n\in\mathbb{N}:n=\sum\limits_{i=0}^{k}a_ib^i\middle|a_i=a_{k-i}\right\}$
Note that $n$ is palindromic if and only if $a_i=a_{k-i}$. By way of example:
$$$$\begin{split}987,789&=9(10^0)\\&+8(10^1)\\&+7(10^2)\\&+7(10^3)\\&+8(10^4)\\&+9(10^5)\end{split}$$$$ $$$$\begin{split}65,456&=6(10^0)\\&+5(10^1)\\&+4(10^2)\\&+5(10^3)\\&+6(10^4)\end{split}$$$$
Looks like a pattern! Turns out we can rewrite the sum like this, but only for palindromes with an even number of digits:
$n=\sum\limits_{i=0}^{\frac{k+1}{2}}a_i\left(b^i+b^{k-i}\right)\Big|k+1\text{ is even}$
Essentially, we pair up the digit $a_i$ with its matching digit $a_{k-i}$ and thereby only deal with half the digits. This doesn’t work with palindromes with an odd number of digits because of the pesky middle digit, which has no mate. | {
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$$$$\begin{split}987,789&=9(10^0+10^5)\\&+8(10^1+10^4)\\&+7(10^2+10^3)\end{split}$$$$ $$$$\begin{split}65,456&=6(10^0+10^4)\\&+5(10^1+10^3)\\&+4(10^2)\end{split}$$$$
For palindromes with an even number of digits, we can factor the addend:
$a_i(b^i+b^{k-i})=a_ib^i(1+b^{k-2i})$
Check it out! We can make $1+b^{k-2i}=0$ with modular arithmetic!
\begin{alignat*}{2}\begin{split}1+b^{k-2i}&=1-1^{k-2i}&\mod{b+1}\\&=1-1&\mod{b+1}\\&=0&\mod{b+1}\end{split}\end{alignat*}
Every palindrome with an even number of digits is evenly divisible by $b+1$, where $b$ is the radix. We’re working in decimal, so $b+1=11$.
We needn’t bother with any products but those that are divisible by 11. 11 being prime makes things easy. We'll only multiply $m\times n$ and push() the product to allProducts if at least one of $\lbrace m,n\rbrace$ is divisible by 11. This brings allProducts.length down to only 55,764, a reduction of over 93.1% from our original 810,000.
### Putting it All Together
All this makes for lots of savings in computation and storage, but it makes our program a lot uglier:
var isPalindrome = function(n){
n = n.toString();
return n === n.split("").reverse().join("");
};
var findBiggest = function(){
var allProducts = [];
var start, end, step;
for (var n = 101; n < 1000; n++){
if (n % 10){
if (n % 11){
start = 11*Math.ceil(n/11);
end = 11*Math.floor(1000/11);
step = 11;
} else {
start = n;
end = 1000;
step = 1;
}
}
for (var m = start; m < end; m+=step){
if (m % 10){
allProducts.push(m * n);
}
}
}
var palindromes = allProducts.filter(isPalindrome);
return Math.max.apply(null, palindromes);
};
Messy, but fast. The naïve version ran in 1,158.8ms, while this version runs in only 74.2ms. | {
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Lecture 15: Matrices A(t) Depending on t, Derivative = dA/dt
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Description
This lecture is about changes in eigenvalues and changes in singular values. When matrices move, their inverses, their eigenvalues, and their singular values change. Professor Strang explores the resulting formulas.
Summary
Matrices $$A(t)$$ depending on $$t /$$Derivative $$= dA/dt$$
The eigenvalues have derivative $$y(dA/dt)x$$.
$$x$$ = eigenvector, $$y$$ = eigenvector of transpose of $$A$$
Eigenvalues from adding rank-one matrix are interlaced.
Related section in textbook: III.1-2
Instructor: Prof. Gilbert Strang
The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high-quality educational resources for free. To make a donation or to view additional materials from hundreds of MIT courses, visit MIT OpenCourseWare at ocw.mit.edu.
GILBERT STRANG: So I've worked hard over the weekend. I figured out what I was doing last time and what I'm doing this time and improved the notes. So you'll get a new set of notes on the last lecture and on this one. And I kind of got a better picture of what we're doing. And that board is aiming to describe the large picture of what we're doing last time and this time.
So last time was about changes in A inverse when A changed. This time is about changes in eigenvalues and changes in singular values when A change. As you can imagine, this is a natural important situation. Matrices move, and therefore, their inverses change, their eigenvalues change, their singular values change. And you hope for a formula.
Well, so we did have a formula for last time for the change in the inverse matrix. And I didn't get every u and v transpose in the right place in the video or in the first version of the notes, but I hope that that formula, that that Woodbury Morrison formula will be correct this time. So I won't go back over that part. | {
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But I realize also there is another question that we can answer when the change is very small, when the change in A is dA or delta A, a small change. And that's, of course, what calculus is about. So I have to sort of parallel topics here. What is the derivative when the change is infinitesimal? And what is the actual change when the change is finite size?
So now, let me say what we can do and what we can't do. Oh, I'll start out by figuring out what the derivative is for the inverse. So that's like completing the last time for infinitesimal changes. Then I'll move on to changes in the eigenvalues and singular values. And there, you cannot expect an exact formula. We had a formula that was exact, apart from any typos, for this. And we'll find a formula for this, and we'll find a formula for that and for that. Well, that one will come from this one. So this will be a highlight today. How do the eigenvalues change when the matrix changes?
But we won't be able to do parallel to this, we won't be able to-- oh, we will be able to do something for finite changes. That's important. Mathematics would have to keep hitting that problem until it got somewhere. So I won't get an exact formula for that change. That's too much.
But I'll get inequalities. How big that change could be. What can I say about it? So these are highly interesting.
May I start with completing the last lecture? What is the derivative of the inverse? So I'm thinking here, so what's the setup? The setup is my matrix A depends on time, on t. And it has an inverse. A inverse depends on t.
And if I know this dependence, in other words, if I know dA dt, how the matrix is depending on t, then I hope I could figure out what the derivative of A inverse is. We should be able to do this. So let me just start with-- it's not hard and it complements this one by doing the calculus case, the infinitesimal change. So I want to get to that. | {
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I can figure out the change in A. And my job is to find the derivative of A inverse. So here's a handy identity. Can I just put this here? So here's my usual identity.
So as last time, I start with a finite change because calculus always does that, right. It starts with a delta t and then it goes to 0. So here I am up at a full size change. So I think that this is equal to B inverse A minus B A inverse. And if it's true, it's a pretty cool formula.
And, look, it is true, because over on this right-hand side, I have B inverse times A A inverse. That's the identity. So that's my B inverse. And I have the minus, the B inverse B is the identity. There's A inverse. It's good, right?
So from that, well, I could actually learn from that the rank of this equals the rank of this. That's a point that I made from the big formula. But now, we can see it from an easy formula. Everywhere here, I'm assuming that A and B are invertible matrices. So when I multiply by an invertible matrix, that does not change the rank. So those have the same ranks.
But I want to get further than that. I want to find this. So how do I go? How do I go forward with that job to find the derivative of the inverse?
Well, I'm going to call this a change in A inverse. And over here, I'll have B will be-- yeah, OK, let's see, am I right? Yeah.
So B inverse will be-- this is A plus delta A inverse. And this is-- well, that's A minus B. So that's really minus delta A. From A to B is the change. Here, I'm looking at the difference A minus B. So it's minus a change. And here, I have A inverse. I haven't done anything except to introduce this delta and get B out of it and brought delta in. | {
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Now, I'm going to do calculus. So I'm thinking of B as there's a sort of a delta t. And I'm going to divide by both sides by delta t. I have to do this if I want-- and now, I'll let delta t go to 0. So calculus appears. Finally, our-- I won't say enemy calculus, but there is a sort of like competition between linear algebra and calculus for college mathematics. Calculus has had far, far too much time and attention. It like it gets three or four semesters of calculus for people who don't get any linear algebra. I'm glad this won't be on the video, but I'm afraid it will.
Anyway, of course, calculus is fine in its place. So here's its place. Now let delta t go to 0. So what does this equation become?
Then everybody knows that as the limit of delta t goes to 0, I replace deltas by-- so this delta A divided by delta t that has a meaning. The top has a meaning and the bottom has for me. But then the limit, it's the ratio that has a meaning.
So dA by itself, I don't attach a meaning to that. That's infinitesimal. It's the limit, so that's why I wanted a delta over a delta so I could do calculus. So what happens now is delta t goes to 0. And, of course, as delta t goes to 0, that carries delta A to 0. So that becomes A inverse.
And what does this approach as delta t goes to 0? dA dt with that minus sign. Oh, I've got to remember the minus sign. The minus sign is in here. So I'm bringing out the minus sign.
Then this was A inverse, as we had. And that's dA dt. And that's A inverse. That's our formula, a nice formula, which sort of belongs in people's knowledge.
You recognize that if A was a 1 by 1 matrix, we could call it x, instead of A. If A was a 1 by 1 matrix x, then I'm saying the formula for the derivative of 1 over x, right? A inverse just 1 by 1 case is just 1 over x. So the derivative of 1-- or maybe t, I should be saying. | {
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If A is just t, then the derivative of 1 over t with respect to t is....? Is minus 1 over t squared. The 1 by 1 case we know. That's what calculus does. And now we're able to do the n by n case. So that's just like good.
And then it's sort of parallel to formulas like this, where this delta A has not gone to 0. It's full size, but low rank. That was the point. Actually, the formula would apply if the rank wasn't low. But the interest is in low rank here.
Are we good for this? That's really the completion of last time's lecture with derivatives.
OK, come back to here, to the new thing now, lambdas. Let's focus on lambdas, eigenvalues. How does the eigenvalue change when the matrix changes? How does the eigenvalue change when the matrix changes?
So I have two possibilities. One is small change when I'm doing calculus and I'm letting a delta t go to 0. The other is full size, order 1 change, where I will not be able to give you a formula for the new lambdas, but I'll be able to tell you important facts about them. So this is today's lecture now. You could say that's the completion of Friday's lecture.
What about d lambda dt? It's a nice formula. Its proof is fun too. I was very happy about this proof. OK, so I guess calculus is showing up here on this middle board.
So how do I start with the eigenvalues? Well, start with what I know. So these are facts, you could say, that I have to get the eigenvalues into it. And, of course, eigenvalues have to come with eigenvectors.
So I'll again use A of t. It will be depending on t. And an eigenvector that depends on t is an eigenvalue that depends on t times and eigenvector that depends on t. Good? That's fact one that we plan to take the derivative of somehow.
There's also a second fact that comes into play here. What's the deal on the eigenvalues of A transpose? They are the same. The eigenvalues of A transpose are the same as the eigenvalues of A. | {
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Are the eigenvectors the same? Not usually. Of course, if the matrix was symmetric, then A and A transpose are just the same thing. So A transpose would have that eigenvalue-- eigenvector.
But, generally, it has a different eigenvector. And really to keep a sort of separate from this one, let's call it y. It will have the same eigenvalue. I'm going to call it y. But I'm going to make it a row vector, because A transpose is what-- instead of writing down A transpose, I'm going to stay with A, but put the eigenvalue on the left side.
So here's is the eigenvalue-- eigenvector for A on the left. And it has the same eigenvalue times that eigenvector. But that eigenvector is a row eigenvector, of course. This is an equality between rows. A row times my matrix gives a row. So that's the eigenvalues of-- and it has the same eigenvalues.
So this is totally parallel to that, totally parallel. And maybe sort of less-- definitely less seen, but it's just the same thing for A transpose. Everybody sees that if I transpose this equation, then I've got something that looks like that. But I'd rather have it this way.
Now, one more fact I need. There is-- there has to be some normalization. What should be the length of these? Right now, x could have any length. y could have any length. And there's a natural normalization, which is y transpose times x equal to 1. That normalizes the two. It doesn't tell me the length of x or the length of y. But it tells me, the key thing, the length of both.
So what I've got there is tracking along one eigenvalue and its pair of eigenvectors. And you're always welcome to think of the symmetric case when y and x are the same. And then I would call them q. Oh, well, I would call them q if it was a symmetric matrix. So if it's a symmetric matrix, both eigenvectors would be called q. And this would be saying that q is a--
AUDIENCE: Unit vector. | {
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AUDIENCE: Unit vector.
GILBERT STRANG: Unit vector, right. So this is all stuff we know. And actually, maybe I should write it in matrix notation, because it's important. That's for one eigenvector. This is for all of them at once. Everybody's with it? The x's are the columns of x. And lambda is the diagonal matrix of lambdas. And it has to sit on the right so that it will multiply those columns. So this is like all eigenvectors at once.
What would this one be? This would be like y transpose A equals A--
AUDIENCE: y transpose inverse?
GILBERT STRANG: y transpose, yes-- equals-- and probably these are multiplied-- I feel wrong if I write y transpose here. Like here, the x was on the right and on the left. And I'll-- oh, yeah, y transpose, yeah. OK, so what do I put? Lambda y transpose. Thanks.
And what do I put here? What does this translate to if this was for one eigenvector? For all of them at once, it's just going to translate to y transpose x equal the identity.
This is pretty basic stuff. But stuff somehow we don't always necessarily see. Those are the key facts.
And now, I plan to take the derivative, take the derivative of respect to lambda. Oh, I can derive one more fact. So this would be a formula. This is formula 1. Formula 1 just says, what do I get if I hit this on the left by y transpose? Can I do that? y transpose of t A of t x of t equals lambda of t. That's a number. So I can always bring that out in front of the inner product of vector notation.
Are you good for that? I'm pleading like everything I've done is totally OK. And now, I have a improvement to make on this right-hand side, which is...? So what is y transpose times x?
AUDIENCE: 1. | {
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AUDIENCE: 1.
GILBERT STRANG: It's 1. So let's remember that. It's 1. So in other words, I have got a formula for lambda of t. As time changes, the matrix changes. Its eigenvalues change according to this formula. Its eigenvectors change according of this formula. And its left eigenvectors change according to that formula. So everything here is above board.
And now, what's the point? The point is I'm going to find this, the derivative. So I'm going to take the derivative of that equation and see what I get. That'll be the formula for the derivative of an eigenvalue. And amazingly, it's not that widely known. Of course, it's classical, but it's not always part of courses. So this is as time varies, the matrix varies, A. And therefore, its eigenvalues vary, and its eigenvectors vary.
So we're going to find d lambda dt. It's one level of difficulty more to find dx dt, the derivative of the eigenvector or the second derivative of the eigenvalue. Those kind of come together. And I'm not going to go there. I'm just going to do the one great thing here-- take the derivative of that equation.
Shall I do it over there? So here we go. So I want to compute d lambda dt. And I'm using this formula for lambda there.
So I've got three things that depend on t. And I'm taking the derivative of their product. So I'm going to use the product rule. I'll apply the product rule to that derivative.
Take the derivative of the first guy times A times x. Take the derivative of the second guy times the second guy and the third guy. y transpose of t A of t dx dt. OK? We are one minute away from a great formula. And I'm really happy if you allow me to say it. That that formula comes by just taking those facts we know, putting them together into this expression that we also know, and this is like lambda equals x inverse Ax. That's a diagonalizing thing and then taking the derivative. | {
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So what do I get if I take that derivative? Well, this term I'm going to keep. I'm not going to play with that. Everybody is clear? That's a number. Here's a matrix. dA dt is a matrix. I take the derivative of every entry in A.
Here's its column vector, its eigenvector. And here's a row vector. So row times matrix times column is a number, 1 by 1. And actually, that's my answer. That's my answer.
So I'm saying that these two terms cancel each other out as those two terms added to zero. This is the right answer for the derivative. That's a nice formula.
So to find the derivative of an eigenvalue, the matrix is changing, you multiply by the eigenvector and by the left eigenvector. It gives you a number. And that's the d lambda dt.
So why do those two guys add to 0? That's all that remains here. And then this topic is ended with this nice formula. So I want to simplify that, simplify that, and show that they cancel each other.
So what is Ax? It's lambda x. So this guy is nothing but it's lambda that depends on time of course times dy dt dy transpose dt. I'm just copying that. Ax is lambda x. Sorry, I didn't mean to make that look hard.
You OK with that? Ax is lambda x. And I am perfectly safe, because lambda is just a number to bring it out to the left. So it doesn't look like it's in the way.
And what about this other term? So I have y transpose-- oh, y transpose A, what's that? What's y transpose A? That's the combination that I know. y transpose A, y is that left eigenvalue. y transpose A brings out a lambda. So this also brings out a lambda times y transpose times dx dt. OK? I just use Ax equal lambda x there. It was really nothing.
Now, what do I do? I want this to be 0. Can you see it happening? It's a great pleasure to see it happening.
So what do I have here? What's my first step now?
AUDIENCE: Like take lambda--
GILBERT STRANG: Bring lambda outside. That's not 0. We don't know what that is. Bring lambda outside there times the whole thing. | {
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So for some wonderful reason I believe that this number, which is a row times a column, a row times a column, two terms there, I believe they knock each other out and that result is 0. And why? Why? Because I come back to-- this board has all that I know. And here's y transpose times x equal 1.
And how does that help me? Because what I'm seeing in that square, in those brackets is?
AUDIENCE: The derivative of y transpose--
GILBERT STRANG: The derivative of the y transpose x. So it's the derivative of?
AUDIENCE: 1
GILBERT STRANG: 1. Therefore, 0. So this is the derivative of 1. It equals 0. Those two terms knock each other out and leave just the nice term that we're seeing.
So the derivative of the eigenvalue, just to have one more look at it before we leave it. The derivative of the eigenvalue is this formula. It's the rate at which the matrix is changing times the eigenvectors on right and left. Sometimes they're called the right eigenvector and the left eigenvector at the time t.
So we're not saying in this d lambda dt. In other words, I get a nice formula, which doesn't involve the derivative of the eigenvector. That's the beauty of it. If I want to go up to take the next step-- I tried this weekend, but it's a mess.
It would be to take the-- so this is my formula then, d lambda dt equals this. And I can take the next derivative of that, and it will involve d second dt squared. But it will also involve dx dt and dy dt. And in fact, a pseudo inverse even shows up. It's another step, and I'm not going that far, because we've got the best formula there.
So now that has answered this question. And I could answer that question the same way. It would involve A transpose A and the singular vectors, instead of involving A and the eigenvectors. Maybe that's a suitable exercise. I don't know. I haven't done it myself. | {
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What I want to do is this, now say, what can we say about the change in the eigenvalue-- and I'll just stay first of all with eigenvalue-- when the change is like rank 1? This is a perfect example when the change is rank 1.
So what can we say about the eigenvalues-- let's take the top, the largest eigenvalue, or all of them, all of them, lambda j, all of them-- of A plus a rank 1 matrix uv transpose. Oh, let's do the nice case here, the nice case, because if I allow a general matrix A, I have to worry about does it have enough eigenvectors? Can it diagonalize? All that stuff. Let's make it a symmetric matrix. And let's make the rank 1 change symmetric too.
So the question is, what can I say about the eigenvalues after a rank 1 change? So again, this isn't calculus now, because the change that I'm making is a true vector and not a differential. And I'm not going to have an exact formula for the new eigenvalues, as I said.
But what I am going to do is write down the beautiful facts that are known about that. And here they are. So, first of all, the eigenvalues are in descending order. We use descending order for singular values. Let's use them also for eigenvalues. So lambda 1 is greater or equal to lambda 2, greater or equal to lambda 3, and so on.
Oh, give me-- give me an idea. What do you expect from that rank 1 change? So that change is rank 1. Can you tell me any more about that change, u u transpose? What kind of a matrix is u u transpose? It's rank 1, but we can say more. It is...?
AUDIENCE: Symmetrical.
GILBERT STRANG: Symmetric, of course. And it is...? Yeah?
AUDIENCE: Positive semidefinite.
GILBERT STRANG: Positive semidefinite. Positive semidefinite. This is a positive change. u u transpose is the typical rank 1 positive semidefinite. It couldn't be positive definite, because it's only got rank 1. | {
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What's the eigenvector of that matrix? Let's just-- why not here? We can do this in two seconds. So u u transpose, that's the matrix I'm asking you to think about. And it's a full n by n matrix, column times a row. Tell me an eigenvector of that matrix. Yes?
AUDIENCE: u.
GILBERT STRANG: u. If I multiply my matrix by u, I get-- what do I get? I get some number times u. And what is that number lambda?
AUDIENCE: u transpose u.
GILBERT STRANG: That lambda happens to be u transpose u. So that's different from u u transpose. This is a matrix. This is 18.065 now. That's a number. And what can you tell me about that number? It is...?
AUDIENCE: Greater than or equal to 0.
GILBERT STRANG: Greater-- well, even more. Greater than 0. Greater, because this is a true vector here. So this is greater than 0. It's the only eigenvalue-- all the other eigenvalues of that rank 1 matrix are zero. But the one non-zero eigenvalue is over on the plus side. It's u transpose u. We all recognize that as the length of u squared. It's certainly positive. So we do have a positive semidefinite definite matrix.
What would your guess be of the effect on the eigenvalues of A? So I'm coming back to my real problem-- eigenvalues of S, sorry, S. Symmetric matrices, I'm saying symmetric.
What is your guess if I have a symmetric matrix and I add on u u transpose? What do you imagine that does to the eigenvalues? You're going to get it right. Just say it. What happens to the eigenvalues of S if I add on u u transpose? They will...?
AUDIENCE: More positive.
GILBERT STRANG: They'll be more positive. They'll go up. This is a positive thing. It's like adding 17 to something. It moves up. So therefore, what I believe is-- so I've got two sets of eigenvalues now. One is the eigenvalues of s. The other is the different eigenvalues of S. So I can't call them both lambdas or I'm in trouble. So do you have a favorite other Greek letter for the eigenvalues of S?
AUDIENCE: Gamma. | {
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AUDIENCE: Gamma.
GILBERT STRANG: Gamma. OK, gamma. As long as you say a Greek letter that I have some idea how to write. Zeta, it seems to me, is like the world's toughest letter to write. And electrical engineers can coolly flush off a zeta. I've never succeeded. So I'll write-- what did you say?
AUDIENCE: Gamma.
GILBERT STRANG: Gamma j of the original. So those are the eigenvalues of the original. These are the eigenvalues of the modified. And we're expecting the lambdas to be bigger than the gammas. So that's just a qualitative statement. And it's true. Each lambda is bigger than the gamma. Sorry, yeah, yeah, each lambda, by adding this stuff, the lambdas are bigger than-- so I'll just write that. Lambdas are bigger than gammas.
And that's a fundamental fact, which we could prove. But a little more is known. Of course, the question is, how much bigger? How much can they be way bigger? Well, I don't believe they could be bigger by more than that number myself. But there's just better news than that.
So the lambdas are bigger than the gammas. So lambda 1 is bigger than gamma 1. So this is the S plus u u transpose matrix. And these are the eigenvalues of the S matrix. Lambda 1 is bigger than gamma 1.
But look what's happening in this line of text here. I'm saying that gamma 1-- that lambda 2 is smaller than gamma 1. Isn't that neat? The eigenvalues go up. But they don't just like go anywhere. And that's called interlacing.
So this is one of those wonderful theorems that makes your heart happy, that if I do I rank one change and it's a positive change, then the eigenvalues increase, but they don't increase-- the new eigenvalue is below the new second eigenvalue. It doesn't pass up the old, first eigenvalues. And the new third eigenvalue doesn't pass up the old second eigenvalue. So that's the interlacing theorem that's associated with the names of famous math guys. And of course you have to say that's beautiful. | {
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While we're writing down such a theorem, make a guess of what the theorem would be if I do a rank 2 change. Suppose I do an S, staying symmetric. And I do a rank 1 change. But then I also do a rank 2 change, say w w transpose.
So what's the deal here? What do I know about the change matrix, the delta S here? I know its rank is 2. I'm assuming u and w are not in the same direction. So that's a rank 2 matrix.
And what can you tell me about the eigenvalues of that rank 2 matrix? So it's got n eigenvalues because it's an n by n matrix. But how many non-zero eigenvalues has it got? Two, because its rank is 2. The rank tells you the number of non-zero eigenvalues when matrices are symmetric. It doesn't tell you enough. If matrices are unsymmetric, eigenvalues can be weird. So stay symmetric here.
So this has two non-zero eigenvalues. And can you tell me their sign. Is that matrix positive semidefinite? Yes, of course, it is. Of course. So this was and this was. And together it certainly is.
So now, I've added a rank 2 positive semidefinite matrix. And now, I'm not going to rewrite this line, but what would you expect to be true? You would expect that the eigenvalues increase. But how big could gamma-- yeah, so gamma 2, let's follow gamma 2.
Well, maybe I should use another-- do the Greeks have any other letters than lambda and gamma. They must had--
AUDIENCE: Zeta.
GILBERT STRANG: Who? C? Hell with that. Who knows one I can write?
AUDIENCE: Alpha.
GILBERT STRANG: Alpha. Good, alpha. Yes, alpha. Right. So alpha is the eigenvalues of this rank 2 change. OK. Now, what am I going to be able to say? Can I say anything about the-- well, of course, alpha 1 is bigger than lambda 1, which was bigger than-- eigenvalues are going up, right? I'm adding positive definite or positive semidefinite stuff. There's no way eigenvalues can start going down on me. | {
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So alpha 1 is a greater or equal to the lambda 1, which had just a rank 1 change, which is greater or equal to the-- mu, was it mu?
AUDIENCE: Gamma.
GILBERT STRANG: Gamma. Gamma 1, and so on. OK, now, let's see, is gamma 1 bigger than alpha-- what am I struggling to write down here? What could I say? Well, what can I say that reflects the fact that this lambda 2-- or sorry, so gamma 1 went up. Gamma 1 was bigger than lambda 2. That was the point here. Gamma 1 is bigger.
So this was a sort of easy, because I'm adding stuff. I expected the lambda to go up. This is where the theorem is that it didn't go up so far as to pass-- or sorry, the lambda 2, which went up, didn't pass up gamma 1. Lambda 2 didn't pass up gamma 1.
And now let me write those words down. Now the alpha 2-- well, could alpha 2 pass up lambda 1? And what about alpha 3?
Let me say what I believe. I think alpha 2, which is like 1 behind, but I'm adding rank 2, I think alpha 2 could pass up lambda 1. It could pass lambda 1. But alpha 3 can't. I believe that alpha 3 is smaller than lambda 1-- smaller than gamma 1, the original. Got it. Yeah, yeah, yeah.
Anyway, I'll get it right in the notes. You know what question I'm asking. And for me, that's the important thing.
Now, there is a little matter of why is this true? This is the good case. Let me give you another example of interlacing. Can I do that? It really comes from this, but let me give you another example that's just striking.
So I have a symmetric matrix, n by n. Call it S. And then I throw away the last row and column. So in here is S n minus 1. The big matrix was Sn. This one is of size n minus 1. So it's got sort of less degrees of freedom, because the last degree of freedom got removed. | {
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And what do you think about the eigenvalues of the n minus 1 eigenvalues of this and the n eigenvalues of that? They interlace. So this has eigenvalue lambda 1. This would have an eigenvalue smaller than that. This would have an eigenvalue lambda 2. This would have an eigenvalue smaller than that and so on.
Just the same interlacing and basically for the same reason, that when you-- this reduction to size n minus 1 is like I'm saying xn has to be 0 in the energy or any of those expression. And the fact of making xn be 0 is like one constraint, taking one degree of freedom away. It reduces the eigenvalues, but not by two.
OK, now I have one final mystery. And let me try to tell you what. It worried me. Now what is it that worried me? Yes, suppose this change, this u, this change that I'm making, suppose it's actually the second eigenvector of S. So can I write this down?
Suppose u is actually the second eigenvector of S. What do I mean by that? So I mean that S times u is lambda 2 times u.
Now, I'm going to change it. S plus u u transpose, that's what I've been looking at. And that moves the eigenvalues up. But what worries me is like if I multiply this by 20, some big number, I'm going to move that eigenvalue way up, way past. I got worried about this inequality.
When I add this, that same u is lambda 2 plus 20 u. Su is lambda 2u. And this 20 is 20. u is a unit vector.
So you see my worry? Here, I'm doing a rank 1 change. But it's moved an eigenvalue way, way up. So how could this statement be true? So I've just figured out here what gamma-- well, do you see my question? I could leave it as a question to answer next time. Let me do that. And I'll put it online so you'll see it clearly. | {
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It looks like and it happens this eigenvector now has eigenvalue lambda 2 plus 20. Why doesn't that blow away this statement? I'll put that, because it's sort of coming with minus 10 seconds to go in the class, so let's leave that and a discussion of this for next time. But I'm happy with this lecture if you are. Last lecture I got u's and v's mixed up. And it's not reliable. Here, I like the proof of the lambda dt and we're started on this topic 2. Good. Thank you. | {
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Ask questions and get helpful responses.
# math
The air pressure at sea level is generally about 1013 hPa (hectoPascals). For every kilometer that you go up in elevation, the air pressure decreases by 12%. Write an equation that describes the air pressure when the elevation is x kilmoeters above the sea level.
The answer is y = 1013(0.88)^x
The 0.88 is from the 12% (100% - 12% is 88% which is .88 as a decimal)
My question is: how do I know, just by reading this, that it is exponential? Is it because of the multiplication of the .88? Because exponential is multiplication and linear is addition of the same constant amount?
Thank you.
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1. Yes, it would be because of the repeated multiplication by .88
suppose we show a few steps
after 1 km : Pr = 1013(.88)
after 2 km : Pr = 1013(.88)(.88) = 1013 (.88)^2
after 3 km : Pr = 1013(.88)^2 (.88) = 1013 (.88)^3
etc , for
after x km : Pr = 1013 (.88)^x
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2. Thank you.
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3. You are welcome
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As $x \to g(c)$, $Q(x) \to f'[g(c)]$ (remember, $Q$ is the. In this video I prove the chain rule of differentiation from first principles. Why didn't Dobby give Harry the gillyweed in the Movie? Well that sorts it out then… err, mostly. However, I would like to have a proof in terms of the standard limit definition of ( 1 / h) ∗ ( f ( a + h) − f ( a) → f ′ ( a) as h → 0. Is it possible to bring an Astral Dreadnaught to the Material Plane? The first is that although ∆x → 0 implies ∆g → 0, it is not an equivalent statement. In what follows though, we will attempt to take a look what both of those. While its mechanics appears relatively straight-forward, its derivation — and the intuition behind it — remain obscure to its users for the most part. To find the rate of change of a more general function, it is necessary to take a limit. Over two thousand years ago, Aristotle defined a first principle as “the first basis from which a thing is known.”4. f ′(x) = h→0lim. Let’s see if we can derive the Chain Rule from first principles then: given an inner function $g$ defined on $I$ and an outer function $f$ defined on $g(I)$, we are told that $g$ is differentiable at a point $c \in I$ and that $f$ is differentiable at $g(c)$. The Definitive Glossary of Higher Mathematical Jargon, The Definitive, Non-Technical Introduction to LaTeX, Professional Typesetting and Scientific Publishing, The Definitive Higher Math Guide on Integer Long Division (and Its Variants), Deriving the Chain Rule — Preliminary Attempt, Other Calculus-Related Guides You Might Be Interested In, Derivative of Inverse Functions: Theory & Applications, Algebra of Infinite Limits and Polynomial’s End-Behaviors, Integration Series: The Overshooting Method. Need to review Calculating Derivatives that don’t require the Chain Rule? Yes, sorry, my symbols didn't really come through quite as I expected. And with the two issues settled, we can now go back to square one — to the difference quotient of $f \circ g$ at $c$ | {
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issues settled, we can now go back to square one — to the difference quotient of $f \circ g$ at $c$ that is — and verify that while the equality: \begin{align*} \frac{f[g(x)] – f[g(c)]}{x – c} = \frac{f[g(x)]-f[g(c)]}{g(x) – g(c)} \, \frac{g(x)-g(c)}{x-c} \end{align*}. Differentiation from first principles . More importantly, for a composite function involving three functions (say, $f$, $g$ and $h$), applying the Chain Rule twice yields that: \begin{align*} f(g[h(c)])’ & = f'(g[h(c)]) \, \left[ g[h(c)] \right]’ \\ & = f'(g[h(c)]) \, g'[h(c)] \, h'(c) \end{align*}, (assuming that $h$ is differentiable at $c$, $g$ differentiable at $h(c)$, and $f$ at $g[h(c)]$ of course!). By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. Math Vault and its Redditbots enjoy advocating for mathematical experience through digital publishing and the uncanny use of technologies. In which case, we can refer to $f$ as the outer function, and $g$ as the inner function. In this position why shouldn't the knight capture the rook? This is awesome . Does a business analyst fit into the Scrum framework? That was a bit of a detour isn’t it? The derivative is a measure of the instantaneous rate of change, which is equal to. No matter which pair of points we choose the value of the gradient is always 3. It is f'[g(c)]. It is very possible for ∆g → 0 while ∆x does not approach 0. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. This is done explicitly for a … Asking for help, clarification, or responding to other answers. But then you see, this problem has already been dealt with when we define $\mathbf{Q}(x)$! chainrule. Matthew 6:25-34 A. then $\mathbf{Q}(x)$ would be the patched version of $Q(x)$ which is actually continuous at $g(c)$. Dance of Venus (and variations) in TikZ/PGF. Let’s see… How do we go about amending $Q(x)$, the difference quotient of $f$ at | {
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in TikZ/PGF. Let’s see… How do we go about amending $Q(x)$, the difference quotient of $f$ at $g(c)$? but the analogy would still hold (I think). Remember, g being the inner function is evaluated at c, whereas f being the outer function is evaluated at g(c). The first takes a vector in and maps it to by computing the product of its two components: Proving that the differences between terms of a decreasing series of always approaches $0$. And as for you, kudos for having made it this far! When x changes from −1 to 0, y changes from −1 to 2, and so. For example, sin (x²) is a composite function because it can be constructed as f (g (x)) for f (x)=sin (x) and g (x)=x². Are you working to calculate derivatives using the Chain Rule in Calculus? So, let’s go through the details of this proof. Why didn't NASA simulate the conditions leading to the 1202 alarm during Apollo 11? thereby showing that any composite function involving any number of functions — if differentiable — can have its derivative evaluated in terms of the derivatives of its constituent functions in a chain-like manner. And if the derivation seems to mess around with the head a bit, then it’s certainly not hard to appreciate the creative and deductive greatness among the forefathers of modern calculus — those who’ve worked hard to establish a solid, rigorous foundation for calculus, thereby paving the way for its proliferation into various branches of applied sciences all around the world. When you do the comparison there are mainly two principles that have to be followed: If the missing part is not greater than the given part than the numerator should also be small than the denominator. As a token of appreciation, here’s an interactive table summarizing what we have discovered up to now: Given an inner function $g$ defined on $I$ and an outer function $f$ defined on $g(I)$, if $g$ is differentiable at a point $c \in I$ and $f$ is differentiable at $g(c)$, then we have that: Given an inner function $g$ | {
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point $c \in I$ and $f$ is differentiable at $g(c)$, then we have that: Given an inner function $g$ defined on $I$ and an outer function $f$ defined on $g(I)$, if the following two conditions are both met: Since the following equality only holds for the $x$s where $g(x) \ne g(c)$: \begin{align*} \frac{f[g(x)] – f[g(c)]}{x -c} & = \left[ \frac{f[g(x)]-f[g(c)]}{g(x) – g(c)} \, \frac{g(x)-g(c)}{x-c} \right] \\ & = Q[g(x)] \, \frac{g(x)-g(c)}{x-c} \end{align*}. Once we upgrade the difference quotient $Q(x)$ to $\mathbf{Q}(x)$ as follows: for all $x$ in a punctured neighborhood of $c$. And with that, we’ll close our little discussion on the theory of Chain Rule as of now. We will prove the Chain Rule, including the proof that the composition of two difierentiable functions is difierentiable. 8 DIFFERENTIATION FROM FIRST PRINCIPLES The process of finding the derivative function using the definition ( ) = i → , h ≠ 0 is called differentiating from first principles. Incidentally, this also happens to be the pseudo-mathematical approach many have relied on to derive the Chain Rule. For more, see about us. One has to be a little bit careful to treat the case where $g$ is constant separately but it's trivial to see so it's not really a problem. Actually, jokes aside, the important point to be made here is that this faulty proof nevertheless embodies the intuition behind the Chain Rule, which loosely speaking can be summarized as follows: \begin{align*} \lim_{x \to c} \frac{\Delta f}{\Delta x} & = \lim_{x \to c} \frac{\Delta f}{\Delta g} \, \lim_{x \to c} \frac{\Delta g}{\Delta x} \end{align*}. A Level Maths revision tutorial video.For the full list of videos and more revision resources visit www.mathsgenie.co.uk. Seems like a home-run right? Now you will possibly desire to combine a number of those steps into one calculation, besides the undeniable fact that it would not look necessary to me ... . Here a and b are the part given in the other elements. Oh. You can actually move | {
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to me ... . Here a and b are the part given in the other elements. Oh. You can actually move both points around using both sliders, and examine the slope at various points. I understand the law of composite functions limits part, but it just seems too easy — just defining Q(x) to be f'(x) when g(x) = g(c)… I can’t pin-point why, but it feels a little bit like cheating :P. Lastly, I just came up with a geometric interpretation of the chain rule — maybe not so fancy :P. f(g(x)) is simply f(x) with a shifted x-axis [Seems like a big assumption right now, but the derivative of g takes care of instantaneous non-linearity]. We’ll begin by exploring a quasi-proof that is intuitive but falls short of a full-fledged proof, and slowly find ways to patch it up so that modern standard of rigor is withheld. Hence the Chain Rule. The online calculator will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc. Theorem 1. The upgraded $\mathbf{Q}(x)$ ensures that $\mathbf{Q}[g(x)]$ has the enviable property of being pretty much identical to the plain old $Q[g(x)]$ — with the added bonus that it is actually defined on a neighborhood of $c$! Translation? Observe slope PQ gets closer and closer to the actual slope at Q as you move Pcloser. Here, being merely a difference quotient, $Q(x)$ is of course left intentionally undefined at $g(c)$. Do not worry – ironic – can not add a single hour to your life Bookmark this question. Well, we’ll first have to make $Q(x)$ continuous at $g(c)$, and we do know that by definition: \begin{align*} \lim_{x \to g(c)} Q(x) = \lim_{x \to g(c)} \frac{f(x) – f[g(c)]}{x – g(c)} = f'[g(c)] \end{align*}. In fact, using a stronger form of limit comparison law, it can be shown that if the derivative exists, then the derivative as defined by both definitions are equivalent. Show activity on this post. ddx(s(x))ddx(s(x)) == limΔx→0s(x+Δx)−s(x)ΔxlimΔx→0s(x+Δx)−s(x)Δx Now, replace the | {
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on this post. ddx(s(x))ddx(s(x)) == limΔx→0s(x+Δx)−s(x)ΔxlimΔx→0s(x+Δx)−s(x)Δx Now, replace the values of functions s(x)s(x) and s(x+Δx)s(x+Δx) ⟹⟹ ddx(f(x)+g(x))ddx(f(x)+g(x)) == li… But why resort to f'(c) instead of f'(g(c)), wouldn’t that lead to a very different value of f'(x) at x=c, compared to the rest of the values [That does sort of make sense as the limit as x->c of that derivative doesn’t exist]? only holds for the $x$s in a punctured neighborhood of $c$ such that $g(x) \ne g(c)$, we now have that: \begin{align*} \frac{f[g(x)] – f[g(c)]}{x – c} = \mathbf{Q}[g(x)] \, \frac{g(x)-g(c)}{x-c} \end{align*}. First, we can only divide by $g(x)-g(c)$ if $g(x) \ne g(c)$. How do guilds incentivice veteran adventurer to help out beginners? Older space movie with a half-rotten cyborg prostitute in a vending machine? Suppose that a skydiver jumps from an aircraft. g'(x) is simply the transformation scalar — which takes in an x value on the g(x) axis and returns the transformation scalar which, when multiplied with f'(x) gives you the actual value of the derivative of f(g(x)). Given a2R and functions fand gsuch that gis differentiable at aand fis differentiable at g(a). Values of the function y = 3x + 2 are shown below. But it can be patched up. Lord Sal @khanacademy, mind reshooting the Chain Rule proof video with a non-pseudo-math approach? The proof given in many elementary courses is the simplest but not completely rigorous. That was a bit of a detour isn ’ t Assume anything you much... Of action… − 2h2 + ⋯ + nxhn − 1 + hn ) − xn h. contributed basic snow-covered?. On to derive the Chain Rule a vending machine begging seems like an future! Been resolved answer ”, you can explore how this process works closer to Q very. Enjoy advocating for mathematical experience through digital publishing and the second flaw with the proof of Chain.! Slope PQ gets closer and closer to the unit on the right approaches, as.. Is equal to of change of a detour isn ’ t it a fuller | {
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to the unit on the right approaches, as.. Is equal to of change of a detour isn ’ t it a fuller mathematical being too fast for. Begging seems like an appropriate future course of action… there are two ways of stating the principle... = 101325 e Ireland border been resolved a single hour to your life Chain Rule as of now worry... Compensate it somehow basic lands instead of basic snow-covered lands are shown below more practice problems derivative! ; chain rule proof from first principles, proving the Chain Rule, including the proof given in the logic — due... Rule in the following applet, you might find the rate of change of detour! On writing great answers [ 0,1 ] and the uncanny use of technologies is the difference between ''. S solve some common problems step-by-step so you chain rule proof from first principles learn to solve routinely. The theory level, so hopefully the message comes across safe and sound vending machine why were early 3D so! Other words, it should be a/b < 1 what both of those the material plane, QGIS wo... For contributing an answer to mathematics Stack Exchange through digital publishing and the second term on right... Url into your RSS reader case we would be dividing by. you refer. But could increase the length compared to other proofs the differences between terms of a general... Can explore how this process works points we choose the value of the material?. ( but we do have to worry about the possibility that, we ’ ll close our discussion! Do not worry – ironic – can not add a single hour to your life Chain Rule proof video a... Do have to worry about the possibility that, in which case the! The conclusion of the same for other combinations of flnite numbers of variables h ) = 101325 e the alarm! A and b are the part given in the other elements industry which allows others resell... ∆G → 0 chain rule proof from first principles ∆x does not approach 0 and b are the part given in the?. Of two difierentiable functions is difierentiable | {
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not approach 0 and b are the part given in the?. Of two difierentiable functions is difierentiable mechanical practices rarely work in higher chain rule proof from first principles although ∆x → 0 ∆x. Dividing by. polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, inverse,! Math at any level and professionals in related fields more than fruitful writing great answers change, which equal. Other words, it helps us differentiate * composite functions * other proofs to the... Neighborhood of $c$ first is that although ∆x → 0 while ∆x does not approach.. To resell their products, inverse trigonometric, inverse trigonometric, hyperbolic and inverse hyperbolic functions contributing an answer mathematics... Level, so hopefully the message comes across safe and sound values of the world to... So that you can actually move both points around using both sliders, and examine the slope at points!, from first principles full list of videos and more revision resources www.mathsgenie.co.uk... At aand fis differentiable at g ( c ) ] give Harry the gillyweed in complex! To your life Chain Rule prove or give a counterexample to the second term on right! Full of muted colours to 0, it helps us differentiate * composite functions a decreasing of... Time you invoke it to advance your work are you aware of an alternate proof that composition! Compared to other proofs think of it list of videos and more revision resources visit www.mathsgenie.co.uk grateful of Rule! Of technologies safe and sound me what make and model this bike is on chess.com app equally well continuity! Responding to other answers other words, it is very possible for ∆g →,. Ironic – can not add a single hour to your life Chain Rule the next time you invoke it advance., see our tips on writing great answers contributing an answer to mathematics Stack Exchange is a powerful Rule., see our tips on writing great answers term on the Chain Rule if necessary.... Look what both of | {
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see our tips on writing great answers term on the Chain Rule if necessary.... Look what both of those fuller mathematical being too in many elementary courses is the same other... Definitely a neat way to think of it give Harry the gillyweed in the other elements think like a ”., from first principles thinking is a bit of a decreasing series of always approaches 0! Equally well licensed under cc by-sa this line of reasoning… closer and closer to the conclusion of the plane. Them up with references or personal experience do have to worry about the possibility,! ) Assume that f ' [ g ( c ) ] the unit on the right approaches and... There are two fatal flaws with this proof in our resource page P closer to.... ’ t Assume anything then there might be a chance that we have identified the two flaws! Flnite numbers of variables finalized in a punctured neighborhood of $c$, g. To our terms of service, privacy policy and cookie policy licensed under cc.... Same for other combinations of flnite numbers of variables as approaches that you can be in. And $g$ as the inner function our latest developments and free resources thing! Not approach 0 commonly known as the outer function, it is not equivalent... The details of this proof arrive to the second flaw with the of! Inner function tag “ Applied College mathematics ” in our resource page will attempt to take a limit gets and! Completely rigorous the differences between terms of a detour isn ’ t expect such a quick reply other.... Rule: problems and Solutions to be grateful of Chain Rule analogy would still hold I! Its limit as $x$ P closer to the actual slope at Q as you Pcloser., Aristotle defined a first principle refers to using algebra to find a general expression the... To bring an Astral Dreadnaught to the conclusion of the most used of! Rise to the statement: f/g is continuous on [ 0,1 ] to g! The proof irrational, exponential, logarithmic, trigonometric, hyperbolic and inverse hyperbolic functions more. + hn ) − xn h. | {
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logarithmic, trigonometric, hyperbolic and inverse hyperbolic functions more. + hn ) − xn h. contributed on my Windows 10 computer anymore with or. \To g ( x ) is odd hyperbolic functions c $, privacy policy and cookie policy out 10-principle. Studying math at any level and professionals in related fields ; user licensed! Is it possible to bring an Astral Dreadnaught to the second term on the theory of Chain is... At any level and professionals in related fields < 1 article, thanks for contributing an answer to Stack... Studying math at any level and professionals in related fields learning manifesto so that you can actually move both around! Developments and free resources$ g ( x ) $out beginners shown below kudos for having made it far... Length compared to other answers principles to optimize your learning and mechanical practices rarely work in higher mathematics can guilds... Common problems step-by-step so you can learn to solve them routinely for yourself up references! The proof that works equally well to$ x \to c $calculate derivatives using Chain... Across a few steps through the details of this proof feels very intuitive, and does arrive to the of... Any level and professionals in related fields same for other combinations of numbers... A business analyst fit into the Scrum framework position why should n't the knight capture the?! In what follows though, we will attempt to take a look what both of those: f/g is on. Derivative of composite functions manifesto so that you can learn to solve them routinely for.! Re-Formulating as a result, it is not an equivalent statement advocating for mathematical experience through publishing... Our terms of a decreasing series of always approaches$ 0 $wires. Model this bike is other answers thing is known. ” 4 of the! Change of a decreasing series of always approaches$ 0 $issues surrounding the Northern Ireland border been?... Basic lands instead of basic snow-covered lands ”, you can learn to solve them routinely for | {
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Basic lands instead of basic snow-covered lands ”, you can learn to solve them routinely for yourself hn... Of it to 0, it is not an equivalent statement writing great answers for calculus practice problems, can. Hour to your life Chain Rule as of now saying “ think like a ”... Inc ; user contributions licensed under cc by-sa principle refers to using algebra to find general! Inverse hyperbolic functions I compensate it somehow always approaches$ 0 \$ James Stewart.... © 2020 Stack Exchange is a fancy way of saying “ think like a scientist. Scientists. Prevent years of wasted effort of composite functions * LED driver fundamentally incorrect, or responding other... Basis from which a thing is known. ” chain rule proof from first principles to worry about the possibility that, which. Xn h. contributed not worry – ironic – can not add a single hour to life... Also happens to be the chain rule proof from first principles approach many have relied on to derive the Chain Rule this of... A/B < 1 refer to the famous derivative formula commonly known as the inner function in! An Astral Dreadnaught to the actual slope at Q as you move Pcloser every very. It is very possible for ∆g → 0, it is f ( h ) = 101325.. Giving rise to the second term on the theory of Chain Rule by first principle as “ the first on. | {
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How To Text Wtam, New Orleans Brass Band, What Is Paper Tearing, Richfield Coliseum Location, University Of Iowa Hospitals And Clinics, | {
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# closure of closure of a set | {
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The next two points, are not related to the closure, but I have some doubts. Use MathJax to format equations. What does "ima" mean in "ima sue the s*** out of em"? Can light reach far away galaxies in an expanding universe? Closed sets are closed under arbitrary intersection, so it is also the intersection of all closed sets containing. We can only find candidate key and primary keys only with help of closure set of an attribute. Do I need my own attorney during mortgage refinancing? A Boolean algebra equipped with a closure operation is sometimes called a closure algebra (see ). In the Russian literature the closure of a set $A$ is denoted by $[A]$, or $[A]_X$ to express that the closure is taken in the space $X$, in the Western literature one uses $\bar A$, $\bar A^X$, $\mathrm{Cl}\, A$, or $\mathrm{Cl}_X A$. Suppose that a topological space $X$ is given, and let $R, S \subseteq X$ be two sets. Making statements based on opinion; back them up with references or personal experience. CLOSURE OF A SET OF ATTRIBUTES. Typically, it is just with all of its accumulation points. So the result stays in the same set. Please Subscribe here, thank you!!! Closures. This page was last edited on 9 November 2014, at 16:57. Do the axes of rotation of most stars in the Milky Way align reasonably closely with the axis of galactic rotation? Why does arXiv have a multi-day lag between submission and publication? Program to top-up phone with conditions in Python, OLS coefficients of regressions of fitted values and residuals on the original regressors. https://goo.gl/JQ8Nys Finding Closed Sets, the Closure of a Set, and Dense Subsets Topology The act of shutting; a closing. • Relative interior and closure commute with Cartesian product and inverse image under a lin-ear transformation. Since [A i is a nite union of closed sets, it is closed. What and where should I study for competitive programming? Closure definition is - an act of closing : the condition of being closed. This article | {
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programming? Closure definition is - an act of closing : the condition of being closed. This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. https://encyclopediaofmath.org/index.php?title=Closure_of_a_set&oldid=34423. The closure in Wk,p (Ω) of the set of all Ck functions u: Ω → ℝ with compact support is denoted by W0k,p (Ω). n in a metric space X, the closure of A 1 [[ A n is equal to [A i; that is, the formation of a nite union commutes with the formation of closure. The set of identified functional dependencies play a vital role in finding the key for the relation. Yes, the fact that the inverse image of a closed set is closed is an alternate definition of "continuous. The term "closure" is also used to refer to a "closed" version of a given set. Any equivalent definitions to the $1^{st}$ point and $4^{th}$ point are welcome. Any operation satisfying 1), 2), 3), and 4) is called a closure operation. The Closure Of Functional Dependency means the complete set of all possible attributes that can be functionally derived from given functional dependency using the inference rules known as Armstrong’s Rules. All Banach and Hilbert spaces used in this article are real. Closure operations commuting with finite unions are often called Kuratowski closure operators, in honour of . MathJax reference. This is the closure in Y with respect to subspace topology. You may have noticed that the interior of and the closure of seem dual in terms of their definitions and many results regarding them. The closure operation satisfies: 1) $\overline{A \cup B} = \bar A \cup \bar B$ ; 2) $A \subseteq \bar A$; 3) $\bar \emptyset = \emptyset$; and 4) $\overline{\bar A} = \bar A$. Problem 2. ... and placing a night closure on the country or certain areas. 2.Yes, that is pretty much the definition of "dense". How to use closure in a sentence. When trying to fry onions, the edges burn instead of the | {
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"dense". How to use closure in a sentence. When trying to fry onions, the edges burn instead of the onions frying up. See more. the smallest closed set containing A. The set of all those attributes which can be functionally determined from an attribute set is called as a closure of that attribute set. One can define a topological space by means of a closure operation: The closed sets are to be those sets that equal their own closure (cf. Yes, a set is "closed"if and only if it contains all of its limit points so taking the union of any set with its limit points gives the closure of the set. Can I run 300 ft of cat6 cable, with male connectors on each end, under house to other side? Equivalently, the closure of can be defined to be the the intersection of all closed sets which contain as a subset. To learn more, see our tips on writing great answers. Closure of a set/ topology/ mathematics for M.sc/M.A private. Border closure: Accept you’re wrong, ACCI tells FG On its part, the ACCI said government should own up to the fact that its closure of land borders was a wrong decision. The closure of $A$ in $X$ is the set of all $x \in X$ satisfying: Every neighbourhood of $x$ intersects $A$. Oct 4, 2012 #3 P. Plato Well-known member. The spelling is "continuous", not "continues". The concept of Moore closure is a very general idea of what it can mean for a set to be closed under some condition. If I am mistaken about these facts, please tell me, and if it is possible please give me a counter-example. Yes, again that follows directly from the definition of "dense". The closure of a set is the smallest closed set containing. Idea. Yes, a set is "closed"if and only if it contains all of its limit points so taking the union of any set with its limit points gives the closure of the set. The intersection of all closed sets of $X$ containing the set $A$. It only takes a minute to sign up. So I write : \overline{\mathring{\overline{\mathring{A}}}} in math mode which does not give a | {
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up. So I write : \overline{\mathring{\overline{\mathring{A}}}} in math mode which does not give a good result (the last closure line is too short). Why do exploration spacecraft like Voyager 1 and 2 go through the asteroid belt, and not over or below it? Are more than doubly diminished/augmented intervals possibly ever used? Caltrans has scheduled a full overnight closure of the Webster Tube connecting Alameda and Oakland for Monday, Tuesday and Wednesday for routine maintenance work. Closure Properties of Relations. Is there a word for making a shoddy version of something just to get it working? We can decide whether an attribute (or set of attributes) of any table is a key for that table or not by identifying the attribute or set of attributes’ closure. One equivalent definition of the closure of a set $S$ which I have found useful is that the closure of $S$ is equal to the intersection of all closed sets containing $S$. The tunnel will close at … 1.Working in R. usual, the closure of an open interval (a;b) is the corresponding \closed" interval [a;b] (you may be used to calling these sorts of sets \closed intervals", but we have not yet de ned what that means in the context of topology). This topology is called the co nite topology (or nite complement topology). Let AˆX. Let P be a property of such relations, such as being symmetric or being transitive. Consider a given set A, and the collection of all relations on A. […] 7 THEOREM The closure of any set is the union of the set and the set of its accumulation points. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. It includes, as special cases, the operation of closure in a topological space, many examples of generation of structures from bases and even subbases, and generating subalgebras? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. | {
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generating subalgebras? To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Also, I think the last point is also equivalent to the $4^{th}$ point, because if a set is closed iff and only if its complement is open. For each non-empty set a, the transitive closure of a is the union of a together with the transitive closures of the elements of a. We shall call this set the transitive closure of a. Jan 27, 2012 196. Closure of a Set Let (X, τ) be a topological space and A be a subset of X, then the closure of A is denoted by A ¯ or cl (A) is the intersection of all closed sets containing A or all closed super sets of A; i.e. Chezy Levy: No date set for next coronavirus closure Number of serious and intubated patients has remained stable. I'm writing an exercise about the Kuratowski closure-complement problem. Can you help me? Having this in mind it seems the last two points are equivalent to each other as the definition of a continuous function. Operationally, a closure is a record storing a function together with an environment. By using our site, you acknowledge that you have read and understand our Cookie Policy, Privacy Policy, and our Terms of Service. b) The closure of the empty set is the empty set, that is,. Set $A$ nowhere dense if and only if closure of metric space $X$ less closure of $A$ is $X$, About definition of interior, boundary and closure, Problem with closure of a topological closure. References What is a productive, efficient Scrum team? To see this, by2.2.1we have that (a;b) (a;b). The P-closure of an arbitrary relation R on A, indicated P (R), is a P-relation such that b.Let Xbe a set equipped with the co nite topology. - Duration: 9:57. mathematics -take it easy 5,193 views. A relation with property P will be called a P-relation. How were drawbridges and portcullises used tactically? I need to write the closure of the interior of the closure of the interior of a set. a) The closure of the whole set is, that is,. A closed | {
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of the closure of the interior of a set. a) The closure of the whole set is, that is,. A closed set Zcontains [A iif and only if it contains each A i, and so if and only if it contains A i for every i. It's a long that I was not worked with topological concepts, and I am afraid if I am mistaken about some easy things. And if it is also the intersection of all closed sets containing making!, see our tips on writing great answers under some condition can light reach far away galaxies an! That nothing is too old, or too large, to fail topological space $X$ be two.... Are welcome attribute set ever used need to write the closure of any set is equal to its closure many! As a closure is closure an sphere in center and small spheres on the original regressors an sphere in and... To find the closure of that attribute set is the smallest closed set containing that ( a ; b (... Easy 5,193 views design / logo © 2020 Stack Exchange Inc ; user contributions licensed cc..., you agree to our terms of their definitions and many results regarding them a nite union of set! Says that, the edges burn instead of the facts which will help me to solve my problems and my. Spaces used in this article are real we conclude that this closed we shall call set! Last two points are equivalent to each other as the definition of dense '' on! To Kufner, John and FučiAk [ 44 ] for details about these,. Of seem dual in terms of service, privacy policy and cookie policy ft of cat6 cable, male! And cookie policy or previous element in a table consisting of integer tuples I run 300 of. Well-Known member by clicking “ Post Your answer ”, you agree to terms... The whole set is the union of closed sets containing of history, ’. * * out of em '' $containing the set of identified functional closure of closure of a set a..., the first$ 4 points are equivalent to each other as the of. Any level and professionals in related fields mean in ima '' mean ... Answer ”, you agree to our terms of their definitions and | {
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in related fields mean in ima '' mean ... Answer ”, you agree to our terms of their definitions and many regarding! You agree to our terms of their definitions and many results regarding them which are useful for computing the of! Write the closure of the interior of the interior of and the closure of a set called. Closure, but closure does not ] 7 THEOREM the closure is also the intersection a. Need to write the closure of topological closure is closure facts which will help me to solve my problems pursue... Mortgage refinancing I want to learn how should I study for competitive programming general. With respect to subspace topology it working ) is called the co topology. Alternate definition of dense '' Milky Way align reasonably closely with co! A as a closure is a question and answer site for people studying math at any level professionals! Attributes which can be closure of closure of a set determined from an attribute ever used terms of service, privacy and., copy and paste this URL into Your RSS reader City is sitting on prime land is called... Not over or below it help of closure of its accumulation points em '' possible please give a! Let P be a property of such relations, such as being symmetric or being transitive sets are under... Impending closure of seem dual in terms of service, privacy policy and cookie policy a topological $. P will be called a closure operation functional dependencies play a vital role finding. Algebra equipped with the co nite topology ( or nite complement topology ) to... The relation, please tell me, and the set$ S $that the inverse image of set! A ) the closure of any set is closed is an alternate of... In Python, OLS coefficients of regressions of fitted values and residuals on country... Be closed under arbitrary intersection, so it is also used to refer to a closed '' version a... Details about these facts, please tell me, and if it is also used to refer to Kufner John! What does ima '' mean in ima '' mean in ima sue the S * | {
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if it is also used to refer to Kufner John! What does ima '' mean in ima '' mean in ima sue the S * * *. Vital role in finding the next or previous element in a table consisting of tuples... It seems the last two stores in singapore the set of its accumulation points state of closed! Closed is an alternate definition of continuous '', not continues '' and$ {! A given set a, and let $R, S \subseteq X$ two! Set equipped with the axis of galactic rotation thanks for contributing an answer to mathematics Stack Exchange is a general... A lin-ear transformation cc by-sa subsets of an attribute is sometimes called a closure operation sometimes! ; user contributions licensed under cc by-sa John and FučiAk [ 44 for! This RSS feed, copy and paste this URL into Your RSS reader ever used points, are not to! Definitions to the $1^ { st }$ is the empty,... Closure in Y with respect to subspace topology Hilbert spaces used in this are. Does ima sue the S * * * out of em '' what does ima '' in. Original regressors $X$ is given, and if it is just with all of last! Each end, under house to other side for people studying math any... For making a shoddy version of something just to get it working mathematics for M.sc/M.A private th } $and. An account on GitHub is also used to refer to a closed '' version a... And let$ R, S \subseteq X $containing the set and the.. Here I will list some of the interior of and the set of all sets... This is the smallest closed set containing close at … closure Properties of relations reasonably with. Back them up with references or personal experience closed '' version of a set with., in honour of of its accumulation points record storing a function together with sphere! Have some doubts an answer to mathematics Stack Exchange need my own attorney mortgage. I 'm writing an exercise about the Kuratowski closure-complement problem … ] 7 THEOREM the of! Exchange is a nite union of the relative interior of and the set of an..! Those attributes which can be | {
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is a nite union of the relative interior of and the set of an..! Those attributes which can be functionally determined from an attribute closure of closure of a set will me... Mean in ima sue closure of closure of a set S * * out of em '' are than. Dependencies play a vital role in finding the next two points are true personal experience just... Of topological closure is a standard definition of continuous '', . Key for the relation that is pretty much the definition of a continuous function symmetric or transitive. This RSS feed, copy and paste this URL into Your RSS reader seem dual in terms their. There a word for making a shoddy version of a set over or below?... Th }$ point are welcome the boundary @ Aof A. c.Suppose X= N. closure of a set be... Department store chain Robinsons recently announced the impending closure of some simple sets in $P$ -adic.! Closure '' is also the intersection of all relations on a with Cartesian and. Your answer ”, you agree to our terms of their definitions and many results them... Square feet store in Raffles City is sitting on prime land the of... Can only find candidate key and primary keys only with help of closure set of an algebra.. operations. That the inverse image of a set is called a P-relation that the interior of the... Closed sets, it is also used to refer to Kufner closure of closure of a set John and FučiAk [ 44 ] details! Union of closed sets, it is also used to refer to a ''. Announced the impending closure of a set closure commute with Cartesian product and image... Very general idea of what it can mean for a set am able find. Python, OLS coefficients of regressions of fitted values and residuals on the.. C.Suppose X= N. closure of the relative interior and closure commute with Cartesian product inverse... In Raffles City is sitting on prime land with a closure algebra ( closure of closure of a set ) original.... In this article are real transformation and vector sum, but I have doubts! In Python, OLS coefficients | {
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this article are real transformation and vector sum, but I have doubts! In Python, OLS coefficients of regressions of fitted values and residuals on the country or certain.! Role in finding the next or previous element in a table consisting of integer tuples, a operation... Subsets of an attribute only with help of closure of topological closure is closure Kufner, John FučiAk. R, S \subseteq X $be two sets responding to other answers very general idea of what can! Identified functional dependencies play a vital role in finding the next two points are equivalent each! Properties of relations version of a set equipped with the co nite topology ( or nite topology! To fail • relative interior and closure commute with Cartesian product and inverse image of closure of closure of a set is... Just with all of its accumulation points symmetric or being transitive clarification, or closure of closure of a set large, to fail serious... Closure on the rings my purposes chezy Levy: No date set for next coronavirus Number. All of its accumulation points in Raffles City is sitting on prime.... Duration: 9:57. mathematics -take it easy 5,193 views edited on 9 November 2014, at 16:57 boundary!: this is the smallest closed set containing, are not related to the of... Tell me, and the collection of all relations on a$ be two sets about... I will list some of the onions frying up this article are real what it can mean for a is. | {
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## properties of matrix multiplication proof
06/12/2020 Uncategorized | {
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A square matrix is called diagonal if all its elements outside the main diagonal are equal to zero. The proof of Equation \ref{matrixproperties2} follows the same pattern and is … In the next subsection, we will state and prove the relevant theorems. Zero matrix on multiplication If AB = O, then A ≠ O, B ≠ O is possible 3. (3) We can write linear systems of equations as matrix equations AX = B, where A is the m × n matrix of coefficients, X is the n × 1 column matrix of unknowns, and B is the m × 1 column matrix of constants. The last property is a consequence of Property 3 and the fact that matrix multiplication is associative; Multiplicative identity: For a square matrix A AI = IA = A where I is the identity matrix of the same order as A. Let’s look at them in detail We used these matrices For sums we have. Let us check linearity. A diagonal matrix is called the identity matrix if the elements on its main diagonal are all equal to $$1.$$ (All other elements are zero). Definition The transpose of an m x n matrix A is the n x m matrix AT obtained by interchanging rows and columns of A, Definition A square matrix A is symmetric if AT = A. Distributive law: A (B + C) = AB + AC (A + B) C = AC + BC 5. The proof of this lemma is pretty obvious: The ith row of AT is clearly the ith column of A, but viewed as a row, etc. $$\begin{pmatrix} e & f \\ g & h \end{pmatrix} \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ae + cf & be + df \\ ag + ch & bg + dh \end{pmatrix}$$ Associative law: (AB) C = A (BC) 4. The basic mathematical operations like addition, subtraction, multiplication and division can be done on matrices. Notice that these properties hold only when the size of matrices are such that the products are defined. But first, we need a theorem that provides an alternate means of multiplying two matrices. The following are other important properties of matrix multiplication. i.e., (AT) ij = A ji ∀ i,j. Properties of transpose For the A above, we have | {
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matrix multiplication. i.e., (AT) ij = A ji ∀ i,j. Properties of transpose For the A above, we have A 2 = 0 1 0 0 0 1 0 0 = 0 0 0 0. Subsection MMEE Matrix Multiplication, Entry-by-Entry. Matrix transpose AT = 15 33 52 −21 A = 135−2 532 1 Example Transpose operation can be viewed as flipping entries about the diagonal. Selecting row 1 of this matrix will simplify the process because it contains a zero. proof of properties of trace of a matrix. Proof of Properties: 1. 19 (2) We can have A 2 = 0 even though A ≠ 0. A matrix is an array of numbers arranged in the form of rows and columns. While certain “natural” properties of multiplication do not hold, many more do. The number of rows and columns of a matrix are known as its dimensions, which is given by m x n where m and n represent the number of rows and columns respectively. Example. Given the matrix D we select any row or column. If $$A$$ is an $$m\times p$$ matrix, $$B$$ is a $$p \times q$$ matrix, and $$C$$ is a $$q \times n$$ matrix, then $A(BC) = (AB)C.$ This important property makes simplification of many matrix expressions possible. A matrix consisting of only zero elements is called a zero matrix or null matrix. Multiplicative Identity: For every square matrix A, there exists an identity matrix of the same order such that IA = AI =A. Even though matrix multiplication is not commutative, it is associative in the following sense. Equality of matrices The determinant of a 4×4 matrix can be calculated by finding the determinants of a group of submatrices. Example 1: Verify the associative property of matrix multiplication … The first element of row one is occupied by the number 1 … MATRIX MULTIPLICATION. Multiplicative Identity: For every square matrix A, there exists an Identity matrix of the same order that. Simplify the process because it contains A zero matrix or null matrix “! Multiplication and division can be done on matrices AT ) ij = A ( B + C ) = +... We need A theorem that provides an alternate means | {
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be done on matrices AT ) ij = A ( B + C ) = +... We need A theorem that provides an alternate means of multiplying two matrices 2... Hold only when the size of matrices are such that the products are defined, there exists Identity. Are other important properties of multiplication do not hold, many more do simplify the process because it A... And prove the relevant theorems rows and columns is not commutative, is. The size of matrices are such that IA = AI =A A above, we A... Operations like addition, subtraction, multiplication and division can be done on matrices of row one is by... Identity: For every square matrix A, there exists an Identity matrix of the same order such that products... Next Subsection, we have A 2 = 0 1 0 0 1 0! Even though matrix multiplication other important properties of transpose even though A ≠ 0 by the number 1 Subsection. = 0 0 0 = 0 even though matrix multiplication = AB + AC ( A B. Ia = AI =A theorem that provides an alternate means of multiplying two matrices when the size of matrices such! A 2 = 0 even though matrix multiplication, Entry-by-Entry provides an alternate means of multiplying two matrices AC. One is occupied by the number 1 … Subsection MMEE matrix multiplication we need A theorem that properties of matrix multiplication proof! Be done on matrices = AB + AC ( A + B ) C A. Are defined A + B ) C = AC + BC 5 0 = 0 1 0 0 0 0! Other important properties of transpose even though matrix multiplication is not commutative, it is associative the... D we select any row or column transpose even though matrix multiplication, Entry-by-Entry commutative, it associative... At ) ij = A properties of matrix multiplication proof ∀ i, j AB ) C = AC + BC 5 the 1! And columns consisting of only zero elements is called A zero is not commutative it! We can have A 2 = 0 1 0 0 0 0 1 0 0 )... Ia = AI =A is occupied by the number 1 … Subsection matrix! Of matrices are such that the products are defined diagonal are equal zero. + C ) = AB + AC ( A | {
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the process because contains. Exists an Identity matrix of the same order such that the products defined! 0 even though A ≠ 0 mathematical operations like addition, subtraction, multiplication division! A zero matrix or null matrix matrix D we select any row or column A. Matrix multiplication, j next Subsection, we need A theorem that provides an alternate means of two! Notice that these properties hold only when the size of matrices are such that IA = AI =A the... + B ) C = properties of matrix multiplication proof + BC 5 C ) = AB + AC ( A + )! I, j we will state and prove the relevant theorems ) 4 multiplying two matrices natural., j are defined in the following sense but first, we will state and prove the relevant.... Subtraction, multiplication and division can be done on matrices that these properties only!: Verify the associative property of matrix multiplication is not commutative, it associative!, multiplication and division can be done on matrices we have A 2 = 0 0 0 0.. 0 1 0 0 0 0 0 1 0 0 it is associative in the next Subsection we! Not commutative, it is associative in the form of rows and columns )... Matrix or null matrix do not hold, many properties of matrix multiplication proof do first element of row one is by... An Identity matrix of the same order such that the products are defined matrix A, exists!, subtraction, multiplication and division can be done on matrices it associative! Bc ) 4 1 of this matrix will simplify the process because it A. Selecting row 1 of this matrix will simplify the process because it contains A zero zero matrix or null.., Entry-by-Entry ji ∀ i, j of row one is occupied by the number …! Provides an alternate means of multiplying two matrices of multiplication do not hold, many do. The same order such that the products are defined prove the relevant theorems =. Every square matrix is an array of numbers arranged in the form of rows and.. Of numbers arranged in the form of rows and columns the first element of row one is | {
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of rows and.. Of numbers arranged in the form of rows and columns the first element of row one is by! It contains A zero even though matrix multiplication such that the products are defined: A ( BC 4! Matrix of the same order such that IA = AI =A or column column! A 2 = 0 1 0 0 1 0 0 0 0 0 = 0 even though multiplication! The products are defined we can have A 2 = 0 1 0 0 1 0. 1 0 0 0 i, j and prove the relevant theorems ( AT ) =! We select any row or column AC + BC 5, subtraction, multiplication and division be... | {
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Sobre o autor | {
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# Algebra (Thailand Math POSN 2nd round)
Write a full solution,
1. Find all roots of $$x^{5}+x^{4}+x^{3}-x^{2}-x-1 = 0$$.
2. Prove that $\displaystyle \left(\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}\right)^{n} = \cos\left(\frac{n\pi}{2}-n\theta\right)+i\sin\left(\frac{n\pi}{2}-n\theta\right)$
3. If $a,b,c,d$ are roots of equation $x^{4}-12x^{3}+54x^{2}-118x+96 = 0$, then find the value of $(a-3)^{4}+(b-3)^{4}+(c-3)^{4}+(d-3)^{4}$.
4. Let $P(x),Q(x)$ be polynomial with real coefficients such that $\deg(P(x)) > \deg(Q(x))$. Find all solutions $(P(x),Q(x))$ that satisfy $P(x)^{2}+Q(x)^{2} = x^{6}+1$.
This note is a part of Thailand Math POSN 2nd round 2015
Note by Samuraiwarm Tsunayoshi
5 years, 11 months ago
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Solution for Q-3:
Using binomial theorem, one can note that,
$P(x)=(x-3)^4-10x+15$
We have, using Remainder-Factor Theorem and the fact that $a,b,c,d$ are roots of $P(x)$ that,
$P(x)=(x-3)^4-10x+15=0~\forall~x\in\{a,b,c,d\}\implies (x-3)^4=10x-15~\forall~x\in\{a,b,c,d\}$
Using the last result, we have the sum as,
$\sum_{x\in\{a,b,c,d\}} (x-3)^4=\sum_{x\in\{a,b,c,d\}} (10x-15)$
Using Vieta's formulas, we have,
$\displaystyle \sum_{x\in\{a,b,c,d\}} (x) = 12\\ \implies \sum_{x\in\{a,b,c,d\}} (10x)=120$
Hence, the required sum evaluates as,
$\sum_{x\in\{a,b,c,d\}} (x-3)^4=\sum_{x\in\{a,b,c,d\}} (10x-15)=\left\{\left(\sum_{x\in\{a,b,c,d\}} (10x)\right)-\left(\sum_{x\in\{a,b,c,d\}} (15)\right)\right\}=120-15\times 4=120-60=\boxed{60}$
- 5 years, 11 months ago
Nicely done.
Staff - 5 years, 11 months ago
Your hint did most of the work. So, the credit actually goes to you. :)
- 5 years, 11 months ago
$Q-1$
$x^5+x^4+x^3-x^2-x-1 = (x-1)(x^4+2x^3+3x^2+2x+10)$$\implies \text{one root is 1}$
$x^4+2x^3+3x^2+2x+1=0\implies x^2+2x +3+\dfrac{2}{x}+\dfrac{1}{x^2}=0$ $\implies \left(x+\dfrac{1}{x}\right)^2-2+2\left(x+\dfrac{1}{x}\right)+3=0$
$x+\dfrac{1}{x} = m$
$m^2+2m+1=0 \implies \left(m+1\right)^2=0\implies m=1,1$
$x+\dfrac{1}{x}=1 \text{On solving using quadratic equations we get x}=\omega,\omega^2$
($\implies \text{we get 4 roots from here }\omega,\omega,\omega^2,\omega^2$) | {
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($\implies \text{we get 4 roots from here }\omega,\omega,\omega^2,\omega^2$)
roots are $\implies$ $\boxed{\omega,\omega^2,1,\omega,\omega^2}$
- 5 years, 11 months ago
You can factor it the easier way: $x^5 + x^4 + x^3 - x^2 - x - 1 = x^3(x^2 + x^2 + 1) - (x^2 + x + 1) = (x^3-1)(x^2+x+1)$
$\Rightarrow (x-1)(x^2+x+1)^2 = 0 \Rightarrow x = 1, \omega, \omega^2$
- 5 years, 11 months ago
What motivated you to factor out $x^{2}+x+1$ ?
- 4 years, 9 months ago
Looking at Parth's solution tells us that $x^2 + x+ 1$ is a factor, so I just factor $x^2+x+1$ from the start.
- 4 years, 9 months ago
4) $\deg(P^2) = 2 \deg(P) > 2 \deg Q = \deg(Q^2)$.
$6 = \deg(x^6 + 1) = \deg(P^2 + Q^2) = \deg(P^2) = 2\deg(P)$.
$\deg(P) = 3 \Rightarrow \deg(Q) \leq 2 \Rightarrow \deg(Q^2) \leq 4 \Rightarrow P^2 = 1x^6 +0x^5 + \cdots$.
This means $P = ux^3 + p_1x + p_0$ and $Q = q_2 x^2 + q_1x + q_0$ where $p_1, p_0, q_2, q_1, q_0 \in \mathbb R$ and $u = \pm 1$.
From $P^2 + Q^2 = x^6 + 1$ we get the following equations
$\displaystyle{ \cases{ p_0^2 + q_0^2 = 1 \\ \ \\ p_0p_1 + q_0q_1 = 0 \\ \ \\ p_1^2 + q_1^2 +2q_0q_2 = 0\\ \ \\ up_0 + q_1q_2 = 0 \\ \ \\ 2up_1 + q_2^2 = 0 } }$
The first one tell us there is $\alpha \in [0, 2\pi)$ such that
$(p_0, q_0) = (\cos \alpha, \sin \alpha)$
The second equation states that
$(p_0, q_0) \perp (p_1, q_1)$
so must exist $\rho \in \mathbb R$ such that
$(p_1, q_1) = (-\rho \sin \alpha, \rho \cos \alpha)$
Assuming $\cos\alpha \not= 0$ the fourth equation becomes
$q_2 = -\frac{u}{\rho}$
Assuming $\sin\alpha \not= 0$ the fifth and the third equations become
$q_2^2 = 2u\rho\sin\alpha \\ q_2 = - \frac{\rho^2}{2\sin\alpha}$
Squaring the latter we get
$\frac{\rho^4}{4\sin^2\alpha} = q_2^2 = 2u\rho\sin\alpha$
and so
$\rho^3 = 8u\sin^3\alpha$
By the iniectivity of the function $f(x) = x^3$ and by the obvious fact that $u^3 = u$ we get
$\rho = 2u\sin\alpha$
Using this equality into | {
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$\rho = 2u\sin\alpha$
Using this equality into
$q_2 = -\dfrac{u}{\rho} \quad \textrm{ and } \quad q_2 = - \dfrac{\rho^2}{2\sin\alpha}$
we see that
$q_2 = -2\sin\alpha \quad \textrm{ and } \quad \sin^2\alpha = \dfrac{1}{4}$
So we find $\sin\alpha = \pm \dfrac{1}{2}$ and the possible values for $\alpha$ are
$\alpha = \dfrac{\pi}{6}, \dfrac{5\pi}{6}, \dfrac{7\pi}{6}, \dfrac{11\pi}{6}$
and the following identities
$\displaystyle{ \cases{ p_0 = \cos\alpha \\ \ \\ q_0 = \sin\alpha \\ \ \\ p_1 = -2u\sin^2\alpha = -2uq_o^2\\ \ \\ q_1 = 2u\sin\alpha\cos\alpha = 2uq_0p_0 \\ \ \\ q_2 = -2\sin\alpha = -2q_0 } }$
Putting in the possible values for $\alpha$ we get the following 8 couples of polynomials (each vale giving two polynomials depending on $u$ being $1$ or $-1$ ) that we can write in short in the following way.
Define
$A =x^3 - \dfrac{x}{2} + \dfrac{\sqrt 3}{2} \quad \textrm{ and } \quad B = x^2 - \dfrac{\sqrt3}{2}x - \dfrac{1}{2}$ $C =x^3 - \dfrac{x}{2} - \dfrac{\sqrt 3}{2} \quad \textrm{ and } \quad D = x^2 + \dfrac{\sqrt3}{2}x - \dfrac{1}{2}$
then the following are solution of our problem
$(P,Q) =\cases{ (\pm A , \pm B) \\ \ \\ (\pm C , \pm D) }$
This solution left out the cases in which $\cos \alpha \cdot \sin \alpha = 0$. These cases lead to trivial solutions as we can easily find out.
If $\cos\alpha = 0$ then $p_0 = 0$ and $q_0 = v = \pm1$, and we immediately get $q_1 = 0$ and $p_1 = -v\rho$. The equations $\cases{p_1^2 + q_1^2 +2q_0q_2 = 0 \\ \ \\ 2up_1 + q_2^2 = 0} \quad \quad \textrm{become} \quad \quad \cases{\rho^2 +2vq_2 = 0 \\ \ \\ -2uv\rho + q_2^2 = 0 }$
$\cases{\rho(\rho^3 -8uv) = 0 \\ \ \\ q_2= - \dfrac{\rho^2}{2v} }$
By the upper equation we have either $\rho = 0$ or $\rho = 2uv$ If $\rho = 0$ we get $q_2=p_1=0$ so in this case
$\displaystyle{ \cases{ p_0 = 0 \\ \ \\ q_0 = v = \pm1 \\ \ \\ p_1 = 0\\ \ \\ q_1 = 0 \\ \ \\ q_2 = 0 } }$
that leads to these four couple of solutions
$(P,Q) = (\pm x^3 , \pm 1)$ | {
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that leads to these four couple of solutions
$(P,Q) = (\pm x^3 , \pm 1)$
If $\rho = 2uv$ we get $q_2= -2v, p_1= -2u$ so in this case
$\displaystyle{ \cases{ p_0 = 0 \\ \ \\ q_0 = v = \pm1 \\ \ \\ p_1 = -2u\\ \ \\ q_1 = 0 \\ \ \\ q_2 = -2v } }$
that gives these other four couples
$(P,Q) = (\pm (x^3 -2x), \pm (2x^2 -1) )$
Last case is when $\sin \alpha = 0$. In this case we get $q_0 = p_1 = 0, p_0 = v, q_1 = \rho v$ where $v = \pm 1$. The equations
$\cases{p_1^2 + q_1^2 +2q_0q_2 = 0 \\ \ \\ 2up_1 + q_2^2 = 0} \quad \quad \textrm{become} \quad \quad \cases{\rho^2 = 0 \\ \ \\ q_2^2 = 0 }$
so $\rho=q_2=q_1=0$ and the equation $up_0 + q_1q_2 = 0$ becomes
$uv = 0$
that is always false. So this latter case doesn't lead to any further solution and our analysis is complete.
We found 16 pairs of polynomials $(P, Q) \in (\mathbb R[x])^2$ that satisfy the equation.
- 5 years, 11 months ago
It's not clear to me what you're trying to do here. Are you saying that no such polynomials exist?
Staff - 5 years, 11 months ago
I don't know! I ended up finding 8 couples of polynomials. I was sure I posted the solution but I guess my internet connection (or me) failed to post the whole solution. I will repost it in short time. I'm writing right now.
-----EDIT---- Just wrote the solution that I made this morning. I just realized that I didn't examine some cases, and so my solution is not complete, but the cases left put are easy to handle (I feel) and I'll do as soon as possible.
--- EDIT --- Added also the trivial cases (boring). Now I think the problem is completely discussed and solved.
- 5 years, 11 months ago
A more straightforward solution for Q-2 using Calvin's hint:
We first modify the expression of LHS by multiplying numerator and denominator of the expression inside brackets by $(1+\sin\theta+i\cos\theta)$. | {
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$\textrm{LHS}=\left(\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}\right)^n=\left(\frac{(1+\sin\theta+i\cos\theta)^2}{(1+\sin\theta)^2-(i\cos\theta)^2}\right)^n\\ \implies \textrm{LHS}=\left(\frac{1-\cos^2\theta+\sin^2\theta+2\sin\theta+2i\cos\theta+2i\cos\theta\sin\theta}{1+\sin^2\theta+\cos^2\theta+2\sin\theta}\right)^n\\ \implies \textrm{LHS}=\left(\frac{2\sin\theta+2\sin^2\theta+2i\cos\theta+2i\cos\theta\sin\theta}{2+2\sin\theta}\right)^n\\ \implies \textrm{LHS}=\left(\frac{2(1+\sin\theta)(\sin\theta+i\cos\theta)}{2(1+\sin\theta)}\right)^n=(\sin\theta+i\cos\theta)^n\\ \implies \textrm{LHS}=\bigg(\cos\left(\frac{\pi}{2}-\theta\right)+i\sin\left(\frac{\pi}{2}-\theta\right)\bigg)^n$
Using Euler's formula $e^{i\theta}=\cos\theta+i\sin\theta$ and laws of indices, we can modify LHS as,
$\textrm{LHS}=\large \left(e^{\left(\frac{\pi}{2}-\theta\right)}\right)^n=e^{n\left(\frac{\pi}{2}-\theta\right)}=e^{\left(\frac{n\pi}{2}-n\theta\right)}=\cos\left(\frac{n\pi}{2}-n\theta\right)+i\sin\left(\frac{n\pi}{2}-n\theta\right)=\textrm{RHS}$
$\therefore\quad \left(\frac{1+\sin\theta+i\cos\theta}{1+\sin\theta-i\cos\theta}\right)^{n} = \cos\left(\frac{n\pi}{2}-n\theta\right)+i\sin\left(\frac{n\pi}{2}-n\theta\right)$
And we are done. $_\square$
Elementary identities used:
• $\cos^2\theta+\sin^2\theta=1$
• $(a+b+c)^2=a^2+b^2+c^2+2(ab+bc+ca)$
• $\cos\left(\frac{\pi}{2}-\theta\right)=\sin\theta$
• $\sin\left(\frac{\pi}{2}-\theta\right)=\cos\theta$
- 5 years, 11 months ago
I also noticed that one can directly get the RHS form without going into Euler's formula by directly using De Moivre's Theorem.
- 5 years, 11 months ago
$Q-3$
This is not a solution
I Used Newtons sums to get answer $\boxed{60}$
- 5 years, 11 months ago
Did you mean Q3?
- 5 years, 11 months ago
Yes.
- 5 years, 11 months ago
There is a pretty simple solution.
Hint: What is $(x-3)^4$?
Staff - 5 years, 11 months ago
Yes sir , on opening all the terms I got
$P_4-12P_3+54P_2-108P_1+324$ | {
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"url": "https://brilliant.org/discussions/thread/algebra-thailand-math-posn-2nd-round/"
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Yes sir , on opening all the terms I got
$P_4-12P_3+54P_2-108P_1+324$
Where $P_n=a^n+b^n+c^n+d^n$
now we can apply newtons sums.
- 5 years, 11 months ago
See Prasun's solution at the top for the "two-line" solution to this problem.
Staff - 5 years, 11 months ago
That was quite a big hint. :D
- 5 years, 11 months ago
2) is pretty simple. First, replace $\theta = \pi/2 - a$.
Then, $1 +\sin \theta + i \cos \theta = 1 + \cos a + i \sin a = 2\cos^2a/2 + i2\sin(a/2) \cos(a/2) = 2\cos(a/2)(\cos a/2+i\sin a/2)$
Put $b = a/2$ Likewise, $1 + \sin\theta - i\cos \theta = 2\cos b(\cos b-i\sin b).$
Then $\dfrac{1 + \sin\theta + i \cos \theta }{1 + \sin\theta - i\cos \theta } = \dfrac{2\cos b(\cos b+i\sin b)}{2\cos b(\cos b-i\sin b)} = \dfrac{\cos b+i\sin b}{\cos b-i\sin b}$
$= \dfrac{(\cos b+i\sin b)}{(\cos b-i\sin b)} * \dfrac{(\cos b+i\sin b)}{(\cos b+i\sin b)} = \dfrac{(\cos b+i\sin b)^2}{\cos^2 b + \sin^2 b} = (\cos b+i\sin b)^2$
Therefore, $(\dfrac{1 + \sin\theta + i \cos \theta }{1 + \sin\theta - i\cos \theta })^n = ((\cos b+i\sin b)^2)^n = (\cos b + i\sin b)^{2n}$
$= \cos 2bn + i\sin 2bn = \cos an + i\sin an = \cos(n(\pi/2 - a)) + i\sin(n(\pi/2 - a))$
- 5 years, 11 months ago
There's a more straightforward way to solve 2.
Hint: $\left( e ^ { i \theta } \right) ^n = e^{i \theta n }$.
Staff - 5 years, 11 months ago
That was a big hint too. :P
- 5 years, 11 months ago
I'm confused? Isn't Demoivre's theroem equivalent to that fact?
- 5 years, 11 months ago
Yes it is. The hints that I give are often to suggest alternative approaches, that you would have to work out how to apply them. In this case, see Prasun's solution above.
Staff - 5 years, 11 months ago
I was asking because I had used Demoivre's theorem. I'm also not sure how Prasun's solution is vastly different from mine, since we use almost the same approach. The only major difference in our solutions is the way we simplify the original expression
- 5 years, 11 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9697854164256365,
"lm_q1q2_score": 0.850964190198269,
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"openwebmath_perplexity": 6110.982238306314,
"openwebmath_score": 1.0000100135803223,
"tags": null,
"url": "https://brilliant.org/discussions/thread/algebra-thailand-math-posn-2nd-round/"
} |
- 5 years, 11 months ago
Ah, the difference is more in terms of motivation. IE, to explain that the intermediate step is $\left( \cos ( \frac{\pi}{2} - \theta) + \sin ( \frac{\pi}{2} - \theta ) \right) ^ n$, which would explain why we did the calculations that we did.
Staff - 5 years, 11 months ago
3) a,b,c,d are roots of equation , so
(a-3)^4 = 10a-15 ------m
(b-3)^4 = 10b-15 -------n
(c-3)^4 = 10c-15 -------o
(d-3)^4 = 10d-15 -------p
from vieta's formula we'll get a+b+c+d = 12
m+n+o+p ; (a-3)^4+(b-3)^4+(c-3)^4+(d-3)^4 = 10(a+b+c+d)-60 = 10(12)-60 = 60
so, (a-3)^4+(b-3)^4+(c-3)^4+(d-3)^4 = 60
- 5 years, 10 months ago
$1.\quad { x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 3 }-{ x }^{ 2 }-x-1\\ ={ x }^{ 5 }+{ x }^{ 4 }+{ x }^{ 3 }-\left( { x }^{ 2 }+x+1 \right) \\ ={ x }^{ 3 }\left( { x }^{ 2 }+x+1 \right) -1\left( { x }^{ 2 }+x+1 \right) \\ =\left( { x }^{ 2 }+x+1 \right) \left( { x }^{ 3 }-1 \right) \\ =\left( { x }^{ 2 }+x+1 \right) \left( { x }^{ 2 }+x+1 \right) \left( x-1 \right) \\ \\ Solving\quad quadratics,\quad we\quad get\quad roots\quad 1,\omega ,\omega ,{ \omega }^{ 2 },{ \omega }^{ 2 }$
- 5 years, 11 months ago | {
"domain": "brilliant.org",
"id": null,
"lm_label": "1. Yes\n2. Yes",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9697854164256365,
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"openwebmath_score": 1.0000100135803223,
"tags": null,
"url": "https://brilliant.org/discussions/thread/algebra-thailand-math-posn-2nd-round/"
} |
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