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http://mathhelpforum.com/math-topics/18740-plz-check-my-work.html	# Math Help - plz check my work.. 1. ## plz check my work.. Fine two vectors in opposite directions that are orthogonal to the vector u. u= <((1/2)i,(-2/3)j Solution:Two vectors are orthogonal if they are perpendicular. There fore a vector v =(x,y) is orthogonal to u if (x,y) . (1/2,-2/3) =1/2x-2/3y=0 now I have to find values of x and y that makes 0..? I am stuck here...plz help? not getting right ans.... 2. Originally Posted by harry Fine two vectors in opposite directions that are orthogonal to the vector u. u= <((1/2)i,(-2/3)j Solution:Two vectors are orthogonal if they are perpendicular. There fore a vector v =(x,y) is orthogonal to u if (x,y) . (1/2,-2/3) =1/2x-2/3y=0 now I have to find values of x and y that makes 0..? I am stuck here...plz help? not getting right ans.... (1/2)x=(2/3)y. Choose any value you like for x, say x=1, then y=3/4, and so on. RonL 3. Hello, Harry! Find two vectors in opposite directions that are orthogonal to the vector: . $\left\langle \frac{1}{2},\:-\frac{2}{3}\right\rangle$ Solution: Two vectors are orthogonal if they are perpendicular. Therefore, a vector $\vec{v} \,=\,\langle x,y\rangle$ is orthogonal to $\vec{u}$ if: . . $\langle x,\,y\rangle\cdot\left\langle\frac{1}{2},\,-\frac{2}{3}\right\rangle \;=\;\frac{1}{2}x - \frac{2}{3}y \;=\;0$ now I have to find values of x and y that makes 0 ? . . . . yes! I am stuck here ...plz help? not getting right ans.... . . What is given as "the right answer" ? You have: . $\frac{1}{2}x - \frac{2}{3}y \:=\:0\quad\Rightarrow\quad 3x \:=\:4y$ As CaptainBlack pointed out, use any pair of values that satisfy the equation. The most obvious is: . $x=4,\:y=3\quad\Rightarrow\quad \vec{v} \:=\:\langle 4,\,3\rangle$ . . An opposite vector would be: . $-\vec{v} \;=\;\langle-4,\,-3\rangle$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 7, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9395280480384827, "perplexity": 1472.894918766474}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500825567.38/warc/CC-MAIN-20140820021345-00205-ip-10-180-136-8.ec2.internal.warc.gz"}
https://onionesquereality.wordpress.com/page/2/	Feeds: Posts ## The Evolution of Programming Languages I found these images on twitter via (Paige Bailey) @DynamicWebPaige Found them so hilarious that I thought they deserved a blog post (Needless to say, click on each image for a higher resolution version). PS: A quick search indicates that these are from “Land of Lisp: Learn to Program in List, One Game at a time” by M. D. Conrad Barski. The 40s and 50s 60s and 70s 80s and 90s 2000 ## On the two notions of “Information” Recently I was going through Shannon’s original 1948 Information Theory paper and a paper by Kolmogorov from 1983 that places the differences between “Shannon Information” and “Algorithmic Information” in sharp relief. After much information diffusion over the past decades the difference is quite obvious and particularly interesting to contrast nonetheless. Nevertheless I found these two paragraphs from these two papers interesting, if for nothing then for historical reasons. “The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. Frequently the messages have meaning; that is they refer to or are correlated according to some system with certain physical or conceptual entities. These semantic aspects of communication are irrelevant to the engineering problem. The significant aspect is that the actual message is one selected from a set of possible messages. The system must be designed to operate for each possible selection, not just the one which will actually be chosen since this is unknown at the time of design.” (C. E. Shannon, A Mathematical Theory of Communication, 1948). “Our definition of the quantity of information has the advantage that it refers to individual objects and not to objects treated as members of a set of objects with a probability distribution given on it. The probabilistic definition can be convincingly applied to the information contained, for example, in a stream of congratulatory telegrams. But it would not be clear how to apply it, for example, to an estimate of the quantity of information contained in a novel or in the translation of a novel into another language relative to the original. I think that the new definition is capable of introducing in similar applications of the theory at least clarity of principle.” (A. N. Kolmogorov, Combinatorial Foundations of Information Theory and the Calculus of Probabilities, 1983). Note the reference that Shannon makes is to a message selected from a set of possible messages (and hence the use of probability theory), his problem was mainly concerned with the engineering application of communication. While Kolmogorov talks of individual objects and is not directly concerned with communication and hence the usage of computability (the two notions are related ofcourse, for example the expected Kolmogorov Complexity is equal to Shannon entropy). ## (Very) Informal thoughts on the No-Free-Lunch Theorem The No-Free-Lunch theorems (Wolpert and Macready) are one of the mainstays in Machine Learning and optimization. Crudely, the NFL says that “for every learner there exists a distribution in which it fails”, giving a precise handle to reason about why a given learning algorithm might fail on certain tasks (or, “specialization” of a learning algorithm might just be accidental). It gives a good justification why it is useful to incorporate prior informaion about the problem at hand to perform better. Oftentimes the NFL is used to argue that a general purpose learning algorithm is theoretically impossible. This is incorrect. Indeed, universal learning algorithms exist and there is a (possibly very small) free lunch, at least for the case of “interesting problems” (thanks to Vick for pointing out these papers). The view “for every learner there exists a distribution in which it fails” is very useful. However I had recently been thinking that is particularly instructive to think of No Free Lunch as a kind of diagnolization argument (this is just a different way of thinking about the prior statement). I am not completely sure if this works but it sounds plausible and makes me see NFL in a different light. It might also be that there is some subtle absrudity or flaw in this kind of thinking. However, I think the NFL theorem can be seen as a simple corollary of the following incompleteness theorem in Kolmogorov Complexity. Theorem (Chaitin): There exists a constant L (which only depends on the particular axiomatic system and the choice of description language) such that there does not exist a string s for which the statement the “The Kolmogorov Complexity of s is greater than L” (as formalized in S) can be proven within the axiomatic system S. Stated informally: “There’s a number L such that we can’t prove the Kolmogorov complexity of any specific string of bits is more than L”. In particular, you can think of any learning machine as a compression scheme (since compression implies learning, we can think of any learning algorithm as a way to compress the data) of some fixed Kolmogorov Complexity (consider the most optimal program for that machine, the programming language is irrelevant as long as it is Turing Complete. Consider the description length of it in bits. That is its Kolmogorov Complexity). Now for any such learning machine (or class of learning machines w.l.o.g) you can construct a data string which has Kolmogorov Complexity greater than what this learning machine has. i.e. this data string is random from the point of view of this learning machine as it is incompressible for it. In short IF the data string has some Kolmogorov Complexity K and your learning machine has Kolmogorov Complexity k and k < K then you will never be able to find any structure in your data string i.e. can never show that it has a KC greater than k. Thus the data string given to our learner would appear to be structureless. Thus no free lunch is just a simple diagonalization argument in the spirit of incompleteness results. One can always construct a string that is Kolmogorov Random or structureless w.r.t our encoding scheme. Ofcourse the Kolmogorov Complexity is uncomputable. However I think the above argument works even when one is talking of upper bounds on the actual Kolmogorov Complexity, but I am not very sure about it or if there are any problems with it. Also, if one assumes that that there is no reason to believe that structureless functions exist (if we assume that the data is generated by a computational process then there is probably no reason to believe that it is structureless i.e. the problem is “interesting”). And if this is the case, then you would always have a machine in your library of learning algorithms that will be able to find some upper bound on the Kolmogorov Complexity of the data string (much less than the machine itself). This search might be done using Levin’s universal search, for example. I think this line of reasoning leads naturally, to thinking why NFL doesn’t say anything about whether universal learning algorithms can exist. PS: This thought train about NFL as diagonalization was inspired by a post by David McAllester on what he calls the “Free Lunch Theorem” over here. Some of this appeared as a comment there. However, now I have been inspired enough to study some of the papers concerned including his. (Unrelated to all this. I particularly found likening Chomsky’s Universal Grammar as simply an invocation of NFL on information theoretic grounds in his post very enlightening). ## Some Proofs of the Cauchy-Schwarz Inequality Over the past 4-5 months whenever there is some time to spare, I have been working through The Cauchy-Schwarz Master Class by J. Michael Steele. And, although I am still left with the last two chapters, I have been reflecting on the material already covered in order to get a better perspective on what I have been slowly learning over the months. This blog post is a small exercise in this direction. Ofcourse, there is nothing mysterious about proving the Cauchy-Schwarz inequality; it is fairly obvious and basic. But I thought it still might be instructive (mostly to myself) to reproduce some proofs that I know out of memory (following a maxim of my supervisor on a white/blackboard blogpost). Although, why Cauchy-Schwarz keeps appearing all the time and what makes it so useful and fundamental is indeed quite interesting and non-obvious. And like Gil Kalai notes, it is also unclear why is it that it is Cauchy-Schwarz which is mainly useful. I feel that Steele’s book has made me appreciate this importance somewhat more (compared to 4-5 months ago) by drawing to many concepts that link back to Cauchy-Schwarz. Before getting to the post, a word on the book: This book is perhaps amongst the best mathematics book that I have seen in many years. True to its name, it is indeed a Master Class and also truly addictive. I could not put aside the book completely once I had picked it up and eventually decided to go all the way. Like most great books, the way it is organized makes it “very natural” to rediscover many susbtantial results (some of them named) appearing much later by yourself, provided you happen to just ask the right questions. The emphasis on problem solving makes sure you make very good friends with some of the most interesting inequalities. The number of inequalities featured is also extensive. It starts off with the inequalities dealing with “natural” notions such as monotonicity and positivity and later moves onto somewhat less natural notions such as convexity. I can’t recommend this book enough! Now getting to the proofs: Some of these proofs appear in Steele’s book, mostly as either challenge problems or as exercises. All of them were solvable after some hints. ________________ Proof 1: A Self-Generalizing proof This proof is essentially due to Titu Andreescu and Bogdan Enescu and has now grown to be my favourite Cauchy-Schwarz proof. We start with the trivial identity (for $a, b \in \mathbb{R}, x >0, y>0$): Identity 1: $(ay - bx)^2 \geq 0$ Expanding we have $a^2y^2 + b^2x^2 - 2abxy \geq 0$ Rearranging this we get: $\displaystyle \frac{a^2y}{x} + \frac{b^2x}{y} \geq 2ab$ Further: $\displaystyle a^2 + b^2 + \frac{a^2y}{x} + \frac{b^2x}{y} \geq (a+b)^2$; Rearranging this we get the following trivial Lemma: Lemma 1: $\displaystyle \frac{(a+b)^2}{(x+y)} \leq \frac{a^2}{x} + \frac{b^2}{y}$ Notice that this Lemma is self generalizing in the following sense. Suppose we replace $b$ with $b + c$ and $y$ with $y + z$, then we have: $\displaystyle \frac{(a+b+c)^2}{(x+y+z)} \leq \frac{a^2}{x} + \frac{(b+c)^2}{y+z}$ But we can apply Lemma 1 to the second term of the right hand side one more time. So we would get the following inequality: $\displaystyle \frac{(a+b+c)^2}{(x+y+z)} \leq \frac{a^2}{x} + \frac{b^2}{y} + \frac{c^2}{z}$ Using the same principle $n$ times we get the following: $\displaystyle \frac{(a_1+a_2+ \dots a_n)^2}{(x_1+x_2+ \dots + x_n)} \leq \frac{a_1^2}{x_1} + \frac{a_2^2}{x_2} + \dots + \frac{a_n^2}{x_n}$ Now substitute $a_i = \alpha_i \beta_i$ and $x_i = \beta_i^2$ to get: $\displaystyle ( \sum_{i=1}^n \alpha_i \beta_i )^2 \leq \sum_{i=1}^n ( \alpha_i )^2 \sum_{i=1}^n ( \beta_i )^2$ This is just the Cauchy-Schwarz Inequality, thus completing the proof. ________________ Proof 2: By Induction Again, the Cauchy-Schwarz Inequality is the following: for $a, b \in \mathbb{R}$ $\displaystyle ( \sum_{i=1}^n a_i b_i )^2 \leq \sum_{i=1}^n ( a_i )^2 \sum_{i=1}^n ( b_i )^2$ For proof of the inequality by induction, the most important thing is starting with the right base case. Clearly $n = 1$ is trivially true, suggesting that it is perhaps not of much import. So we consider the case for $n = 2$. Which is: $\displaystyle ( a_1 b_1 + a_2 b_2 )^2 \leq ( a_1^2 + a_2^2 ) (b_1^2 + b_2^2)$ To prove the base case, we simply expand the expressions. To get: $\displaystyle a_1^2 b_1^2 + a_2^2 b_2^2 + 2 a_1 b_1 a_2 b_2 \leq a_1^2 b_1^2 + a_1^2 b_2^2 + a_2^2 b_1^2 + a_2^2 b_2^2$ Which is just: $\displaystyle a_1^2 b_2^2 + a_2^2 b_1^2 - 2 a_1 b_1 a_2 b_2 \geq 0$ Or: $\displaystyle (a_1 b_2 - a_2 b_1 )^2 \geq 0$ Which proves the base case. Moving ahead, we assume the following inequality to be true: $\displaystyle ( \sum_{i=1}^k a_i b_i )^2 \leq \sum_{i=1}^k ( a_i )^2 \sum_{i=1}^k ( b_i )^2$ To establish Cauchy-Schwarz, we have to demonstrate, assuming the above, that $\displaystyle ( \sum_{i=1}^{k+1} a_i b_i )^2 \leq \sum_{i=1}^{k+1} ( a_i )^2 \sum_{i=1}^{k+1} ( b_i )^2$ So, we start from $H(k)$: $\displaystyle ( \sum_{i=1}^k a_i b_i )^2 \leq \sum_{i=1}^k ( a_i )^2 \sum_{i=1}^k ( b_i )^2$ we further have, $\displaystyle \Big(\sum_{i=1}^k a_i b_i\Big) + a_{k+1}b_{k+1} \leq \Big(\sum_{i=1}^k ( a_i )^2\Big)^{1/2} \Big(\sum_{i=1}^k ( b_i )^2\Big)^{1/2} + a_{k+1}b_{k+1} \ \ \ \ (1)$ Now, we can apply the case for $n = 2$. Recall that: $\displaystyle a_1 b_1 + a_2 b_2 \leq (a_1^2 + a_2^2)^{1/2} (b_1^2 + b_2^2)^{1/2}$ Thus, using this in the R. H. S of $(1)$, we would now have: $\displaystyle \Big(\sum_{i=1}^{k+1} a_i b_i\Big) \leq \Big(\sum_{i=1}^k ( a_i )^2 + a_{k+1}^2\Big)^{1/2} \Big(\sum_{i=1}^k ( b_i )^2 + b_{k+1}^2\Big)^{1/2}$ Or, $\displaystyle \Big(\sum_{i=1}^{k+1} a_i b_i\Big) \leq \Big(\sum_{i=1}^{k+1} ( a_i )^2 \Big)^{1/2} \Big(\sum_{i=1}^{k+1} ( b_i )^2 \Big)^{1/2}$ This proves the case $H(k+1)$ on assuming $H(k)$. Thus also proving the Cauchy-Schwarz inequality by the principle of mathematical induction. ________________ Proof 3: For Infinite Sequences using the Normalization Trick: Problem: For $a, b \in \mathbb{R}$ If $\displaystyle \Big(\sum_{i=1}^{\infty} a_i^2 \Big) < \infty$ and $\displaystyle \Big(\sum_{i=1}^{\infty} b_i^2 \Big) < \infty$ then is $\displaystyle \Big(\sum_{i=1}^{\infty} \|a_i\|\|b_i\|\Big) < \infty$ ? Note that this is easy to establish. We simply start with the trivial identity $(x-y)^2 \geq 0$ which in turn gives us $\displaystyle xy \leq \frac{x^2}{2} + \frac{y^2}{2}$ Next, take $x = \|a_i\|$ and $y = \|b_i\|$ on summing up to infinity on both sides, we get the following: $\displaystyle \Big( \sum_{i=1}^{\infty} \|a_i\|\|b_i\| \Big)^2 \leq \frac{1}{2}\Big(\sum_{i=1}^{\infty} a_i^2\Big) + \frac{1}{2} \Big(\sum_{i=1}^{\infty} b_i^2\Big)\ \ \ \ \ \ \ \ (2)$ From this it immediately follows that $\displaystyle \Big(\sum_{i=1}^{\infty} \|a_i\|\|b_i\|\Big) < \infty$ Now let $\displaystyle \hat{a}_i = \frac{a_i}{\Big(\sum_j a_i^2\Big)^{1/2}}$ and $\displaystyle \hat{b}_i = \frac{b_i}{\Big(\sum_j b_i^2\Big)^{1/2}}$; substituting in $(2)$, we get: $\displaystyle \Big( \sum_{i=1}^{\infty} \|\hat{a}_i\|\|\hat{b}_i\| \Big)^2 \leq \frac{1}{2}\Big(\sum_{i=1}^{\infty} \hat{a}_i^2\Big) + \frac{1}{2} \Big(\sum_{i=1}^{\infty} \hat{b}_i^2\Big)$ or, $\displaystyle \Bigg(\sum_{i = 1}^{\infty}\frac{a_i}{\Big(\sum_j a_j^2\Big)^{1/2}}\frac{b_i}{\Big(\sum_j b_j^2\Big)^{1/2}}\Bigg)^2 \leq \frac{1}{2} + \frac{1}{2}$ Which simply gives back Cauchy’s inequality for infinite sequences thus completing the proof: $\displaystyle \Big(\sum_{i=1}^{\infty} a_i b_i\Big)^2 \leq \Big(\sum_{i=1}^{\infty}a_i^2\Big) \Big(\sum_{i=1}^{\infty}b_i^2\Big)$ ________________ Proof 4: Using Lagrange’s Identity We first start with a polynomial which we denote by $\mathbf{Q}_n$: $\mathbf{Q}_n = \big(a_1^2 + a_2^2 + \dots + a_n^2 \big) \big(b_1^2 + b_2^2 + \dots + b_n^2 \big) - \big(a_1b_1 + a_2b_2 + \dots + a_nb_n\big)^2$ The question to now ask, is $\mathbf{Q}_n \geq 0$? To answer this question, we start of by re-writing $\mathbf{Q}_n$ in a “better” form. $\displaystyle \mathbf{Q}_n = \sum_{i=1}^{n}\sum_{j=1}^{n} a_i^2 b_j^2 - \sum_{i=1}^{n}\sum_{j=1}^{n} a_ib_i a_jb_j$ Next, as J. Michael Steele puts, we pursue symmetry and rewrite the above so as to make it apparent. $\displaystyle \mathbf{Q}_n = \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n} \big(a_i^2 b_j^2 + a_j^2 b_i^2\big) - \frac{2}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} a_ib_i a_jb_j$ Thus, we now have: $\displaystyle \mathbf{Q}_n = \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n} \big(a_i b_j - a_j b_i\big)^2$ This makes it clear that $\mathbf{Q}_n$ can be written as a sum of squares and hence is always postive. Let us write out the above completely: $\displaystyle \sum_{i=1}^{n}\sum_{j=1}^{n} a_i^2 b_j^2 - \sum_{i=1}^{n}\sum_{j=1}^{n} a_ib_ib_jb_j = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} \big(a_ib_j -a_jb_i\big)^2$ Now, reversing the step we took at the onset to write the L.H.S better, we simply have: $\displaystyle \sum_{i}^{n} a_i^2 \sum_{}^{n} b_i^2 - \big(\sum_{i=1}^n a_ib_i\big)^2 = \frac{1}{2}\sum_{i=1}^{n}\sum_{j=1}^{n} \big(a_ib_j -a_jb_i\big)^2$ This is called Lagrange’s Identity. Now since the R.H.S. is always greater than or equal to zero. We get the following inequality as a corrollary: $\displaystyle \big(\sum_{i=1}^n a_ib_i\big)^2 \leq \sum_{i}^{n} a_i^2 \sum_{}^{n} b_i^2$ This is just the Cauchy-Schwarz inequality, completing the proof. ________________ Proof 5: Gram-Schmidt Process gives an automatic proof of Cauchy-Schwarz First we quickly review the Gram-Schmidt Process: Given a set of linearly independent elements of a real or complex inner product space $\big(V,\langle\cdot, \cdot\rangle\big)$, $\mathbf{x_1}, \mathbf{x_2}, \dots, \mathbf{x_n}$. We can get an orthonormal set of $n$ elemets $\mathbf{e_1}, \mathbf{e_2}, \dots, \mathbf{e_n}$ by the simple recursion (after setting $\displaystyle \mathbf{e_1 = \frac{\mathbf{x_1}}{\\|x_1 \\|}}$). $\displaystyle \mathbf{z_k} = \mathbf{x_k} - \sum_{j=1}^{k-1} \langle\mathbf{x_k},\mathbf{e_j}\rangle\mathbf{e_j}$ and then $\displaystyle \mathbf{e_k} = \frac{\mathbf{z_k}}{\\|\mathbf{z_k}\\|}$ for $k = 2, 3, \dots, n$. Keeping the above in mind, assume that $\\| x\\| = 1$. Now let $x = e_1$. Thus, we have: $\mathbf{z} = \mathbf{y} - \langle \mathbf{y}, \mathbf{e_1}\rangle\mathbf{e_1}$ Giving: $\displaystyle \mathbf{e_2} = \frac{\mathbf{z}}{\\|\mathbf{z}\\|}$. Rearranging we have: $\displaystyle \\|\mathbf{z}\\|\mathbf{e_2} = \mathbf{y} - \langle \mathbf{y}, \mathbf{e_1}\rangle\mathbf{e_1}$ or $\displaystyle \mathbf{y} = \langle \mathbf{y},\mathbf{e_1}\rangle \mathbf{e_1} + \\|\mathbf{z}\\|\mathbf{e_2}$ or $\displaystyle \mathbf{y} = \mathbf{c_1} \mathbf{e_1} + \mathbf{c_2}\mathbf{e_2}$ where $c_1, c_2$ are constants. Now note that: $\displaystyle \langle\mathbf{x},\mathbf{y}\rangle = c_1$ and $\displaystyle \langle \mathbf{y},\mathbf{y}\rangle = \|c_1\|^2 + \|c_2\|^2$. The following bound is trivial: $\displaystyle \|c_1\| \leq (\|c_1\|^2 + \|c_2\|^2)^{1/2}$. But note that this is simply $\langle x,y \rangle \leq \langle y,y \rangle^{1/2}$ Which is just the Cauchy-Schwarz inequality when $\\|x\\| = 1$. ________________ Proof 6: Proof of the Continuous version for d =2; Schwarz’s Proof For this case, the inequality may be stated as: Suppose we have $S \subset \mathbb{R}^2$ and that $f: S \to \mathbb{R}$ and $g: S \to \mathbb{R}$. Then consider the double integrals: $\displaystyle A = \iint_S f^2 dx dy$, $\displaystyle B = \iint_S fg dx dy$ and $\displaystyle C = \iint_S g^2 dx dy$. These double integrals must satisfy the following inequality: $\|B\| \leq \sqrt{A} . \sqrt{C}$. The proof given by Schwarz as is reported in Steele’s book (and indeed in standard textbooks) is based on the following observation: The real polynomial below is always non-negative: $\displaystyle p(t) = \iint_S \Big( t f(x,y) + g(x,y) \Big)^2 dx dy = At^2 + 2Bt + C$ $p(t) > 0$ unless $f$ and $g$ are proportional. Thus from the binomial formula we have that $B^2 \leq AC$, moreover the inequality is strict unless $f$ and $g$ are proportional. ________________ Proof 7: Proof using the Projection formula Problem: Consider any point $x \neq 0$ in $\mathbb{R}^d$. Now consider the line that passes through this point and origin. Let us call this line $\mathcal{L} = \{ tx: t \in \mathbb{R}\}$. Find the point on the line closest to any point $v \in \mathbb{R}^d$. If $P(v)$ is the point on the line that is closest to $v$, then it is given by the projection formula: $\displaystyle P(v) = x \frac{\langle x, v \rangle }{\langle x, x \rangle}$ This is fairly elementary to establish. To find the value of $t$, such that distance $\rho(v,tx)$ is minimized, we can simply consider the squared distance $\rho^2(v,tx)$ since it is easier to work with. Which by definition is: $\displaystyle \rho^2(v,tx) = \langle v - x, v - tx \rangle$ which is simply: $\displaystyle \rho^2(v,tx) = \langle v, v \rangle - 2t \langle v, x \rangle + t^2 \langle x, x \rangle$ $\displaystyle = \langle x, x \rangle \bigg( t^2 -2t \frac{\langle v, x \rangle}{\langle x, x \rangle} + \frac{\langle v, v \rangle}{\langle x, x \rangle}\bigg)$ $\displaystyle = \langle x, x \rangle \bigg\{ \bigg(t - \frac{\langle v,x \rangle}{\langle x,x \rangle}\bigg)^2 - \frac{\langle v,x \rangle^2}{\langle x,x \rangle^2}\bigg\} + \frac{\langle v, v \rangle}{\langle x, x \rangle}\bigg)$ $\displaystyle = \langle x, x \rangle \bigg\{ \bigg(t - \frac{\langle v,x \rangle}{\langle x,x \rangle}\bigg)^2 - \frac{\langle v,x \rangle^2}{\langle x,x \rangle^2} + \frac{\langle v, v \rangle}{\langle x, x \rangle} \bigg\}$ $\displaystyle = \langle x, x \rangle \bigg\{ \bigg(t - \frac{\langle v,x \rangle}{\langle x,x \rangle}\bigg)^2 - \frac{\langle v,x \rangle^2}{\langle x,x \rangle^2} + \frac{\langle v, v \rangle \langle x, x \rangle}{\langle x, x \rangle^2} \bigg\}$ So, the value of $t$ for which the above is minimized is $\displaystyle \frac{\langle v,x \rangle}{\langle x,x \rangle}$. Note that this simply reproduces the projection formula. Therefore, the minimum squared distance is given by the expression below: $\displaystyle \min_{t \in \mathbb{R}} \rho^2(v, tx) = \frac{\langle v,v\rangle \langle x,x \rangle - \langle v,x \rangle^2}{\langle x,x \rangle}$ Note that the L. H. S is always positive. Therefore we have: $\displaystyle \frac{\langle v,v\rangle \langle x,x \rangle - \langle v,x \rangle^2}{\langle x,x \rangle} \geq 0$ Rearranging, we have: $\displaystyle \langle v,x \rangle^2 \leq \langle v,v\rangle \langle x,x \rangle$ Which is just Cauchy-Schwarz, thus proving the inequality. ________________ Proof 8: Proof using an identity A variant of this proof is amongst the most common Cauchy-Schwarz proofs that are given in textbooks. Also, this is related to proof (6) above. However, it still has some value in its own right. While also giving an useful expression for the “defect” for Cauchy-Schwarz like the Lagrange Identity above. $P(t) = \langle v - tw, v - tw \rangle$. Clearly $P(t) \geq 0$. To find the minimum of this polynomial we find its derivative w.r.t $t$ and setting to zero: $P'(t) = 2t \langle w, w \rangle - 2 \langle v, w \rangle = 0$ giving: $\displaystyle t_0 = \frac{\langle v, w \rangle}{\langle w, w \rangle}$ Clearly we have $P(t) \geq P(t_0) \geq 0$. We consider: $P(t_0) \geq 0$, substituting $\displaystyle t_0 = \frac{\langle v, w \rangle}{\langle w, w \rangle}$ we have: $\displaystyle \langle v,v \rangle - \frac{\langle v, w \rangle}{\langle w, w \rangle} \langle v,w \rangle - \frac{\langle v, w \rangle}{\langle w, w \rangle} \langle w,v \rangle + \frac{\langle v, w \rangle^2}{\langle w, w \rangle^2}\langle w,w \rangle \geq 0 \ \ \ \ \ \ \ \ (A)$ Just rearrangine and simplifying: $\displaystyle \langle v,v \rangle \langle w,w \rangle - \langle v, w \rangle^2 \geq 0$ This proves Cauchy-Schwarz inequality. Now suppose we are interested in an expression for the defect in Cauchy-Schwarz i.e. the difference $\displaystyle \langle v,v \rangle \langle w,w \rangle - \langle v, w \rangle^2$. For this we can just consider the L.H.S of equation $(A)$ since it is just $\displaystyle \langle w,w \rangle \Big(\langle v,v \rangle \langle w,w \rangle - \langle v, w \rangle^2\Big)$. i.e. Defect = $\displaystyle \langle w,w \rangle \bigg(\langle v,v \rangle - 2 \frac{\langle v,w \rangle^2}{\langle w,w \rangle} + \frac{\langle v,w \rangle^2}{\langle w,w \rangle}\bigg)$ Which is just: $\displaystyle \langle w,w \rangle\bigg(\Big\langle v - \frac{\langle w,v \rangle}{\langle w,w \rangle}w,v - \frac{\langle w,v \rangle}{\langle w,w \rangle}w \Big\rangle\bigg)$ This defect term is much in the spirit of the defect term that we saw in Lagrange’s identity above, and it is instructive to compare them. ________________ Proof 9: Proof using the AM-GM inequality Let us first recall the AM-GM inequality: For non-negative reals $x_1, x_2, \dots x_n$ we have the following basic inequality: $\displaystyle \sqrt[n]{x_1 x_2 \dots x_n} \leq \Big(\frac{x_1 + x_2 + \dots x_n}{n}\Big)$. Now let us define $\displaystyle A = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2}$ and $\displaystyle B = \sqrt{b_1^2 + b_2^2 + \dots + b_n^2}$ Now consider the trivial bound (which gives us the AM-GM): $(x-y)^2 \geq 0$, which is just $\displaystyle \frac{1}{2}\Big(x^2 + y^2\Big) \geq xy$. Note that AM-GM as stated above for $n = 2$ is immediate when we consider $x \to \sqrt{x}$ and $y \to \sqrt{y}$ Using the above, we have: $\displaystyle \frac{1}{2} \Big(\frac{a_i^2}{A^2} + \frac{b_i^2}{B^2}\Big) \geq \frac{a_ib_i}{AB}$ Summing over $n$, we have: $\displaystyle \sum_{i=1}^n\frac{1}{2} \Big(\frac{a_i^2}{A^2} + \frac{b_i^2}{B^2}\Big) \geq \sum_{i=1}^n \frac{a_ib_i}{AB}$ But note that the L.H.S equals 1, therefore: $\displaystyle \sum_{i=1}^n \frac{a_ib_i}{AB} \leq 1$ or $\displaystyle \sum_{i=1}^n a_ib_i \leq AB$ Writing out $A$ and $B$ as defined above, we have: $\displaystyle \sum_{i=1}^n a_ib_i \leq \sqrt{\sum_{i=1}^na_i^2}\sqrt{\sum_{i=1}^nb_i^2}$. Thus proving the Cauchy-Schwarz inequality. ________________ Proof 10: Using Jensen’s Inequality We begin by recalling Jensen’s Inequality: Suppose that $f: [p, q] \to \mathbb{R}$ is a convex function. Also suppose that there are non-negative numbers $p_1, p_2, \dots, p_n$ such that $\displaystyle \sum_{i=1}^{n} p_i = 1$. Then for all $x_i \in [p, q]$ for $i = 1, 2, \dots, n$ one has: $\displaystyle f\Big(\sum_{i=1}^{n}p_ix_i\Big) \leq \sum_{i=1}^{n}p_if(x_i)$. Now we know that $f(x) = x^2$ is convex. Applying Jensen’s Inequality, we have: $\displaystyle \Big(\sum_{i=1}^{n} p_i x_i \Big)^2 \leq \sum_{i=1}^{n} p_i x_i^2$ Now, for $b_i \neq 0$ for all $i = 1, 2, \dots, n$, let $\displaystyle x_i = \frac{a_i}{b_i}$ and let $\displaystyle p_i = \frac{b_i^2}{\sum_{i=1}^{n}b_i^2}$. Which gives: $\displaystyle \Big(\sum_{i=1}^n \frac{a_ib_i}{\sum_{i=1}^{n}b_i^2}\Big)^2 \leq \Big(\sum_{i=1}^{n}\frac{a_i^2}{\sum_{i=1}^{n}b_i^2}\Big)$ Rearranging this just gives the familiar form of Cauchy-Schwarz at once: $\displaystyle \Big(\sum_{i=1}^{n} a_ib_i\Big)^2 \leq \Big(\sum_{i=1}^{n} a_i^2\Big)\Big(\sum_{i=1}^{n} b_i^2\Big)$ ________________ Proof 11: Pictorial Proof for d = 2 Here (page 4) is an attractive pictorial proof by means of tilings for the case $d = 2$ by Roger Nelson. ________________ ## Gian-Carlo Rota on Combinatorics The writings (and even papers/technical books) of Gian-Carlo Rota are perhaps amongst the most insightful that I have encountered in the past 3-4 years (maybe even more). Rota wrote to provoke, never resisting to finish a piece of writing with a rhetorical flourish even at the cost of injecting seeming inconsistency in his stance. I guess this is what you get when you have a first rate mathematician and philosopher endowed with an elegant; at times even devastating turn of phrase, with a huge axe to grind. The wisdom of G. C. Rota is best distilled in his book of essays, reviews and other thoughts: Indiscrete Thoughts and to some extent Discrete Thoughts. Perhaps I should review Indiscrete Thoughts in the next post, just to revisit some of those writings and my notes from them myself. Rota in 1962 This post however is not about his writing in general as the title indicates. I recently discovered this excellent dialogue between Rota and David Sharp (1985). I found this on a lead from this László Lovász interview. Here he mentions that Rota’s combinatorics papers were an inspiration for him in his work to find more structure in combinatorics. From the David Sharp interview, here are two relevant excerpts (here too the above mentioned flourish is evident): “Combinatorics is an honest subject. No adèles, no sigma-algebras. You count balls in a box, and you either have the right number or you haven’t. You get the feeling that the result you have discovered is forever, because it’s concrete. Other branches of mathematics are not so clear-cut. Functional analysis of infinite-dimensional spaces is never fully convincing; you don’t get a feeling of having done an honest day’s work. Don’t get the wrong idea – combinatorics is not just putting balls into boxes. Counting finite sets can be a highbrow undertaking, with sophisticated techniques. […] Much combinatorics of our day came out of an extraordinary coincidence. Disparate problems in combinatorics, ranging from problems in statistical mechanics to the problem of coloring a map, seem to bear no common features. However, they do have at least one common feature: their solution can be reduced to the problem of finding the roots of some polynomial or analytic function. The minimum number of colors required to properly color a map is given by the roots of a polynomial, called the chromatic polynomial; its value at N tells you in how many ways you can color the map with N colors. Similarly, the singularities of some complicated analytic function tell you the temperature at which a phase transition occurs in matter. The great insight, which is a long way from being understood, was to realize that the roots of the polynomials and analytic functions arising in a lot of combinatorial problems are the Betti numbers of certain surfaces related to the problem. Roughly speaking, the Betti numbers of a surface describe the number of different ways you can go around it. We are now trying to understand how this extraordinary coincidence comes about. If we do, we will have found a notable unification in mathematics.” ________________ While having an axe to grind is a fairly common phrase. I got the idea of using it from an amazon review of Indiscrete thoughts (check link above). Because I really do think that that is the best description for a lot of his writings! ## Book Review: “Probably Approximately Correct: Nature’s Algorithms for Learning and Prospering in a Complex World” by Leslie Valiant Darwinian Evolution is a form of Computational Learning The punchline of this book is perhaps: “Changing or increasing functionality of circuits in biological evolution is a form of computational learning“; although it also speaks of topics other than evolution, the underlying framework is of the Probably Approximately Correct model [1] from the theory of Machine Learning, from which the book gets its name. “Probably Approximately Correct” by Leslie Valiant [Clicking on the image above will direct you to the amazon page for the book] I had first heard of this explicit connection between Machine Learning and Evolution in 2010 and have been quite fascinated by it since. It must be noted, however, that similar ideas have appeared in the past. It won’t be incorrect to say that usually they have been in the form of metaphor. It is another matter that this metaphor has generally been avoided for reasons I underline towards the end of this review. When I first heard about the idea it immediately made sense and like all great ideas, in hindsight looks completely obvious. Ergo, I was quite excited to see this book and preordered it months in advance. Before attempting to sketch a skiagram of the main content of the book: One of the main subthemes of the book, constantly emphasized is to look at computer science as a kind of an enabling tool to study natural science. This is oft ignored, perhaps because of the reason that CS curricula are rarely designed with any natural science component in them and hence there is no impetus for aspiring computer scientists to view them from the computational lens. On the other hand, the relation of computer science with mathematics has become quite well established. As a technology the impact of Computer Science has been tremendous. All this is quite remarkable given the fact that just about a century ago the notion of a computation was not even well defined. Unrelated to the book: More recently people have started taking the idea of digital physics (i.e. physics from a solely computable/digital perspective) seriously. But in the other natural sciences its usage is still woeful. Valiant draws upon the example of Alan Turing as a natural scientist and not just as a computer scientist to make this point. Alan Turing was more interested in natural phenomenon (intelligence, limits of mechanical calculation, pattern formation etc) and used tools from Computer Science to study them, a fact that is evident from his publications. That Turing was trying to understand natural phenomenon was remarked in his obituary by Max Neumann by summarizing the body of his work as: “the extent and the limitations of mechanistic explanations of nature”. ________________ The book begins with a delightful and quite famous quote by John von Neumann (through this paragraph I also discovered the context to the quote). This paragraph also adequately summarizes the motivation for the book very well: “In 1947 John von Neumann, the famously gifted mathematician, was keynote speaker at the first annual meeting of the Association for Computng Machinery. In his address he said that future computers would get along with just a dozen instruction types, a number known to be adequate for expressing all of mathematics. He went on to say that one need not be surprised at this small number, since 1000 words were known to be adequate for most situations in real life, and mathematics was only a small part of life, and a very simple part at that. The audience responded with hilarity. This provoked von Neumann to respond: “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is” Though counterintuitive, von Neumann’s quip contains an obvious truth. Einstein’s theory of general relativity is simple in the sense that one can write the essential content on one line as a single equation. Understanding its meaning, derivation, and consequences requires more extensive study and effort. However, this formal simplicity is striking and powerful. The power comes from the implied generality, that knowledge of one equation alone will allow one to make accurate predictions about a host of situations not even connected when the equation was first written down. Most aspects of life are not so simple. If you want to succeed in a job interview, or in making an investment, or in choosing a life partner, you can be quite sure that there is no equation that will guarantee you success. In these endeavors it will not be possible to limit the pieces of knowledge that might be relevant to any one definable source. And even if you had all the relevant knowledge, there may be no surefire way of combining it to yield the best decision. This book is predicated on taking this distinction seriously […]” In a way, aspects of life as mentioned above appear theoryless, in the sense that there seems to be no mathematical or scientific theory like relativity for them. Something which is quantitative, definitive and short. Note that these are generally not “theoryless” in the sense that there exists no theory at all since obviously people can do all the tasks mentioned in a complex world quite effectively. A specific example is of how organisms adapt and species evolve without having a theory of the environment as such. How can such coping mechanisms come into being in the first place is the main question asked in the book. Let’s stick to the specific example of biological evolution. Clearly, it is one of the central ideas in biology and perhaps one of the most important theories in science that changed the way we look at the world. But inspite of its importance, Valiant claims (and correctly in my opinion) that evolution is not understood well in a quantitative sense. Evidence that convinces us of its correctness is of the following sort: Biologists usually show a sequence of evolving objects; stages, where the one coming later is more complex than the previous. Since this is studied mostly via the fossil record there is always a lookout for missing links between successive stages (As a side note: Animal eyes is an extremely fascinating and very readable book that delves with this question but specifically dealing with the evolution of the eye. This is particularly interesting given that due to the very nature of the eye, there can be no fossil records for them). Darwin had remarked that numerous successive paths are necessary – that is, if it was not possible to trace a path from an earlier form to a more complicated form then it was hard to explain how it came about. But again, as Valiant stresses, this is not really an explanation of evolution. It is more of an “existence proof” and not a quantitative explanation. That is, even if there is evidence for the existence of a path, one can’t really say that a certain path is likely to be followed just because it exists. As another side note: Watch this video on evolution by Carl Sagan from the beautiful COSMOS Related to this, one of the first questions that one might ask, and indeed was asked by Darwin himself: Why has evolution taken as much time as it has? How much time would have sufficed for all the complexity that we see around us to evolve? This question infact bothered Darwin a lot in his day. In On the Origins of Species he originally gave an estimate of the Earth’s age to be at least 300 million years, implying indirectly, that there was enough time. This estimate was immediately thrashed by Lord Kelvin, one of the pre-eminent British physicists of the time, who estimated the age of the Earth to be only about 24 million years. This caused Darwin to withdraw the estimate from his book. However, this estimate greatly worried Darwin as he thought 24 million years just wasn’t enough time. To motivate on the same theme Valiant writes the following: “[…] William Paley, in a highly influential book, Natural Theology (1802) , argued that life, as complex as it is, could not have come into being without the help of a designer. Numerous lines of evidence have become available in the two centuries since, through genetics and the fossil record, that persuade professional biologists that existing life forms on Earth are indeed related and have indeed evolved. This evidence contradicts Paley’s conclusion, but it does not directly address his argument. A convincing direct counterargument to Paley’s would need a specific evolution mechanism to be demonstrated capable of giving rise to the quantity and quality of the complexity now found in biology, within the time and resources believed to have been available. […]” A specific, albeit more limited version of this question might be: Consider the human genome, which has about 3 billion base pairs. Now, if evolution is a search problem, as it naturally appears to be, then why did the evolution of this genome not take exponential time? If it would have taken exponential time then clearly such evolution could not have happened in any time scale since the origin of the earth. Thus, a more pointed question to ask would be: What circuits could evolve in sub-exponential time (and on a related note, what circuits are evolvable only in exponential time?). Given the context, the idea of thinking about this in circuits might seem a little foreign. But on some more thought it is quite clear that the idea of a circuit is natural when it comes to modeling such systems (at least in principle). For example: One might think of the genome as a circuit, just as how one might think of the complex protein interaction networks and networks of neurons in the brain as circuits that update themselves in response to the environment. The last line is essentially the key idea of adaptation, that entities interact with the environment and update themselves (hopefully to cope better) in response. But the catch is that the world/environment is far too complex for simple entities (relatively speaking), with limited computational power, to have a theory for. Hence, somehow the entity will have to cope without really “understanding” the environment (it can only be partially modeled) and improve their functionality. The key thing to pay attention here is the interaction or the interface between the limited computational entity in question and the environment. The central idea in Valiant’s thesis is to think of and model this interaction as a form of computational learning. The entity will absorb information about the world and “learn” so that in the future it “can do better”. A lot of Biology can be thought of as the above: Complex behaviours in environments. Wherein by complex behaviours we mean that in different circumstances, we (or alternately our limited computational entity) might behave completely differently. Complicated in the sense that there can’t possibly be a look up table for modeling such complex interactions. Such interactions-responses can just be thought of as complicated functions e.g. how would a deer react could be seen as a function of some sensory inputs. Or for another example: Protein circuits. The human body has about 25000 proteins. How much of a certain protein is expressed is a function depending on the quantities of other proteins in the body. This function is quite complicated, certainly not something like a simple linear regression. Thus there are two main questions to be asked: One, how did these circuits come into being without there being a designer to actually design them? Two, How do such circuits maintain themselves? That is, each “node” in protein circuits is a function and as circumstances change it might be best to change the way they work. How could have such a thing evolved? Given the above, one might ask another question: At what rate can functionality increase or adapt under the Darwinian process? Valiant (like many others, such as Chaitin. See this older blog post for a short discussion) comes to the conclusion that the Darwinian evolution is a very elegant computational process. And since it is so, with the change in the environment there has to be a quantitative way of saying how much rate of change can be kept up with and what environments are unevolvable for the entity. It is not hard to see that this is essentially a question in computer science and no other discipline has the tools to deal with it. In so far that (biological) interactions-responses might be thought of as complicated functions and that the limited computational entity that is trying to cope has to do better in the future, this is just machine learning! This idea, that changing or increasing functionality of circuits in biological evoution is a form of computational learning, is perhaps very obvious in hindsight. This (changing functionality) is done in Machine Learning in the following sense: We want to acquire complicated functions without explicitly programming for them, from examples and labels (or “correct” answers). This looks at exactly at the question at how complex mechanisms can evolve without someone designing it (consider a simple perceptron learning algorithm for a toy example to illustrate this). In short: We generate a hypothesis and if it doesn’t match our expectations (in performance) then we update the hypothesis by a computational procedure. Just based on a single example one might be able to change the hypothesis. One could draw an analogy to evolution where “examples” could be experiences, and the genome is the circuit that is modified over a period of time. But note that this is not how it looks like in evolution because the above (especially drawing to the perceptron example) sounds more Lamarckian. What the Darwinian process says is that we don’t change the genome directly based on experiences. What instead happens is that we make a lot of copies of the genome which are then tested in the environment with the better one having a higher fitness. Supervised Machine Learning as drawn above is very lamarckian and not exactly Darwinian. Thus, there is something unsatisfying in the analogy to supervised learning. There is a clear notion of a target in the same. Then one might ask, what is the target of evolution? Evolution is thought of as an undirected process. Without a goal. This is true in a very important sense however this is incorrect. Evolution does not have a goal in the sense that it wants to evolve humans or elephants etc. But it certainly does have a target. This target is “good behaviour” (where behaviour is used very loosely) that would aid in the survival of the entity in an environment. Thus, this target is the “ideal” function (which might be quite complicated) that tells us how to behave in what situation. This is already incorporated in the study of evolution by notions such as fitness that encode that various functions are more beneficial. Thus evolution can be framed as a kind of machine learning problem based on the notion of Darwinian feedback. That is, make many copies of the “current genome” and let the one with good performance in the real world win. More specifically, this is a limited kind of PAC Learning. If you call your current hypothesis your genome, then your genome does not depend on your experiences. Variants to this genome are generated by a polynomial time randomized Turing Machine. To illuminate the difference with supervised learning, we come back to a point made earlier that PAC Learning is essentially Lamarckian i.e. we have a hidden function that we want to learn. One however has examples and labels corresponding to this hidden function, these could be considered “queries” to this function. We then process these example/label pairs and learn a “good estimate” of this hidden function in polynomial time. It is slightly different in Evolvability. Again, we have a hidden “ideal” function f(x). The examples are genomes. However, how we can find out more about the target is very limited, since one can empirically only observe the aggregate goodness of the genome (a performance function). The task then is to mutate the genome so that it’s functionality improves. So the main difference with the usual supervised learning is that one could query the hidden function in a very limited way: That is we can’t act on a single example and have to take aggregate statistics into account. Then one might ask what can one prove using this model? Valiant demonstrates some examples. For example, Parity functions are not evolvable for uniform distribution while monotone disjunctions are actually evolvable. This function is ofcourse very non biological but it does illustrate the main idea: That the ideal function together with the real world distribution imposes a fitness landscape and that in some settings we can show that some functions can evolve and some can not. This in turn illustrates that evolution is a computational process independent of substrate. In the same sense as above it can be shown that Boolean Conjunctions are not evolvable for all distributions. There has also been similar work on real valued functions off late which is not reported in detail in the book. Another recent work that is only mentioned in passing towards the end is the study of multidimensional space of actions that deal with sets of functions (not just one function) that might be evolvable together. This is an interesting line of work as it is pretty clear that biological evolution deals with the evolution of a set of functions together and not functions in isolation. ________________ Overall I think the book is quite good. Although I would rate it 3/5. Let me explain. Clearly this book is aimed at the non expert. But this might be disappointing to those who bought the book because of the fact that this recent area of work, of studying evolution through the lens of computational learning, is very exciting and intellectually interesting. The book is also aimed at biologists, and considering this, the learning sections of the book are quite dumbed down. But at the same time, I think the book might fail to impress most of them any way. I think this is because generally biologists (barring a small subset) have a very different way of thinking (say as compared to the mathematicians or computer scientists) especially through the computational lens. I have had some arguments about the main ideas in the book over the past couple of years with some biologist friends who take the usage of “learning” to mean that what is implied is that evolution is a directed process. It would have been great if the book would have spent more time on this particular aspect. Also, the book explicitly states that it is about quantitative aspects of evolution and has nothing to do with speciation, population dynamics and other rich areas of study. However, I have already seen some criticism of the book by biologists on this premise. As far as I am concerned, as an admirer of Prof. Leslie Valiant’s range and quality of contributions, I would have preferred if the book went into more depth. Just to have a semi-formal monograph on the study of evolution using the tools of PAC Learning right from the person who initiated this area of study. However this is just a selfish consideration. ________________ References: [1] A Theory of the Learnable, Leslie Valiant, Communications of the ACM, 1984 (PDF). [2] Evolvability, Leslie Valiant, Journal of the ACM, 2007 (PDF). ________________ ## “Order and Chaos” On a humorous note: “In every chaos there is an order: A very good writer wrote a rather nice book. But there was no title to it yet. He asked his friend for suggestions. The friend asked him if there was either a drum or a guitar mentioned in the book, to which the author replied in the negative. 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https://math.stackexchange.com/questions/2589385/can-we-prove-this-factorial-interpolation-process-converges-to-the-gamma-functio	# Can we prove this factorial interpolation process converges to the Gamma function? I've been thinking of using polynomial interpolation combined with least squares optimization to approximate various functions of a single variable, defined by linear differential or functional equations. For differential equations the problem is rather simple and straighforward, since they are local. On the other hand, the cases such as Gamma function can be more complicated. Gamma function is defined by a non-local equation: $$\Gamma(x+1)=x \Gamma(x)$$ Additionally, we know its values at positive integer points: $$\Gamma(n)=(n-1)!$$ There's also the condition of log-convexity, which I will return to later. I have applied the following approximation scheme. It can be used on any interval $[n,n+2]$, which is good enough for practical applications. Let a polynomial $P_k(x)$ be an approximation for the function on the interval: $$P_k(x)=a_k x^k+a_{k-1}x^{k-1}+\dots+a_1 x+a_0$$ First, it has to satisfy the constraints at integer points: $$\begin{cases} P_k(n)=(n-1)! \\ P_k(n+1)=n! \\ P_k(n+2)=(n+1)! \end{cases}$$ Second, it has to approximately satisfy the defining equation, in some sense. I chose the least squares method. In other words, it has to minimize the following "Lagrangian": $$L(a_j,\lambda_j)=\int_n^{n+1} \left(P_k(x+1)- x P_k(x) \right)^2 dx-\lambda_1 \left(P_k(n)-(n-1)! \right)- \\ - \lambda_2 \left(P_k(n+1)-n! \right)- \lambda_3 \left(P_k(n+2)-(n+1)! \right)$$ It's just the "norm" of the square of the defining equation, with constraints added with the use of Lagrange multipliers. To minimize it, we equate all the partial derivatives to zero. Which, thanks to linearity of the defining equation, reduces to solving a system of linear equations, which has only one solution. Thus, we find the coefficients $a_0,a_1,\dots,a_k$ and can approximate Gamma function. Or can we? Does this algorithm allow us to approximate Gamma function on the interval $[n,n+2]$ for any positive integer $n$ with any degree of accuracy for large enough $k$? Does this approximation converge to Gamma function for $k \to \infty$? As far as I understand, without the condition of log-convexity, Gamma function can't be uniquely defined, which is one of the reasons I have doubts. Obviously, each approximation is not log-convex, since the polynomials oscillate a lot. Here's the example for $n=2$. I plot the error $\Gamma(x)-P_k(x)$ on the interval $[2,4]$ and list the coefficients in descending order (from $k$ to $0$). • $k=2$, in this case we have just enough variables to accomodate the constraints at integer points, so no optimization is involved. $$\{a_j\}=\{3/2, -13/2, 8\}$$ • $k=3$. $$\{a_j\}=\{0.623249, -4.10924, 9.70447, -6.95797\}$$ • $k=4$. $$\{a_j\}=\{0.243426, -2.24618, 8.32714, -13.7811, 9.32815\}$$ • $k=5$. $$\{a_j\}=\{0.0762967, -0.874414, 4.20777, -10.0217, 11.8993, -4.82474\}$$ The method appears to converge to the Gamma function, and quite fast, since the maximum error falls by a factor of approximately $5$ with increasing $k$. Edit I'm not claiming this method is in any way optimal! I'm interested in it for other, more general, purposes. • Sorry, I'm an old man, could you try and concentrate your question a bit? Yes, $\Gamma(x+1)=x\Gamma(x)$ doesn't define it uniquely without the assumption of log-convexity, but who cares, there are other properties of that function. Whatever, your approximation scheme looks far too complicated, I use far simpler methods in my (arbitrary precision) library. – Professor Vector Jan 2 '18 at 21:06 • @ProfessorVector, my question is highlighted. I'm not sure what kind of clarification you are asking for, I would be happy to provide any. You may also be interested in my edit at the end of the post. Thank you for your comment. – Yuriy S Jan 2 '18 at 21:10 • Sorry, this may sound mean, but... there are well-known, established methods to compute the gamma function. Ok, in my own library, I use another one, solving a functional equation for the digamma function with polynomial approximation, integrating.... But I don't have to ask at MSE if that converges. You seem to use another scheme, and don't have a proof that converges, though your numerical evidence seems to indicate it, did I get that right? – Professor Vector Jan 2 '18 at 21:31 • @ProfessorVector, I am not going to compute Gamma function with this method. I am interested in the method itself. This particular question deals with Gamma function, because it is well known. Sorry, for your last question, yes, this is what I'm asking – Yuriy S Jan 2 '18 at 21:39 • It doesn't break the official rules. But there are others, violating of which guarantees zero answers. Look, people here volunteer... if they are interested. So you might want to add some motivation before the tedious staff. – Professor Vector Jan 2 '18 at 21:44	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9092510938644409, "perplexity": 358.5433472472022}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-35/segments/1566027316549.78/warc/CC-MAIN-20190821220456-20190822002456-00530.warc.gz"}
http://mathhelpforum.com/number-theory/147752-riemann-s-transform.html	# Math Help - Riemann's Transform? 1. ## Riemann's Transform? In his 1859 paper, Riemann discussed what is today known as the Inverse Mellin Transform, in orderto obtain his explicit formula for the prime counting function. If this is the case, why hasn't the transform been credited to Riemann's name, as Mellin was only 5 years old at the time of publication of "On the Number of Primes..."? 2. Originally Posted by raheel88 In his 1859 paper, Riemann discussed what is today known as the Inverse Mellin Transform, in orderto obtain his explicit formula for the prime counting function. If this is the case, why hasn't the transform been credited to Riemann's name, as Mellin was only 5 years old at the time of publication of "On the Number of Primes..."? I'm not sure but probably because Riemann only used this transform for one function, whereas Mellin defined it for all functions. 3. good point! 4. ## Comment In math history there's a number of cases where credit should have gone to someone else. 5. @wonderboy1953 I know...this happens quite alot. This case particularly caught my eye because I'm studying Riemann's Paper as part of my course, and we keep referring to the Mellin Transform and Inverse Mellin Transform...and I'm thinking, 'surely Riemann should get credit for this?' He was really intuitive and alot of the steps in his paper are done without much explanation or proof, so that could be why he didn't get much recognition...plus, his hypothesis kind of overshadowed his transform! 6. ## I'm willing to bet you that... Gauss should receive the credit for many theorems that others have gotten credit for.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8529308438301086, "perplexity": 1075.0829932720944}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395560.69/warc/CC-MAIN-20160624154955-00174-ip-10-164-35-72.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/find-postition-on-third-charge-question.209408/	# Find postition on third charge question 1. Jan 17, 2008 ### flaming Find postition of third charge question 1. The problem statement, all variables and given/known data A charge of 1.57E-9 C is placed at the origin, and a charge of 4.23E-9 C is placed at x = 1.51 m. Find the position at which a third charge of 3.11E-9 C can be placed so that the net electrostatic force on it is zero. Please help me out with this one 2. Relevant equations 3. The attempt at a solution Last edited: Jan 17, 2008 2. Jan 17, 2008 ### Tom Mattson Staff Emeritus First question: Given that the charges are all positive, and that positive charges repel, do you think the 3rd charge should be placed to the left of both charges, to the right of both charges, or inbetween the two charges? Why? 3. Jan 17, 2008 ### flaming i guess it would be at the left of the two charges to counter them..i guess 4. Jan 17, 2008 ### Tom Mattson Staff Emeritus Don't guess. Think. If the 3rd charge were to the left of the other two, and the other two were both repelling it, then is it even possible that the net force on the 3rd charge is zero? Think about the directions of the forces on the 3rd charge. Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook Have something to add? Similar Discussions: Find postition on third charge question	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9320725798606873, "perplexity": 1290.3812383961395}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542712.49/warc/CC-MAIN-20161202170902-00127-ip-10-31-129-80.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/density-matrix-on-a-diagonal-by-blocks-hamiltonian.905486/	# I Density matrix on a diagonal by blocks Hamiltonian. Tags: 1. Feb 25, 2017 ### voila If I have a Hamiltonian diagonal by blocks (H1 0; 0 H2), where H1 and H2 are square matrices, is the density matrix also diagonal by blocks in the same way? 2. Feb 27, 2017 ### Staff: Mentor The density matrix describes the state of the system, so its exact form will depend on the state of the system. If H1 and H2 correspond to different particles, then the density matrix will be block diagonal only if the particles are not entangled (corresponding to a product state). Draft saved Draft deleted Similar Discussions: Density matrix on a diagonal by blocks Hamiltonian.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9408272504806519, "perplexity": 1280.5005579683404}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084887067.27/warc/CC-MAIN-20180118051833-20180118071833-00385.warc.gz"}
http://mathrefresher.blogspot.com/2009_11_15_archive.html	## Wednesday, November 18, 2009 ### Bertrand's Postulate (Theorem) In today's blog, I will present a proof for Bertrand's Postulate which is really a theorem. The content is taken primarily from an excellent Wikipedia article. Lemma 1: If n is greater than 5, then 2n/3 is greater than 2n Proof: (1) For n ≥ 5, 2n(2n-9) is greater than 0 since 2n ≥ 10 and 2n-9 ≥ 1. (2) 2n(2n - 9) = 4n2 - 18n. (3) Since 4n2 - 18n is greater than 0, 4n2 is greater than 18n. (4) Dividing both sides by 9 gives us 4n2/9 is greater than 2n. (5) Taking the square root of both sides gives us 2n/3 is greater than 2n QED Lemma 2: Let p be a prime number that is greater than 2n Let x be the highest nonnegative integer such that px divides 2n!/[(n!)(n!)] then: Proof: (1) We know that in general, the maximum power of p that divides n! is (see Theorem, here): (2) So the highest power of p that divides (2n)!/[(n!)(n!)] is: (3) Since p is greater than 2n, it follows that p2 is greater than 2n. So we only need to consider the case where i=1. (4) This leaves us with: QED Lemma 3: Bertrand's Posulate is true for n less than 2048 Proof: Between any n less than 2048 and its double, one will find one of the following primes: 3, 5, 7, 13, 23, 43, 83, 163, 317, 631, 1259 and 2503 since: n = 2 --> 3 n between 3 and 4 --> 5 n between 5 and 6 --> 7 n between 7 and 12 --> 13 n between 13 and 22 --> 23 n between 23 and 42 --> 43 n between 43 and 82 --> 83 n between 83 and 162 --> 163 n between 163 and 316 --> 317 n between 317 and 630 --> 631 n between 631 and 1258 --> 1259 n between 1259 and 2047 --> 2503 QED Definition 1: R(p,n): maximum divisor of 2n!/(n!)(n!) Let R(p,n) be the function that returns the maximum power of a given prime that divides 2n!/(n!)(n!) For any prime p, pR(p,n) divides 2n!/(n!)(n!) but pR(p,n)+1 does not. Example 1: R(3,5) (25)!/(5!)(5!) = 109876/5432 = 292722 = 24327. R(3,5) = 2 since 32 divides 24327 but 33 = 27 does not. Lemma 4: For all p, pR(p,n) ≤ 2n Proof: (1) The multiplicity m of p in a factorial n! (see Theorem 1, here) is: so that pm divides n! but pm+1 does not. (2) Based on this, the formula for R(p,n) is: which is equivalent to: (3) We can see that if pi is greater than 2n, then: floor(2n/pi) = 0 and floor(n/pi) = 0 so that we have: floor(2n/pi) - 2(n/pi) = 0 so that we have: (4) Using the Division Algorithm (see Theorem 1, here), there exists q,r such that: n = qpi + r where r is less than pi (5) From this, we can see that: since: Case 1: r = 0 In this case, floor(2n/pi) = 2q and 2floor(n/pi) = 2q so that: floor(2n/pi) - 2floor(2n/pi) = 0 Case 2: r is less than (1/2)pi In this case, floor(2n/pi) = 2q and 2floor(n/pi) = 2q so that: floor(2n/pi) - 2floor(2n/pi) = 0 Case 3: r ≥ (1/2)pi In this case, floor(2n/pi) = 2q+1 and 2floor(n/pi) = 2q so that: floor(2n/pi) - 2floor(2n/pi) = 1 (5) So that we have: pi ≤ plogp(2n) = 2n. QED Lemma 5: For all i: 0 ≤ i ≤ 2n → (2n!)/(n!)(n!) ≥ (2n!)/(i!)(2n-i)! Proof: (1) We know that (see Definition 3, here for review of C(n,r) if needed): C(2n,i-1) = (2n!)/(i-1)!(2n-i+1)! C(2n,i) = (2n!)/(i!)(2n-i)! (2) So, we see that: C(2n,i) = (2n-i+1)/(i)C(2n,i-1) and C(2n,i-1) = (i)/(2n-i+1)C(2n,i) (3) If i ≤ n, it follows that: 2n - i + 1 is greater than i since 2n - i + 1 ≥ n+1 So up to i=n, C(2n,i) is greater than C(2n,i-1) (5) If i ≥ n+1, it follows that: i is greater than 2n - i + 1 since 2n - i + 1 ≤ n So down to i=n+1, C(2n,i-1) is greater than C(2n,i) (6) So, we have shown that C(2n,n) is greater than C(2n,i) for i less than n and C(2n,n) is greater than C(2n,i) for i greater than n. QED Corollary 5.1: Proof: (1) Using the Binomial Theorem (see Theorem, here): (2) Using Lemma 5 above, we know that: (3) So that we have: (4) And dividing both sides by (2n+1) finishes the proof. QED Lemma 6: If 2t is less than 8t, then t is less than 6 Proof: (1) If t=6, then 2t is greater than 8t since: 26 = 64 8(6) = 48 (2) Assume it is true up to some t ≥ 6 so that: 2t is greater than 8t. (3) Then 2(2t)=2t+1 is greater than 28t = 16t (4) 16t is greater than 8t+8 since 16=8t + 8t and for t ≥ 6, 8t is greater than 8. (5) So that we have 2t+1 is greater than 8t+8 = 8(t+1). QED Lemma 7: If: Then: n is less than 2048. Proof: (1) Let n = 22t-1 (2) So 2n = 22t (3) 22t = 2t (4) So 2nln(2) = 2tln(2) (5) (4)ln(2n) = 4ln(22t) = 2t4ln(2) (6) Dividing both sides by ln(2) gives us: 2t is less than 8t (7) Using Lemma 6 above, we conclude that t is less than 6. (8) So, it follows that n is less than 22(6)-1 = 211 = 2048. QED Theorem: Bertrand's Postulate For any n, there exists a prime p such that p is greater than n and less than 2n. Proof: (1) We know that Bertrand's Postulate is true for n less than 2048 [see Lemma 3 above] (2) Assume for some n ≥ 2048, there is no prime between n and 2n. (3) Let p be the largest prime factor of (2n!)/[(n!)(n!)] that is less than or equal to n. (4) Using Lemma 1 above, we know that p ≤ 2n/3 since: (a) Assume that that p is greater than 2n/3. (b) Since n ≥ 2048, using Lemma 1 above, p is also greater than 2n (c) This means that the highest power of x is given by (see Lemma 2 above): (d) Since 2n/3 is less than p, it follows that 2n is less than 3p, and further that 2n/p is less than 3 so that 2n/p ≤ 2. (e) But from step #3 above, we know that n ≥ p so that we have n/p ≥ 1. (f) This gives us that 2n/p ≤ 2 and n/p ≥ 1 so that we have: x = floor(2n/p) - 2floor(n/p) ≤ 2 - 21 = 0. (g) So, under these circumstances, the highest power of x is 0 which means that there is no such p and we can conclude that p ≤ 2n/3. (5) From step #4, we have: (2n!)/(n!n!) = ∏ (p ≤ 2n/3) pR(p,n) where R(p,n) is a function on p,n such that pR(p,n) divides (2n!)/(n!n!) but pR(p,n)+1 does not. [see Definition 1 above] (6) From Lemma 2 above, we know that if p is greater than 2n, then R(p,n)=1 so that we have: (7) Since there are less than 2n primes less than 2n, using Lemma 4 above, we have: (8) If we use a limit for primorials (see Theorem, here), we have: (9) Using Corollary 5.1 above, we also have: (10) So, putting it all together, we have: (11) If we multiply 4(-2n/3) to both sides we get: (12) Now, since n ≥ 1, we note that 4n2 is greater than 2n+1 (a) For n=1, 4(1)2 is greater than 2(1)+1 (b) Assume that 4n2 is greater than 2n+1 (c) Since n ≥ 1, 4n2 + 8n is greater than 2n+1+2 (d) 4n2 + 8n + 1 is greater than 2(n+1) + 1 (e) And: 4(n+1)2 is greater than 2(n+1)+1 (13) So it follows that: (14) Since n is greater than 18, it follows that (4/3)√2n is greater than 2 + √2n since: (a) 2(n) is greater than 6 (b) (4/3)√2n = (1/3)√2n + √2n (c) And (1/3)√2n is greater than (1/3)6 = 2. (15) This gives us: (16) Putting it all together gives us: (17) Taking the logarithm of both sides, gives us: (18) We can simplify this further by multiplying 3 to both sides and noting that 4=22 so that we get: (19) We can further simplify it by dividing both sides by 2n to get: (20) But now we use Lemma 7 above to reach a contradiction. QED References ### Primorial The primorial is the product of primes less than or equal to a number n. Definition: Primorial: n# The primorial n# is defined as follows: Example: 1# = 1 3# = 3(2#) = 32(1#)=32 6# = 5# = 532 Theorem: if n≥ 1, then n# is less than 4n Proof: (1) It is true for (n=1 and n=2) For n=1, 1#=1 is less than 4. For n=2, 2#=2 is less than 42 = 16 (2) If n is even and n is greater than 2, then n is composite and we have: n# = (n-1)# which is proven if we show that (n-1)# is less than 4n-1 since 4n-1 is less than 4n. (3) To complete the proof, we show that n# is less than 4n if n is odd: (a) By the inductive hypothesis, we can assume that it is true for all integers less than n and we assume that n is an odd integer where n=2x+1 and x ≥ 1 so that n ≥ 3. (b) Now, we note that 4m = (1+1)2m = (2-1)(1+1)2m+1 = (1/2)(1+1)2m+1 (c) Using the Binomial Theorem (see here), we note that: (1+1)2m+1 = ∑ (i=0, 2m+1) (2m+1!)/[(i!)(2m+1-i)!] (d) So, (1/2)(1+1)2m+1 = (1/2) ∑ (i=0, 2m+1) (2m+1)!/[(i!)(2m+1-i)!] is greater than: (1/2) [(2m+1)!/(m!)(m+1)! + (2m+1)!/(m+1)!(m)!] = (2m+1)!/(m!)(m+1)! (e) So, from (b) and (d), we can conclude that: 4m is greater than (2m+1)!/(m!)(m+1)! since 4m = (1/2)(1+1)2m+1 which is greater than (2m+1)!/(m!)(m+1)! (f) Each prime p that is greater than m+1 and less than 2m+1 divides (2m+1)!/(m+1)!(m!) since: (2m+1)!/(m+1)!(m!) is an integer. [see Lemma 5, here] Let u = (2m+1)!/(m+1)!(m!) (2m+1)! = u(m+1)!(m!) Since p does not divide (m+1)! and does not divide (m!), by Euclid's Generalized Lemma (see Corollary 2.1, here), p must divide u. (g) This gives us: From the definition of primorial [see Definition above], step #3f above, and step #3e above. (h) From the inductive hypothesis in step #3a, we can assume that: (m+1)# is less than 4m+1 (i) So, we have (using step #3g above): n# = (2m+1)# which is less (m+1)#4m (j) Using step #3h above, we have: (2m+1)# is less than 4m+14m = 42m+1 QED References ### Pascal's Rule Theorem: Pascal's Rule C(n,k) + C(n,k-1) = C(n+1,k) where: Proof: (1) We can find a common denominator for C(n,k) and C(n,k-1) so that we have: (2) So that: (3) To complete the proof, we note that: n-k+1 = n+1-k so that we have: = C(n,k+1) QED References ## Monday, November 16, 2009 ### Multiplicity of a Prime Factor Definition 1: Multiplicity of a prime factor The multiplicity of a prime factor of a given integer is the highest power of that factor that divides the integer. Example 1: 60 The prime factorization of is 2235 The multiplicity of 3 is 1. The multiplicity of 5 is 1. The multiplicity of 2 is 2. Adrien-Marie Legendre came up with the following theorem regarding the multiplicity of a prime for n!. (For review of factorials (n!), see here) Theorem 1: The multiplicity of a given prime p in n! is: That is, let x = the multiplicity of a prime p. Then px divides n! but px+1 does not. Proof: (1) If p is greater than n, then p doesn't divide n! and the answer is 0. (2) So, we can assume p ≤ n. (3) Let k be the largest power of p such that pk ≤ n. (4) For each 1 ≤ i ≤ k, the number of integers from 1 to n that are divisible by pi but not divisible by pi+1 is: If no numbers are divisible by pi, then floor(n/pi) = 0 [as does floor(n/pi+1)]. floor(n/pi+1) equals the number of integers that are greater than pi+1 and divisible by pi+1. So the number of integers that are uniquely divisible by pi is the difference of these two floor functions. (5) The value of px which divides n! is equal to the product of all the powers of p that divide any of the integers in the sequence from 1 .. n. (6) So, the we can compute the value of x with: (7) Putting this together, we see the following pattern: (8) We can see for i=1, we have floor(n/p1), for i=2, we have 2floor(n/p2) - 1floor(n/p2), and for i=3, we have 3floor(n/p3) - 2floor(n/p3) and so on. (9) The net result of this is that we rewrite the equation in step #7 to the following: QED References:	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.966621458530426, "perplexity": 1550.956207965811}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-17/segments/1492917121778.66/warc/CC-MAIN-20170423031201-00334-ip-10-145-167-34.ec2.internal.warc.gz"}
http://mathhelpforum.com/calculus/82414-challenging-calculus-problem.html	1. ## Challenging calculus problem the series (n=1 to infinity) (-1)^(n+1)/n and (n=1 to infinity) 1/n*2^(n) both converge to ln 2. 1) How do we know? 2) Which converges "faster" (i.e. gives a more efficient way to get a numerical approx.) To justify this, compute how many terms of each series must be added uup to approximate ln 2 with max. error 10^(-6)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9953746795654297, "perplexity": 4103.8499903647535}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886109525.95/warc/CC-MAIN-20170821191703-20170821211703-00096.warc.gz"}
https://infoscience.epfl.ch/record/151118	Infoscience Journal article # The nuMSM, leptonic asymmetries, and properties of singlet fermions We study in detail the mechanism of baryon and lepton asymmetry generation in the framework of the νMSM (an extension of the Standard Model by three singlet fermions with masses smaller than the electroweak scale). We elucidate the issue of CP-violation in the model and define the phase relevant for baryogenesis. We clarify the question of quantum-mechanical coherence, essential for the lepton asymmetry generation in singlet fermion oscillations and compute the relevant damping rates. The range of masses and couplings of singlet leptons which can lead to successful baryogenesis is determined. The conditions which ensure survival of primordial (existing above the electroweak temperatures) asymmetries in different leptonic numbers are analysed. We address the question whether CP-violating reactions with lepton number non-conservation can produce leptonic asymmetry below the sphaleron freeze-out temperature. This asymmetry, if created, leads to resonant production of dark matter sterile neutrinos. We show that the requirement that a significant lepton asymmetry be produced puts stringent constraints on the properties of a pair of nearly degenerate singlet fermions, which can be tested in accelerator experiments. In this region of parameters the νMSM provides a common mechanism for production of baryonic matter and dark matter in the universe. We analyse different fine-tunings of the model and discuss possible symmetries of the νMSM Lagrangian that can lead to them. #### Reference Record created on 2010-09-07, modified on 2017-05-12	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9071717262268066, "perplexity": 1566.001856896015}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-30/segments/1500549429417.40/warc/CC-MAIN-20170727182552-20170727202552-00052.warc.gz"}
http://residuetheorem.com/2016/02/	## Monthly Archives: February 2016 ### A sum I can only imagine being evaluated using the residue theorem The challenge this time is to evaluate the following sum: $$\sum_{n=1}^{\infty} \frac{1}{n^3\sin{\left (\sqrt{2} \, n\pi \right)}}$$ NB (20 Nov 2016) I just got word that Wolfram fixed the problem described below. See below for details. It should be clear that the sum converges…right? No? Then how do we show this? Numerical experiments are more or […]	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8979482054710388, "perplexity": 844.1681286159726}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-39/segments/1505818695375.98/warc/CC-MAIN-20170926085159-20170926105159-00123.warc.gz"}
https://forum.support.xerox.com/t5/Printing/Pale-horizontal-line-with-small-dots-above-it/m-p/226525/highlight/true	cancel Showing results for Did you mean: New Member ## Pale horizontal line with small dots above it Product Name: WorkCentre 7830/7835/7845/7855 Operating System: Windows 10 Hey there, I have a problem with the WorkCentre 7835, it prints on every single page a pale horizontal line with 5 or 6 small dots above the line. Here is an example picture: I have already tried to clean the IBT belt but the problem persist. If R1 and R3 are switched, the dots become magenta and the line more visible. Here is a picture: Can someone tell me where my problem is and how can I resolve it, please? Thank you! 9 Replies ## Re: Pale horizontal line with small dots above it Hi, If the issue moves to another color replace the affected drum (R1-R4). Gabi New Member ## Re: Pale horizontal line with small dots above it First of all, thank you for the fast response. So, if the dots are black before switching the drums and they become magenta after switching them, is R1 the affected drum? If not, can you please tell me how can I troubleshoot this issue and find out the affected drum? ## Re: Pale horizontal line with small dots above it The affected drum is the one that was initially in position R1 when it was getting black dots, so that drum you must replace. New Member ## Re: Pale horizontal line with small dots above it Thank you so much for the support! Have a nice day. New Member ## Re: Pale horizontal line with small dots above it How do they look? New Member How does it look New Member ## Re: Pale horizontal line with small dots above it Hi! thanks you so much again for resolving my problem with the dots, can you tell me which can the reason be for the double black line? Because the dots desappear but the line stills a little bit.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.897859513759613, "perplexity": 3395.6726503454483}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154878.27/warc/CC-MAIN-20210804142918-20210804172918-00150.warc.gz"}
http://mathhelpforum.com/discrete-math/158406-empty-cartesian-product-implies.html	Math Help - Empty Cartesian product implies...? 1. Empty Cartesian product implies...? Suppose that A x B = null set, where A and B are sets. What can you conclude? 2. By definition, $A\times B=\{(x,y)\mid x\in A\mbox{ and }y\in B\}$. Now, when does it happen that any set of the form $\{f(x,y)\mid A(x,y)\}$ is empty? This happens when it is not the case that there exist x and y such that A(x,y), or, equivalently, when for all x and y, it is not the case that A(x,y). Can you state this more simply for this particular A? 3. I am not sure. Can one be a set and another the null? I am very confused on the null set. 4. Suppose that A x B = null set, where A and B are sets. Can one be a set and another the null? It is given that both A and B are sets, though one or both can be empty. The easiest way to find the answer is to consider several cases. Suppose both A and B are nonempty. Then there exists an $x\in A$ and a $y\in B$. By the definition of $A\times B$, then $(x,y)\in A\times B$; therefore, $A\times B\ne\emptyset$. Suppose that A is empty, regardless of B. Then you can't find a single pair of $x$ and $y$ such that $x\in A$ and $y\in B$. Therefore, $A\times B=\emptyset$. The same happens when B is empty. The same can be shown more formally. I'll write $\exists$ for "there exists", $\land$ for "and", $\lor$ for "or" and $\neg$ for negation. Then for any function $f(x,y)$ and a predicate $P(x,y)$, $\{f(x,y)\mid P(x,y)\}\ne\emptyset$ iff $\exists x\exists y.\,P(x,y)$. In our case, $P(x,y)$ is $x\in A\land y\in B$, and $f(x,y)$ is the function that returns a pair $(x,y)$. So $A\times B=\{(x,y)\mid x\in A\land y\in B\}\ne\emptyset$ iff $\exists x\exists y.\,x\in A\land y\in B$. This in turn is equivalent to $(\exists x.\,x\in A)\land(\exists y.\,y\in B)$, or $A\ne\emptyset\land B\ne\emptyset$. Taking negation, $A\times B=\emptyset$ iff $\neg(A\ne\emptyset\land B\ne\emptyset)$ iff $(\neg A\ne\emptyset)\lor (\neg B\ne\emptyset)$ iff $A=\emptyset\lor B=\emptyset$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 32, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9742823839187622, "perplexity": 117.1476962898428}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1404776435808.92/warc/CC-MAIN-20140707234035-00004-ip-10-180-212-248.ec2.internal.warc.gz"}
https://www.math-only-math.com/coordinate-geometry-graph.html	# Coordinate Geometry Graph Here we will learn how to draw coordinate geometry graph. When two variables x, y are related, the value(s) of one variable depends on the value(s) of the other variable. Let x, y be two variables related by 9x – 3y + 4 = 0. Then, y = 3x + $$\frac{4}{3}$$. This gives the value of y corresponding to a value of x. For x = 0, 1, -1, etc, we get y = $$\frac{4}{3}$$, $$\frac{13}{3}$$, $$\frac{-5}{3}$$, etc. Therefore, (0, $$\frac{4}{3}$$), (1, $$\frac{13}{3}$$), (-1, - $$\frac{5}{3}$$), etc,. are all values of the ordered pair (x, y) given by the relation y = 3x + $$\frac{4}{3}$$. Every ordered pair of real numbers represents the coordinates of a point. So, we can always plot a point corresponding to a ordered pair of values of (x, y) satisfying a given relation between x and y. The collection of all the points corresponding to the ordered pairs (x, y) satisfying the relation between x and y, is the graph of the relation. Graph of Linear Relations Between x, y: The points with coordinates (x, y) satisfying the linear relation ax + by + c = 0 lie on a straight line. So, the graph of a linear relation ax + by + c between the variables x and y is a straight line. The graph can be constructed by plotting at least two points corresponding to two ordered pairs of values of (x, y) satisfying the relation and then joining the two points by a straight line.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9562276005744934, "perplexity": 154.46527367161198}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500058.1/warc/CC-MAIN-20230203154140-20230203184140-00114.warc.gz"}
https://www.physicsforums.com/threads/a-doubt-on-the-fundamental-unit-of-charge.983835/	# A doubt on the fundamental unit of charge • #1 48 3 "a charge smaller than e has not been found. if one determines the amount of charge on any charged body like a charged sphere or charged drop) or any charged particle (like positron, a-particle) or any ion, then its charge is always found to be an integral multiple of e, i.e., e,3e; 4e,... No Charge will be fractional multiple of e like 0.7e or 2.5e." the book(notes actually) also gave this side note- "The existence of charged particles called,quarks whose electric charges come in multiples of e/3, would not alter the fact that charge is quantized- it would merely reduce the size of the basic unit from e to e/3" my question is wouldn't that still imply then that a charge smaller than e does exist since charge of quarks comes in multiples of e/3. it would make sense if you add something more like an isolated quark doesn't exist like they always come in triplets or of that sort such that the sum of charges always adds up to e anyways.am i getting it right.do correct if wrong. • #2 scottdave Homework Helper 1,814 778 "a charge smaller than e has not been found. if one determines the amount of charge on any charged body like a charged sphere or charged drop) or any charged particle (like positron, a-particle) or any ion, then its charge is always found to be an integral multiple of e, i.e., e,3e; 4e,... No Charge will be fractional multiple of e like 0.7e or 2.5e." the book(notes actually) also gave this side note- "The existence of charged particles called,quarks whose electric charges come in multiples of e/3, would not alter the fact that charge is quantized- it would merely reduce the size of the basic unit from e to e/3" my question is wouldn't that still imply then that a charge smaller than e does exist since charge of quarks comes in multiples of e/3. it would make sense if you add something more like an isolated quark doesn't exist like they always come in triplets or of that sort such that the sum of charges always adds up to e anyways.am i getting it right.do correct if wrong. I'm not sure which book it is. From what you stated, it appears the author of the book does not know what the charge on a quark is, but is stating that if quarks have a fractional charge, they will still be quantized. Like you couldn't split a proton and get 1 over sqrt(2) as one of the "pieces". For "regular" particles, we have only encountered integer multiples of e. Likes ohwilleke • #3 mfb Mentor 35,720 12,303 All elementary particles have an electric charge that is a multiple of e/3. All particles (elementary or composite) that can exist freely have an electric charge that is a multiple of e, as quarks always come in combinations that add up to a multiple of e. Likes ohwilleke, vanhees71 and scottdave • #4 mathman 8,005 518 Proton have 2 up quarks (charges at 2/3e) and one down quark (charge at -1/3e). Neutrons have one up and two down quarks. Mesons have various combinations, such as a quark plus anti-quark of the same kind. Likes ohwilleke • Last Post Replies 3 Views 2K • Last Post Replies 18 Views 2K • Last Post Replies 3 Views 2K • Last Post Replies 8 Views 10K • Last Post Replies 6 Views 1K • Last Post Replies 13 Views 11K • Last Post Replies 5 Views 3K • Last Post Replies 14 Views 1K • Last Post Replies 6 Views 1K • Last Post Replies 12 Views 4K R	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9370395541191101, "perplexity": 2675.1512572588463}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964362230.18/warc/CC-MAIN-20211202145130-20211202175130-00417.warc.gz"}
http://clay6.com/qa/25996/which-of-the-following-statements-about-magnetic-properties-is-are-incorrec	# Which of the following statements about magnetic properties is/are incorrect? (1) The magnetic field produced by unpaired electrons is due to their spin and orbital motion (2) Paramangetism is when the inducted magnetic moment in the compound acts in the opposite direction of the applied field (3) Ferromagnetism occurs mainly in metal alloys and oxides of transition elements (4) Ferromagnetic substances show normal paramagnetic behaviour above a certain temperature called Neel Point $\begin {array} {1 1} (1), (3) and (4) are\; incorrect \\ Only (2)\; is\; incorrect \\ Only (2)\; and\; (4)\; are\; incorrect \\ Only\; (3) \;and\; (4)\; are \;incorrect \end {array}$ Answer: Only (2) and (4) are incorrect (2) Paramangetism is when the inducted magnetic moment in the compound acts in the opposite direction of the applied field: $\quad$ Diamagnetism is when a magnetic field is applied to a substance in which all electrons are paired and the inducted magnetic moment in the compound acts in the opposite direction. (4) Ferromagnetic substances show normal paramagnetic behaviour above a certain temperature called Neel Point $\quad$ For ferromagnetic substances, this is called Curie Point, and for anti-ferromagentic substances, its called Neel Point. edited Aug 6, 2014	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9341214895248413, "perplexity": 1548.5720959360633}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886103270.12/warc/CC-MAIN-20170817111816-20170817131816-00332.warc.gz"}
http://slideplayer.com/slide/4564443/	# Section IX Electroweak Unification. 221 Electroweak Unification  Weak Charged Current interactions explained by W  exchange.  W bosons are charged, ## Presentation on theme: "Section IX Electroweak Unification. 221 Electroweak Unification  Weak Charged Current interactions explained by W  exchange.  W bosons are charged,"— Presentation transcript: Section IX Electroweak Unification 221 Electroweak Unification  Weak Charged Current interactions explained by W  exchange.  W bosons are charged,  couple to photon. Consider 2 diagrams (+interference)  Cross-section DIVERGES at high energy  Divergence cured by introducing Z 0  Extra diagram  Only works if , W , Z 0 couplings are related  ELECTROWEAK UNIFICATION ee e+e+ e+e+ ee e+e+ ee  WW (pb)  222 The GWS Model The Glashow, Weinberg and Salam model treats EM and WEAK interactions as different manifestations of a single UNIFIED ELECTROWEAK force (Nobel Prize 1979) Basic Idea: Start with 4 massless bosons {W +,W 0, W  } and B 0. The neutral bosons MIX to give physical bosons (the particles we see), i.e. the W , Z 0 and . Physical fields: W +, Z 0, W  and A (photon) WEAK MIXING ANGLE W , Z 0 “ acquire” mass via the HIGGS MECHANISM. 223 The beauty of the GWS model is that it makes EXACT predictions of the W  and Z 0 masses and couplings with ONLY 3 free parameters. Couplings given by  em and Masses given by G F and From Fermi theory If we know {  em, G F, sin } (from experiment) everything else is FIXED. WW Z0Z0  224 Evidence for GWS Model  Discovery of Neutral Currents (1973) The process was observed. ONLY possible Feynman diagram (no W  diagram) Indirect evidence for Z 0  Direct Observation of W  and Z 0 (1983) First DIRECT observation in collisions at GeV via decays into leptons.  Precision Measurements of the Standard Model (1989-2000 ) LEP e + e - collider (see later) provided many precision measurements of the Standard Model.  Wide variety of different processes consistent with GWS model predictions and measure SAME VALUE of 225 The Weak NC Vertex All weak neutral current interactions can be described by the Z 0 boson propagator and the weak vertices:  Z 0 NEVER changes type of particle  Z 0 NEVER changes quark flavour  Z 0 couplings are a MIXTURE of EM and WEAK couplings and therefore depend on (exact form treated in Part III Particles course) +antiparticles Z0Z0 STANDARD MODEL WEAK NC LEPTON VERTEX Z0Z0 STANDARD MODEL WEAK NC QUARK VERTEX Z0Z0 226 Examples: 227  Summary of Standard Model Vertices ELECTROMAGNETIC STRONG WEAK CC WEAK NC (QED) (QCD)  g WW WW Z0Z0 Z0Z0 +antiparticles 228 Drawing Feynman Diagrams A Feynman diagram is a pictorial representation of the matrix element describing particle decay or interaction To draw a Feynman diagram and determine whether a process is allowed, follow the FIVE basic steps: Write down the initial and final state particles and antiparticles and note the quark content of all hadrons. Draw the SIMPLEST Feynman diagram using the Standard Model vertices. Bearing in mind: - Similar diagrams for particles/antiparticles - NEVER have a vertex connecting a LEPTON to a QUARK - Only the WEAK CC vertex changes FLAVOUR within generations for leptons within/between generations for quarks 1 2 229 - Particle scattering  If all particles (or all antiparticles), only SCATTERING diagrams involved e.g.  If particles and antiparticles, can have SCATTERING and/or ANNIHILATION diagrams e.g. Initial state Final state 230 Check that the whole system CONSERVES - Energy, momentum (trivially satisfied for interactions) - Charge - Angular Momentum Parity - CONSERVED in EM/STRONG interaction - CAN be violated in the WEAK interaction Check SYMMETRY for IDENTICAL particles in the final state - BOSONS - FERMIONS Finally, a process will occur (in order of preference) via the STRONG, EM and WEAK interaction if steps 1 - are satisfied. 3 4 5 1 5 231 Examples: 1. 2. 3. 4. 232 Experimental Tests of the Standard Model The Large Electron Positron (LEP) collider at CERN provided precision measurements of the Standard Model (1989-2000). Designed as a Z 0 and W  boson factory Precise measurements of the properties of Z 0 and W  bosons provide the most stringent test of our current understanding of particle physics. e+e+ ee f = fermion 233 LEP France Switzerland  LEP is the highest energy e + e - collider ever built.  Large = 27 km circumference  4 experiments combined: 16,000,000 Z 0 and 30,000 W  events 234 A LEP Detector: OPAL OPAL was one of the 4 experiments at LEP. Size = 12 m x 12m x 15m Muon Chambers Hadron Calorimter Electromagnetic Calorimeter Tracking Chambers 235 Particle Identification Different particles leave different signals in the various detector components allowing almost unambiguous identification. e    EM energy + track  : EM energy, no track      track + small energy deposit + muon      decay, observe decay products : not detected Quarks: seen as jets of hadrons Hadrons: energy deposit+ track (charged only) ee   JET Electron Photon Muon Pion Neutrino Jet 236 Typical Events 237 In the event, the tau leptons decay within the detector (lifetime ~ 10 -13 s). Here, and 238 The Z 0 Resonance Consider the process of  Previously,, only considered an intermediate photon  At higher energies also have the Z 0 exchange diagram (plus Z 0  interference)  The Z 0 is a decaying intermediate massive states (lifetime ~ 10 -25 s)  BREIT-WIGNER RESONANCE  At the Z 0 diagram dominates. (see pages 123 and 100) 239 240 BREIT-WIGNER formula for (where is ANY fermion-antifermion pair) Centre-of-mass energy with giving   Z is the TOTAL DECAY WIDTH, i.e. the sum over the partial widths for different decay modes. 241 At the peak of the resonance : Hence, for ALL fermion/antifermion pairs in the final state Compare to the QED cross-section at 242 Measurement of M Z and  Z  Run LEP at various centre-of-mass energies ( ) close to the peak of the Z 0 resonance and measure  Determine the parameters of the resonance: Mass of the Z 0, M Z Total decay width,  Z Peak cross-section,   One subtle feature: need to correct measurements for QED effects due to radiation from the e + e - beams. This radiation has the effect of reducing the centre-of-mass energy of the e + e - collision which smears out the resonance.  had [nb] E CM [GeV] 243 M Z measured with precision 2 parts in 10 5  To achieve this required a detailed understanding of the accelerator and astrophysics! Tidal distortions of the Earth by the Moon cause the rock surrounding LEP to be distorted. The nominal radius of LEP changes by 0.15 mm compared to radius of 4.3 km. This is enough to change the centre-of-mass energy !  Also need a train timetable ! Leakage currents from the TGV rail via lake Geneva follow the path of least resistance…using LEP as a conductor. Accounting for these effects (and many others): 244 Number of Generations  Currently know of THREE generations of fermions. Masses of quarks and leptons INCREASE with generation. Neutrinos are ~ massless (?)  Could there be more generations ? e.g.  The Z 0 boson couples to ALL fermions, including neutrinos. Therefore, the total decay width,  Z, has contributions from all fermions with m f < M Z / 2 with  If there were a FOURTH generation, it seems likely that the neutrino would be light, and, if so would be produced at LEP  The neutrinos would not be observed directly, but could infer their presence from the effect on the Z 0 resonance curve. 245 At the peak of the Z 0 resonance A FOURTH generation neutrino would INCREASE the Z 0 decay rate and thus INCREASE  Z. As a result a DECREASE in the measured peak cross-sections for the VISIBLE final states would be observed.  Measure the cross-sections for all visible decay modes (i.e. all fermions apart from ) Examples: 246  Have already measured M Z and  Z from the shape of the Breit- Wigner resonance. Therefore, obtain from the peak cross- sections in each decay mode using Note, obtain from  Can relate the partial widths to the measured TOTAL width (from the resonance curve) where N is the NUMBER OF NEUTRINO SPECIES and  is the partial width for a single neutrino species. 247 The difference between the measured value of  Z and the sum of the partial widths for visible final states gives the INVISIBLE WIDTH In the Standard Model, calculate Therefore  THREE generations of light neutrinos 2494  4.1 MeV 83.7  0.2 MeV 84.0  0.3 MeV 83.9  0.4 MeV 1745.3  3.5 MeV 497.3  3.5 MeV 248 Most likely that ONLY 3 GENERATIONS OF QUARKS AND LEPTONS EXIST In addition:  are consistent  universality of the lepton couplings to the Z 0  is consistent with the expected value which assumes 3 COLOURS – yet more evidence for colour 249 W + W - at LEP  e + e - collisions W bosons are produced in pairs.  Standard Model 3 possible diagrams:  LEP operated above the threshold for W + W - production (1996-2000)  Cross-section sensitive to the presence of the Triple Gauge Boson vertex  WW [pb] ee e+e+ e+e+ ee e+e+ ee 250 W + W - Decay at LEP In the Standard Model and couplings are  equal. EXPECT (assuming 3 colours): Branching fractions QCD corrections ~  3 FOR COLOUR 10.5% 43.9% 45.6% 251 W + W - Events in OPAL 252 Measurement of M W and  W  Unlike, W boson production at LEP is NOT a resonant process  M W measured by reconstructing the invariant mass on an event-by- event basis. 4-momenta 253 W Boson Decay Width In the Standard Model, the W boson decay width is given by  -decay: LEP: Therefore, predict partial width: Total width is the sum over all partial widths: IF the W coupling to leptons and quarks is EQUAL, and there are 3 colours: Compare with measured value (LEP):  Universal coupling constant  Yet more evidence for colour !  3 FOR COLOUR 254 Summary Now have 5 precise measurements of fundamental parameters of the Standard Model In the Standard Model, ONLY 3 are independent. Their consistency is an incredibly powerful test of the Standard Model of Electroweak Interactions ! (at q 2 =0) Download ppt "Section IX Electroweak Unification. 221 Electroweak Unification  Weak Charged Current interactions explained by W  exchange.  W bosons are charged," Similar presentations	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9627393484115601, "perplexity": 4478.507135675877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986676227.57/warc/CC-MAIN-20191017200101-20191017223601-00356.warc.gz"}
http://physics.stackexchange.com/questions/9840/detection-of-w-and-z-bosons?answertab=oldest	# Detection of W and Z bosons What specific behaviour confirmed the existence of the W and Z bosons at the UA1 and UA2 experiments? Thanks! - See an UA2 paper from 1983 about Z0, ccdb4fs.kek.jp/cgi-bin/img/allpdf?198308375 , as an example. They found 8 events with the right mass around 91 GeV that decayed into e+ e-, relatively to the non-Z background of 0.7 events or so only. So clearly there had to be something at the mass indicated and it agreed with the electroweak theory. – Luboš Motl May 14 '11 at 6:32 @ Luboš Motl: most enlightening, thank you very much. – Richard Terrett May 14 '11 at 6:47	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9017248749732971, "perplexity": 2186.2590753232594}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1414637899632.42/warc/CC-MAIN-20141030025819-00144-ip-10-16-133-185.ec2.internal.warc.gz"}
https://essayhelpp.com/linear-algebra-questions-and-answers/	# Linear Algebra questions and answers • Please show all your work. 6. Solve this linear system using a method of your choice. DO NOT CHANGE FRACTIONS TO DECIMALS! 3x X _-10 4 6 12 3x +2y=-1 @ (5) 7. When solving a linear system, Alice got t… • Please show all your work. Proper mathematical form is expected. You must show your work for full marks. 1. Without solving, determine if this linear system has one solution, no solution, or an infini… • Please show each step, thank you.. E Day20-Activity.pdf 1 / 1 100% + 1. Given and R= ‘ ‘] a. Compute the eigenvalues of M. b. Compute eigenvectors for each eigenvalue of M c. M is reflection across y … • R= o -1 Â Please show all steps. Thank you. Â Â Â 1 Â 0. E Day19-Activity.pdf 1 1 – 125% + Given v = [2, -1 62 a. Compute Tv to get coordinates in the standard basis B1 b. Compute R(Tv) to get coo… • Please show through parameterization.. 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Ex: 1.2345 • This is wrong – need correct answer. Question 8 In each case, compute the required projection of one vector along another: Tries remaining: 1 proj(-28,5)(-18, -22) -1103 1970 Marked out of 10.00 1792/… • Determine by inspection whether the vectors are linearly independent. Justify your answer. Â Â . Determine by inspection whether the vectors are linearly independent. Justify your answer. 6 4 Select … • Find theâ€‹ value(s) of h for which the vectors are linearly dependent. Justify your answer. Â Â . Find the value(s) of h for which the vectors are linearly dependent. Justify your answer. 10 The val… • Determine if the columns of the matrix form a linearly independent set. Â Â . Determine if the columns of the matrix form a linearly independent set. 1 3 -3 3 2 7 -4 2 2 9 2 -10 Select the correct ch… • Determine if the columns of the matrix form a linearly independent set. Â Justify your answer. Â . Determine if the columns of the matrix form a linearly independent set. 2 -1 O Justify your answer. … • . Consider the simplex tableau: x y u U M b 2 3 1 0 O 12 1 1 0 1 0 10 â€”10 â€”20 0 0 1 0 Using the rules, enter the value of the correct pivot element in the space below: Answer:. Continuing with t… • Describe all solutions of A x = 0 in parametric vectorâ€‹ form, where A is row equivalent to the given matrix.. O Points: 0 of 1 Describe all solutions of Ax = 0 in parametric vector form, where A is … • Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below.. Describe the sol… • . Which of the following shows the name of these properties respectively: (g (f (a) ) ) = dg df df dx and a (f (ac) + g (a) ) = if (a) + ag(x) O Product rule and Sum rule O Sum rule and Chain rule O… • Considering d/dx means derivative with respect to x, which of the following is correct? (choose one) O d/dx[f(x).g(x)]= d/dx(f(x)) . d/dx(g(x)) O d/dx[f(x).g(x)]= g(x).d/dx(f(x)) + d/dx(g(x)) O d/dx[f… • Describe all solutions of A x = 0 in parametric vectorâ€‹ form, where A is row equivalent to the given matrix.. Describe all solutions of Ax = 0 in parametric vector form, where A is row equivalent to… • Which of the following is answer to dig (at) where f(w) = 3:132 â€” 5.5!: + 2? xx. 6X /"’\_ 6X’ 5 3x-5 None of the above. • Suppose a rocket is being launched into outer space. On its way up, several different forces act on the rocket. One of these forces, thrust, is applied by the rocket’s engines. The rocket’s engines … • . The constraint relating to Choose… his total exercise: The constraint related to the Choose… swimming restriction:. Continuing with this same question: Use the Simplex Method to solve this lin… • . Continuing with this same question, enter the value at "A" after you have completed the next pivot: x y u 1 1 1 3 E 0 g A 6 4 0 ‘3 3 Enter this as a fraction: Answer:. Use the Simplex Me… • . Consider the simplex tableau: x y u v M b 2 3 1 0 O 12 1 1 0 1 O 10 â€”10 â€”20 0 0 1 0 Using the rules, enter the value of the correct pivot element in the space below: Answer:. Consider the simp… • . Find the particular solution corresponding to the tableau below: x y u v 1 0 3 11 0 1 10 17 0 0 5 â€”1 X: Choose… Â¢ M = Choose… Â¢ u = Choose… Â¢ y= Choose… Â¢ 0 v = Choose…. Restate th… • 2) a) Determine the matrix X if it satisfies the matrix equation [8 8 ] [1 0]- [8 :] b) Given A, B and C that are three n by n invertible matrices. A and B have inverses A , B-respectively. Solve for… • Good evening, can you please include the work for question #4.. 2. Factorize the expression yx 4. You sell 600 shares in a company via a stockbroker who charges a flat 35$commission rate on all trans… • Good evening, can you please include the work as well. The example on my textbook isn’t enough.. 1. In the demand function below, a) Find the elasticity if the price goes down from 15 to 10$. b) Find … • What’s the initial height & calculate the height after 2 seconds. Question 4 (6 points) Juanita is a competitive diver. One of her dives can be modelled by the quadratic relation h – 2t- – 18, whe… • please show me how in gauss jordan form. Do the three lines $1-412 = 1, 201-12 = -3, and -21-312 = 4 have a common point of intersection? Explain. • 5 solution Sets of Linear Systems.Â Please complete question 10 found between pages 48-49. Show all work explicitly stated or mathematically derived in nature. I’ve also added contextual pages whic… • . 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Show that the area of R is given by the formula to the right. X1 y1 {area of triangle} = ? det X2 y2 1 X3 /3 1 . . . Trans… • Please answer question 5 on page 40 – 42. There are additional pages for context.. EXAMPLE 2 For v1, V2, V3 in R", write the linear combination 3v1 – 5v2 + 7V3 as a matrix times a vector. 36 CHAP… • . Find an explicit description of Nul A by listing vectors that span the null space. 130 2 O A=001-3 O 000 0-1 A spanning set for Nul A is D. (Use a comma to separate answers as needed.) • . Find an explicit description of Nul A by listing vectors that span the null space. _137 0 014-8 A: A spanning set for Nul A is U. (Use a comma to separate vectors as needed.) • . A set in R is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of R. (For instance, find two vectors in H whose sum is not… • how do I find and describe the solution of b?. 2 Let A = and b = b1 3 b2 Show that the equation AT = b does not have a solution for all possible b, and describe the set of all b for which it does have… • Which of the following pairs of vectors are linearly dependent? (choose one only) Â a=[1,2,3] and b=[5,0,19] Â a=[12,6,2] and b=[-6,-3,-1] Â a= [7,1,3] and b=[14,2,-1] Â None of the above • What is the inner product of the following vectors? (enter the numerical value, e.g., 8) x=[-1, 0, 14], and y= [2Â 17 1] x.y=? • From Lecture 6 the price changed for soy from$30/acre to $50/acre and the price of corn from$40/acre to $60/acre. Redo this problem using the procedure outlined in the lecture notes. Show all ste… • Fred Budgette has a limited food budget and wants to minimize his food costs while stillÂ meeting his daily nutritional requirements. He can buy two foods. Food1 sells for$7.00Â per pound and ea… • Which of the following show the commutative and associative properties of vector addition? Â a + b = b + a and a+0=0+a Â a-a=0 and (a + b) + c = a + (b + c) Â a + b = b + a and (a + b) + c = a + (b… • Which of the following pairs of vectors are equal? a=[1,5] and b=[1,5] a=[1,2,5] and b=[4,2,0] a=[7,0,4,5] and b=[9,1] • . Compute the product AB by the definition of the product of matrices, where Ab, and Ab are computed separately, and by the row-column rule for computing AB. 4 3 A = 3 5 , B = 2 4 Set up the product… • . 2 -4 Let A = Compute 212 – A and (212 ) A. 3 2 212 – A = (Type an integer or decimal for each matrix element.) • Any help at all would be appriciated.. f(x, x’ , yly? ) = Eil(yly?) (1/2) x2 Prove that Eil(yly?) (1/2) ac!) where 1 and 2 refer to periods 1 and 2 respectively; satisfies the condition that: fk(y 2X… • . Assume that T is a linear transformation. Find the standard matrix of T. T: R2 – R, T(e1 ) = (5, 1, 5, 1), and T (e2) = (- 7, 9, 0, 0), where e, = (1,0) and ez = (0, 1). A = (Type an integer or d… • Solve this problem using R code.Â . c. (This problem in similar except the last two rows have been changed by changing the last two RHS.) Min z = 1X1 + 4X2 ST: Nut A: 10X1 + 10X2 = 40 Nut B: 1X1 + 0X2… • Let X be a set, and let F be a field. Let V be the set of functions from X to F , with addition and scalar multiplication defined as in class. Show that V is a vector space, by checking that these two… • Problems 1. Recall from Tutorial 1 that the set of continuous functions C from R to R form a vector space. Prove or disprove the following: (a) The set { sin? x, cos2 x} is linearly independent in C. … • . 3 0 2 a Let A = , and define T : R – R< by T(x) = Ax. Find the images under T of U = and V = 0 3 – 4 b T(u) = • Let F be a finite field of cardinality p, and let V be a vector space containing a basis (el, , en) of cardinality n. Show that V is a finite set, and calculate its cardinality in terms of n. Use this… • Make the following changes (where X1 = Feed1 and X2 = Feed 2) and determine theÂ solution which includes the decision varies, disposal variables, and the cost (includeÂ units). Â Using R code. a. Mi… • . Question 5. The equation of a line in the plane is an equation of the form ax + by + c = 0 where not both a, b are zero. The equation of a conic in the plane is an equation of the form ax2 + by + … • . Question 4. Let A be an m x 71 matrix and 1â€”; E Rm an mâ€”vector. (a) Assume that 151,162 6 R" are two solutions to the equation A5? = I; Show that (I = 171 â€” 172 is a solution to the equ… • . (1 point) Let V be the vector space of symmetric 2 X 2 matrices and W be the subspace W = spant 3 _. [9 2]. a. Find a nonzero element X in W. 1 -2 X = -2 -6 b. Find an element Y in V that is not i… • Not sure how it is wrong. (1 point) Given the augmented matrix U – A = 4 L L W N 4 perform each row operation in the order specified and enter the final result. First: -4R1 + R2 – R2, Second: 2R1 +… • Thank you. X – + Z. W + 2z + w – Z W = N – X + 2y – 3z + 5w = X = , y= , Z= , W = • THANK YOU. -6×2 +2×3 +2X4 = 4 5×1 +x2 +2×3 +3×4 = 3 -X1 4×1 -5×2 +4×3 +5X4 = 7 -2×1 +2×2 +4×3 +6X4 = 6 X ] X2 +S +t = X3 X4 • Problem 9-09 (Algorithmic) Epsilon Airlines services predominantly the eastern and southeastern united States. The vast majority of Epsilon’s customers make reservations through Epsilon’s website, but… • suppose the system below is…. Suppose the system below is consistent for all possible values of f, g. What can you say about the coefficients of c and d? Justify your answer. x1 + 3×2 = f cr1 + dx2 … • Let A = […… Let A = [6 3] and 6 – [box Show that the equation AT = b does not have a solution for all possible b, and describe the set of all b for which it does have a solution. • Option 2: Point of Intersection [6 points] Line A has the same slope as 3x – 7y = 21, and the same y-intercept as y = 3x – 1. Line B passes through point (-4,1) and has a slope of -2. a) Determine the… • Supposons que (a, b, c) et (u, v, w) etre deux vecteurs, et supposons que a, b, c, u, v w Que pouvons-nous conclue ? (a, b, c) (u, v, w) 2 1 1 (a, b, c) x (u, v, w) 1 (a, b, c) and (u, v, … • Determine (without solving each system) if the two systems below have the same solution set. âˆ’x + 7y âˆ’ 2z = 4 5x âˆ’ 15y + 10z = 20 x âˆ’ 3y + 2z = 4 4y = 8. 2. Determine (without solving each s… • please help me. Write the augmented matrix for the linear system that corresponds to the matrix equation AT = b. Then solve the system and write the solution as a vector. A = -4 -3 • Hi can someone solve this problem and explain it clearly because it doesn’t explain in our class well. Â Â 1. Maximization Problem Â Â Â Â Â 2. Minimization Problem Â Â . To make one unit of p… • Part a,b,c. 3. Consider the linear system X1 – 2×2+ X3 = 3 X1 +x2=0 X1 + 4×2 – X3 = -4. (a) Write the linear system in the matrix form Ax = b. (b) Create the augmented matrix [A b] and bring it into t… • Part a,b,c,d. 2. Consider the linear system -X1 +x2 =6 X1 + 2×2 = 3. (a) Write the linear system in the matrix form Ax = b. (b) Create the augmented matrix [A b] and bring it into the upper triangular… • Part a,b,c. 1. Calculate Ax and describe using as many of the keywords as appropriate how the matrix A acts on the vector x = [x, ] to produce Ax. For each of parts (a)-(c), there is at least one keyw… • Q5 1 Point Let A be an m X n matrix. Which of the following statements is not logically equivalent to the others? O For each b in R, the equation Ax = b has a solution O Each b in R" is a linear … • Q4 1 Point The collection of all vectors that can be written in the form y. = CIV1 + . . . CpVp is called the O augmented form of V1, V2, . . ., Vn O span of V1, V2, . . ., Vn O set V1, V2, . . ., Vn • Q3 1 Point Given a collection of vectors V1, V2, .. ., Vp, and scalars C1, C2, . . ., Cp, the vectors y defined by y = CIV1 + . . . CpVp is called a O output vector O linear combination O distributed … • Based on my solution shown below, what am I missing to solve this question? How can I obtain torque u(t) corresponding to x 1 â€‹ = Î˜ 1 â€‹ . What do I need to do after the last line I have written t… • . 1. Which of the following subsets of R2 are subspaces? (a) ((x, y) \|x = 3y) (d) {(x, y) \|x = y3 ] 7. Which of the following subsets of the space M2z of 2 X 2 matrices are subspaces? Prove that you… • . Question 3: Suppose 4 – 35] Verify that the rank A and A" are equal (i.e. calculate both). Question 4: Find a 2 x 4 matrix , that is not equal to zero, such that – 0. Literally just state an … • . 1. Let S = Which of the following vectors are linear combinations of the vectors in S? (a) 1 (b) 2 (c) 2 2 • Hi can someone solve this problem and explain how to get the answer of this problem? If you can just explain it in step by step. This will help me to study in advance. Â Â Â 1.) Maximization Proble… • Please help me with the problem below. Â It has multiple parts but it’s one problem. Â Thanks! Â . Let F be a field of characteristic zero and let D be the formal polynomial differentiation map, so th… • . Question 1: Are there real numbers 21, 72, 23, Is such that 31 -12 = 02 – $3 = 03-D = 14-0 = 1? (Hint: recognize this as a system of linear equations and solve). Question 2: Find the general solut… • Tutors, please help me out with this theory Linear Algebra problem. Â Thx!. Let D be an integral domain and x an indeterminate. a. Describe the units in D[x]. b. Find the units in Z[x]. c. Find the un… • Please help me with the problem above. Â Only brief explanations are necessary since it’s just work toÂ prep me for the midterm. Â Thanks! • . 20. Consider the element f (x, y) = (3×3 + 2x) y3 + (x2 – 6x + 1)y2 + (x4- 2x)y + (x4- 3×2+2) of (Q[x])[y]. Write f(x, y) as it would appear if viewed as an element of (Qly])[x]. • Assume the training feature matrix X is a (p+1)N matrix, where p is the number of predictors and N is the sample size of training data, and p>N . Assume X is of full rank , i.e. rank(X)=r=N<… • Given matrix and its corresponding linear map M: . Show that M is not surjection nor injection. • please prove or disprove the following statement Â . Let L : V -> V be a linear operator and let v E V such that v * Ov. If v E Range(L) then v & ker(L). • helpppppppp/linear algebra Â . Problem 2. Consider the following system of linear equations in ï¬ve variables: 22:2 â€” 6% = â€”12 2:31 â€” 105% + 4:53 + 6:34 + 12% = 28 2551 â€” 5332 + 45.53 + 6554 … • Do the three lines…. Do the three lines x1 -4×2 = 1, 2×1-12 = -3, and -21-312 = 4 have a common point of intersection? Explain. • What are the emerging trends in structured programming. • . Let L be the line through the points (D, {1) and (1, 2]. Let T : R2 â€”> R2 be a linear transformation such that for each :3 E R2, T reï¬‚ects if over L and then rotates the reï¬‚ection by Â«,3… • 3 Linear Algebra Questions Â . Linear Algebra Assignment 3 Solve the following problems (and show work). 1. Compute the determinant of the following matrices using cofactor expansion onlyâ€”no row/col… • prove or disprove the following Â . Let L : V -> V be a linear operator and let v E V such that v * Ov. If v E Range(L) then v & ker(L). • How do I do this?Â 1(a, d). 7(d) 7.Â Â . (a) {(x, y) /x = 3y]. (d) (x, y)x =. 7. Which of the following subsets of the space M22 of 2 X 2 matrices are subspaces? Prove that your answer is correct.. … • (2) Suppose T : R3 – R2 is a linear transformation and we know that T(:)-. NO 12 T 13 ([:]) Find the standard matrix A of T. (Hint: you need to find what T does to the standard ordered basis. You have… • on a certain day 85 cars and trucks got gasoline from a gasoline station Each car got 10 liters of gas while each truck got 18 liters each The station sold 1122 liters of gasoline How many cars and ho… • . 7. Given matrix M that represents the dynamics of the marble system in Figure 1. O Oo 0 0 0 0 DHOOO 0 HOO OOH HOOO OOOH O 0 O O Calculate M2 and M3. If all the marbles start at vertex 2, where wil… • . Handin C7b Handin C7a Name Foundations 20 Foundations 20 Please show ALL of your work! Name : Please show ALL of your work! 1. A quadratic function is defined by y = -x2 + bx + 3. Make a predicaio… • Question 3 [10 points] Given the following set S of matrices , find a matrix A in M3.2 so that SU { A} spans M3,2 and a non-zero matrix B in M3.2 so that SU{B; does not span M3,2 3 5 8 -5 -10 5 0 2 -1… • Feed Mix problem: The manager of a milkdiary decides that each cow should get at least15, 20 and 24 units of nutrients A, B and Crespectively.Twovarietiesoffeedareavailable. In feed of variety 1(varie… • .. In Vancouver, if it rains today, there is a 40% chance it will rain tomor- row. If it doesn’t rain today, there is a 50% chance it will rain tomorrow. (I made these numbers up, but it really does… • . -s + 3t t Which of the following is a spanning set of V = :STER 2s – t S Select one: O a . V – span – \| 2 -/ Ob. V – span . \| 3 – – } O c. V = span \| 2 .} O d. W O V = span • Question 3 [10 points] Given the following set S of matrices , find a matrix A in M3 2 so that SU{A} spans M3.2 and a non-zero matrix B in M3 2 so that SU {B} does not span M3,2. 3 5 8 -5 -10 5 0 2 -1… • LetÂ S Â be the subspace ofÂ M 2,2 Â defined by Give a basis forÂ S . • . 5. Show that the matrix 1+i 3+i 2 2V15 4+3i Could . co – colla NIHIL 215 52 2V15 is a unitary matrix. • . 2. Show that the set of vectors 1 2 is not linearly independent. • . Determine the currents I,, I,, and I, for the electrical network shown in the figure. (Assume V, = 3 V and V, = 4 V.) A A I, = A VI I R =40 R2 = 30 R3 = 10 V2 • . Consider the following points. (-1, 4), (0, 0), (1, 1), (4, 52) (a) Determine the polynomial function of least degree whose graph passes through the given points. P (x) : • please tell me how to Parameterize the infinitely many solutions. (using Â r, s, and/or t for the parameters.) Â ***************************************************** x + 2 y + 3 z = 6 4 x + 5 y + 6… • . Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite number of solutions, se… • Consider the following linear system. 3x + 6y = 24 -3x – 6y = -24 Create the augmented matrix of this system. (Do not perform any row operations.) Use elementary row operations to rewrite the matrix i… • Please show work.. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, express x, , x,, and x, in terms of the… • . Find the steady-state vector for the matrix below. 0.2 0.4 0.4 P = \| 0.4 0.2 0.4 0.4 0.4 0.2 q (Type an integer or simplified fraction for each matrix element.). The weather in Columbus is either … • Next question You invested$11,000 in two accounts paying 5% and 9% annual interest, respectively. If the total interest earned for the year was $790, how much was invested at each rate? The amount in… • . (20 marks) 2. Solve the following system of equations by computing the inverse of the coefficient matrix. No credit will be given for solutions obtained by any other method. -x + 31 +2z =2 2x+ 1+ … • determine a cubic spline approximation that fits through data pts:Â P1(1; 1); P2(2; 2); P3(3; 1:5); P4(4; 3); P5(5; 2) Spline built with two functions: i) Formulate 3 linear algebraic equations so y1… • How do I do this?Â Â . 18. Two matrices A and B are said to anticommute if AB = – BA. If A and B are anticommuting 3 X 3 matrices, show that one or the other is noninvertible.. 5. Three bodies having… • Question Help: Video Question 11 [ 0/1 pt 5 3 2 19 Based on the data shown below, calculate the correlation coefficient (to three decimal places) X y 3 90.4 4 88.8 5 86.4 6 82.6 7 79.8 8 74.2 Question… • Question 10 0/1 pt 5 3 19 Which of the following pairs of variables is likely to have a positive correlation? Check all that apply. The square footage of a home and its price Years of education and sa… • . Find formulas for the voltages x, and x2 (as functions of time t) for the following circuit represented by the 2 x2 matrix A. Assume Ry = 1/17 ohms, R = 1 / 15 ohms, Cy =4 farads, C2 = 15 farads, … • 1/26/22, 11:35 PM MyOpenMath VUCSLIVII 7 UVIIPL UJ UIZ Which of the following pairs of variables is likely to have a negative correlation? Check all that apply. A person’s height and their favorite co… • Match each scatterplot shown below with one of the four specified correlations. a. -0.03 O b. -0.94 C. 0.88 d. -0.35 O OO O OO O O O O O Question is O Question 10 O O which of O O O O O O OO O O O cul… • 1/26/22, 11:35 PM MyOpenMath Match each scatterplot shown below with one of the four specified correlations. O a. 0.00 OO b. 0.89 O C. -0.50 d. -0.93 O Q O O OO OO 8 OO OO O O O O O O O O O O O O O O … • Question 6 0/1 pt 5 3 19 Here are the scatter plots for two sets of bivariate data with the same response variable. The first compares the variables x & y. The second compares the variables w &amp… • MyOpenMath Here are the scatter plots for two sets of bivariate data with the same response variable. The first compares the variables x & y. The second compares the variables w & y O OO OO 8 … • . Problem 1 (5 points). Prove that R(zw) = R(2)R(w) – 3(z)3(w) and 3(zw) = R(z)3(w) + 3(z) R(w) where z and w are complex numbers. • A regression model capturing trend and seasonality is represented by the equation below with X1X1 representing time (in months), X2X2 representing the February dummy variable, X3X3 representing the Ma… • I’m working on a linear algebra multi-part question and need an explanation and answer to help me learn. solve all the questions Â Do not skip any steps in your calculations or arguments. Justify you… • A customer requires during the next four months 50, 65, 100, and 70 units of a commodity respectively. Production costs are$5, $8,$4, $7 per unit during these months. Storage cost per unit is$2 … • Please show detailed steps. Thank you:) Â . Q4 Geometry of linear transformations 6 Points (a) Let A = 1 1 and let T() = Are be the linear transformation T : R" – R?. What does the set with 0 &lt… • 6 -2 2 Using orthogonal diagonalization Diagonalize matrix A= -2 3 -1 .. 2 -1 3 • Given H(p) =_ 8p evaluate Need Help? Read It Watch It Submit Answer 12. [-/1 Points] DETAILS AUFINTERALG9 3.2.0 Given H(p) = 3p evaluate. P + 2′ H(-8) Need Help? Read It • Linear Regression The table below shows the value, V, of an investment (in dollars) n years after 1987. n 1 3 7 12 14 19 V(n) 10330 9594.2 8710 7580.8 6820 5196.5 Determine the linear regression equat… • 11:35 PM MyOpenMath Linear Regression The table below shows the value, V, of an investment (in dollars) n years after 1990. n 1 3 7 12 14 19 V(n) 8152 8678.88 10264 12384.72 12728 13763.6 Determine th… • Use your graphing calculator to find the line of best fit for the given data. Use the indicated variables and proper function notation in your answer. Round to 3 decimal places as needed. a 2 3 4 5 6 … • Question 1 0/1 pt 9 3 19 Linear Regression Use linear regression to find the equation for the linear function that best fits this data. Round both numbers to two decimal places. Write your final answe… • Problem 3 (8 marks)The following matrix is the adjacency matrix of graph 1 2 3 1 tea-tom calâ€”Ilâ€”Io: Claw-“H 1. Is there an isolated vertex? Please explain your answer. 2. Find the degree of the … • Let V1= (a) What does it mean to say that the vectors v1, v2, v3 are linearly independent? (b) Find all values of the constant b so that the vectors v1, V2, V3 are linearly dependent. • envellu mathxl.com/Student/PlayerHomework.aspx?homeworkld=615973348&questionld=1&flushed=false&cld=6793725&centerwin=yes h MTH 154 – Spring 2022 – 16 weeks Lisa Robertson 01/30/22 11:2… • Let K3,2 be a complete bipartite graph on (3, 2) vertices. Draw K3,2 and find the total degree of the graph. • Use the Handshake theorem to show that there is no simple graph with four vertices of degrees 1, 1, 3, and 3. • Suppose that T is a linear transformation from IR2 to IR3. Also suppose that (1:1)- 🙂 and – (14)-8) (a) Write ON as a linear combination of 1 and 1, ] N (b) Find T =? O (c) Find T ( [2]) – (d) Fin… • E Homework: Section 2.4 Question 1, 2.4.1 HW Score: 7.69%, 1 of 13 Homework: Financial Literacy Part 1 of 3 points O Points: 0 of 1 Save Recall that the PE ratio is the ratio of the price per share to… • MTH 154 – Spring 2022 – 16 weeks LISd R HW Score: 7.69%, 1 of 13 h E Homework: Section 2.4 Question 1, 2.4.1 points Homework: Financial Literacy Part 1 of 3 Points: 0 of 1 Save Recall that the PE rati… • . 3. Given the following system of linear equations -x+ 2y+3z =1 2x+ 1- z=1 3x + 1+ 2z =4 (a) Solve the system of equations by performing elementary row operations on the augmented matrix. (b) Based… • . 1. Given the linear system: ax+ 4y=3 2x+51+8z =3 x+2y+z = B Find the values of a and B, if any, such that the system: a) has one single solution, and give the solution. b) has infinite solutions, … • These questions are related to Systems of Equations, Algebraic Procedures (stuff like augmented matrix, coefficient matrix, elementary row equations, row- echelon form, zero row, row- echelon matrix, … • . Show that EXT(XEXT + 020-1 = [XT(a2I)-1X + 2’1]’1XT(02I)’1, Where X is an n x p matrix with n > p, and Z is a nonsingular and symmetric matrix. Note that the right hand side only requires… • . For an n x m matrix X, show that the column space of X, C(X ), is a vector space. • Alena is 40 years older than her grandson Mariano. Ten years from now, Alena will be thrice as old as Mariano will be ten years from now. How old are they now? • SolveÂ the followingÂ Linear Programming model using the graphical methodÂ (USING EXCEL) {Write the steps of construction} Â Q1) Maximize Â HÂ =Â xÂ +Â 3y Â Â Â Objective Â Â function subject to Â … • . – 2 (1 point) Are the vectors -4 Choose If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linearly independent, find the only sca… • . 2. Which of the following equality of determinants are true? (Select all correct answers.) (1 mark) 0 O V W a ac 0 10 0 b OO O 0 d e O N . = 6 2 8 O b = b 6 O C O C C • I have trouble with these question Â . Exercise 1.2.22 Row reduce the following matrix to obtain the row-echelon form. Then continue to obtain the reduced row-echelon form. Exercise 1.2.23 Row reduce … • I have trouble with these question Â These questions are related to Systems of Equations, Algebraic Procedures (stuff like augmented matrix, coefficient matrix, elementary row equations, row- echelon… • I have trouble with these questions Â . Exercise 1.2.3 Do the three lines, x + 2y = 1,2x -y = 1, and 4x + 3y = 3 have a common point of intersection? If so, find the point and if not, tell why they do… • Use substitution method. -21x + 9y = 7 9x – 12y = 6 • . 5. Which of the following linear spans contain the vector (1,2,2, 1) ? (Select all the correct options) (1 mark) . Response to this question requires at least one option. spanf (1,2), (2,1) } span… • . 3. Given that A is a 5 x 4 matrix and is not full rank. Which of the first four statements below is not true? (If they are true, choose option 5). (1 mark) Any row echelon form of A has at least 1… • . Probleml let I be a finite dimensional vector space over F. let V, and Vy be subspaces. Show that it B, and By are buses of V, and Vz , then their Union $3 = BUB , can be extended to a basis of V … • I 3.9.3: Determinants and two-dimensional coordinate geometry. CHALLENGE mm” 32′ 59-1-3234 392 Jericho 3;! Jump to level 1 Fill in the matrix in the equation below to ï¬nd the line that passes thro… • (c) (4 points) Suppose that S: R3 – R2 is a linear transformation such that S 2 and What is the matrix corresponding to S? • Previous Problem Problem List Next Problem (1 point) Determine the value of k for which the system x ty +2z = 2 x +2y -3z = 3 3x +7y +kz = 11 has no solutions. k = • -4 -1 -4 (1 point) If A = 0 4 4 and B = -1 0 -1 , then 3 1 3 -2 N 4A – 3B = and A = • 3 Read the following definition carefully, and then answer the question below. Def We define an n x n matrix B to be meek if B2 = I. Remember, if the statement is True, explain why. If it’s False expl… • If B is an n Ã— n meek matrix, then the homogeneous system Bx = 0 has only the trivial solution • . Here we’ve stopped the animation at a second value of theta for which the unique solution is again if = [:3]. What value of theta resuits in this frame? The slope of the blue line is exactly m = x… • . Student x ENG 102-IN-V5 – Lesso X Microsoft Word – Voice Spoof Ad Options X Edready – /assessment X Homework Help – Q& x + V X C https://maricopa.edready.org/assessment/startAssessment?assessm… • can you please type the answer so i could see clear thankyouÂ . 4) Provide a proof for the following: Let A, B, and C be any n x n matrices: a. Show that trace(ABC) = trace(CAB) = trace(BCA) (10 point… • nvellu mathxi.com/Student/ PlayerHomework.aspx?homeworkid=6159 / 354 1 &questionid= Toflushed= MTH 154 – Spring 2022 – 16 weeks Lisa Robertson 01/29/22 2:30 PM ? HW Score: 72.22%, 2.89 of 4 E Home… • Problem 5. Let M be a module over the integral domain R. (a) Let r E M be a torsion element. Show that r is linearly dependent. Conclude that the rank of Tor(M) is 0, so that in particular any torsion… • Linear Algebra Question (complex numbers) Solve for the complex value z = a + bi AND z = \|z\|cis(Î¸) (that is, show solution for both forms) (a) z 3 = âˆ’1 + i Â (Please show full solution don’t skip … • Solve for the complex value z = a + bi AND z = \|z\|cis(Î¸) (that is, show solution for both forms) Â a) z 3 = -8 Â (Please show full answer and dont skip any calculations, Also can you please show de… • . Let A and B be n x n matrices with AB = 0. Each question below is 5 points. Provide a proof or counterexample for each of the following: a) BA = 0 b) Either A = 0 or B = 0 (or both) c) If det(A) =… • . 4) Provide a proof for the following: Let A, B, and C be any n x n matrices: a. Show that trace(ABC) = trace(CAB) = trace(BCA) (10 points) b. trace(ABC) = trace(BAC). Provide a proof or a countere… • Linear Algebra , Complex Numbers Question 4. Solve for the complex value z = a + bi or z = \|z\|cis(Î¸) (that is, you may give your answer in either form) a) z 4 = -1 Â (Please show full solution) • give a counterexample. 2) Assume that we have two (2) dâ€”dimensional real vectors x and y. And denote by x.- (or y.) the value in the i-th coordinate of x (or y). Prove or disprove the following s… • Prove the following two matrix derivative properties for square symmetric matrices A, B, a log(\|A\|) = 2A-1 – diag(A-1) OA a tr(AB) = 2B – diag(B) • Sea Re = R y p(x) 2+2-x/3/3/3/2 =1 determine Ap(x) • CROP PLANNING. A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is$40/acre whereas the cost of cultivating Crop B is $60/acre. The farmer has a maximum of$7400 available fo… • Is it best to wait for stock prices/ REIT prices to dip before you purchase it? If not, then when do you think is the best time to purchase ? • How do you best explain the ways of scammers in the industry? • help please. Perform the row operations indicated beside the matrix. 1 4 -3 e -4R2 -+ R1 – R1. -7R2 + R3 .R3 01 1 Write the matrix obtained • Find the area of the parallelogram with vertices K(2, 3, 2), L(2, 6, 5), M(7, 10, 5), and N(7, 7, 2). • Compute the expressions below (where 8 is the Kronecker delta). Explain your work in detail. 8,6, = ? ki = ? • Expand the following using standard (longhand) notation A. B. jk = C ik A. = B. C • Given subsets X and S of the vector space V, compute the exchange T from the Exchange Theorem by mirroring Example 1.37. (a) x = { (;), ( 8 ) }, s = {i,j, k}, v=3 (b) X = {1 – x, 2 + x2, 1 + x3}, S… • . Problem 5) Taylor approximation of norm. Find a formula for the Taylor approximation of the function f (x) = [\|x\|\| near a given nonzero vector 2. Check your approximation numerically as follows. F… • Letting Ax=b and our solution set (for this case lets call it G = {x=x1 +xg :xg is in G} . So G is the solution set for Ax=0Â we can use fact that If A âˆˆ MnÃ—n(F) then A is invertible if and only i… • if a set of vectors (the reduced row echelon is linearly independent with pivots in each column) does that mean that it spans R^4? But how would we know if it spans the C^4 (complex space)? • (1 point) The reduced row-echelon forms of the augmented matrices of four systems are given below. How many solutions does each system have? ms O A. Unique solution B. No solutions O C. Infinitely man… • = -4X5 -3×6 (1 point) Solve the system -2×2 +2×3 -2×6 -7 X 1 -X4 -4×5 +6×5 +5×6 = -3 X1 -2×2 X1 X2 tu +t X3 +S X4 X5 X6 • (1 point) Solve the equation -4x – 9y +9z = -10. X y E +s +t Z • . 5. (Optional ) Give a simple expression for a a a a abb b det ab cc abcd • . 3. If A + jB is Hermitian, A, B real, then A = A, B =-B. • D Question 5 1 pts 5. Consider the following system of linear equations in the five variables r, y, z, u, and v. How many free variables will there be for the solution set? a + 2y + 3z + 5u + 70=11 2y… • Activity for $3. The following problems refer to the 2 x 2 matrices: A B 6 c = 4 1. Test whether the usual laws of arithmetic apply to these three matrices. o Associativity: Calculate and compare (AB)… • Homework: Section 2.2 HW Score: 22.22%, Homework: Weighted Question 2, 2.2.2 > 0.89 of 4 points Part 2 of 2 Averages O Points: 0 of 1 Save our GPA is a weighted average, your grades are weighted by… • weighted Part 2 of 2 0.09 01 4 points Averages O Points: 0 of 1 Save Your GPA is a weighted average, your grades are weighted by the credits for Quality Credits Grade 4.0 Scale each course. … • E Homework: Weighted < Question 2, 2.2.2 Averages Part 1 of 2 0.89 of 4 points O Points: 0 of 1 Save each course. Quality points are the product of the credits for a course and the Credits Graue 4…. • oning – MTH.1 X Assignments X + X https://ope Do Homework – Section 2.2 Homework: Weighted Averages . Personal – Microsoft Edge O X https://www.mathxl.com/Student/PlayerHomework.aspx?homeworkid=615973… • A survey asks students how many children are in their Complete the table household. Complete the table to compute the average number of children per household (Simplify your answers.) Children Frequen… • MTH 154 – Spring 2022 – 16 weeks Lisa Robertson 01/28/22 5:51 PM ? Homework: Section 2.2 Question 1, 2.2.1 HW Score: 0%, 0 of Homework: Weighted > 4 points Part 1 of 8 Averages O Points: 0 of 1 Sav… • . For the matrix, list the real eigenvalues, repeated according to their multiplicities. 9 -30 2 O 4 4 5 0 0 89 0 0 0 9 The real eigenvalues are (Use a comma to separate answers as needed.). For the… • ive Reas X Assignments X PeopleSoft sessic x ] PeopleSoft session X _ Pay or Apply for ! x M [TCC] Student Mix + Do Homework – Section 2.3 Homework: Proportionality – Personal – Microsoft Edge 0 X htt… • . O 2 A = 3 4 – 6 B = 0 5 2 LI 1 2 – – 2 3 Find The diagonal form and model matrices . • Solve the system I + 20 + 32 + 31 = 2 21 + + + 32 + 31 = 3 3x + 27 + 52 +1 =5 Your solution should include an augmented matrix, identification of free/ dependent variables, and a complete description … • B https://www.mathxl.com/Student/PlayerHomework.aspx?homeworkld=615973337&questionld=9&flushed=false&cld=6793725&centerwin=yes MTH 154 – Spring 2022 – 16 weeks Lisa Robertson \|01/28/22… • 10) The equation gives the volume, V, of a sphere with radius r. Determine the radius of a sphere with volume 500 cm3. options: 4 8x-4 64 11 Express the right side of as a power of 4, the… • . #/test Question Evaluate the determinant: 02 On -y1 0 0 Activate Windows to action pdf pdf pdf 1) ENG • #/test Question Evaluate the determinant: 02 On -y1 0 0 Activate Windows to action pdf pdf pdf 1) ENG • Homework: Section 2.1 Question 15, HW Score: 61.76%, Homework – Ratios and 2.1.52 10.5 of 17 points Proportions Part 1 of 2 Points: 0 of 1 Save The nutrition information on the cereal box says that a … • MTH 154 – Spring 2022 – 16 weeks Lisa Robertson 01/28/22 1:36 PM ? Homework: Section 2.1 HW Score: 61.76%, E Homework – Ratios and < Question 14, 10.5 of 17 points Proportions 2.1.51 Points: 0 of 1… • (a) (2 points) Find constants a and b such that (b) (3 points) Suppose that T: R2 – R2 is a linear transformation such that and 10 00 What is the matrix corresponding to T? (Hint: Your answer to pa… • . 2 Read the following definition carefully, and then answer the questions below. Def We define an n x n matrix A to be passive if A2 = A. Note: before diving in to solving the questions below, you … • . Exercise 2.3.13 Compute the following using block multiplication (all blocks are k x k). I X – X a. b. 1 . [IX ] [ IX ]] d. [ I X ] [ – X 1 ] X n e. 0 -1 any n > 1. Exercise 2.3.12 In each case… • Homework: Section 2.1 E Homework – Ratios and Question 10, HW Score: 38.24%, > Proportions 2.1.24 6.5 of 17 points O Points: 0 of 1 Save Which equality represents the proportion$8 is to 13 cans as… • . Exercise 2.3.13 Compute the following using block multiplication (all blocks are k x k). I X – X a. b. 1 . [IX ] [ IX ]] d. [ I X ] [ – X 1 ] X n e. 0 -1 any n > 1. 2 Read the following definit… • Answer all parts: 1. On the surface of Io, a moon of Jupiter, the gravitational field strength has a magnitude of 2.2 N/kg.Â Io’s mass isÂ 8.93 x 1022 kg.Â What is the radius of Io? Â -What is the ma… • Find A given the following: (2AT)âˆ’1=[60] Â Â Â Â Â Â Â Â [ 1âˆ’2] T is Transpose and -1 is inverse of 2A transpose Â Â 4. Find all values for m and n for which matrix A and B are both no… • .. What is the minimum value of n for which R" contains two planes that intersect only at the origin? You may take the definition of a plane to be "the span of two linearly independent vecto… • A una tobera entra aire constantemente a 2.25 kg/m3 y 43 m/s, y sale a 0.780 kg/m3 y 190 m/s. Si el Ã¡rea de entrada de la tobera es 85 cm2, determine a) la tasa de flujo mÃ¡sico por la tobera, y b) e… • For what a in R is a diagonalizable?. * Aufgabe 4. 0 1\ a Fur welche a E R ist A := -1 a E R2X2 diagonalisierbar? b) Fur welche a E C ist A := E C2x2 diagonalisierbar? • is this matrix diagonalizable?. 3 2 O 3×3 d) -2 -1 0 2 -1 0 • Hi there would you please solve both questions in detail as they are similar, that would be very helpful and appreciated. Thank you. And I won’t forget to give you a good feedback too.. 31. Explain wh… • For the 4 nodes shown in the figure, what is the 3×8 strain displacement matrix (B), using the 2×2 gauss quadrature integration points?. 4 ( 2 , 5 ) 3 ( 8 , 5 ) 741 1 (0, 0) 2 ( 10, 0 ) Fig. 2 • A reason for your answer in each of 1-12 in that all the variable represent integers is 52 and divisible by 13 • . In Exercises 77-84, solve the given equations. 84. Given f(x) solve for x if 3f(x) + = 2. + 1 • Choose one of the 8 linear data sets from the Seattle Central College website (https://seattlecentral.edu/qelp/Data_MathTopics.html#Linear) Â Â Data sets #1, 11, 14, 18, 22,Â 44, 73, 77 Â Explain t… • . 1. A brewery produces beer and ale. Beer sells for $5 per barrel, and ale for$ 2 per barrel. The production of a barrel of beer requires 5 pounds of corn and 2 pounds of hops. The production of … • Assume that you have 4 samples each with dimension 3, described in the data matrix X, 2 2 4 5 X = 2 3 2 For the problems below, you may do the calculations in python. Describe the commands you used. (… • . 1. A brewery produces beer and ale. Beer sells for $5 per barrel, and ale for$ 2 per barrel. The production of a barrel of beer requires 5 pounds of corn and 2 pounds of hops. The production … • Would you solve it real quick.Thank you .. The statement is either true in all cases or false. If false, construct a specific example to show that the statement is not always true. If v1, …, v4 are … • Evaluate the following expression, given x = 2 and y = 3. (6x)2(10y) / 6y Evaluate the following expression, given x = 5 and y = 4. 7xâˆ’5y /3 Simplify and evaluate the following expressions, where x=… • State why the system of equations must have at least one solution. (Select all that apply.) 8x + 3y + 13z = 0 7x + 8y + 242 = 8x + 2y + 17z = 0 The system contains three equations and three variables…. • Solve the system of linear equations. (Enter your answers as a comma-separated list. If there is no solution, enter NO SOLUTION. If the system has an infinite number of solutions, set x3 = t and solve… • (4) Find the inverse (if it exists) of the matrix 0 0 5 -3 2 3 I+ A = 0 + 15 -9 6 15 0 10 6 4 10 6 Hint: you should probably compute this directly. But! there is an algebraic trick worth doing, as wel… • . LetA be a 3×2 matrix. Explain why the equation Ax: b cannot be consistent for all h in R3. Generalize your argument to the case of an arbitraryAwith more rows than columns. Why is the equation Ax:… • Consider the following linear program:. Min 24 + 2B s.t. 1A + 3B < 12 3A +1B > 13 1A – 1B = 3 A, B 2 0 a. Show the feasible region. b. What are the extreme points of the feasible region? c. Find… • The American Safety Council has allocated $500,000 for projects designed to prevent autoÂmobile accidents. Four proposals were submitÂted: (a) TV advertisements, (b) teenage safety education, (c) im… • A set of 4 6-tuplets(sextuplets) is linearly independent :(always),(never),(sometimes) • I’ve been stuck on the basismatrix problem and would like some help Â Â . Utilize the following set of vectors in R* for this activity. Script Save C Reset LL MATLAB Documentation XEnter vectors v1, … • Math 340 linear Algebra (Module 4 discussion) Â 1) What was the most interesting thing you learned in the modules or created for the project? Â Â Â ?… • Homework: HW Lesson 2.3 Question 1, Let f(x) = 300x – 5x + 10. Find the maximum value of f to four decimal places graphic .. . The maximum value of f is (Round to four decimal places as needed.) • . Submit to Class pra… a https://www.amazo… G Plagiarism Checker… Microsoft Word – A… PDFfille HW2: Problem 15 Previous Problem Problem List Next Problem (1 point) Find a 2 x 2 matrix such t… • . Calculating by hand, find the characteristic polynomial, eigenvalues and the eigenvectors of the following matrix: N P -3 – 6 • how many solutions in the row echlon matrix. 100-3-5 010 0 -5 0 01 5 • . Below is a network of one-way streets and traffic flow rates in vehicles per hour in each section of the roads. For instance, traffic comes into this area on Huntington Ave. at 400 vehicles per ho… • Construct a 3x 3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the columns of A. • What is the area of parallelogram ABCD (from problem 3) whose height = 4″ ?Â Â question #3 What is the perimeter of parallelogram ABCD if AB = 5″ and BC = 8″ ? • What is the perimeter of parallelogram ABCD if AB = 5″ and BC = 8″ ? • why is quadratic equation the best method to solve a maths problem instead of squareroot and factorizing? • why is quadratic equation the best method to a maths problem instead of square root and factorizing? • . Let V be the vector space of functions spanned by NTT B = {b1 = 5 sin x + 8 cos x, b2 = 7 sin x + 6 cos x } where a 2, NEZ NTT C = {c1 = sin x, c2 = cos x } where x ,, n E Z is also a basis for V…. • Bonjour j’ai besoin d’aide pour cet exercice merci d’avance • the equation y = 10x represents the flight of a snowbird at the Hamilton air show. graph the flight of the snowbird and label it S Â b. at the air show another snowbird flies along the path descri… • . Let V be the vector space of functions spanned by NTT B = {b1 = 6 sin x + 2 cos x, b2 = 8 sin x + 7 cos x } where x * J, NEZ NTT C = {c1 = sin x, c2 = cosa } where x , n E Z is also a basis for V…. • . Consider the image below. 9 1 2 9 (1) 1 6 7 3 3 5 Show the results of 3×3 median filtering if the following masks are used. A 50" is a mask position means that the corresponding pixel is not … • Activity for$4. 1. Write down the 4 x 4 matrices corresponding to the row operations: (a) R2 + R2 -5R1 (b) R4 + R4 + 7R2 (C) R2 <> R4 2. For each system of equations, write the augmented matrix… • Activity for $4. 1. Write down the 4 x 4 matrices corresponding to the row operations: (a) R2 + R2 – 5R1 (b) R4 + R4 + 7R2 (C) R2 4 R4 2. For each system of equations, write the augmented matrix versi… • Describe linear algebra as used for LAN designÂ and advantages and disadvantages Â of using algebraic calculation in LAN design and to identify And use linear algebra to calculate size of LAN • . Define T : R< – R’ by T(x) =Ax. Find a basis B for R with the property that is diagonal. 3 – 6 A= – 5 4 A basis for R with the property that [T], is diagonal is (Type a vector or list of vector… • Density of states for a 2D electron gas Consider a 2D electron gas in which the electrons areÂ restricted to move freely within a square area a 2 in the xy plane. Following the procedure in Section 4… • . Given the following bases for P2, ï¬nd I(3.33.13, the change of coordinates matrix from B to C. {â€” 13: {â€”3332 +2mâ€”5,6:c2 +8m+9,â€”4:s2 â€”4mâ€”12} C={m2+m+lamâ€”iâ€”1,l} EX: 7′ P = (34â€”5 • Tutors, Please help me with this problem. Â Promptness is especially appreciated. A long explanation is not necessary since I just with to check my own answer against a tutor’s. Thanks!Â Â Â Â . Le… • Tutors, Please help me with this multiple choice problem. Â Promptness is especially appreciated. No work necessary since I just want to compare my answers to someone knowledgeable. Thanks!Â Â Â Â … • . Find all subspaces of R- which are stable under the linear transformation T : R2 – R2, T ( x, y ) = ( x ty, -x+2y). • . Find the general solution of the system whose augmented matrix is given below. 2 -5 90 -20 36 0 6 – 15 27 0 Choose the correct answer below. O A. O B. O c. X1 = -2×2 OD. 5 X 1 = 7X 2 – NO x1 = 2 T… • . Row reduce the matrix to reduced echelon form. Identify the pivot positions in the final matrix and in the original matrix, and list the pivot columns. 2 3 4 4 5 6 7 8 9 10 11 Row reduce the matri… • . Determine if the given system is consistent. Do not completely solve the system. 2X1 â€” 4×4 2 -10 6×2 +6×3 : 0 x3 + 4×4 : 0 â€” 3×1 + 5×2 + 3×3 + x4 : 13 Choose the correct answer below. 0 A. The… • solve the systemÂ Â . Solve the system. X1 – 6X3 = 21 4×1 + 2×2 – 9X3 = 49 X7 + 4X3 = -7 Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice. O A. The … • . Given the following bases for IR, find P , the change of coordinates matrix from 3 to C. C+B B Ex: 9 P = C+B Ex: 1 If [v]B = , then [v]c = • . Given the following bases for IR, find P , the change of coordinates matrix from 3 to C. C-B 22 18 B = 6 C-{[2] . 12]} Ex: 7 P C+B 7 Ex: 7 If [ VB = , then [v]c = • . Consider a splitâ€”plot experiment With m Whole plots each having exactly n treatment com- binations. Let W be the model matrix for the Whole plot effects, and let S be the model matrix for the su… • solve it clearly. 5. * The Petersen Puzzle Before you attempt this problem, look at the article about Ringel’s conjecture in Quanta Mag- azine: https://www.quantamagazine.org/mathematicians-prove-ring… • Listen A line passes through A(8, 5) and Z(8, 4). Which of the following coordinates are on a line parallel to to AZ? (3, 2) and (3, 11) (0, 0) and (3, 0) (8, 5) and (8, 4) (5, 8) and (4, 8) • () Listen Work out the response to this question on a piece of paper and show all your work. Answer with a sentence and units (where applicable). Scan or take a picture of your work and upload it to t… • Q1 only need part c and d solutions..send me ASAP THANKS. 1. (a) The linear programming problem in canonical form has a series of constraints of the form Ax = b where A is an m x n matrix with m < … • Q1 only need part A and B solution in handwriting ASAP thanks… 1. (a) The linear programming problem in canonical form has a series of constraints of the form Ax = b where A is an m xn matrix with m… • 2 DUE THURSDAY 27 JANUARY 2022, AT 12PM (2) For each of the systems of equations, write the corresponding augmented matrix. Use Gaussian elimination to solve the sys- tem, or explain why there is no s… • Solve the system of linear equations xty + z=6 x – 2y = 5 y – 3z = -23 (2) Explain carefully why there is no system of two linear equations in two unknowns with coefficients in R that has exactly … • Please solve 4 and 5. Thank you for your help!. 4. S (1 5 V ( 4 , 6 ) , T ( 3 , 3) are midpoints of the sides of 4ABC . Find the coordinates of points AB and C 5 P (5, 4 ) I ( 1 6 G ( 7 , 1) are midpo… • The following data were taken from the books of Denver Company which manufactures a single product through a two-department manufacturing process – machining and finishing. In the production process, … • A Linear Algebra(Julia code) question needs help! Thanks!. T&B Exercise 8.2. Write a julia function Q, R = mgs (A) that computes a reduced QR factorization A = Q R of an m X n matrix A wi… • Find the general solution of the system whose augmented matrix is given below. 134 9 263 3 cti Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice. O … • Homework: Homewrk 1 Question 2, 1.1.7 points O Points: 0 of 1 Save nm The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operati… • HW Score: 40.28%, 4.83 of 12 Homework: Homewrk 1 Question 3, 1.1.10 points O Points: 0 of 1 Save he augmented matrix of a linear system has been reduced by row operations to the form shown. Continue t… • HW Score: 40.28%, Homework: Homewrk 1 Question 6, 1.1.35 points S Points: 0 of 1 m Find an equation involving g, h, and k that makes this augmented matrix correspond to a consistent system. – 6 7 g 0 … • Directed Graphs Â Write down the edge set of the graph: Â Who are the out-neighbors of node F ? Â Who are the in-neighbors of node D ? Â What is the intersection of the out-neighborhoods of vertic… • Suppose S is an m-dimensional hyperplane in R n . Â Let Â X n Ã— m â€‹ = [ V 1 â€‹ , V 2 â€‹ , . . . , V n â€‹ ] be an Â n Ã— m Â matrix whose columns are a linearly independent spanning set for S …. • Considering the ratio of age 65-and-over to the total population of the Midwest, what kind of ratio is this: a part-to-part or part-to-whole? • as shown. (Lange Exercise 9.3, modified) It can be shown that the matrix norm induced by the Euclidean (L2) norm of matrix A is equal to the largest singular value of this matrix. Now, let A be an inv… • as shown,Â Â . (Lange Exercise 8.5, modified) Suppose A is an eigenvalue of the orthogonal matrix O with corresponding eigenvector v. Show that if v has real valued entries, then 1 = 11. Hint: use pr… • as shown. ( Lange Exercise 8.4, modified) Show that the reflection matrix cos(0) sin(0) A = sin(0) -cos(0) is orthogonal and find its eigenvalues and eigenvectors. • as shown,Â Â . (Lange Exercise 7.8) Suppose the matrix A = {ajj } is banded in the sense that aij = 0 when i – j > d. Prove that the Cholesky decomposition B = {bij } also satisfies the band condi… • solve this questionï¼š Â Â Â . Question 6 (10 marks) Both antibiotics were prescribed in high dosage slow release capsules. The function C(x) =5log(x+1) +10 models the concentration of levofloxacin … • The raw materials needed to manufacture each type of perfume can be purchased for$3 per pound. Processing 1 lb of raw material requires 1 hour of laboratory time. Each pound of processed ra… • Define and explain substitution method Solve this system of equation (in two variables) using substitution method 4x+3y=-23 Y=-7x-2. 9201998 4x + 3y = -23 y = – 7X – 2 • Please fill in the blanks. Thank you! Â Â . 1. [-/1 Points] DETAILS HOLTLINALG2 1.1.031B. The linear system is not in echelon form but can be put in echelon form by reordering the equations. Write th… • Define and explain substitution Solve the equation using substitution method. 4x + 3 -23 • 1,000 km 2,000 + 3,000 (each dash is 500 km apart) – 4,000 – 5.000 -6,000 inner core 6,500 • My courses / MATH 102 (LEC E850 EQ1 ER1 ES1 ET1 EU1 EV1 EW1 Winter 2022) / Online Assignments / OA3 An orthogonal set of vectors is a set of vectors in which every pair of vectors within the set are o… • Question 2 With the complex numbers Tries . a =-5.1-5 remaining: 2 . w =3-7. i Marked out of 10.00 . z =2 – 7. i Flag question a( w – z)2 + a( w + z)2 = Check Save Progress • . Question 4: Give an example of a 3 x 3 matrix A with every entry being nonzero such that A has rank 2 but Ar = has no solutions. • . CHALLENGE ACTIVITY 4.9.1: Coordinatiation. 376936.973580.qx3zqy7 V Jump to level 1 1 < 2 Given the ordered basis -9 B =\u1: 3 U2 = -7 13 = O 9 35 find [v] , the coordinates of V = 18 with respe… • Tried to do these questions but I’m not sure of What I’m doing and would like some guidance • 5. Find a unit vector that is perpendicular to vectors v1 and 122 Where: 5 â€”3 #1 3 and “5.2 = â€”1 â€”3 4 • 3. Compute the eigenvalues and eigenvectors of the matrix N A = N 4. One important use of matrices is to represent linear operators (maps). A linear operator F maps vectors v1, V2 and v3 into ve… • 1. Consider the following four vectors: CT H U1 = U2 = U3 = NHN UA = N CT CO Compute: (a) v1 + 12; (b) v1 + 13; (c) 12 . V3; (d) v1 . U3; (e) U1 X U3; (f) 1/3All; (g) the angle 024 between vecto… • . 1. Consider the matrices and vectors given below: b -2 a. Show that AB # BA b. Compute uv and uv. c. Calculate det (A) d. Compute (A+AT)(B-B!). e. Find the inverse of A by using Gauss-Jordan elimi… • The determinant of the matrix is A. 5 B. C. 3 D. 4 7. The original matrix for the simultaneous equations is B. [2 21 CC 31 D. 132 1 g If X and Y are matrices of the same order, then which of the fo… • Dear Tutors, Please help me with this HW. Â Thanks! Â . Please write down the definition of the Euler’s y function and compute p(7) and (20). • Dear Tutors, Please help me with this HW. Â . Explain the Chinese remainder theorem and use it to solve x= 3 mod 5 = 4 mod 11 • Please help with the following. I just want to compare against my own answers so no explanation, or only brief explanation needed. Â This is actually one problem only but with multiple parts. Â Thanks… • Provide workings for both. Thanks! Â Â . Question 1 Find the value of k which results in the system having no solutions. x ty + 4z = -2 x + 2y – 3z = 0 5x + 13y + kz = 7 Question 2 Solve the system X… • Solve the following system using either the elimination or the substitution method. Show all work. {3ð‘¥ âˆ’ 4ð‘¦ = 10 2ð‘¥ + ð‘¦ = 3 • Please show each step. Thank you. E Day27-Activity1.pdf 1 125% + Go through the same steps for the form defined by Q(x, y) = 2×2 + 2xy + 2y2 = [x 3] 3 3] That is: a. Compute eigenvalues b. Compute eig… • Solve the following system graphically. Show all work. y = 5x + 2 12x + y + 5 =0 10 4V 8 6 2 -10 -8 -6 4 -2 0 2 4 6 8 10 -2 10 • Please show each step. Thank you.. E Day27-Activity1.pdf 1 125% + Go through the same steps for the form defined by Q(x, y) = 2×2 + 2xy + 2y2 = [x 3] 3 3] That is: a. Compute eigenvalues b. Compute ei… • A line which has a y-intercept of 5 is parallel to the line that passes through the points (-2, 3) and (4, 2). Determine the equation of the line. Show all work. • Determine if the following sets of lines are parallel, perpendicular, or neither. Show all work to justify your answer. a. {ð‘¦âˆ’5=4(ð‘¥+3) ð‘¥ + 4ð‘¦ + 16 = 0 b. { ð‘¦=âˆ’6ð‘¥+5 ð‘¥ + 6ð‘¦ +… • . Given the following bases for IR, find P , the change of coordinates matrix from B to C. C-B B = Ex: 3 P = C+B Ex: 1 If V B = then [v]c = • Use simplex method to solve the following problem! Min.Z=-X1+2X2 Â Â Â Â Â St: Â Â Â Â Â -X1+3X2 < 26 Â Â Â Â Â Â X1 +X2 Â < 6 Â Â Â Â Â Â X1-X2< 2 Â Â Â Â Â Â Â ?… • . The change of coordinates matrix from basis B to basis C of R is given by P = 9 C-B -J -9 The coordinates of v E R relative to the basis B are given by \|v B = 8 .Find vic, the coordinates of V rel… • Determine if the following sets of lines are parallel, perpendicular, or neither. Show all work to justify your answer. a. – 5 =4(x + 3) ix + 4y + 16 =0 y = -6x+5 b. x + by+ by + 12 = 0 2. A line w… • . Given the ordered basis B = {p1 = -4 + 7x, p2 = 5 + 6x} find p , the coordinates of p = 15 + 18x with respect to B. Ex: 5 [p] B = • . Assume that E is a symmetric second order diagonal tensor and C =I + 2E . Show that Ic = 3+21E, IIc = 3+41g + 4Ilg, IIIc =1+21E + 4Ilg + 8IIIg Are these relations valid if E is not diagonal? Expla… • solve it please. 5. * The Petersen Puzzle Before you attempt this problem, look at the article about Ringel’s conjecture in Quanta Mag- azine: https://www.quantamagazine.org/mathematicians-prove-ringe… • A total of 3500 tickets were sold to spectators for the curling championship. Tickets were $55.00 for adults and$40.00 for seniors/children. Ticket sales totaled $173 000.00. Define the variables, wr… • please solve it. 1. (Matrix construction). (4.1 #3) In each part, construct a matrix that has the desired properties, or explain why it is impossible. If it is possible, you must show that your matrix… • The sum of two numbers is 249. Twice the larger number plus three times the smaller number is 591. Define the variables, write the system of equations, and determine the two numbers by using the metho… • Madeline has a pile of quarters and dimes she has been saving so she can buy her friend’s birthday present. When she counts them, she knows she has 210 coins. The bank told her she had a total of$33…. • A thin triangular plate of uniform density and thickness has vertices at v1 = (1,1), v2 = (7,1), vg =(3,7), as in the figure to the right, and the mass of the plate is 3 g. Complete parts a and b belo… • Find a unit vector that is perpendicular to vectors v1 and v2 where: Â . 19 and U = 71 = 4 • Can someone explain and also Graph and share the feasibility region of the following linear inequalities and/or system of linear inequalities. • doing past papers. 1.1 Gegee / Given: f(x) = -2×2 + x+6 1.1.1 Bereken die koordinate van die draaipunt van f. Calculate the coordinates of the turning point of f. 1.1.2 Bepaal die y-afsnit van f. Dete… • Find the dimensions of the domain space and of the range space, respectively, of the operator F.Â (b) Find the operator F.Â . A linear operator F maps vectors v1, v2 and ’03 into vectors 11:1, w;… • working through a past paper. 1.1 Gegee / Given: f(x) = -2×2 + x+6 1.1.1 Bereken die koordinate van die draaipunt van f. Calculate the coordinates of the turning point of f. 1. 1.2 Bepaal die y-afsnit… • . 3. Let a = X1 22 23 X5 . Find det (I + aa ). Find lal/2. 4. Let X E Rnxm. Show that X" X _ 0. 5. Let y = f(x) = Arth "Txtd; where A E Rmam is non-singular, b, r, C’ E RM, d E R. Find the… • five minutes please answer. vector spaces Let v be an endonmorphism O necessary and sufficient condition for it to be a projection a diagonal whose diagonal elements are for 0 8 the existence of a bas… • .. Laura and Marty solved the same nonhomogeneous linear system. Laura’s answer, in parametric vector form, was 2 2 + t 5 Marty’s answer was 4 12 + t 10 17 -14 Could Laura and Marty both be right? • . Let F E {R, C) and let V be a nonzero vector space over F. Suppose that V is the union of finitely many subspaces of V. Prove that one of these subspaces is V. • kindly correctly. 03 Vector and Matrix Equations and Applications: Problem 17 (1 point) Consider an economy with three sectors: Fuels and Power, Manufacturing, and Services. Fuels and Power sells 75% … • kindly correctly. 03 Vector and Matrix Equations and Applications: Problem 2 (1 point) Write a system of equations that is equivalent to the vector equation: 5 9 +y 3 8 • just tell me the true and false but please correctly. Are the following statements true or false? 1. The first entry in the product As is a sum of products. w 2. The solution set of a linear system wh… • kindly solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 12 (1 point) Rewrite the matrix equation below as a vector equation. Recall that to enter a vector we are using th… • Linear Algebra Questions Â . 2. For each of the following matrices, use our formula ï¬nding the inverse of a 2 x 2 matrix to determine the inverse (or to show that the inverse does not exist). a. [:3… • Can someone show me how to do these questions? Thank you.. 98. Dimensions and Bases. (a) Show that any four polynomials in P2 are linearly dependent. (b) Show that two polynomials cannot span P… • . 6. Suppose that A is a 3 x 3 matrix with eigenvalues A1 = 2, 12 = -1, and 13 = 4 with corresponding cigenvectors vi = 0 Find the matrix A. • . 2. Let A = C d. where a, b, c, d E R. Show that if (a – d) + 4bc > 0, then A is diagonalizable. 7 5 3. Let A = 10 8 (a) Compute A 100 (b) Let To = 0 and define the sequence Ik+1 = Ack for k = 0… • 5 marks For given two nonempty sets S1, S2 C R", define the operation S1 + S2 as follows: Sit S2 := {z ER" \| there exist some x E S1 and some y E S2 such that z = x ty} that is, each poin… • . 4. Let A be an invertible and diagonalizable n x n matrix. Show that Ais also invertible and diagonal- izable. 5. (a) Let A be a 6 x 6 matrix with two eigenvalues. Suppose one eigenspace is two di… • . 1. Find the eigenvalues and eigenvectors of each of the following matrices. If possible, diagonalize the matrices. If it is not possible to diagonalize, explain why not. (a) A = 4 4 (b) A = 10 NHO… • . In each of problems [12.3:3,4), find the eigenvalues and corresponding eigenvectos of the given symmetric matrix. Show that eigenvectos corresponding to distinct eigenvalues are orthogonal. In the… • . 12.2.12 Use the idea of Problem 11 to compute the indicated power of the matrix. [Reference] Problem 11: Let A have eigenvalues A1,…,An and suppose that P diagonalizes A. Show that for any posit… • . In each of Problems [12.2:2,8], find a matrix P that diagonalizes the given matrix, or show that this matrix is not diagonalizable. • im not sure which one is incorrect.. Two systems of equations are given below. For each system, choose the best description of its solution. If applicable, give the solution. O The system has no solut… • . = 31..= 2 , and w= 2. 5. (a) Describe all vectors u = (u1, u2) that are perpendicular to u = (1, 1). (b) Describe all vectors v = (v1, v2, v3) such that v1 + 12 + Us = 0. • . 6. Let 1 . .= 1, and w (a) Show that w is not a linear combination of u. (b) Show that u and u are linearly dependent. (c) Show that u is a linear combination of v and w. (d) Find the span of {u, … • . 4. Find two vectors 1t and tr such that: {i} both 1.: and v are perpendicular ten in = [1, â€”1, â€”1} and (ii) it is perpendicular to 1:. • . 2. Let in, o be two unit vectors. (a) Compute the dot product between a and i. (b) Compute the dot product between a – 40 and u + 40. • . 3. Let 1.: =(u1,ug] and a = {111,113} and let "1’1″: 6 and \|\|s\|\|= 2. {:1} Find the smallest and the largest pessihle 1values sf T33). Us} Find the smallest and the largest pessihle values s… • A = [1 4, 2 5, 3 6] Find a left inverse. Can you find another one? Explain why or why not. Please explain in detail. If there are multiple approaches to a question, try to answer using the SVD (Si… • The answer should be in this form: Â . Find the inverse of the given matrix, if it exists. Use the algorithm for finding A ‘ by row reducing [A I]. 1 0 2 A = – 2 -4 4 -3 -4 … Set up the matrix [A… • CO b1 1. Let A = -2 6 1 . Describe the set of all b = b2 for which AT = b has a solution. 0 0 b3 2. Solve the linear system below. Describe the solution set in parametric vector form. 2×1 + 22 + 24 = … • . Consider the feasible set given in the picture below. If the Objective Function is M = x + ky, match the value of the slope of each ofthe lines: (05) (3.4) feasible set Slope 0f Line 1 Choose… c… • Explain three contribution to rapid population growthÂ Â . 6239 p p P 0 .III : OI. ( See more. She spoke on the late Koï¬ Annan and Jerry John Rawlings. She respects these men so much according to h… • solve this question Â Â Â Â Â . Question 36 (1 point) Is it possible to estimate the instantaneous rate of change at x: â€”2 for the function 0 c) Impossible to tell from the equation alone • solve these two questions: Â Â Â . Question 43 (1 point) What type of function is f(x)=â€”x’ +J5x” +1 O a) Polynomial O b) Rational O c) Exponential Q d) None of the above • Determine the eigenspace basis of the matrix 3 2 41 A = 0 2 4 Include complete calculation details at each stage. (Hint: Find the characteristic equation, solve the characteristic equation to get the … • solve these two questions: Â Â Â . Question 31 (1 point) ~/ Saved Express 3m'(4x)-35i.'(4x) as a single trig function in simplest form.. Question 30 (1 point) J Saved In f(x)=asin[l(x-d)]+c, the… • sole these two question Â Â . Question 30 (1 point) J Saved In f(x)=asin[l(x-d)]+c, the c represents 0 a) The amount of vertical translation 6) b) The axis of the curve 0 c) The average between the m… • (a) (2 points) Consider a triangle with one vertex at (0, 0) and with the other two vertices at the points Pi = (1, 1) and P2 = (-3, 2). Explain why the area of this triangle is (b) (4 points) Now … • Let T: R3 – R3 be the linear transformation that (in order) . rotates (counterclockwise) about the x-axis by : radians, then . reflects about the yz-plane (a) (3 points ) Find the matrix of T. (b) … • (a) (2 points) Find constants a and I} such that [(3] = a [5] + b [g] . (b) (3 points) Suppose that T: R2 s R2 is a linear transformation such that 1 2 3 1 T [0] = H T [2] = [4] – What is the matri… • Are the following functions linear transformations? If yes, then find the matrix correspond- ing to the transformation; if not, then explain why. x (a) (2 points) T: R3 – R2 4x – Z T = atytz (b) (2… • Let A = of -2 (a) (2 points) Describe the effect of the linear transformation T(x) = Ax on the square with vertices (0, 0), (1, 0), (0, 1), (1, 1). (A drawing is OK.) (b) (2 points) Use your answer… • Consider the following formulations of the two-variable PRF: ModelI: Yi=Î²1+Î²2Xi+ui ModelII: Yi=Î±1+Î±2ô°€Xiâˆ’X Ì„ô°+ui (a) Find the estimators of Î²1 and Î±1. Are they identical? (b) Find the es… • . A major coffee supplier has warehouses in Seattle and San Jose. The coffee supplier receives orders from coffee retailers in Salt Lake City and Reno. The retailer in Salt Lake City needs 400 pound… • . Given the ordered basis 2 4 3 BA1 = 6 3 , A2 = 9 7 6 4 -4 A3 = 5 , A4 = -3 8 18 -4 find A B, the coordinates of A = with respect to B. 12 17 Ex: 5 [VB = • . (1 point) Which of the following subsets of P2 are subspaces of P2? A. {p(t) \| So p(t) dt = 0] VB. {p(t) \| p(3) = 0} c. {p(t) \| p(7) = 3} OD. {p(t) \| p(-t) = -p(t) for all t} VE. {p(t) \| p’ (t) + … • 6 7 3 Let u = – 8 and A = \| 0 2 – 6 . Is u in the subset of RS spanned by the columns of A? Why or why not? 2 – 2 Select the correct choice below and fill in the answer box to complete your choice. (T… • . R3 – R2 We know that linear function T T (7,4,1) = (9,8), T (5,3,1) = (7,6), T (3,2,1) = (5,4). Find linear mapping T corresponding matrix [T]. • . Please enter the optimal solution to the dog food problem in the space below. What is the minimum cost? Answer:. Mr. Smith decides to feed his dog a combination of two dogs foods, Brand A and Bran… • the locations of the main areas of the park are: A – Origin of Fun: (0, 0) B – Quadratics Village: (-7, 6) C – Towers of Trigonometry: (-2, 14) D – Food Functions: (5, 8) You decide the best place for… • . Given the ordered basis 5 2 B = U1 = -7 1, 12 2 , Ug = -6 8 find V B, the coordinates of v = 12 with respect to B. 8 Ex: 5 [v]B = • . Determine if the columns of the matrix form a linearly independent set. -4 -3 0 Justify your answer. 0 – 1 4 1 1 – 4 2 1 -8 . . . O A. If A is the given matrix, then the augmented matrix represent… • The class is asked to decide upon the location of the first aid station. You start by placing a grid over a map of the park. The class decides to place the entrance to the park at the origin. As a res… • . Given the ordered basis – B = u1 2 , uz = 93} 3 find V B, the coordinates of V= with respect to B. 6 Ex: 5 [vJB = • An investor has two money-making activities, coded Alpha and Beta, available at the beginning of each of the next four years. Each dollar invested in Alpha at the beginning of a year returns $1.40 two… • . Continuing with this same question, upload a scan or a photo of: (a) Your graph of the feasible set (b) Your calculation of the vertices of the feasible set () A table, evaluating the Objective Fu… • R3 – R2 We know that linear function T T (7,4,1) = (9,8), T (5,3,1) = (7,6), T (3,2,1) = (5,4). Find linear mapping T corresponding matrix [T]. • . 1. Consider a randomized complete block design Where the treatments are categorical and the blocks are random. Assume an appropriate less than full rank model. (a) Show that the estimate of any co… • . Consider the problem: Maximize M = 7a + 4y subject to: 3x + 2y – 36 Constraint 1 2 + 4y < 32 Constraint 2 3, y 20 Match the each vertex with the value of the Objective Function at the Point (0,… • . Find a basis for the subspace w={i A basis for W is {\|: 7a 5b 305 + 61′) BE) Ex: 1 3 E sz2 3H • .. Laura and Marty solved the same nonhomogeneous linear system. Laura’s answer, in parametric vector form, was 2 + t 5 -J Marty’s answer was 3 4 12 + t -10 17 14 Could Laura and Marty both be right? • i need explanantion, answer and working and steps. Andy is thrice as heavy as Paul and twice as heavy as Bobby. Given that the average mass of the three boys is at least 33 kg, find the minimum mass o… • hi could you help me with question 3 and 4. 3. [2 marks] Does (6, 6, 6) belong to span{ (2, 0, 1), (0,3, -1), (2, 3, 0) }? (This is the same as asking whether (6, 6, 6) is a linear combination of (2, … • . e) Prove Lemma 18 in the slide. (HINT: If it is not a chain, there is r* E X such that {f(x) \| x < r} is a chain but {f(r) \| x < x} is not.). Define o : X x J – X as follows: For x E X … • 2 Linear Algebra Questions Â . 2. For each of the following matrices, use our formula finding the inverse of a 2 x 2 matrix to determine the inverse (or to show that the inverse does not exist). â€”9 … • A 50-ft rope is cut into two pieces. The length of one piece is 9 times the length of the other. What is the length of the longer piece? • show the work, explain. (1) Modify Newton’s method for the case that both ï¬‚ï¬} = 0 and f'(ï¬) = 0, but f”(Â§:) 75 U (i.e., f has a maximum or minimum at the root 5:), by showing the modiï¬e… • kindly correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 3 (1 point) Let th – -1 and ex – [ ]- List five vectors in Span (1, 12). (Use the notation we have been using… • kindly correctly. 03 Vector and Matrix Equations and Applications: Problem 12 (1 point) Rewrite the matrix equation below as a vector equation. Recall that to enter a vector we are using the linear al… • let A be the matic A: 2 -8 10 Find inverse of It is it is inversible • Find a nonzero 3×3 matrix M that has b as a solution to Mu= 0 5 M= C • find the parametric equation of line I through a and parallel to b where a : 6 : – 3 • Compute the intersection of the planes S and T in parametric vector form 5: 42 + 8y-2=15 T : Z – 34 + 32 : 2 SAT = write answers in boxes . • please correctly and asap. 03 Vector and Matrix Equations and Applications: Problem 17 (1 point) Consider an economy with three sectors: Fuels and Power, Manufacturing, and Services. Fuels and Power s… • just tell me the true and false but please asap and correctly. True False 6. When u and v are nonzero vectors, Span { , } contains only the line through u and the origin and the line through @ and the… • just tell me the true and false but please correctly and asap. 03 Vector and Matrix Equations and Applications: Problem 9 (1 point) Are the following statements true or false? v 1. Any list of five re… • . A Moving to the next question prevents changes to this answer. Question 7 of 20 > Question 7 16 points Save Answer Using MOSA find the polynomials that will approximate the solution to: y’ = x2… • 9) just tell me the true and false but please correctly. 03 Vector and Matrix Equations and Applications: Problem 9 (1 point) Are the following statements true or false? 2 v 1. Any list of five real n… • Bill has a patent for the production of a new coffiee machine but doesn’t have any money to start production. He is unable to borrow the funds he needs.He has a close friend Ramu Â who has offered to … • To draw a graph for y = 3/4x + 7, a person can draw a point at x of 0 and y of ___, a second point by going over 3 and up ____ • . 5. How many functions f : [5] – [5] satisfy f(1) = f(2) or f(1) = f(3)? 6. How many bijections f : [5] – [5] satisfy f(1) = 1, f(2) = 2, or f(3) = 3? • 6) kindly clearly define your answer. 03 Vector and Matrix Equations and Applications: Problem 6 (1 point) Determine if b is a linear combination of the vectors formed from the columns of the matrix A… • Angles and Triangle Relationships Quiz Three triangles are shown. a a xly b b X b xy x represents the measure of an angle, y represents the measure of its supplement, and a and b represent the measure… • I want all solutions. Let A(2, -1, 1), B(3, 2, -1) and C(7, 0, -2) be three points in space. 1. Find the vectors AB, BC and CA, and calculate their sum. Give a geometric argument for your answer. [7p]… • Given a 3 x 4 augmented matrix, R2 <3R1 + Ry is an elementary row operation. O True OFalse 4. A system of linear equations with 4 equations and 4 variables can have 4 parameters in its general s… • Could someone show me how to do this question? Thank you. • . The expansion of a 3 x 3 determinant can be remembered by this device. Write a second copy of the first two columns to the right a12 a11 a12 213 a11 of the matrix, and compute the determinant by m… • Hello! I am really struggling with (b). I can’t figure out what the correct answer is. Would it be possible to help please? :). (2 points) Write each of the given numbers in the polar form re", -… • . Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. 10 0 4 8 8 3 -5 200 0 7 2 1 6 10 0 4 8 8 3 -5 = (Simplify your an… • Could someone show me how to do this? Tbakhyou.. 82. Spanning Sets with Parameters. Find spanning sets for the following subspaces of R3: (a) { (2a, b, 0) : a, b ER} (b) {(a + c,c- b, 3c) : a, b,c… • Could someone show me how to do this? Thank you?. 83.. Spanning sets obtained below are not the only ones possible. (a) { (1, 0, 1, 0), (0, 1, 0, 1) } (b) { (1, 0, 1, 1), (0, 1, -1, 1) } (c) { (-1… • 7) kindly clearly define your answer. 03 Vector and Matrix Equations and Applications: Problem 7 point) Determine if b is a linear combination of the vectors formed from the columns of the matrix A. 3… • 4) kindly clearly define the answer. 03 Vector and Matrix Equations and Applications: Problem 4 (1 point) Determine if b is a linear combination of a1, a2, and a3. a = -3 = -6 ? v a linear combination… • HelloÂ Can you help me with this exercises? Â . Note: For full marks, provide detailed solutions in the space provided for each question. Complete questions 4 and 5 using SCILAB. Screenshot the outpu… • 18) just tell me the true and false but please correctly and all. 03 Vector and Matrix Equations and Applications: Problem 16 (1 point) Are the following statements true or false? w 1. The first entry… • please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 1 (1 point) Given that U = Find u + . Find 2u – 30. • 12) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 12 (1 point) Rewrite the matrix equation below as a vector equation. Recall that to enter a vector we are… • 11) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 11 (1 point) Compute the product below. If the product is not defined, you can enter DNE (for "does … • 17) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 17 (1 point) Consider an economy with three sectors: Fuels and Power, Manufacturing, and Services. Fuels … • 10) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 10 (1 point) Compute the product below. If the product is not defined, you can enter DNE (for "does … • 18) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 18 (1 point) Balance the chemical equation MnO, + HCI -> MnCl, + H,0 + Clz and give your answer in lowest t… • 14) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 14 (1 point) Write the system below as a vector equation. Recall that to enter a matrix or column vector we ar… • 10) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 10 (1 point) Compute the product below. If the product is not defined, you can enter DNE (for "does not e… • 7) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 7 (1 point) Determine if b is a linear combination of the vectors formed from the columns of the matrix A. 3 4 … • 8) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 8 (1 point) Let an – 4 . 4- 8– – – [-$50] For what value of h is b a linear combination of a] and a2? • please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 8 (1 point) Let an – 4 . ex – 8- -ex – [-$50] For what value of h is b a linear combination of a, and a2? • 6) please solve this correctly. 03 Vector and Matrix Equations and Applications: Problem 6 (1 point) Determine if b is a linear combination of the vectors formed from the columns of the matrix A. 2 5 … • please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 3 (1 point) Let Uj = List five vectors in Span {v1, 12}. (Use the notation we have been using. For example… • please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 4 (1 point) Determine if b is a linear combination of a1, a2, and a3. 23 21 b a1 = -3 a2 = a3 = -6 -3 0 ? … • please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 5 (1 point) Determine if b is a linear combination of al, a2, and a3. 2 al = a2 = ? v a linear combination… • 2) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 2 (1 point) Write a system of equations that is equivalent to the vector equation: • 15) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 15 (1 point) Let 8 11 A = b = 16 2 8 12 Write the augmented matrix for the linear system that corresponds… • 13) please correctly and handwritten. 03 Vector and Matrix Equations and Applications: Problem 13 (1 point) Rewrite the vector equation below as a matrix equation. Recall that to enter a matrix we are… • just tell me true and false but please correctly. 03 Vector and Matrix Equations and Applications: Problem 16 1 point) Are the following statements true or false? 1. The first entry in the product 42 … • just tell me true and false but please correctly. 03 Vector and Matrix Equations and Applications: Problem 16 (1 point) Are the following statements true or false? ? w 1. The first entry in the produc… • In the healthcare field, the professionals take steps to verify dosage before medication is given to a patient. Whose responsibility is to protect the patients from medical error • Consider the following inequality: -2z + 3>-7 Step 1 of 2: Solve the linear inequality for the given variable. Simplify and express your answer in algebraic notation. Answer 2 Points Keypad Keyboar… • P3 -R2 From this linear function A: , we know that: A(7, 4, 1) = (9,8), A(5, 3, 1) = (7,6), A(3, 2, 1) = (5, 4) Find linear transformation A describing the matrix [A]. • linear algebra recursive. 3. I want to construct a train of total length 20. I have three kinds of cars, A and B cars each have length 2 and C cars have length 3. For example, here is one possibility … • linear algebra, recursive equations, see attached. thank you in advance!. 3. I want to construct a train of total length 20. I have three kinds of cars, A and B cars each have length 2 and C cars have… • Explain the answer please. The characteristic equation of A= -4 4 -2 is p ( 1 )= 12 + 5. 1+ 12 Select one: O TrueV 1 2+ 4 OFalse • . Student Portal & X Monday January 2 X B Assignments – MI X MHF4U0- Final St X Solving a Polynon X Ashish X MHF4U0-_Final_S X Homework Help X + X C O File \| C:/Users/joshc/Downloads/MHF4UO-_Fi… • State the exact value, showing ALL proper steps. [5] a) cos- 6 b) sin In 3 c) tan3n 10. If tan 0 = – V3, determine the possible value(s) of 0 for 0 0 S 27 . [2] 11. Calculate the equation of the fo… • A mathematics teacher wanted to see the correlation between test scores and homework. The homework grade (x) and test grade (y) are given in the accompanying table. Write the linear regression equatio… • . A politican has budgeted at most$80,000 for her campaign. She plans to distribute these funds between TV and Radio ads. Each 1-minute TV ad is expected to be seen by 20,000 viewers and costs $800… • . Given the demand in the Transportation Sector is$5 million and the demand in the Energy Sector is $3 million, what should the total production for Energy be? Enter your answer in millions of doll… • . An economy consists of three sectors: manufacturing, transportation, and agriculture. Each dollar of manufacturing requires 40 cents worth of manufacturing, 20 cents worth of transportation, and 1… • Please explain in detail, and show all working. Thank you!. 2. Suppose an exam with m questions worth 10 points each is given to n students. All the grades are collected in an m x n matrix G, where g … • see attached Â . Piazza Question: Define the function L : M2(R) – M2(R) via the map L(A) = A3. (a) Determine whether or not L is a linear mapping. (b) Determine whether or not L is an injective functi… • see attached Â . Let V be a vector space over R and let L : V – V be a linear map. Determine whether or not the set S = {v EVIL(v) = 10v} is a subspace of V. Note: We may also call L here a linear ope… • Show that performing a row operation to a consistent system yields a consistent system.Â Â Explain the details that are missing from the proof in the texbook.Â Â I need help. Thank you! Â . Ev… • L Question 4 (1 mark): Consider the following set of half spaces that belong to constraints: 3/ Tum Y 5 9 Each of the six constraints is an inequality, so the system will automatically add a slack or … • .. 2. A certain species of annual plant has two life stages: seedling and adult. Each year one tenth of the seedlings survive to adulthood and the rest die. Each adult (on average) produces 36 seeds a… • .. 0 2 4 1. For Which value(s) of h, if any, is (10) in the span of (3) and ( 1 ) ? h 1 â€”3 • Show that the projection is a linear transformation :-Â Â . W 2: A slight variation of Example 1 is given by the function P : R3 â€”> lR3 given by P(x, y, z) = (x, y, 0). It is routine to check th… • the first attachment is the simulation of heay load and someone told me to put values of compensation which is in second attachment can any one tell me how these values are selected beacuse my teacher… • Explain the answer please. Which of the following is the eigenvalue 1 of the matrix -8 6 A = -9 7 associated with the eigenvector U = W N None of the others. 0 1=1 01=-2 01=-1 • Question 14 pts] If L:P,â†’P, be linear transformation defined by L(p(1)) =p'(t) – p(1). If p(t) = – 3rÂ² + 5t +4. then L(p(t)) = ? Write an optional reasoning for your answer:. Question \|4 pts] If L … • Can someone help me with this question? Thank you.. 78. Independent or Dependent? Let S = {v1, v2, v3, V4, v5} be vectors in R3 and let A be the 3 x 5 matrix with the ith column given by the vecto… • Vincent and Micaella were saving their allowance Â to but a special gift for their parents’ wedding anniversary. Edward’s savings is Php150 more than twice the value of Allphonse’s savings. If togethe… • Consider f(u, v) = (ue" – 2ve-4)2 i. Compute the gradient Vf and the Hessian Vf evaluated at u = 1, v = 1. ii. Give the second order Taylor series approximation to f at u = 1, v = 1. • How do I do this?Â 1.Â 2.Â notation:Â 3.Â Â . Let x1, X2, x3 be column 3-vectors. Show that the following equation holds. (See the preceding exercise for notation.) det[ x, + x2 x2 + x] x) + x] =… • How do I do this?Â 1.Â Â 2.Â Â . Assume that (7,5.j,,…,j,,) is a permutation of (1,2,…, n) having m inver- sions. How many inversions docs (1’2, j,,…, j"), which is a permutation of (1,… • . 4. [3 marks] (Use MATLAB with format short option to find the answers of this 15 13 51 31 question.) Consider the matrix C = 51 31 15 13 11 35 11 53 (a) Use MATLAB command to find the product CCT…. • 0 O a 1. [3 marks] Let A = b 1 O . Here a and b represent some real numbers that 0 1 O are not zero. Find A’1 in terms of a and b using Gaussâ€”Jordan elimination. (Show your working.) 2. [2 marks] … • How do I do this?Â 1.Â 2.Â Â Â . Let A be a matrix such that each column of A has exactly one nonvanishing entry and that nonvanishing entry is 1. Show that dctA = +1, â€”1, or 0. Exhibit an examp… • For which values of a is the linear system corresponding to the following augmented matrix inconsis- tent? I 11 â€”143\|2 a\|3 • What is the function equation for absolute value regarding the “M” symbol ?. sin, cos, tan quadratic absolute value -30 20 -10 10 20 30 linear exponential square root • Please show detailed steps and explain.. 2 3 â€”5 bl (b) Consider the three vectors {Â£1 = 7 A752 = 2 ,fig = 8 .Let b = b2 be an â€”1 1 â€”5 b3 arbitrary vector in R3. Use Gaussian elimination (Gauss-… • . (5) For any a, b, c, d E C with ad – bc # 0. Define TA : C – {-d/c} – C if c#0, TA : C-C if c=0, with az + b TA(2) = a A = b cz + d’ Prove : (a) If c # 0, then lim TA(z) = ~, lim TA(z) = 00. 27-d/… • . (2) Using that lines and circles in R2 are given by the equation Arc+By+C(332+y2)=D, A,B,C,Dâ‚¬R. Prove that the function f : C â€” {0} â€”> (3 deï¬ned as f (z) = 1/ 2: maps any line and any c… • . (1) Consider f : C – C defined as f (z) = 22, i.e. (x, y) + (u(x, y), v(x, y)) with u(x, y) = Rf(z) = (Real part of f(z)) = x2 – y2, and v(x, y) =$f(z) = (Imaginary part of f (z) ) = 2xy. Prove t… • Please answer in detailed steps and explain. Thank you.. This uses material from Section 2.2; although the vocabulary of Section 2.3 may be helpful for answering part (c). (a) The set P: 272 :2$1â€”$2… • 1 2 4 X1 2 2 3 X2 A= 010 and x = 1 X3 1 0 1 1. Write the matrix equation Ax = 0 as a homogeneous system of linear equations. 2. Use elementary row operations to put A in row echelon form. If you like … • Put each of the following augmented matrices into reduced row echelon form, identify the free variables and the lead variables, and find all solutions. 1 1 0 2 3 -1 2 (a) 3 2 1 1 5 3 6 -1 -1 4 (b) -2 … • Each of the following augmented matrices are in reduced row echelon form. In each case, ï¬nd the solution set to the corresponding linear system. 140\|2 (a)001\|3 000\|0 15â€”20\|3 0001\|6 (b)0000\|0 0000\|… • CHALLENGE 2.11.1: LU decomposition. ACTIVITY 376940.2343920.qx3zqy7 Jump to level 1 Given the matrices below and that A = LU, complete I-1, U-1, and A-1 H 1 0 0 1 1 3 A = -2 -1 L = -2 1 0 U = 0 1 5 2 … • Suppose that 1 3 N 4 X 1 2 2 3 3 X2 A = and x = 1 0 1 X3 0 0 X 4 (a) Are the columns of the matrix A above linearly dependent or not? (b) Give an example of a 3 x 3 matrix C and a vector b so that the… • The general equation for a circle is a(x2 + y? ) + ba+ cy+ d=0. There is exactly one circle passing through the points (2, -2), (-1, 2), and (0, 0). Find an equation for this circle. I ( 2 2 + 3 7 ) +… • Question 3 (2 marks): Consider the following linear programming problem: Max z = 8x, + 5x, -5x, + 2x, < 10 7x, + 15x, $108 10x, + 12x, < 120-8i x – 5x,$5 X, x, 30 Use the graphical solution tec… • . Let V be the set { (x, y) \| x, y ( R) with addition operation (@ defined by (T1, y1 ) D (T2, y2) = (X1 + 2, 914/2) and scalar multiplication O defined by a O (x, y) = (x, 0). Show that I is not a … • Find all the solutions to Ax = 0. Suppose that 2 3 x 1 2 2 3 3 X2 A = and x = 1 0 1 X3 0 1 0 1. Write the matrix equation Ax = 0 as a homogeneous system of linear equations. 2. Use elementary row o… • Please explain and graph. Graph a line with the indicated slope and y-intercept. lope = – 5 and y-intercept (0,9) • Linear equation. Determine the equation of the line with y-intercept at (0, -6) and a slope of – 3. Enter Equation • Linear equation. Find the equation of the line passing through the points (6, -10) and (-4, -3). Input equation • Please solve the below questions Â Show all the necessary steps. Thank you!Â Â . Exercise 3 (Market Inefï¬ciencies) Suppose there are two firms. A chemical firm A produces a saleable good, but prod… • For this assignment you will need to submit 2 documents. A word document with your answers typed out and an excel file with your supporting work. Remember to always start with the original LP for each… • For each of the following systems of equations, write the corresponding augmented matrix describing the system. Put it in reduced row echelon form (Do this by hand and indicate your steps). (a) 1 +… • . hfactorgir manufactures three products, A, B, and C. Each product requires the use of two machines, Machine \| and Machine \|\|. The total hours available, respectively, on Machine \| and Machine \|\| p… • Linear equation. 9 X =-5 10 y = 27105 11 Expression to Evaluate: 5(2x-3)+27096-y 12 • can I please see the correct answer. System A -2x – y =8 -2xty=8 System B 3x +y=9 -3x =y-9 • M Gmail YouTube Maps Find the y-intercept and x-intercept of the following linear equation. 5x – y = -5 Answer Enter the coordinates to plot points on the graph. Any lines or curves will be drawn once… • Question: Â i = 0. Question 4 11 mark 1: Consider the following linear programing problem: P: Max 2 = 5×1â€”2×2 X238 x1+szS (1â€”0.2i)x1â€”x2 S 2 x1, X2 2 0 (a) Plot the graph & determine the opt… • Given any basic feasible solution to a LPP (where the objective is to minimize the objective function), if for some a; (not in the basis), z, – c, > 0 and y S 0 (i = 1,2, …, m), what would be … • Could someone help me with the question? Thank you.. 72. Matrix Linear Combinations. Is 6 a linear combination of the matrices -1 8 4 A = -2 B = 2 3 and C = 4 • A function f : A – B is linear if, for all a, be R, f(a . x t b) = a . f(x) +b. Apply the definition of linear to (a) f(x) = 2x (b) f(x) =12 (c) f(x) = >fair • One Linear Algebra Question Â . 6. Suppose A and B are invertible matrices. a. Rearrange the equality BCB-1A = D to solve for the matrix C in terms of the other matrices (A, B, D, A-1, B-1, etc.). Sim… • 2 Linear Algebra Questions Â . 2. For each of the following matrices, use our formula finding the inverse of a 2 x 2 matrix to determine the inverse (or to show that the inverse does not exist). 107 1… • One Linear Algebra Question Â . Solve the following problems (and show work). 1. Suppose Ax = b represents a linear system of m equations in n unknowns. Also, recall the general fact that any system o… • Could someone help me with this question? Thank you.. 58. Equation of Plane. Let the line & be given by the intersection of the planes 2x-y+z = 0, x+z-1=0, and M be the point (1, 3, -2). Fi… • The graph below does not represent a function. The vertical line is not part of the graph, but may be used to answer the question below. 10 10 0 10 10 List the minimum number of point(s) that you woul… • Determine the inequality shown in the graph below. The inequality symbols can be entered using the toolbar. y 5 -5 -5- • Determine the equation of the line that passes through the point -3, -5) and is perpendicular to the line y = 3. Preview Equation • Let A be the 3×3 matrix associated with the transformation that projects vectors onto the plane 2x+3y-z=0 Let B be the 3×3 matrix associated with the transformation that reflects vectors onto the p… • Determine the equation of the line that passes through the point (6, 10) and is perpendicular to the line x = 5. • Determine the equation of the line that passes through the point 11 9. and is perpendicular to the line y = 5x + 6. 5 • 17 15 2 -1 Question 1: Let A = 16 15 Solve Ar = 0 and express your solution 15 15 6 -3 in vector form. What is the rank of A? Question 2: Find all solutions of 2AT(x + b) – -B(x + c) where 2 ON A = – … • Find the equation of the straight line which makes intercepts of 2a on the x-axis and 3a on the y-axis, given that the straight line passes through the point (14,-9) • Find the value of k if the straight lines X+2y+1=0 and 3x+KY+5=0 are parallel • The point (5,5) divides the interval between (-1,p) and (q,6) in the ratio 2:5. Find the value of p and q • . 2 11 – 5 Exercise 7.10. The vectors v = and w = – -10 span a plane P through the origin in RA. Let 1 4-t 4+ 4t L = 4-t :ER -7 -2t be a line in R4. (a) Consider the displacement vector x between an… • Suppose that U and W are subspaces of a vector space V .Â ï¼ˆi) How can I prove that if W is a subset of UÂ (ii)and how to prove U +W = U. • only this solve this. Multiplique y sume las matrices utilizando los bloques indicados 1 0!-1 1 2 4 ! 1 6 2 1 -3 4 3 -2 5 A = ;B = -2 1 4 6 2 1 1-1 0 C 0 2 3 5 -2 4 13 • CHALLENGE ACTIVITY 2.11.1: LU decomposition. 376940.2343920.qx3zqy7 v Jump to level 1 1 2 Given that A is transformed into U by the elementary row operations below, find L. 3 -3 4 1 -3 4 -3 4 -3R1 + R… • (1 point) Find the quadratic polynomial whose graph goes through the points (-1, 7), (0, 5), and (2, 25). f ( x ) = • So I’ve attempted this question and have found all of them except the answers are wrong where the 12, -7 & -1 is.. (1 point) On the augmented matrix A below , perform all three row operations in t… • (1 point) Solve the system by finding the reduced row-echelon form of the augmented matrix. x – 3y + 4z -1 -3x+1ly – 16z = 11 3x – 8y+ 10z = 1 reduced row-echelon form: How many solutions are there to… • (1 point) Solve the system = X 1 + x 2 = X2 +X3 HUUH X3 +X4 = +X4 = X1 X1 X2 +S = X3 X4 • . 3. Define spanning set and basis of a vector space. Explain the difference between them. Examine whether the set S = {(1, 2, 3, ), (0, 1, 2), (0, 0, 1) } is a basis of R3 or not? • Please show each step.. E Day29-Activity1.pdf 1 125% + Find all least squares solutions to Ax = b in each case given below: a. A = and b = b. A= 61 1 and b = • . Let V = Foo and let W = {(1, . .., 2021, 0, [2023, 2024, . . .) \| I; ( F } CV. That is, W consists of all the elements of Foo whose 2022nd coordinate is zero. Prove that W is a subspace of V. • Un barril lleno de petrÃ³leo hasta la mitad descansa de lado. Si cada extremo es circular, de 10 pies de diÃ¡metro, determine la fuerza total ejercida por el petrÃ³leo contra un extremo. Suponga que l… • Finish simplifying the matrix what is the resulting matrix ?. Note: to proceed to the next step, hit the Enter key while in one of the entry cells, or click the Preview My Answers button. If your answ… • . 8:42 YoE +48 72% coursehero.com/file/76457493, Find study resources Q 5. 4x: 4 = 4:4 ->x=1 GBC \| MATH1102/KG/F20 1 6. X: 1 = 1 ->X=1 7. 24: 12 = a: 6 -> 2=2:6 -> a=12 8. Solve for FP: … • . 4. 7 marks Let C C R" be a given set and C + 0. Let f : R" – R be a given function. Consider the following optimization problem: (Probl) Maximize f (2) under the constraint: x E C. Defin… • (1 point) Find the linear equation of the plane through the origin and perpendicular to the vector (4, 4, 1). Equation: • pint) Find the linear equation of the plane through the point (1, 10, 8) and parallel to the plane x + 6y + 5z + 4 = 0. ation: • . The following LP is formulated in the handout "LP_Formulations,_Cheese_Factory" posted on Learn, under content –>Linear Programming Formulations (Ch.3). The variables and constraints… • Let V be the subspace of R[x] of polynomials of degree at most 3. Equip V with the inner product < f âˆ£ g > = âˆ« 0 1 â€‹ f ( t ) g ( t ) d t (a) Find the orthogonal complement of the subspace … • Please answer and show work, thank you!Â . 10. (four marks) A scholarship of $1500 will be awarded to one eligible high school student on each of the following 20 years. If the money can be invested a… • The relation P=-n^2+58n-480 models the profit, P, and the number of tickets sold, n. Calculate the number of tickets the students must sell to break even. • which of the following is not a restriction for the exponential function?. 11:45 PM M 1 all all 12 21 QUIZ 3: EXPONENTIAL FUNCTION . . . docs.google.com Which of the following is NOT a 1 point restric… • Please i need a screenshot. Below is a video tutorial about sketching a graph of a function using Desmos Graphing Calculator. Watch the video carefully to know how to sketch a graph of a function. Int… • . 2. Give an example of a linear combination of 3 vectors in R’. Justify carefully why your example is a linear combination, using the definition. • . 1. Use set builder notation to describe the following set. U is the set of vectors in Ra, such that the x-coordinate is an integer, and the y-coordinate is a negative multiple of 7. • . Suppose p (@) and q (@) are the functions whose values are given by the following tables. p x -2 -NO NHO – 2 1 Calculate (p o q) (0). • Graph a line whose slope is – and that passes through the point ( -2, -2). y • 5 marks Consider a finite set of nonzero vectors, v1, V2, …, VK E R", with vi\| > 0, i = 1, …, k; k 2 1. Define K C = Hvi. (1) i=1 where we denote Ho = {x ER. \|v .x <1}. a. For the se… • Hillary wants to spend under 30 minutes on printing her portfolio. It takes her 3 minutes to print a colored copy and 1 minute to print a black-and-white one. • . 1. (15 pts) Consider the following two matrices: A: Â«Innâ€”- 1 and B: 1 4 WWW CDC’JDJ 0153M GEM” (a) Write down one way to convert A to B using elementary row operations. There are many corre… • NEED ANSWER AND STEPS THANK YOU Â . 3 Read Section 1.4 (page 27) in our Course Textbook, https://lila1. lyryx. com/ textbooks/OPEN_LAWA_1/marketing/Nicholson-OpenLAWA-2021A . pdf Then consider the fol… • The answer above is NOT correct. (1 point) Determine which of A-D form a solution to the given system for any choice of the free parameter s, . List all letters that apply. If there is more than one a… • see attached Â . 1) In M2(R), define U a 6 C d EM2 (R) atb+c- 2d=0. (a) Prove that U is a subspace of M2 (R). (b) Find a basis of U and determine dim(U). 2) In P3(R), define U = {f(x) E P3(R) \| f(0) =… • . Question 3. We have a plane m â€” 2y â€” z = 2 in R3. Find equation of the line L passing through point (1,1,1) and perpendicular to this plane. Find the distance between the zitâ€”axis and the li… • . Question 2. Let T : R4 -> R4 be a linear transformation 2y + 3w T y – x + 2y x + z W 3x+ y – 52 + w Find the matrix B of transformation T and compute determinant of B. • . Question 8. We know that the 2 x 2 matrix OT A = 3 4 has eigenvectors Diagonalize this matrix: find a 2 x 2 matrix V and a diagonal 2 x 2 matrix D such that A = VDV-1. Use this diagonalization res… • . Question 6. Solve the system of linear equations with complex coefficients (1 + i)x + y = 2i -2ix + (1 – i)y = 1 • . Question 3. We have a plane x – 2y – z = 2 in R3. Find equation of the line L passing through point (1, 1, 1) and perpendicular to this plane. Find the distance between the x-axis and the line L. … • LEE ..\|I\| “HIE”: K. E @ “343% {Moms 6 V 3. A rancher raises milking cows and goats which he feeds with two types of mixes, type x and type y. Each bag of type x costs ?400 and contains 100 units… • . Question 7. (7a) [3 marks] Give example of three vectors ui, U2, us in R4 that are linearly independent. (7b) [2 marks] Give example of three vectors vi, 12, 13 that are different from uj, U2, us … • . Question 7. (7 a) [3 marks] Give example of three vectors 131, u}, u}, in R4 that are linearly independent. (7 b) [2 marks] Give example of three vectors iii, v3, 65, that are different from if}, … • Hi! I was working on my hwk, but I would like to double check my answer. Could you please solve it? Â Â . Consider the system of equations whose augmented matrix is 1 k 1 Lh 0 There is exactly one ch… • help prove this. 2. In the following section, we are going to want to use the fact that if {v1, . .., Uk-1} is linearly independent in vector space V over F and {v1, . .., UK} is linearly dependent in… • pls help solve this. thanks. 2. Which of the vectors vl = (1, 0, 0), U2 = (1, 1, 0), 13 = (1, 1, 1) in F3 are eigenvectors for the linear map T : F3 – F where T(x, y, z) = (-3x + 5y -5z, -7x + 9y – 5z… • umm I need help. Identify a constant 7x – x3 . 90 X 7x • need to know if each step is right or if one is wrong. 5. A student worked the equation below to solve for x. In which step (if any) was an error made? 2x – 13 + x =9(x -1) A [Step 1] > 3x – 13 =… • Hi! I was working on my hwk, but I would like to double check my answer. Could you please solve it? Â . Consider the system of equations whose augmented matrix is [‘1l k 1 0 4 There is exactly one c… • . 10. Show that As contains an element of order 15. 14. Suppose that o is a 6-cvcle and B is a S-cvcle. Determine whether ([5134 Cirâ€”1,8â€”3cr5 is even or odd. Show your reasoning. 26. Let u an… • https://lila1.lyryx.com/textbooks/OPEN_LAWA_1/marketing/Nicholson-OpenLAWA-2021A.pdf Â . Read Section 1.4 (page 27) in our Course Textbook, tittps:,.’,Â»’1ilal.Lynne.emur textbooksx’ DPEH_LAHA_1Â£… • Please help me understand these questions! answer and show work, thank you!Â . 13. (four marks) The table shows the path of a parabolic gateway, where x is the horizontal distance from the left base a… • Please help me understand and answer the questions, thank you!!!. 15. (four marks) The table shows the path of an extraterrestriale, where x is the horizontal distance from the launch pad and h is the… • . 13. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Choose the response below that explains why the system is consistent. a. The system is consist… • bulk food store sells peanuts for$6.50/kg and cashews for $16.00/kg. henri purchases a large bag of each and the total cosy of both bags in$206.at home he mixes them together and the total weight is… • . b. How many bottles of soda have to be sold to breakeven? Interpret the result. c. What should be the monthly breakeven sales volume if monthly fixed costs were Birr 2350? d. Plot the breakeven ch… • Can someone show me how to do this?. (b) All complex 2 X 2 matrices with z1 + 24 = 0. 68. General Subspaces. Use the subspace theorem to decide which of the following are real vector spaces with t… • . Q3-b (4 points) Consider the plane II : 2x – y + z = 1 and the line & : x – y – 3 z+1 -2 (b) Find an equation for the plane II1 through the origin and parallel to the plane II. + Drag and drop… • Here is the questions – please make it with clear hand font and clear letters and numbersÂ . 1. Solve the following system using :a) Gauss-Jordan algorithm, b) the inverse of the corresponding matrix(… • . Q2-a (3 points) Consider the polynomial equation of P(x) = 0 where P(x) = x4 – x3 -x2 – x-2. (a) What are the number of possible positive and negative real zeros of P(x)? + Drag and drop an image … • . Q1 (8 points) Let i be the complex number for which i = -1. Use mathematical induction to prove that 1+ (1 + i) + (1+ 1)2 + (1 + 1)3 + … + (1 +1)" = if1- (1+1)"+1] for all positive int… • Q4 (10 points) Find all basic solutions of the homogeneous linear system -X1 +2×2 +x3 -2×4 OO +x3 -2×4 -2×1 +4×2 +XS lick to browse… • . Q6-b (3 points) 1 N Let A = \|2 -1 11 NW 0 : w X (b) Use part (a) to solve the linear system A -2. Q6-b (3 points) 1 N Let A = \|2 -1 11 NW 0 : w X (b) Use part (a) to solve the linear system A -2 • . 15. Given the following statement, choose the option below that best describes whether it is true or false, and the reason why: The equation Ax = b is consistent if the augmented matrix [A b… • . 12. Suppose a 5 x 7 coefficient matrix for a system has five pivot columns. Is the system consistent? Why or why not? Choose the correct answer below a. There is at least one row of the coeffic… • . Q10-a (6 points) 2 0 0 2 Let A = and B = 4 -1 0 2 – 3 7 3 5 (a) Find all eigenvalues of A and all eigenvalues of B. • . Q8-a (6 points) Let u = (-1, 0, 1) , v = (3, 2, 1) and w = (0, 1, 2). (a) Prove that the vectors u – v, u + v and u X v are linearly independent. + Drag and drop an image or PDF file or click to b… • Q9 (8 points) Let 1 A linear transformation Ti from R2 to R2 maps vectors v = (v1, U2) to v’ = (up, U2) according to U1 = 501 + 302 T1 : Up = 201 + 12 Also a linear transformation T2 from R … • Is there an association between practice time and being a starting player • Q6-b (3 points) 1 0 Let A = \| 2 -1 N WEN (0 3 X (b) Use part (a) to solve the linear system A y -2 • Q7 (7 points) 0 0 1 2 -10 0 Let A = First show that A is invertible and then find the entry of A that lies in 0 2 -1 011 the fourth row and first column. Show your work. (You do not need to … • Q6-a (7 points) 1 0 Let A = 2 -1 N WHEN 10 3 2 (a) Find inverse of AT. • Q5 (7 points) Let a be a negative real number. Use Cramer’s rule to find only the value of z. ax + z = a-2 xty+z=-3. 3x – y+ 2z = 1 • Q9 (8 points) Let 1 A linear transformation Ti from R2 to R2 maps vectors v = (v1, U2) to v’ = (up, U2) according to U, = 501 + 302 T1 : U, = 201 + U2 Also a linear transformation T2 from R2… • N x (x >0} undefined 30 20 25 10 X N (x > 2) undefined undefined 20 30 40 SO 60 70 undefined 0.16565667 0,125 -10 • . 13. Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Choose the response below that explains why the system is consistent. a. The system is cons… • Q4 (10 points) Find all basic solutions of the homogeneous linear system -x1 +2×2 +x3 -2×4 +x3 -2×4 = 0. -2×1 +4×2 +x5 d drop an image or PDF file or click to browse… • All of the types of transformations have at least 2 for each type of function 2. Include all mathematical solutions that illustrate these functions a. Show the correct equations for your piecewise … • . 14. . A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. a. Can such a system be consistent? (Yes or no) b. Illustrate your answer with a … • . 5678 9101112 Given the matrix A = 1234′ a. Row reduce A to reduced echelon form b. Which columns from the original matrix A are the pivot columns?. 10 Given the linear transfor… • . 16. Given the following statement, choose the option below that best describes whether it is true or false, and the reason why: If x and y are linearly independent, and if z is in Span… • . 2. Compute the matrix-vector product \| 2 2] 3. If possible, write the vector 5 as a linear combination of the vectors 4. If possible, write the vector as a linear combination of the vecto… • Let V be a K-Vector space. Prove that the intersection of any set of linear subspaces in V is also a linear subspace. ‘ • Please help answer and show work for the questions below, thank you!. 9 . (four marks) Rani invested $9000 per year in a mutual fund that averaged 13% net growth per year with interest compounded annu… • Please help with these questions! Thank you!. 13. (four marks) Determine the amount if$7995 is invested at 1.25% per year, compounded daily, from September Ist to October 10th inclusive. Y. 1. (four … • Assume you are driving on a highway, and you get a text message from a friend and want to respond Time yourself as you write the following, “Sorry, I’m driving. I Will call you back” Using the spee… • . 8. Let A = 3 3 and define T: R2 – R2 by T(x) = Ax. Find the image under T of: a b. 9. Assume that T is a linear transformation. Find the standard matrix of T: R3 – R2 if T(e,) = . T(e2) = an… • . Name: LAS#3: Part A: Determining Function with Tables and Mappings DIRECTION: Determine if each relation below is a function. 1.Function: YES/ NO _2. Function: YES/NO 3. Function: YES/ NO 4. Funct… • . Given the following matrices: 3 3 â€”2 2 A: â€”2 2,B=â€”1 2,C=[; j ’32] 3 1 â€”2 2 Solve the following (if possible): a) A+B b) A+CT c) (Bâ€”A)T+c d) (AT+BT)+C e) AT+CT f) 3(A+ch)Tâ€”4c • .. 1 1. For which real numbers h, if any, is 2 in the span of the vectors â€”1 1 1 2 l} , h ,and â€”1 ‘1’ 2 3 3 2. Suppose that the vector b E R4 is in the span of the columns of a matrix A. Is the … • . 8. Let A = and define T: R2 – R2 by T(x) = Ax. Find the image under T of: a. b. 9. Assume that T’ is a linear transformation. Find the standard matrix of T: R3 – R2 if T(e,) = . T(e2) = and … • . Rewrite as a matrix-vector product: 3x+4yâ€”z+w= 12 â€”x + 2y â€” z + w = 9 2xâ€”3y+22â€”4w=â€”3 Given the following augmented matrix, list all the variables {x1. , x5] . Indicate which are … • . A mining company has two mines. One day’s operation at mine #1 produces ore that contains 10 metric tons of copper and 600 kilograms of silver, while one day’s operation at mine #2 produces ore… • please explainÂ Â Â Â Â . Given the following matrices: 3 3 â€”2 2 A: â€”2 2,3: â€”1 2],C=[g j ’32] 3 1 â€”2 2 Solve the following (if possible): a) A+B b) A+cT c) (Bâ€”A)T+… • Give a matrix E which, when multiplied to the left of any 3 by 3 matrix A, will modify A by replacing row 1 by row 1 – 4 times row 2. This matrix is called an elementary matrix. E = Check • Check Lawn King sells a push mower at price p1 and a ride-on mower at price p2. The demand function for each mower is a linear function of p1 and p2. Market research shows that if the price of the pus… • Solve the system of linear equations given by the following augmented matrix 13 – 6 3 66 9 – 3 0 36 4 -2 1 21 write answer here • . TMA 03 Cut-off date 1 March 2022 Question 1 – 10 marks This question is based on your work on MU123 up to and including Unit 6. (a) Figure 1 shows two points A and B with a straight line drawn thr… • Hello, can someone help me solve these exercises?. Exercise 1 Consider the following 1-form : w = (1â€” 2.3%)!13’1 â€” 2$1$2d$2. 1) Is there any function In such that db = w? 2) Consider the followi… • hello , can anyone help me with this ?. Exercise 1 Consider the following 1-form : w = (1 – 2x]) dx 1 – 2x142dx2. 1) Is there any function h such that dh = w? 2) Consider the following vector field : … • . 7. Prove that w E (C is in the closure of a set E C (C if and only if there is a sequence {zn} C E such that lim zn = w. Thus, a set E is closed if and only if it contains all limits of convergent… • Can you show and explain the process by solving this Linear equation by substitution • Please help me unstuck solve the following Linear System Â . Solve the following Linear systems See Figure below. What is ab + c? 2x+z=0 â€”4x+y=1 3y+z=â€”7 Ans: (a, b, c) Your answer See Figure below… • I Â need help with problem 3. More so Â showing the followingÂ ***For this value of L, we have the system of equations.. (transform back into a system of equation’s and solve for the unknown … • I am having issues with Problem Three. I am having difficulty with writing out the steps as asked below.Â Â Â **For this value of L , we have the system of equations.. ( transform back into a … • 1/17/22, 12:36 PM 504 W1 Assignment1 – Jupyter Notebook Please use numpy array and kl or k2 matrix you created above to create a matrix /3 that is equivalent to this matrix below: 50. 50. 50.1 k3 = 50… • Please help Â . True or False: The union of any collection of open sets is open. Provide a proof if true, or a counterexample if false. • Please show the working in as much detail as possible, I would like to understand how you arrived at the conclusion. Thank you!. suppose that Av1, . .., AVn E R" are orthonormal vectors. In part … • Please answer and show work for questions 3-6. Thank you!. 3 . (four marks) Mei Ling invested$12 000 in a mutual fund that decreased in value by 1.25% in its first year. The management company charge… • Please show the full working that leads to the answer.. 2 4 (a) Let A = 8 5 1 represent a linear transformation f : R3 -> R2, f(v) = Av with respect to the standard bases of R3 and R2. Find the mat… • A machine can make 5 miles of ribbon in an hour. Graph the length of the ribbon the machine will make in eight hours. Another machine can make 8 miles of ribbon in an hour. Graph the length of the rib… • E Homework: Homework: 2.3 to 2.4 < Question 11, 2.4.4 The sum of and five times a number is equal to subtracted from six times the number. Find the number. • A 19-foot piece of string is cut into two pieces so that the longer piece is 4 feet longer than twice the shorter piece. Find the lengths of both pieces What is the length of the shorter piece? • Â . 3. A thief escapes from jail. When he was 5 kilometres from the prison, the police began following him at a pace of 20 metres per second. The criminal is fleeing at a pace of 10m/s. A police dog… • Liner equations. Â . 2. Mark wishes to scale a 50-mcter-tall pole. He climbs Im at ï¬rst, then slips down 1m to the bottom. Then he climbs 2 metres but slips down 1 metre. He then climbs 3m but slips… • The left and right page numbers of an open book are two consecutive integers whose sum is 293. Find these page numbers. • . Continuing with this same question, choose all the constraints that apply: Select one or more: 0 4x + 3y < 10, 000 100x + 200y < 10, 000 0 4x + 3y < 300 200x + 100y < 10, 000 0 3x + 4y… • Write the following as an equation. Then solve. Twice the sum of – 6 and a number is the same as the number decreased by N / W Find the number. • . A simplified economy consists of two sectors, Transportation and Energy. For each $1worth of output, the transportation sector requires$0.25 worth of input from transportation, and $0.20 of input… • . Find the matrix C so that A . C = B, 3 2 where A = and 4 3 5 6 B = 1 7 First, enter the values for A-1: a b A- = d Note: you can use the shortcut method to find A- • . 0 0 Let A = 2 1 -2 -1 2 1 Use the Gauss-Jordan method to calculate the inverse of this matrix. In other words, start with: 0 0 10 0 2 1 -2 01 0 -1 2 1 0 0 – • 3 5 6 8 10 47:47 Aisha changed 1.45 + 2.38 to 1.5 + 2.4 in order to estimate the sum. What estimation method did she use? front-end O clustering O rounding to the nearest tenth O rounding to the neare… • Please answer and show work, thank you!. 13. (four marks) Payments of$1 17.45 per month are made for 3 years. What is the total amount paid? 14. (four marks) Payments of $682.50 per month are made fo… • Please answer and show work, thank you!!Â . 4. (four marks) Wilson makes monthly payments of$180 into an account that pays 4% per year, compounded monthly. How much money will be in his account in 3 … • Please answer these questions and show work. thank you!Â . 11. (four marks) Determine the amount if $1 150 is invested at 5% per year for 2.5 years, compounded semi- annually. 12. (four marks) Determi… • No explanation needed . Â just want to chech my answers against yours Â Â . ofdatetents in the following. Please answer whether the statement is true or not (you don’t need to give any reasons). … • Please help answer and show work for these questions. Thank you!!Â . 15. The table shows the path of a parabolic arch, where x is the horizontal distance from the left base and h is the height of the … • Please show each step. Thank you!. Complete the square for each of the following quadratic forms a. Q(x, y) = x2 – 8xy + 25yz b. Q(x, y) = 3×2 + 6xy + 5y2 Then do a change of variables to complete the… • -. 2. The line ED is tangent to the circle point A(4,8). If the centre of the circle is C (-6,6), determine the equation of the tangent line BD. Bonus: ï¬nd the diameter of the circle through linear … • . Use the techniques of linear algebra to ï¬nd a quadratic f(x} = a + bx + rm:2 such that the graph 1; = f(x} passes through the points (â€”1, 6), (2, D) and (3, 2). • Please answer and explain question 9-15, Thank you!Â . 15. (four marks) The table shows the path of an extraterrestriale, where x is the horizontal distance from the launch pad and h is the height of … • . Solve the following system of linear equations using an augmented matrix. x+y+22 = â€”l 2x+y+3z = 0 â€”2x+z = 2 • . 11. Find the standard matrix of the linear transformation a. T : R’ – R’, which first rotates points through -37 /4 radian (clock- wise) and then reflects points through the horizontal x-axis. b. … • . 2 â€”6 3 9. Find the value of h for which the vectors [â€”4] , [ 7 😐 and \|:h:\| are lin- â€”3 4 early dependent. • Linear equation any of this substitution, graphing , elimination or typical solution • . Let u and v be eigenvectors of a matrix A, with corresponding eigenvalues ) and u, and let c, and c2 be scalars. Define XK = C1 2ku + C2/v (k = 0, 1, 2, …). a. What is Xk + 1, by definition? b…. • Please show every step! Thank you!. Complete the square for each of the following quadratic forms a. Q(cc,y) = 3:2 â€” Swy + 25312 b. Q(:l:,y) = 32:2 + 62:31 + 53/2 Then do a. change of variables to c… • elementry linear algebra ex 5.2 q no 19…book name hiward anton 11 edition. Question : Find characteristic equation for the given matrix. Find Eigen values for the given matrix. Find Eigen vectors co… • . 5. Let w1, w2, w3, u and v be vectors in R". Suppose the vectors u and v are in Span ( W1, W2, w3}. Show that u + v is also in Span { w1, W2, w3}. • A delivery truck can transport packagesÂ weighing at most 3,800 pounds (lbs)Â andÂ with a volume of no more than 400 cubic feetÂ Â The truckÂ transports only two sizes of packages: a small • ONLY PART C!!!!!!!! Â . a) Find the general solution of the system corresponding to the (augmented) matrix 2 4 3 1 8 A = 3 6 â€”6 0 15 2 4 9 3 12 b) What is the geometrical form of the solution set… • Can someone please help me understand these? Â . Find the equation of the exponential function, in y = a (b) form that passes through the points (-2, 3) and (6, 35) . Round both a and b to the nearest… • Can someone please help me understand these? Â . PART I QUESTIONS: Answer all questions in this part by writing the choice of the appropriate answer in the blank beside the problem. Each question is w… • . Find the solution set of the following system by means of matrices t 2xâ€”3y=â€”1 x+4y=5 2. x+y=2 2xâ€”z=1 2yâ€”32=â€”1 3. xâ€”2y+z=4 3x+yâ€”22=4 yâ€”z=1 4. â€”4x+2yâ€”92=2 3x+4y+z=5 xâ€”3y+2z=8 5… • Hello, can I get help with these Exercises?. Exercise 1 Consider the following 1-form : w = (1â€” 2.3%)!13’1 â€” 2$1$2d$2. 1) Is there any function In such that db = w? 2) Consider the following vec… • Hi there , I need help with these 2 questions . Kindly provide step by step explanation on how to solve these 2 questions . Thank you in advance !. 5_ Which of the following linear spans contain th… • elemnentry linear algebra chapter 5 ex 5.2 q 19..solve by A10. Question : Find characteristic equation for the given matrix. Find Eigen values for the given matrix. Find Eigen vectors corresponding to… • elementry linear algebra chapter 5 ex 5.2 question 19. Question : Find characteristic equation for the given matrix. Find Eigen values for the given matrix. Find Eigen vectors corresponding to each Ei… • Could someone show me how to do a) and b) for both questions as examples?. 62. Subspaces of R3. Decide which of the following are subspaces of RS. Explain your answers. (a) A = {(a, b, 0) ER3 :… • Can someone show me how to do a) and b)? Thank you.. 61. Subspaces of R2. For each of the following subsets of R2 sketch the set, then determine whether it is (i) closed under addition, (ii) closed un… • . 3. (a) Define the following terms An R – basis of a vector space The coordinate of a vector (b) Find an R – basis for the solution space of the system x – Zy + 32 – W = 0 2x+V-2+W=0 (c) Let the se… • Postpaid plans to linear equation in two variables • Proposition 4.1.11. Exercise 4.1. For each of the following collections of vectors, show that either it is a span of finitely many vectors (so it is a linear subspace) or that it violates the conclusi… • . Identify the following matrices whether it is a singular or non- singular. 2 3 -2 0 N 2. N w 3. w 3 Identify the following matrices whether it is symmetric or skewsymmetric. N 6 7 4. 6 -2 3 7 3 0 … • . Find the solution set of the following system by means of matrices 1. 2x – 3y = -1 x + 4y = 5 2. xty =2 2x – Z =1 2y – 3z = -1 3. x – 2y + z=-1 3x + y – 2z = 4 y – Z= 1 4. -4x + 2y – 9z = 2 3x + 4… • . Find the minors and cofactors of all the elements of the given determinants/ matrix. 1. -2 3 2. [4] 2 3 3. M = 2 3 4 W 4 5 For the given determinant below, find what is being asked. N CT 8 Cofacto… • The function f(x) = = + bx, f(2) = 1 has an extremum at x = 2. Determine a and b. Is this point (2,1), a point of maximum or minimum on the graph of f (x)? • This is a MATH related question. Please don’t change the subject. Don’t give wrong answers, some tutors are giving incomplete answers. The answer should include 3 examples of similar problems with ste… • Hi there, would you solve with explanation that would be very helpful and appreciated. Thank you. I will give you a good feedback too.. Could a set of three vectors in R* span all of R? Explain. What… • Hi there if you can solve in details. That would be helpful. Thank you. I won’t forget to give you a good feedback too.. In Exercises 15 and 16, list five vectors in Span { VI, V2}. For each vector, s… • Nestor’s Galore charges PhP 1,500 and an additional rate of PhP 550 per week. â€¢ Jelai’s Rentals charges PhP 1,000 and an additional rate of PhP 750 per week. Your father wants to avail of the rental… • Please help me anyone!! Â Â . Problems 1. (30 points) Consider the following system of linear equations. 3d1 – 3d2 = f1 -3d1 + 5d2 – 2d3 = f2 -2d2 + 2d3 = f3 (a) Rewrite the system of equations in th… • How do you graph a line when you are given the equation of the line? There are three different methods: 1. Plotting points 2. Finding x-intercept and y-intercept 3. Use slope-intercept form of an equa… • Length: 2:00:00 Kendra Conley: Attempt 1 Based on the graph, what is the initial value of the linear relationship? (4 points) 5- ONE 3 -2 0 1) -4 (2) – 3 ( 3) /5 O 5 • Can someone show me how to do b)? Thank you.. 54. Planes. Find the equations of the following planes in both cartesian and (vector) parametric for (a) the plane through the point (1, 4, 5) and perpend… • Could someone help me with parts c&d)? Thank you.. 52. Lines. Write down the equation for the following lines in both vector and cartesian form: (a) the line passing through P(2, 1, -3) and parall… • Hello, can I get help with these two questions?. Exercise 1 Consider the following 1-form : w = (1 – 2x]) dx 1 – 2x142dx2. 1) Is there any function h such that dh = w? 2) Consider the following vector… • Hello, Can I get help solving these exercises?. Exercise 1 Consider the following 1-form : w = (1â€” 2.3%)!13’1 â€” 2$1$2d$2. 1) Is there any function In such that db = w? 2) Consider the following … • [-/20 Points] DETAILS TANFIN12 4.3.013. Use the method of this section to solve the linear programming problem. Maximize P = x – 2y + z subject to 2x + 3y + 2z s 8 x + 2y – 32 2 4 x 20, y 2 0, z 2 … • How do you graph the equation to find the angle? When I tried in desmos it did not work.. Subbing (2) into (1) 0 = 1606.32cosA – 1520.7cos(sin-1( 780-401.58sinA 380.175 )) – 87.7475 By graphing this y… • \I._.l J Exercise 4,-7. Deï¬ne the 3-vectors vâ€”sw-swatâ€”ti no two of which are scalar multiples of each other. Show in both of the following ways that the planes spacn(v, w) and span(v", w”… • . A matrix A is given. Determine if the homogeneous system Ax = 0 (where x and 0 have the appropriate number of components) has any nontrivial solutions. 2 A = 3 5 O Ax = 0 has nontrivial solutio… • need help with these two question regarding linear equationsÂ 1. Eight students in Ms. Reese’s class decided to purchase their textbook from two different sources. Some of the students purchased the … • publisher has order for 600 copies of a certain text from san francisco and 400 copies from sacremanto. the company has 700 copies in a warehouse in novato and 800 copies in a warehouse in lodi. it co… • . Q3. Solve the following system of linear equations using an augmented matrix. cty + 2z -1 2x + y + 3z 0 -2x + z 2 • Please answer and explain questions 4-8. Thank you.. 6. (four marks) The table shows the path of a projectile after it is launched, where x is the horizontal distance from the launch pad and h is the … • Please answer and explain these questions, thank you!. 3. (four marks) The path of a parabolic arch is given by h(d) = -0.025 + d, where h(d) is the height of the arch above ground, and d is the horiz… • Here is Prop 4.1.11 the question is referring to.. Exercise 4.1. For each of the following collections of vectors, show that either it is a span of finitely many vectors (so it is a linear subspace) o… • Without solving, determine which of the following systems have a unique solution (hint: find the determinant). (a) 6x + 3y = 0 4x + 2y = 0 (b ) (c) 2x + 5y = 1 + 2y =12. (2) Evaluate the followi… • . 4 â€”1 Exercise 6.2. Let ‘P be the plane in R3 through 0 spanned by v = \|:U:\| , andw = [â€”1]. 3 (a) Verify that an orthogonal basis for ‘P is given by {v,w’} for w’ = w + v E ‘P (explicitly… • Please help! Can you answer and show work for questions 4-8, please draw a graph if needed to find answer ( do not use excel or desmos) thank you!Â . 6. (four marks) The table shows the path of a proj… • Fill in the missing cells to create a proportional relationship. X Y 1 – Co Blank 1: Blank 2: • Suppose you are given the points (1, -2), (2, 5), and (3,0) in the ry-plane. You would like to find a quadratic polynomial, p(x) = a + bx + cr, that has a graph which passes through all three of th… • (4 points; 1 each) Are the following statements true or false? If the statement is true, then show this fact. If it is false, then provide a counterexample. (a) det (2A) = 2 det(A) (c) det(AB-1) = … • L110 3.LetA=101. U l â€” (a) (2 points) For which values of the constants a and b is the matrix A invertible? (b) (4 points) Use Gaussian Elimination to compute 14. (Tip: You may want to begin by swa… • Exercise 1 Consider the following 1-form : w = (1 -2×7)dx1 – 21 1X2dx2. 1) Is there any function h such that dh = w? 2) Consider the following vector field : fi = 20142 711 ar + (1 -251) 912 Compute w… • Consider an economy with three industries: I1, 12, and 13. Consumer demand is currently . 100 units from /1, . 75 units from 12, . 200 units from 13. Also suppose that . for every unit that /1 prod… • Consider an economy with three industries: I1, I2, and 13. Consumer demand is currently a 100 units from I1, 0 75 units from I2, 0 200 units from 13. Also suppose that o for every unit that I1 prod… • Find the expressions for the dimensions(length,width and height) of the rectangular prism with the volume of 10x+3x-4x • Let v1 and 12 be cigenvectors of the matrix A with corresponding eigenvalues , and 12. For scalars c1 and c2 define the sequence Th = c1X101 + c21202 for k = 0, 1, 2, …. Show that Ek+1 = Ark by sepa… • . Exercise 4.3. For v = and w = , find scalars a, b, c so that an(v , w ) = { [ ] ER’ : as + by (Hint: first find a candidate triple (a, b, c) by solving for b, c in terms of a using that v, w must … • Let A be an nxn real matrix. a. Show that the matrix AA’ is positive-semidefinite. When is it positive definite? b. Suppose A is symmetric. Show that A is positive definite if and only if every eigenv… • Consider the n-dimensional Euclidean matrix. i. Show that any set of n linearly independent vectors in the n-dimensional Euclidean vector is a basis. ii. For a basis, show that every vector is a un… • Need help with this. the first part especially. Â . Let R = M.(C). Let V = C" considered as a left -module in the natural way, i.e.. the action of a matrix A e M.(C) on a column vector x of lengt… • You are required to find any linear function which you can see in your environment.Â You need to mention the limitations of that function and the defined input and output of that function.Â example:… • Given L, a lower triangular matrix, prove, using mathematical induction, that In is lower triangular for any positive integer n. b. Given a sequence of lower triangular matrices L, , L2 , …, In ,… • Solve these problems using graphical linear programming.Â a. Maximize Â Â Â z= 4x 1 + 3x 2 Â Â Subjected to: • Professor Yataha needs to schedule six round-trips between Boston and Washington, D.C. The route is served by three airlines: Continental, Eastern,and US Air, and there is no penalty for the purchase … • Managed to find the eigenvalues and eigenvectors, but struggling with the principal axes of the probability density function.. A probability density function , p(x), x = (x1 2) , has a gaussian distri… • 2 Math 101 Linear Algebra Questions. 8. Using your work from the previous problem, find all solutions to each of the following systems. (This should be very quickly and easy.) a. 3×1 – 5X2 = 0 2×1 – 4… • 4 Linear Algebra Questions. 4. Consider the following augmented matrix. a. Using variables X1, X2, X3 write out the system of linear equations this matrix represents. b. Reduce the augmented matrix to… • 3 Math 101 Linear Algebra Questions. 1. Let matrices A, B, C, D, E, and F be defined by NN INO Compute the following quantities, if the operations are all defined. (Otherwise state why the quantity is… • Let A by an eigenvalue of an invertible matrix A. Sinee A is invertible, what can you say about A? Show that )F] is. an eigenvalue of 14â€”1- 7. Show that A is an eigenvalue of A if and only if A i… • Explain why an n Ã—n matrix can have at most n distinct eigenvalues. Hint : If you form a set of eigenvectors from distinct eigenvalues, what can you say about the set? • Suppose there are 50,000 residences in a town. For television viewing, the residents have three options: they can subscribe to cable/satellite service in the home, stream or download television online… • . 1. Consider the sets B = (x + x’,2 +x -4×2, 1 + x -2x} and C = {1 + 2x -3×2, 2 + 5x – 4x’, 1 + 3.x}. (a) Let E = {1, r, x } be the standard basis for P2 which is the vector space of polynomials of… • . 1. Consider the sets B = (x + x’,2 +x -4×2, 1 + x -2x} and C = {1 + 2x -3×2, 2 + 5x – 4x’, 1 + 3.x}. (a) Let E = {1, r, x } be the standard basis for P2 which is the vector space of… • Consider the sets B = (x + x’,2 +x -4×2, 1 + x -2x} and C = {1 + 2x -3×2, 2 + 5x – 4x’, 1 + 3.x}. (a) Let E = {1, r, x } be the standard basis for P2 which is the vector space of polynomials of deg… • I am having trouble with 42b). Could someone show me how to do it?. (a) 42. Parallel and Perpendicular Vectors. (a) Find two unit vectors parallel to the vector u = 2i – 4j + 4k. (b) Write the vec… • Could someone explain how to do (f). 35. Determinants by Cofactors. Evaluate the determinant of the following matrices using cofactors: 21 1 2 1 (a) [ 3 ] (b ) ( c) 30 -1 Co -1 4 5 2 2 3 4 5 2 4 2 0 3… • A certain tree has a height of 10 ft. Every year it grows by 2 ft. What is the height of the tree after 3 years? after 5 years? after 6 months? • Can someone show me how to do these. 34. More Determinants. Use row operations to evaluate the determinant of the following matrices: 1 2 3 2 1 1 (a) 1 3 7 (b) 3 0 (c) COIN NIH 0710 CO1H 00100 001 1 4… • Hi there i just need the answer for 42. And if you could solve in details that would be appreciated. Thank you. I will give you a good feedback too.. 42. Could a set of three vectors in R span all of… • Hi there if you could solve in detail , that would be helpful and appreciated. Thank you. I will give a good feedback too.. In Exercises 15 and 16, list five vectors in Span { V1, V2}. For each vector… • Given a system of the form â€”m1x1 + 362 = b1 â€”m2x1 + x2 = b2 where m1, m2, b1, and 192 are constants: (a) Show that the system will have a unique solu- tion if m1 9′: m2. (b) Show that if m1 =… • Solve math problem Using financial calculator solve the value of 2756 to the power of 1/15 • Write a system of linear inequalities to model the question.Â Â A camp counselor needs no more than 30 campers to sign up for 2 mountain hikes. The counselor needs at least 10 campers on the low tra… • I in…” I 2.11.1:LUdeoomposition. U 3 f5Â§4\|Â§\|.2343Â§23.q13;cï¬‚ V Given thai A is transformed into U by the elementary row operations below, complete E1â€” 1 and E171. 4 3 1 n+3 443 4 3 1 4441241… • CHALLENGE ACTIVITY 2.5.1: Inverse of a matrix. 376940.2343920.qx3zqy7 V Jump to level 1 – 2 1 0 A = -3 1 3 -3 -2 7 Ex: 5 A-1 = 1 2 Check Next Feedback? • Help needed Consider the following vector and matrix. y 16 {B} = 0 [D] = 15 E 2+r 0 Defined on the rectangle -2 < x < 2, -1 < y <1, where E is a constant. (a) Find g(x, y) = {B} "[DNB… • Please help (Linear algebra) Consider the following system of linear equations.Â Â . 3d] – 3d2 = fl â€”3d1 + 5d2 â€” 2d3 = f; â€”2d2 + 2d3 = Is (a) Rewrite the system of equations in the form [A]{d} … • 1. (i) Determine if the set T := RU {co}, with addition and scalar multiplication defined by vow : min(v, w) cov=ctu for all v, w E T and all c E R, is a real vector space. If it is not, then list… • Question Â I cant understand this problem The SVD algorithm can be used to… Â I cant understand this problem Â The SVD algorithm can be used to solve the Least Square problem introduced in the cl… • Can anybody help me with this.. Function Problem Functions A, B, and C are linear functions. X Some values of Function A are shown in the table to the right. W Function B has a y-intercept of (0, 3) a… • How do I do this?Â . (5) (20 pts) Let V be the real vector space consisting of all real valued func- tions on the real line. Determine whether each of the following subsets is linearly dependent or in… • Match the system type with the number of solutions: Dependent Choose… Inconsistent Choose… One Independent and Consistent zero Infinite • el ejercicio 2,9Â el 2,16 y el 2,6Â me podrian ayudar Â Â . LiE . 78% 11:01 a. m. K enz-hcke-vae G be Acrobat Pro Aruda Perse 116% – Herramientas Firma Besides its intuitive appeal, and its analogy… • A local travel agent is planning a charter trip to a major sea port. The eight day and night package Â»%$are for rouÂ»d trip, board and dging and selected tour options. The charter trip is restricted … • QUESTION I A system of three linear equations is given below: 2×1 + 4×2 + 2×3 = 8 3×1 – 2×3 – 4×2 = 2 5×1 + 3×2 + 5X3 = -1 (a) Write the system of the linear equations above in a matrix form A. x = b … • Prove vectors u, v E Cn parallelogram identity: \|lu + 2/12+ 1/2- \|12 = 21/2112+ 2/12112. Also draw level parallelogram with vertices: O. uv, utvER?. • What is the standard form for linear equations in two variables? • Question 1 The volume of water / litres in a tank at time t minutes is given by the relation 500 -10t for Ost <20 V= 3 300 for 20 < t < 40 900 -15t for 40 S t < 60 a. On the axes provided … • How are lies presented as Â historical truths in chapter 7and 8 of Gulliver’s travels part 3 • Let a, b E C. (i) Prove that if \|a\| < 1 and \|b\| < 1, then (5â€”!) 1â€”be <1. (ii) Prove that if either \|a\| = 1 or \|b\| = 1, then aâ€”b 1â€”be =1. • Prove that 21, 22, 23 EC are the vertices of an equilateral triangle if and only if ZI +. = 2122 + 2223 + 2321 . • Question 2. Is the collection of all polynomials p(x) of the form p(x) = a + x2 with a E R a subspace of P, the collection of all polynomials? • Question 1. Is the collection F of all polynomials p(x) of the form p(x) = ar with a E R a subspace of P, the collection of all polynomials? • . Prove that 21, 22, 23 EC are the vertices of an equilateral triangle if and only if ZI +. = 2122 + 2223 + 2321 . • How do I do this?Â Â . (2) (20 pts) Let V be a vector space containing vectors v1, V2, and v3. • Letter d is a part of the problem.. 1. (20 points) For each of the following cases, let S be the set of vectors (x, y, z) in R3 whose components satisfy the given conditions. Determine whether S is a … • math problems : Â . 1. Solve the system of equations using Gaussian Elimination by hand. Make sure you continue with the process until your augmented system in REEF form (Row Reduced Echelon Fomr). Wr… • Describe going back to school as a mother and full time employee using the five steps to critical thinking and explain in detail how it applies • Vectors (a) What does it mean to say that b is a linear combination of a, and a2? (b) Show how to write as a combination of OH N and • Let T E Â£(V) and let all,” .ek E V eigenvecters with corresponding distinct eigenvalues A], . . . ,Ah cf T. (1) Shew that if W E V is a “JP-invariant subspace, and if 111 + . . .+ tug E W, then a… • Consider a unitary one-qubit gate A . Prove that A âŠ— I = I âŠ— A â€ . HereÂ A â€ is the conjugate transpose of A. • Let A be a nonsingular n x a matrix and let A. be an eigenvalue of A. (a) Show that 2 0. • For this assignment you will need to submit 2 documents. A word document with the linear programs typed out, and your answers typed out, and an excel file with your supporting work. Â A political can… • Please help! Answer and explain, please show work for the questions 7-10, 23-25. Thank you!!!. 25. (four marks) Write an equation for this parabola. YA 8 4- (-4, 0) -8 8 X 4 18 (-8.0) (-1.5,-12.5). 7…. • Please answer and explain questions 16-22. Thank you!. 21. (five marks) The quadratic function y = x2 – 5x – 14 can be factored. a) Find the x-intercepts. b) Find the y-intercept. c) Fine the coordina… • Please answer and explain questions 8-10. Please do not use excel, want to know the proper equation to find the answer. Thank you!Â Â . 9. (four marks) The table shows the path of a ball thrown fro… • B a A Two right triangles each have one angle equal to 59.20. The two smaller sides of the first triangle measure 4.13 in and 7.25 in. The smallest side of the second triangle measures 28.4 in. What i… • The figure to the right shows two poles drawn with correct perspective relative to a single vanishing point. As you canâ€‹ check, the first pole is 1.5 units tall in the drawing and the second pole is… • Please answer the following questions related to analysis (riemann integration). Everything is provided.Â . Problem 2: Decide which of the following functions are integrable on [0, 2], and calculate t… • 19 02.08 Linear Functions Test Part One N/A / 180 0.00 % 60 Segment: 1 N/A • Need help with the right answer and explaining why please.. Where are all points whose x-coordinates equal their y-coordinates? Choose the correct answer below. O A. All points whose x-coordinates equ… • raise to m d) is lim m to infinity,. 1) Show that if j > 0 then the function f (m) = 1+ is an increasing function of m. m ) Show that if j > 0 then the function g(m) = m (1 + )) ~ – 1 is a de… • Let vi and v2 be vectors in an inner product space V. Show that • . 2. Let v, and v2 be vectors in an inner product space V. (a) Is it possible for \| (v1, v2) \| to be greater than Ivill v2 1/? Explain. • The base vectors e; span a three-dimensional space (indices = 1, 2 or 3) as shown in the sketch. This is a non-standard coordinate system in which the magnitudes of e, and es are both 1, but the magni… • LetA2= [â€”2 _1] 2 â€”1 Let A3 = [â€”2 1 ] â€”2 â€”3 In the blank below, enter the 2mI row, 1St column entry for the matrix A. Answer:. â€”2 â€”1 Let A2: [ ] 2 â€”1 Let A3 = [â€”2 1 ] â€”2 â€”3 Which … • Continuing with this same question, if this week, there are 13,000 people who are sick and 35,000 people who are well, how many people were sick in the preceding week? (Enter the number with no spaces… • This week we learned a shortcut formula for the inverse of a 2 x 2 matrix. a b Specifically, if A = , then C d d – b A-I ad-bc -C a Find the Inverse of the given matrix: After you have found the inver… • When you subtract the matrices: 2815 344-}42 1-2 â€”30 The value of the 3rd row, 1st column entry is: Select one: Q 4 Q -2. Perform the following matrix multiplication: 0123-1’5 445022 1304-60 In the … • Using your final augmented matrix for this question, find an expression for y: Select one: O y = 5 – 3w O y = 5+3w O y = 2z + W. Consider the following system of linear equaï¬‚ons: x+yâ€”22+2w= 5 2x+y… • Given the function f(a:) = 35in (cc â€” 1) 4 a. Find the average rate of change of f(a:) from m = 117â€”2" to m = 37", rounded to three decimal places. 171r b. Approximate the instantaneo… • Question 2, please explain in details your answer. O N 09 M O 2. Find the matrix for a 120 rotation about the axis defined by the vector r = (1,1,0). 3 M • For each of the following functions k(a), find two functions f(x ) and g(a) such that k(a) = (f o g) (a). i. (a) = (4x – 1) 3 ii. k(x) = 6x – log(3x – 1) ii. k(a) = sin (a – T) • Could someone help me with 47g?. 3.3 – Cross product in R (AR $3.5) 43. Dot and Cross Products. Let u = (3, -1, 4), v = -i – 3j +k, w = (-1, 1, 2). Find, if they exist: (a) u . v (b) (3u) . (-2v) ( c)… • Consider the functions f (ac ) = ac + 3 and g(ac) = \|ac – 21. a. Identify the intervals on where (f . g) (a) 2 0. b. Sketch the graph of y = (f – g) (x)- c. Identify the intervals where (20) 2 0. … • Given that f(ac) = lac + 21, g(a) = \|x – 31, and h(x) = (f + g) (a) a. Draw the graphs of y = f(x) , y = g(x), and y = h(@) on the same grid. b. The function f (a) , written in piecewise form, is g… • Hi I need help for Question 6 (a),(b) and (c) . Kindly provide step by step solution with explnation . Thank you in advance .. 6. Let A be a 3 X 4 matrix. Suppose that$1 = 1, x2 = 0, x3 = â€”1, x4 = … • Use Gaussian Elimination to find all solutions to the following system: x – 3y+z = 5 – 2x+7y-6x =-9 x – 2y – 3z = 6 The final pivot should be: 1 0 -11 8 0 1 -4 0 0 0 0 The row of zeros in the last row… • Hello, i’m havong hard time solvong this. Can’t come the the right answer. Knowing that: vector u = (0,3,8) vector v = (2,-1,1) vector w = (-4,11,22) express the vector w as a combination of vectors u… • Graph The system of Inequalities. y> 4 x – 2 4 3 -3 + 5y <-2x 2 1 -4 -3 -2 -1 1 2 -1 -2 -3 -4 • Graph The system of Inequalities. y> 4 x – 2 4 W 3 -3 + 5y <-2x 2 -3 -2 -1 0 1 2 3 4 -1 -2 -3 Pear Deck Interactive Slide • Please answer and explain the following questions. 19. (four marks) For a science experiment, a projectile was launched. Its path was given by h(d) = -4.9d’ +55d + 100, where h(d) is the height of the… • Consider the linear system: x – 3y + 42 =1 4x -10y +10z = 4 – 3x + 9y – 5z = -6 After you have finished pivoting and can read the answers off the "b" column, the solution to the system is:. … • -1 12 20 Continuing with this same question, where you start with the matrix: –1 O 5 after one 12 6 pivot you would have: 0 a C 1 d e. 12 -20 the first thing you would do is rearrange the rows Given … • You can only multiply two matrices together if the number of in the first matrix is the same as the number of in the second matrix. The resulting matrix will have dimensions m x n, where m is the numb… • 3 Cauchy-Schwarz inequality for complex vectors. Extend the proof of the Cauchy-Schwarz inequality on page 57 of the textbook to complex vectors, ie., show that \|ba) < \|a\| \|\|b\|\| holds for all com… • Both please. se stion 17 is vet The correct sequence of steps to transform fox) = (= ) to g() – – 4/120x+ ered ed out of Select one: O a. vertically stretch about the x-axis by a factor or 4, reflect … • Central Airlines would like to determine how to partition their new aircraft serving Boston-Chicago passengers into first-class passenger seats, business-class passenger seats, and economy -class pass… • Given two lines: L1 ={a+tb\|tâˆˆR}, L2 ={c+sd\|sâˆˆR}. Â The vectors a, b, c, d are given n-vectors with b != 0, d != 0.Â Â The distance between L1 and L2 is the minimum of \|\|(a+tb)âˆ’(c+sd)\|\| over a… • Linear Programming. 1. We Heart Desserts Bakery sells cookies and mini cakes. For each batch of cookies sold, they make a $35 profit. For each batch of mini cakes sold, they make a$50 profit. The tab… • Linear Programming. 2. Several students are earning extra money detailing cars and doing yard work. It costs $6 in supplies and takes an average of 8 hours to detail a car. It costs$3 in supplies and… • The sum of two numbers is 88. One number is 4 less than 3 times the other. Find the numbers. • . Let V1, . .., VK be vectors, and suppose that a point mass of m1, . .., my is located at the tip of each vector. The center of mass for this set of point masses is equal to V = miv 1 + . . . + … • . Let V1, . .., VK be vectors, and suppose that a point mass of m1, . .., mk is located at the tip of each vector. The center of mass for this set of point masses is equal to V- miv 1 + . . . + mkk … • After solving the problems, please clearly indicate the axioms used in your proofs, as we are required to provide sources for them either from the text or from the internet. Thank you!. 5. For u, v E … • Find the distance from the point P0=(âˆ’4,1,6) to the plane âˆ’2x+3y+4z=2. • Find the linear equation of the plane through the origin and perpendicular to the vector [-2,1,-5] • For the following system, design a state feedback controls using estimated states X1 = X2 2 = 2×1 + 3×2 + u y= x1+ x2 (a) Before you design the state feedback gain and observer gain, write down th… • Data below shows the measurements of sugar levels and BMI for each patient in a hospital. Â Â Â Â Â Â Â a. write the linear regression model between sugar levels and BMI. b. What is the degree o… • Grades Exercise a A basket ball has the shape of ( r … H ) asquare . It’s -perimeter is 368 meters What is the length of its side in meters A garden has the shape of a square calculate the perimet… • Hello, Please I need help with this question. Can someone assist me with it ? Thanks!. 2. Let v, and , be vectors in an inner product Space V. 9 ) Is it possible for ( <V,iVar ) to be greater than … • Can someone please show me how to solve these?. 17. If g (x) = 15 -3 then algebraically determine the solution to the equation g (x) = 22. 18. A bank account’s worth can be modeled using the formula w… • Can someone please show me how to solve these?. Free Response Questions 14. On the grid shown below, the graph of f (x) =2" is shown. (a) On the same graph grid, create an accurate sketch of this… • Can someone please show me how to solve these?. 1 1. The water level in a draining reservoir is changing such that the depth of water decreases by 7.5% per hour. If the water starts at a depth of 45 f… • Can someone please show me how to solve these?. 6. Which of the following could be the equation of the graph shown below? (1) y = 10(0.5) (2) y = 3(0.75)* (3) y=4(1.25)* (4) y = 5(2.2)" 7. Selec… • 5) Sam’s DVD purchases over the same 10 weeks are shown in an input-output table. Does the table represent a linear or nonlinear function? Explain. Number of weeks, x 0 2 6 Sam’s DVDs, y 10 14 22 38 6… • 7:05 PT-Quarter-2-Week-1-FINAL (1) – Saved . . . Second Quarter Week 1 & 2 PERFORMANCE TASK 1 Mathematics 8 NAME: SECTION: Problem Solving Involving Linear Inequalities in Two Variables A. Cite si… • (4 marks) Consider the matrices 53 -5 3 2 4 A – 2 B – 2 -10 C – -1 5 2 2 -4 1 5 where matrices B and O have been obtained from A by applying a single row operation. Find elementary matrices E, and… • (8 marks) For each of the statements below, state whether the statement is true or false. If it is true then give an example. (a) There exist vectors v, w E R5 such that \| \|v + wil > llvl\| + \|wll. … • a (3 marks) Consider a linear map T: R – R. If T T and then what is T (b) (5 marks) Can the maps below be represented by a matrix, and if so what is the size of the matrix? You do not need to justify … • (2 marks) Let A E MA,7(R) and C E MAs (R) such that there exists a matrix B with AB – C. What is the size of matrix B? (b) (6 marks) Find the inverse of the matrix 6 4 • (4 marks) Let x = (1, -3, 2, 5) and y = (0, 2, -1, 1) E R . Find the angle 0 between x and y. (4 marks) Let z = 2 – 2v3i E C. Find 26 in the form a + ib where a, be R. • 2:49 PM O O LTE 4G 4 65 (292 kB) Pre-algebra Multiplication & sub. IMPORT Name: Class: ALGEBRA Solve for the variable. 1. 33 – 5y = 8 2. 3 = 11 – 8y 3. 7y – 7 = 28 4. 9y – 7 = 56 5. 8y – 7 = 25 6…. • (4 marks) Is T, : R’- R’ defined by Ti(r. y. 2) = ( z . y – 2) a linear map (ii) (3 marks) Let T, : R – R be a linear map, with ker T2 – { (r, 0, 0, -TO) :ZE R. Is T2 surjective? (iii) (6 marks) L… • .(5 marks) Show that V – {(z, y,a +y, -y) : z, y c R} is a linear subspace of RA. ii) (4 marks) Find a basis B of the subspace V from part (i), explaining your answer. iii) (8 marks) Show that the… • (i) (3 marks) Calculate the determinant of the matrix A – 3 by using elementary column operations to reduce it to triangular form. (ii) (7 marks) Consider a function f: R x R x R -> R which is m… • (6 marks) Solve the system of equations 2 + 47 + 35 – -2 2.c + 9y + 52 – -7 Ar + 64 + 93 – 9 using Gaussian elimination. (ii) (8 marks) Show that if a system of linear equations has two solutions … • here is the question – please make the slove clear. 5. (5 points) Find the coordinates of vector Q, if it’s length is 75 and it is parallel to vector P=(16,-15,12). • here is the question – please make the slove clear. 6. (3 points) Write down the equation of line, passing through the point M(2,4) being perpendicular to the line 2x+4y=5. 7. (5 points) Find the dist… • here is the question – please make the slove clear. 2. (5 points) Determine if the columns of the matrix form a linearly independent set. Justify your answer. 4 -3 O 0 -1 4 0 3 5 6 • here is the question – please make the slove clear. 3. (5 points) Find A, if the matrix A is -2 12 -1 5 Apply A = PDP representation. • here is the question – please make the slove clear. 1. (15 points) Solve the following system using :a) Gauss-Jordan algorithm, b) the inverse of the corresponding matrix(apply the method you prefer),… • Let [V1, . .., Vn] be a basis for a vector space V, and let Li and L2 be two linear transformations mapping V into a vector space W. Show that if Li(v;) = Ly(v;) for each i = 1, …,n, then L1 = L2; t… • Let L be a linear operator on R’ such that L(1) = a. Show that L(x) = ar for every r e R . • Help me here please kindly answer with clear explanation thank you. • Problem 3 [ 9 points ] Let L be the language over the alphabet )3 = {a,b,c,g} that contains ex- . . INSTRUCTOR’S SOLUTION: actly those strings whose form 1s: 1: (b) Proof that L is not context free:… • How much do you have to deposit today so that exactly 10 years from now you can withdraw $10,000 a year for the next five years? Assume an • Hello, Can someone please help me with this question? Thank you !. 1. Let (c) Verify that the Pythagorean Law holds for X, p, and x – p. X = and y = ONAN NNEP (a) Find the vector projection p of x ont… • Hello, Can someone please assist me with these questions. I only need help with 1,2 and 4. I have done the rest on my own. Thank you!. True or False In each of the following answer true if the stateme… • In mixing aâ€‹ weed-killing chemical, a 25â€‹% solution of the chemical is mixed with a 75â€‹% solution to get 50 L of a 70â€‹% solution. How much of each solution isâ€‹ needed? • Thanks for your help.. Solve the initial value problem y" – 5y’ -6y = 0, y(0) = a, y'(0) = 4. y (t) = Then find a so that the solution approaches zero as t – co. a = • Thanks for your help.. 3. [-/1 Points] DETAILS BOYCEDIFFEQ10 2.6.010.GO. Determine whether the equation is exact. If it is exact, find the solution. (If the equation is not exact, enter NOT EXACT.) y … • Thanks for your help.. 2. [-/1 Points] DETAILS BOYCEDIFFEQ10 2.2.004. Solve the given differential equation. y’ = 11×2 – 1 11 + 4y Submit Answer • Need help with this question. Thank you :). Question 2 (25 marks) Modelling our 5 x 5 x 5 Rubik’s cube The following pictures show a curious Rubik’s-cube-like structure that greets visitors to the Mel… • D14: How do we determine if a relation is a function? Explain how to determine from a graph if y is a function of x. Explain how to determine from an arrow diagram if y is a function of x. Explain how… • A company is producing three types of liquid products: P1, P2, and P3. The production contract is arranged for the next month and assumes the delivery of 5000 litres of the products to a customer. The… • Let V be the set of functions f: R â†’ R which satisfy the differential equation f” – f = 0. Show that V is a vector space over R and, assuming its dimension is 2, find a basis for V. • Please answer and explain 11-15. Thank you!Â . 11. (four marks) Find the minimum value of y = 3x + 12x+ 11 by completing the square. 12. (four marks) Find the maximum value of y= -2x- 16x – 29 by comp… • Please answer and explain 1-10. Thank you!. 1. (two marks) The value of k that makes the expression x + 8x+ & a perfect square trinomial is 2. (two marks) The value of k that makes the expression … • Consider the function h : (0, e2) -> R given by 018 if x E (0, 1 ) h(x) = In x if x e [1, e2 ) where e is the base of the natural logarithms. (i) Does h’ (1) exist? Briefly explain your answer. (ii… • Can you help me please. Problem Solving Involving Linear Inequalities in Two Variables A. Cite situation in real – life where system of linear inequalities in two variables are applied. Formulate prob… • (30 points) Find the solution of the following system of difference equations using the Z-transform method: x [n] 0 x[n – 1] on [a [-1 y [n] = y[n – 1] + where y – 1 = [n] CO 3 zn – 1] Z O • DISCLAIMER: Please don’t just z-transform and leave it there. It says find the solution. So the answer needs to be like x[n] = e^^….. , y[n] = ……, z[n] = …….. Â . 2. (30 points) Find the sol… • Could someone show me how to do d)f) g)I). 43. Dot and Cross Products. Let u = (3, -1, 4), v = -i – 3j +k, w =(-1, 1, 2). Find, if they exist: (a) u . v (b) (3u) . (-2v) (c) uxv (d) (v xu) . (-… • I’m stuck in the question solve this question quickly. Al. (a) 47 What is a symmetric matrix? What is a skew-symmetric matrix? (b) Given any square matrix A, argue that -(A + A") is a symmetric m… • I’m stuck in the question solve this question quickly. A5. (a Write down the matrix A that represents the linear transformation T; that takes the vector [x, y, z] to the vector [x + y, x – 2y – z]]. (… • For the following system: 2 = -2×2 + 0.lu y = x1 + 0.1×2 (a) Please find the controllability matrix and observability matrix of above system. (b) Please check if the system is controllable and/or … • min z = x, + 2×2 2 X3 s.t. X +x, 29 x, + 5x, 215 2x, + x,$16 X 1 , 2 , X, 20 Given the above linear programming model. Find the optimal solution by using Big nethod. • Question 6 Find the numbers of the pivots to the matrix below: 0 2 3 1 1 2 1 2 â€”1 0 1 4 4 4 â€”3 11 02 04 @3 01 • Question 7 Find c such that the number of the pivots to the matrix below is 2: 0 1 c 2 0 2 1 1 â€”1 • Question 8 Given the augmented matrix below, find the solution set. [o 1 1 ) 1 1 1 0 -1 1 0 2 1 Then (21, 2 2, 203 ) = • proof both lemmas.theory of module exact sequences. THE THREE LEMMA Let the following diagram of R-module and their homomorphisms B C B K M. IN be commutative with two exact yous them ( i) If x, 8 and… • Which of the matrices that follow are in row eeh- 2. The augmented matrices that follow are in row elon form? Which are in reduced row echelon echelon form. For each case, indicate whether the form… • Let S1 and S2 be subgroups of (Z, +). Prove that S1 âˆª S2 is a subgroup of (Z, +) if and only if S1 âŠ‚ S2 or S2 âŠ‚ S1. • Hello, can I get help with knowing if this True or False. Any possible explanation would be appreciated, thank you! Â 1) If A is a 3 Ã— 4 matrix and vector V is in R 4 , then vector AV is in R 3 . ?… • 4. 5. Use back substitution to solve each of the following systems of equations: (a) x1â€”3×2=2 (b) x1+x2+ x3=8 2×2=6 2X32+ x3=5 33C3=9 (c) x1+2×2+2×3+ x4: 5 3×2+ x3â€”2×4â€” l â€”x3+2×4=â€”l 4x… • linear programing problems. 3. A junior student earns $20,000 during her summer internship. Since she plans to pursue her graduate study next year, she wants to invest all her earnings in an annual ta… • Need help with question 14.. 14. Let 75 = {0, 1,2,3,4} together with addition and multiplication modulo 5. (a) Prove that every non-zero element of Z5 has a multiplicative inverse (Z5 is a field): for… • kindly correctly and handwritten. 4. (25 points) Solve the following system of equations using LU decomposition method: X1 + 12 + 23 =1 (c+1)x1 + (c+2)12 – 13 = 6 (c+ 1)x1 + 2×2 + 3×3 = 4 . If LU fact… • kindly solve this correctly and handwritten. 4. (25 points) Solve the following system of equations using LU decomposition method: T1 + 12 + 13 = 1 (c+ 1):1 + (c+2):2 – 13 = 6 (c+ 1)x1 + 2:2 + 313 = 4… • Do both parts: a) Evaluate the following real integral: – 8 1 dx a (x – 2) (x2 + 4 ) Roughly sketch the contour over which you are actually integrating and the location of the poles. b) Now assume t… • Evaluate the following real integral using residue theory: 2 TT de 6 + sin(0 Please show ALL details. • Please help me with this study question. Thanks!. (a) Let z = x + iy. Write (1 + 2) z in terms of a and y. (b) What is Re((1 + 2) z)? (c) Using the fact that Jez \| = eRe(2) , describe (or draw) the fo… • Consider the production model x = Cx + d for an economy with two sectors, where C = os 07 and a = so Itcan be shown that the production levelnecesary 115 to satisfy the final demand is x = Complete pa… • 10 6) y =- -+3 7) y = – 5 X+4 x+7 VA: HA: VA: HA D: D: R: Compare: Compare: 8) MC: What are the asymptotes of the graph of y – 3 –3 ? x +8 a) x = 8, y = 3 b) x 8. y = -3 c) x = -8. y =3 d) x= -8. y =… • The function y = 7 + 3 has aln) of all real numbers except 3 and a (n) X +4 of all real numbers except -4. For problems 2 – 7, graph the function (you should have a RST tablel). State the domain an… • Linear Algebra Question Precisely motivate your answer means to show working out., thanks.. 1. (a) For every number t, determine the solutions of the linear system -T1 C3 + 2 -I1 + 212 + 13 + TA = + T… • Which of the following matrices are stochastic and/or unitary? 0 1/3 2 -1 A = , B = 1 2 / 3 , C= 1/2 1/2 , D = ( N. E = 3/5 -4/5 F = 3/5 -41/5 H 4/5 3/5 12/5 3/5 • Week 1: Practice sketching in 3D. In R3 (so using three coordinate axes) sketch the following, making a new sketch for each part (i)-(v): (i) a horizontal plane, (ii) a plane which is not horizontal, … • Hello, Can someone please assist me with these three questions. I do not need help with question number four since I have already answered it on my own. I need help with questions 1, 2 and 3. Thank yo… • Hello, I need assistance answering this question. Can someone please help me? Thank you very much!. 7. Given the table of data points X -1 1 2 y 1 3 3 find the best least squares fit by a linear funct… • Hello, Can someone please help me with these 3 questions. Thank you and I truly appreciate the help.. 3. Let vi and v2 be vectors in an inner product space V. Show that 4. Let A be a 7 x 5 matrix with… • please solve this correctly as the previous tutor did it wrong. 4. (25 points) Solve the following system of equations using LU decomposition method: T1 + 12 + 13 =1 (c+ 1)x1 + (c+2)12 – 13 = 6 (c+ 1)… • Hello, I need assistance with these two questions please. There is a continuation for question number 2. I will also attach it. Thank you for the help!. 1. Let (c) Verify that the Pythagorean Law hold… • Please answer and explain 1-16. Thank you!. 6. (four marks) The vertex form of y = x* + 8x + 5 is 7. (four marks) The vertex form of y = x – 6x + 13 is 8. (four marks) The vertex form of y = -x* + 12x… • Please answer and explain 1-20, Thank you!. 1. (five marks) When Atif rides his dirt bike off a ramp, his path can be modelled by h(d) = -4.9d’ + 25d + 25, where d is the horizontal distance from the … • Please answer and explain 1-8. Thank you!. 6. (four marks) What are the x-intercepts, to the nearest hundredth, of y = 4x + 5x – 6? 7. (four marks) What are the x-intercepts of y = 2x + x+ 1? 8. (four… • please correctly and handwritten. 4. (25 points) Solve the following system of equations using LU decomposition method: 21 +12 + 13 =1 (c+ 1):1 + (c+2)12 – 13 = 6 (c+ 1)x1 + 2×2 + 313 = 4 . If LU fact… • 42W7: The Equation of a Circle Name: Gr. and Sec.: Score: Parent’s Name: Parent’s Signature/Date: WRITTEN WORK 50% / PERFORMANCE TASKS 50% Directions: Solve, then graph the following problems. 1. 8(1…. • Hello, I need assistance with question number 7. Thanks for the help! The directions for the question is: For the statement that follows, answer true if the statement is always true and false otherwis… • Hello, I need help with these three questions. I crossed out questions 2 and 4 because I have already answered them on my own. Thank you for the help. I appreciate it!. For each statement that follows… • 4) Zachary runs steadily at 10km/hr. His distance can be modelled by the equation y = 10x, where y gives his distance (km) in relation to time (hours). The kidnapper runs at 5km/hr (since he’s carryin… • Q No 1. Let ð‘“: ð‘€ â†’ ð‘ be an R-homomorphism. Then if ð´ is submodule of ð‘, the ð‘“ (ð‘“ï»¿ âˆ’1ï»¿ (ð´)) =ï»¿ ð´ â‹‚ ð¼ð‘šð‘”(ð‘“). Q No 2 Let ð‘“: ð¿ â†’ ð‘€ and ð‘”: ?… • A hot liquid starts at 97 Â°C. Its temperature decreases at a constant rate of 8% per minute after it has been placed in a refrigerator. (a) To the nearest degree, what is the temperature temperature … • Explain how you can see on the curve what kind of logarithm it is.. (I) Forklara hur du ser losningarna till ekvationen i uppgift (c) i bilden. 45.18 (a) Rita kurvan y = log15(x) for hand. (b) For… • Proof by induction that 2 + 4 + 6 + 8 + … + 4n = 4n^2 + 2n for positive integer n. Show all your steps. • . BASIS of Vectors My Solutions > A given set of vectors is said to form a basis if the set of vectors are both linearly independent and forms a spanning set for the given space. In this exercise… • Solve the following systems of equation. 6x + /=9 1. 6 x + 2y = 2 2x – y-3 2. 3x + 4y = 10 5x+ y =-1 3. 6 x + 2y = 0 4 1/r – 8/t = 6 2/r – 9/t = 6 (Hint: let x = – and y = – ) • Show that if M and N are Vector subspaces of a Vector space X then M and N is also Vector subspace of X • Exercise 3.10. Up to parallel translation, a plane in R3 is specified by the perpendicular line to the plane. Hence, we can define the "angle" between two non-parallel planes as the nonzero … • I’m trying to solve and understand the steps for the following question • CAN SOMEONE EXPLAIN ME THE ANSWERÂ FROM LINEAR ALGEBRA I PROMISE I WILL UPVOTE IT . • kindly solve this toughest question . i am also giving you chapter link for reference. if you can solve this , i will give you link chapter – 7.7 and 27 problem Â https://drive.google.com/file/d/1quR… • Please answer and explain 1-16. Thank you!. 1. (two marks) The value of k that makes the expression x + 8x+ & a perfect square trinomial is 2. (two marks) The value of k that makes the expression … • Bond Brothers, a real estate developer, builds houses in three states. The projected number of units of each model to be built in each state is given by the matrix. Model I II III IV NY 60 80 120 4D A… • [-5x – 2y – 2z = 1 1) Find the solution of the system -X+ y = 2 . using Gauss-Jordan method of inverse of a matrix. – X + Z=-3. 3} When solving a system of linear equations with the unknowns x, y, and… • Math subjectÂ 1. In P(x) =2xâ´+4xÂ²-x+13 ,what is 13?. A. Directions: Analyze and choose the appropriate answer from the giver options. Write the letter of the answer on the space before each number… • Could someone how me how to solve this with concepts related to determinants? Thanks. 39. Idempotent Matrices. A matrix P is called idempotent if P2 = P. (a) If P is idempotent, show that either det P… • Could someone show me how to do this? I had trouble simplifying the terms.. 37. Properties of Determinants. Verify by direct calculation that if A b d then: (a) detA = det(AT) 1 (b) det(A-) = Jet A In… • hi can you help me with this question?. 8) Line 1 is shown on the grid below. 10 PART D: Communication Find the eguation of the Line 2. in the form 31 = mx + b, so that: 0 Line 2 passes through the po… • What is the distance between P(â€”1, 4) and the line 49: + 3y + 4 = 0? 2. Recall that for any two or threeâ€”dimensional vectors Â§ and E, any = llwllllyIICOSW) Where 0 is the angle formed by the v… • NEED ANSWERED ASAP, next 10 minutes speed please. 3. Evaluate the following using exponent and logarithm rules. Show all work for full marks. 1082(4 X V4) – 3 083(3 -) + 2 1084 76 • need answered fast, asap, next 10 minutes. 1. Given the function f(x) = (4)* a) State the equation of the inverse of the function as y =_ b) Sketch the graph of the function f(x) and its inverse on th… • need asap please. Let V = M22 with inner product < A, B >= 2011bl1 + 012612 + a21621 + 2a22622. If A then \| \| All = 4 N Select one: O V30 O 30 O 35 O V35 • A travel pillow company must determine the production plan for the next production cycle. The company wishes to manufacture at least 300 pieces of each of the three models it offers and no more than 1… • . Killua Premiums .unlock services . X Course Hero X M Inbox (45) – morenojc0912@gma x + stocks – Google Sheets X + V X > C A twitter.com/messages/1322694092027879424-1453617894420467714/media/14… • Math subject 1. Which of the following is not a polynomial function?. 32. If an inscribed angle of a circle intercepts a then the measure of the angle is 90. A. major arc B. minor arc C. segment D. se… • CAN SOMEONE ANSWER THIS QUESTION FROM SETS I PROMISE I WILL UPVOTE IT . • CAN SOMEONE ANSWER THIS QUESTION FROM SETS I PROMISE I WILL UPVOTE IT , • CAN SOMEONE ANSWER THIS QUESTION FROM SETS I PROMISE I WILL UPVOTE IT , LINEAR ALGEBRA • need asap please. 4) Solve the following systems by Gauss-Jordan elimination. a) 2x +2×5 +2x= 0 -2x +5x +2x, =1 87 + +41 =-1 b) 2x – 3x, = -2 2x, + x, =1 3x, + 2x, =1 • Please answer this question in excel format. not handwriting. Linear programming using excel.. 2. The goal of the task: It is necessary to apply the method of sociometric modeling of the ranks of prof… • please asap and correctly handwritten also. 4) Solve the following systems by Gauss-Jordan elimination. a) 2x +2×5 +2x= 0 -2x +51 +2x =1 8 +15+4 1 =-1 b) 2x – 3x, = -2 2x, + x, =1 3x, + 2x, =1 • please asap and correctly. 5 -10 3) Let A= Find all values of a such that A? = A. a • please asap and handwritten also correctly. 2) Consider the matrices 5 5 2 O A = 2 1 ). B = ( 3 2 ) . C = ( 1 3). D= 0 1 2 -1 Compute the followings (where possible) a) CA b) AD C) AT +C • need asap and handwritten. 1) a) Find the 2×2 matrix D – a b c d satisfying D.[ ]- and D. [ ]-13 b) Compute the following multiplication [1 -1 0] -1 5. 2 • kindly tell me the correct answer. (a) If A is a 3 x 5 matrix, what is the largest possible rank of A? (b) If A is an m x n matrix, what is the largest possible rank of A? The largest possible rank is… • Show that the space V = {(x1, X2, 23) ( F3x1 + 2×2 + 2×3 = 0} forms a vector space. • My values are a=8 ,b=-2. ART A: The point A(x,y) will be emailed to you after the break. iven the point A(x,y) answer the following questions: 1. Create a system of linear equations with point A(x,y) … • 1)(Supply Demand Model, 10pt) Your supply/demand model is based on the following two equations: Â s = 23.7 + 10.6 âˆ™ pÂ d = 1712.7 – 13.2 âˆ™ pÂ a) [5pt] Find a solution for this supply/demand mode… • how do i do this ?. 8 Solve the linear system (find x, y, and z). 6 z = 2yx + 3y + 2z =- 115x – y + 8z = 5 • 1)(Supply Demand Model, 10pt) Your supply/demand model is based on the following two equations: s = 23.7 + 10.6 âˆ™ pÂ d = 1712.7 – 13.2 âˆ™ pÂ a) [5pt] Find a solution for this supply/demand model … • The augmented matrix of a linear system has been reduced by row operations to the form shown. Continue the appropriate row operations and describe the solution set of the original system. 1 6 3 -4 0 0… • Exercise 2.10. Let v and w be two nonzero n-vectors. (b) Show _ – _ – if v and w have the same length then v + w and v â€” w are perpendicular. • how do i do this. 5. Olivia has$6000 to invest. She invested a portion that earned 6% interest per year and the remainder at 8% Interest per year. She invested $3000 more at 8% than she did at 6%. Ho… • My LRN is: 108043100007. Instruction: All problems will require data coming from your LRN. The numbers to be used from your LRN are the fifth, sixth, tenth, eleventh, and twelfth numbers, designated a… • 1 -3 – 12 8 If A= and AB = 3 5 1 16 5 . determine the first and second columns of B. Let b, be column 1 of B and by be column 2 of B. by =. 3 – 12 Let A = . Construct a 2 x2 matrix B such that AB is t… • Show that (1) holds.. Consider a matrix U = [u1, u2, …, Un] with column vectors ui E R" and another matrix V = [v1, V2, . . ., Un] with column vectors vi E RP. In class, I claimed that UVT = u1… • Let Â x , y Ïµ R ( n ) . Show that d e t ( I âˆ’ x y T ) = 1 âˆ’ y T x . • I just need answer for what x equals. PLEASE WRITE ANSWER CLEARLY • Find the inverse of the function: f (x) = 5x-7 4) Graph the function. Determine whether the inverse of f is a function. Explain. a) (x) = =x -3 bj (x) = -2/x+31 +5 Is the Inverse a function? Is the… • Here is a table and graph of f (x) = 3x? – 1. x -2 -1012 a) Graph the inverse. (You can make a loble to help) b) Find the domain and range for f(x). R: c) Find the domain and range for the inverse. D:… • need short explanation and clear answer on how you got the answer. Given the matrices below and that A = LU, solve Ax = b. 1 5 -17 23 A = -3 -14 5 b = 67 -6 -35 -3 L 149 1 0 1 5 -1 C L = -3 1 0 U = 01… • Let L be a solvable Lie subalgebra of gl(V ), where V is a finite-dimensional vector space over C. Show that every element of L’ is nilpotent. Â (b) Let L be a finite-dimensional Lie algebra ove… • Explain the process you would use to write an equation of a line when given two points of a line.Describe the similarities and differences between point-slope form, slope-intercept form and general fo… • Give an example of vectors i R 3 that spans R 3.Â (Show work) • Rewrite the following linear system WITHOUT brackets or fractions. The final equation(s) should be written In the form Ax + By = C. Please DO NOT SOLVE. Equation 1: 2 5 Equation 2: 2(x (= 4)- ( + 3… • Express w- (3,6) as a linear combination of u = (2,1) and v (2, -1). • 3. Provide concise answers to the following questions about the practicalities of online learning. (i) Will your internet connection permit you to participate in synchronous lectures and tutorials… • 2. Provide short answers to the following questions. Some internet research may be useful. Remember to cite your sources. (i) In mathematics, what is the difference between an axiom and a theorem?… • how do i do this. Part B: Application 4. A local grocery store has cashews that sell for$6.50 a pound and peanuts that sell for $3.75 a pound. How much of each type of nut would Luca, the employee, h… • 1. Let n be an integer greater than 1. For any (n x n)-matrix A, let adj (A) denote the adjugate of A. Establish the following three equations. (i) We have det (adj(A) ) = (det(A) )"-1. (ii) … • kindly solve this correctly to get a like button instantly. Let u = [ ] andv – and let (u. v) = 24,v1 + 34,v2 be an inner product. Compute the folowing (a) (u, v) 18 ( b ) llu] 10 X (c) d(u, v) 61 X • need asap and handwritten. Let u = -2 and v =$ et ( w . v ) – and let (u, v) – 20. v. + 3u,v, be an inner product. Compute the following. (a) (u, v) -18 (b) lull V 10 X (c) d(u, v) V 61 X • please asap and handwritten. 3. Let D denote the differential operator, that is Df = . Each of the following sets is a basis of a vector space V of functions. Find the matrix representing D in each ba… • i need part d and e solutions but please asap. 2. Consider a linear operator H acting in a linear vector space spanned by vectors e, and e2. The following properties of the operator are known: He1 = e… • consider the vector space M22 of real 2Ã—2 matrices see the below picture. (c) Find the representation (coordinate vector) of the matrix v in this basis. 2 3 V = 4 -7 (8 marks) • consider the vector space M22 of real 2Ã—2 matrices see the below picture. (b) Prove that the following set of matrices, e1 = es = (6 1 ), a = (6 8 ) is a basis in M22. (8 marks) • thx for help. Consider the wave equation where m is a. constant and (b depends on the coordinates (t, :L’).. satisfy the relation w2 = m2 + k2.. b) The general solution of the wave equation is writt… • thx for help. b) The general solution of the wave equation is written in the form oodk $(t, x) = 2 TT p(k) exp(iwt – ikx) , • thx for help. b) The general solution of the wave equation is written in the form (Mm) = / gamma â€” was) , â€”00 where q; is a generic function of k. Show that w and k satisfy the same relation as in • thx for help. 1.3) The Cartesian coordinates (x, y) are expressed in terms of the curvilinear coordinates (u, v) as x = uv and y = u/v. The infinitesimal surface area in the coordinates (u, v) is A as… • thx for help. d) Assume m = 0 and derive an expression for the function o(t, x) when Q ( k ) = 0 if [k – pl > A if \| k – p<4 where p, A and A are constants. Determine the maximum amplitude of… • dk$(t, 20 ) = p ( k) exp(iwt – ikx) , 2 TT. c) Derive an expression for the function (Mt, r) when $013) 2 A(6(k â€” p) + (W: +130) , where A and p are constants. • Hello May you please provide me with the solutions of those problems. I WOULD NEED THE SOLUTIONS. The answer key is 1 – b; 2 – b; 3 – b; 4 – a; 5 – a; 6 – b; 7 – d; 8 – d; 9 – c; 10- c Thanks a lot in… • thx for help. Consider the wave equation where m is a constant and (,6 depends on the coordinates (t, x). a) Show that any function of the form (Mt, :3) = exp(z’wt â€” ikrc) is a solution of the wave … • Be sure to describe your findings in words. Explain your answers. Solve using Excel.. The subscript l tells us that the matrix gives the population after one year. The names of the matrices for subseq… • Use the following table to answer the next question # of Adults Frequency 00 N 15. The table above describes the number of adults living in each house on a city street. i. Mean of the data equals 2 ii… • Solve the following simultaneous equations using the method of substitution. 45 x + 58 y = -335 84 x – 2y = -94 Number y = Number • We have the following vectors in R3 V1 V2 V3 And f : R3 – R3 defined as f (x) = v1(v1, x) + v2(V2, x)+ 13(V3,x). Where (-, -) It is standard inner product. 1. Find the matrix A that represents f with … • It is translated so to english might be all correct : ). We have the following vectors in R3 1 2= 03 = -1 0 And f : R3 – R3 defined as f (x) = v1(v1, x) + v2(V2, x) + v3(V3, x). 0 1. Find the matrix A… • Find the general solution to the given system. 5×1 – 10×3 = 20 -4×2 = 8 X1 Ex: 6 – 3x_2 20 2 Enter a1 as: X_1 Enter 2 1 as: 2x_1 X 3 (Signs should be simplified, so answers such as 6 + – 3x_2 will not… • How do I do this?. A triangle has angles described as follows: The measure of the first angle is four more than seven times a number, the measure of the second angle is four less than the first, and t… • Hello, may you please help me solve those Arithmetic sequence problems?. ARITHMETIC SEQUENCE PRACTICE PROBLEMS 1. Find a, and d for the following arithmetic sequences: a1, 9 , a3, Q4, as, 29 2. Find t… • thx for help. Given the function g(x) = exp(-x2), the Fourier transform of the convolution f(x) = (g * g)(x) is O A. f ( k) = _ dx exp(-2×2 – ikx) O B. None of these answers O C. f ( k) = ([- dx exp(-… • thx for help. The Cartesian coordinates (x, y) are expressed in terms of the curvilinear coordinates (u, U) as x = cos(u) cosh(v) and y = – sin(u) sinh(v). The metric in the coordinate system (u, U) i… • thx for help. t is the time and x is a Cartesian coordinate. The function q = exp(iwot – ikx) , where w and k are positive constants, is a solution of the wave equation a C2 at 2 dx 2 where c and a ar… • thx for help. r is the radial coordinate in spherical coordinates and t is the time, and f(t, r) = T(t) R(r) is solution of the diffusion equation of = DAf. The functions T(t) and R(r) satisfy the fol… • thx for help. In spherical coordinates (r, 0, ), the Laplacian of ys = sin(0)eld is O A. None of these answers O B. Aw = Cos(20)-1 eid 72 sin(0) O C. Ay = sin(20)-1 eip 12 sin(0) O D. Ay = cos(20)-1 e… • Let matrix A ( I"x" with elements ajj. vx E R" is a vector whose entries are all zero except for the k-th entry, whose value is 3. Write down the result Avk. • Explain why the v1=(1 1 1)?? Â Â Â . To this end we calculate 2 Reduced A – 313 = 2 0 2 Echelon Thus our first eigenvector vi is the basic solution to this matrix, which is VII= We now find our seco… • . 1. Calculate f (3.25) using Newton’s interpolating polynomials of order 1 and 2. Choose your base points to attain good accuracy. [30] X 2 4 Y=f(x) 8.6785 56.7854 • . 4. [15 points] Compute the number of free parameters in the following models. In each case, assume the number of classes is C and the dimensionality of input features is D. a) [5 points] We f… • Solve it step by step.. 23 Let V = Pn p(z) = ER=opk* and q(2) = >=0 9Z, define (p. q) = Piq i+j+1′ ij=0 (a) Show that (, .) : V x V – C is an inner product. (b) Deduce that the matrix A = [(i + … • . Find the angle between the vectors v = (2,-3) and w = (3,2)T. 75 23 1.00 Let L : Rn – Rn be a linear transformation. If L(x1) = L(x2), then the vectors x1 and x2 must be equal. 186 53 1.00 164 In:… • Kindly solve this step by step. 33 Let A E M, be a Markov matrix. If A is normal, show that AT is a Markov matrix. • please solve this step by step , i am confused. 27 In this problem, we approach the theory of least squares approximation from a different perspective. Let V, W be finite- dimensional F-inner product … • Graph the following inequalities: Line 1: x + 3y S 21 Line 2: x + y$9 Line 3: 3x + y $19 Line 4: x 20 Line 5: y 2 0 (2) Shade the overlapping region (the feasible set) (3) Determine the coordina… • Every set of n+1 vectors in an n-dimensional vector space is linearly dependent • + Midterm 2 – C A Not secure \| system.na-edu.com/mod/quiz/attempt.php?attempt=3512&cmid=11995 N North America Education Online #x (zh_cn) " MATH 026-21-12 Linear Algebra MEN / WE / MATH 026-2… • SUBJECT: LINEAR ALGEBRA QUESTION: Solve every question on a separate sheet. The questions are placed in attachment below kindly refer them and provide best solution. Â . Q1: 1. Let R have the weighted… • SUBJECT: LINEAR ALGEBRA QUESTION: Solve every question on a separate sheet. The questions are placed in attachment below kindly refer them and provide best solution.. Q1: Find an initial point P of a … • Hi!Â I was wondering if I could please get some help with the following maths question? Thank you so much in advance for your help! Â Â . 3. (3 marks) Happy new year! Let f(x) = x2 cos(xx), to celeb… • Hi!Â I was wondering if I could please get some help with the following integration-related question? Thank you so much in advance for your help! Â Â Â Â Â . 4. Evaluate the following integrals: … • Hi!Â I was wondering if I could please get some help with the following maths question? Thank you so much in advance for your help! Â . 2. Let f(:1:) = â€”ln(1 â€” x). (a) (3 marks) Determine the l\?… • Hi!Â I was wondering if I could please get some help with the following maths question? Thank you so much in advance for your help!. 6. Let L be the line given by the equation r = (1,2,3) + /$$â€”1,1… • Hi! I was wondering if I could please get some help with the following maths question?Â Thank you so much in advance for your time!. 8. There has been a natural disaster on an island off the coast of… • Can you please help with q 5 and q7 please. 5. Consider a triangle formed by connecting the three points p1 = (0, 0,0), p2 = (1, 0,0) and p3 = (1, 1, 1). (a) Find the area of the surface of this trian… • Thanks for your help. 4 2 1 3 Let A = and B = What value(s) of k, if any, will make AB = BA? -2 2 3 k Select the correct choice below and, if necessary, fill in the answer box within your choice. OA. … • Thanks for your help. Determine if b is a linear combination of a1, a2, and a3- Choose the correct answer below. O A. Vector b is not a linear combination of a1, a2, and a3. O B. Vector b is a linear … • Thanks for your help. 1 . 2 b1 Let A = 4 0 and b = b2 Show that the equation Ax = b does not have a solution for all possible b, and describe the set of all b for which Ax = b does have a solution. 3 … • Thanks for your help. Evaluate the expressions found in the previous steps. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. Ax = (Simplify your … • Thanks for your help. Diagonalize the following matrix, if possible. 10 3 – 3 4 Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O O A. For P = D= 0 -… • Thanks for your help. Determine by inspection whether the vectors are linearly independent. Justify your answer. 3 3 3 Choose the correct answer below. O A. The set of vectors is linearly independent … • Thanks for your help. Determine if b is a linear combination of a1, a2, and a3 Choose the correct answer below. O A. Vector b is a linear combination of a1, a2, and a3. The pivots in the corresponding… • Thanks for your help. Find the determinant by row reduction to echelon form. 1 5 â€”7 1 6 5 â€”2 â€”8 7 Use row operations to reduce the matrix to echelon form. 1 5 â€”7 1 5 5 ~D â€”2 â€”8 7 Find the … • Thanks for your help. Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? 3 2 1 4 0 5 – 6 1 – 2 – 5 – 9 – 2 0 5 11 27 11 6 A basis for the subspace is gi… • E HW71.pdf 1 / 2 175% + Moral: quadratic forms are not linear. 2. Given the quadratic form Let 12 X = = 12 -6 and Q(x) = x"Ax = 3x] + 24x1x2 – 612. Do the following steps: a. Compute the eigenval… • /"’\ 0 Find the maximal and minimal value of z = 3x + 4y subject to the following constraints: x+2y514 3xâ€” ya 0 xâ€” 1:5 2 • Set up (do not solve!) a linear equation to numerically solve the following ODE:Â u â€²â€²(x) âˆ’ 3u(x) = 2 , for all x in (0, 1) , with u(0) = u(1) = 0 ,Â using a partition of [0, 1] with m = 5 (an… • 75 Exercise 8: 1. A company has a total of 3,000 shares, – of these shares are owned by Mr. Vitale, 350 are owned by Mr. Cruz, 200 are owned by Mr. Velasquez and the rest are owned by Sudalga group of… • Exercise 9: I. An amortized loan of P250,000 with an interest rate of 9 % will be paid monthly for 3 years. Find the monthly amortization. 2. A high-end laptop computer was bought at an amortized loan… • . â€”1 4 1 1 ï¬‚ 4 1 Let A = l ] [ I I I {The product {if these matrices.) 2 1 ï¬‚ â€”1 2 1 Find A15 using the abnve. Write 5mm answer as {me 2 X 2 matrix. • . 1 4 4 diagonalizable? (I need you to FULLY show and justify your work.) Is the matrix A = 4 1 4 1 4 4 1 • . Work Problem 1 [21] Points] 1 4 Let A = [ ] . 4 1 Find a 2 x 2 matrix P and a diagonal matrix D such that D : P_1AP. (I need you to Show you: FULL work.) • Show that T is a contraction. iii. Does there exist a function X e X such that (1 ) = 12+ e" x(s) ds for every r E [0, 1]? JO Justify your answer, • discrete math subject: graph and trees topic. 10. Consider the graph in the figure below and answer the following questions. (a) Give a proper coloring with minimal number of colors. You can use label… • The part I outlined please! Thank you!. E HW6.pdf 3 / 3 – 100% + plane amounts to rotation by 90 degrees counterclockwise. We saw that by thinking of the complex plane as a real vector space, 1 and i … • 2 0 3. For a discrete time system: X(+1) = 0 0.5 X (4) +u(k) (a) X(0) = [1 0] , find a control sequence u(0) ~ w/(1) such that X"(2) =[4 5] (b) x(0) = [1 0] , find a control sequence w/(0) ~w/… • For the following system, design a state feedback controls using estimated states 1 = X2 2 = 2x, + 3×2 + u y= x1 + X2 (a) Before you design the state feedback gain and observer gain, write down t… • For the following system: i =-X+xz +u 2 = -2×2 + 0.lu y = x1 + 0.1×2 (a) Please find the controllability matrix and observability matrix of above system. (b) Please check if the system is controll… • #4 please. 1. A rnath student deep in thought is pacing back and forth along a line going East and West. There is a 1/3 probability that the student will start off walking west (W) and 2/3 probability… • Find the stable state vector for the Schoenberg piano sonata in example P = NIHWINDIN Interpret this in terms of which chords are favored over the whole course of the sonata. 2 (t). Activity Find t… • Given the transition matrices 0 A = B = C = – ONI- NIH ON O O ON- OHO OHO NIH ONI 0 determine whether or not the corresponding markov chains are regular. • Which of the following is a linear program? Option 3 is not correct. The rest of option 4 is x_2=free. min 0. 5×1 + 3×2 (0. 99×1 \| 22) /22 2 2.7 X1 – X2 <8 x120 x2 = free min 0. 5×1 + 3×2 (0. 99×1 … • Vector space. 5. For which value of x do the following vectors form a linearly dependent set in R M =x. I • Tunjukkan f(x) = x dan g (x) = x-1 naik murni pada I = (0,1), akan tetapi hasil kali fg tidak naik pada I • please do full calculation i need to understand this. 1. Let G be the group defined by the following Cayley’s table 1 2 3 4 5 6 5 6 – N INDA 6 OUTAWN. OUTAWN -WNCIA ODWA AND NA – i. Find the order of … • If c < d and < >1, which of the following must be true? O A log.(x) > log (x) O B. log.(x) < log.(x) O c. log.(x) = log(x) O D. none of the above • Bank A charges 10 per month service charge as well as 25 cents for each transaction. Bank B changes 15 per month, and 20 cents for each transaction. Let x = the number of transactions per month Let … • Refer to the profit model above and your equation for P(x). A company is said to ‘break even’ if P(x) = 0. Find the number of tables the company needs to make to break even. Enter that number in the s… • I need assistance. * * T aph the following linear equ 2 Y= – X – 3 5 • Please answer and explain 13-25, thank you!!Â . 13. (two marks) What is the value of this expression (3x – 1)(3x + 7) after expanding and simplifying. 14. (two marks) Write this equation is standard f… • Let Line 1 be given by the equation: x – 4y = 5 Let Line 2 be given by the equation 4x + y V = 7 The slope of Line 1 is The slope of Line 2 is Therefore these lines are • Please answer and explain answer 6-11, thank you!Â . 6. (two marks) The quadratic function y =-x2 – x + 1 passes through the point a. (-1, 3) c. (-1, -1) b. (-1, 1) d. (1, -3) 7. (three marks) What is… • Find five solutions for the liner equation x+y=3, and plot the solutions as on a coordinate plan • Suppose an economy has four sectors: Mining, Lumber, Energy, and Transportation. Mining sells 20% of its output to Lumber, 70% to Energy, and retains the rest. Lumber sells 15% of its output to Mining… • Analytical Methods in Economics Exercises on vectors and matrices – 2 1. If A = 3 5 and B 0 4 2 7 from the following (if they exist) (a) A’, (b) 2A, (c) 2B, (d) A+B, (e) 2(A+B), (f) 2A+2B 2 2. A = 5 1… • How do I do this?Â . (l) (20 pts) Show that there exists at most one way to put the structure of a ï¬eld on the three element set F = {0, 17 a} where 0 and I are the additive and multiplicative ident… • Please answer and explain, thank you!. 9. (two marks) The table represents a quadratic function. What is the missing value? x 0 6 8 N 14 w 38 5 56 • For what value of h and k is the following system consistent ? 2x 1 – x 2 =h -6x 1 +3x 2 =k • Please answer and explain 6-8 please. Thank you!. 6. (two marks) What is the range of the graph? -2 O A NO 7. (three marks Which equation is a quadratic function? Explain how you know. a) yz = 3(x – 2… • Please explain and answer 24 & 25 please. Thank you!. 24. (three marks) What is the equation of the graph? 12 -8 4 8 12 X 25. (three marks) What is the equation of the graph? YA 12- 8 4- -6 -4 -2 … • Please explain and answer 1-5. thank you!. MCF3M Lesson Assignment 12 In this assignment you are going to focus on: The Introduction to Quadratic Functions. Total Marks: 63 marks these steps: After co… • I really don’t know how to solve it. .7b 4. MCC.8.EE.7b The perimeter is 22. How much bigger is the longest side than the shortest side? 20 – 3 2a 3a + 1 E.7b 6 MCC.8.EE.7b • Every matrix A = (ajj) defines the following game. In each round, the row player selects one of the rows i = 1, 2, …, m, and the column player selects one of the columns j = 1, 2, …, n, the result… • LO 2 A = 0 -1 If A e omxn is an m x n matrix, and all the nonzero singular values of the matrix A are 01, 02,"..,0 , show that (i) rank(A) = r; (ii) A = • Question No. 4: One part of Lahore’s network of traffic is given with the number of vehicles that enter and leave during a typical rush hours as shown below. All the lanes are one-way in the direction… • apartment of computer Science LINEAR ALGEBRA ASSIGNMENT-II Total Marks: 5 (FALL-2021) Submission date: 23-12-2021 Resource Persons: DR. M. SHAFIQUE BAIG ID # B- NAME: Semester # Question No. 1: Solve … • Suppose we are tracking the spread of a virus through the population and there are two genetic variants of this virus in circulation: type a and type b. Let a; denote the number of active cases that a… • Let XI,…, Xn be i.i.d. R’-valued random variables distributed as A(0, E1). Find the asymptotic variance of eigenvectors by hand calculation. • Let W be the set of all vectors p(t) in P 2 such that p(0) = 0. Show that W is a subspace of P 2 and find a basis for W. Â Please solve I Will give Good rating thank u • Graph and shade the feasible region of the following systems of linear inequalities. 5x +Â Â y Â â‰¤Â 10Â Â Â Â Â Â Â 3x + 2y â‰¤ 12 Â Â Â Â Â Â x â‰¥ 0 Â Â Â Â Â Â y â‰¥ 0 Â Â Â … • 7-5h-2-k , 5-4x-8y , 9k+7-k+4 I need to know the terms, variables, coefficients, constants to all Three questions • 7-5h-2-k Â I need the terms, variables Â ,coefficients, constants • Just do question 8 ignore the rest.. (b) u1 = (1, 2, -1) , u2 = (0, 1, 1), us = (5, 1, 7) ; (c) u1 = (1,2, -1, a) , u2 = (6, 0, 1, 1) , us = (5, 1, c, 2) , u4 = (0, 0, 0, 0). 8. Let u = (1, 2, -1,5), … • Let A, B and C be n x n invertible matrices such that \|A\| = 2 , [B\| = -3 and \|A + I\| =6. Find determinant of each of the following: (a) A2 BT: (b) (C-1A)-2 B3C-2; (c) B-1(I + BA-1 B-1) BA. 7. Deter… • Use Cramer’s rule to find only the value of 2, where a is a positive real number. 212 – 13 + 14 = -4 413 + aT4 = 0 T2 – 513 + 2×4 = -2 C1 + 712 + 8×3 + 64 = 10 1 2a a -a -2a 4. Let A = -2 -2 a ) a… • Attempt all questions and show all your work. Some or all questions will be marked. 1. Solve each of the following system of linear equations. (a) Use Gauss-Jordan elimination to solve 3×1 -912 +13 +4… • Question 1 (Unit C1) – 19 marks (a) Solve the following system of linear equations by reducing its augmented matrix to row-reduced form. + 3 + 1 = 10 201 -0 + 403 – 04= 0 4×1 + 2 + 203 – 204 = 6 9 (b… • need to be done using excel solver. Case Study 2 – To determine optimal number of calculators to be produced (15 Marks) Read the case scenario carefully and submit your linear programming model (in ex… • . Find the maximal and minimal value of z = 3x + 4y subject to the following constraints: x +2> <14] 3x – 120 x- 15 2 • let v be the set of all real numbers define by u+v=uv-1 and c.v=v is v is a vector space • Production Linear Programming Problem. A company in its production process uses two production processes (Normal Production) and (Overtime Production) that produce X1 and X2 units of product respectiv… • Are the vectors (2,1,0) (0,-2,4) and (1,0,1) linearly independent or linearly dependent? Explain why • Are the vectors (2,1,0), (0,-2,4) , (1,0,1) linearly independent or linearly dependent? Explain why. • Q9 (0 points) Let 1 -2 A= 6 4 and B = -6 -2 -3 -3 2 5 Find the inverse of the matrix A + B – I. + Drag and drop an image or PDF file or click to browse… • Q7-b (0 points) Determine if each of the following set of vectors are linearly dependent or linearly independent. Show your work. (b) u1 = (1, 2, -1) , uz = (0, 1, 1) , us = (5, 1, 7) ; + Drag and dro… • Find the indicated matrix product or state that the product is undefined. A = [_8]. B = [-3 3] AB O [-89 -10 ] Undefined O -59 15 O -35 20 -54 -30 • Find the value(s) of h for which the vectors are linearly dependent. Justify your answer. 6 3 4 11 h The value(s) of h which makes the vectors linearly dependent is(are) because this will cause to be … • Find the indicated matrix. Let A and B Find 4A + B. • . Multiple linear regression solves the minimisation problem agility" ‘ erii; – X3), 1’ T wheretheveotorsï¬‚=(ï¬n 1131 ,8?) 1y=(y1 y”) andXisannx[p+1} matrix with 1″" 1’0″! (1 1-5… • Intersections in England are often constructed as one-way 130 130 "roundabouts," such as the one shown in the figure. Assume that traffic must travel in the directions shown. Find the genera… • can you help me solve this problem please. and the numbers to graph.. deltamath.com/app/student/solve/ 14333295/twoVariableSystemsWordGraphically Easton is working two summer jobs, making 18 per hour… • is the graph of a linear function always a straight line?. 5. The graph of a meal straight lines true B false • y Assume: rp = 0.18 y = 0.55 A = 5.75 sigma = 0.22 What is the value of rf? • Suppose U = V = NON For what value of z is u equal to its projection onto v? • Determine whether triangles are similar or not similar. If they are similar, write a similarity statement. D A E 770 C 48 48 55 B F Are the similar? (Yes or No) If yes, then AABC ~ 4 (If not put &quot… • This is a MATLAB Activity for Linear Combination.. Linear Combination My Solutions > Given the following vectors: v. =< l,4,5 >__ v; =4 3,1,6 >) and v3 =< â€”2, i,â€”2> Perform the i… • Solve each system of linear equations. 1 . fy = 2x – 3 do.1 bry ly – 2x =-3 is melave aint vileeslost yns H noiluloe • Using the Information provided, fill In the missing table values. 1. a) Original table Inverse table -5 2 4 9 Y -376 -4 23 191 2186 Original table Inverse table X -23 5 -11 29 17 -7 0 6 3 2. Given the… • Suppose a firm produces three types of output using two types of inputs. Its outputÂ quantities are given by column vector ð‘ž and the unit price of these outputs are given by rowÂ vector ð‘. The… • Gaussian elimination is the name of the method we use to perform the three types of matrix row operations on an augmented matrix coming from a linear system of equations in order to find the solutions… • a piece of rope k meters long is cut into 8 partsso what is the length of each part • . Multiple linear regression solves the minimisation problem min (y – XB)(y – XB), T T where the vectors B = ( Bo Bi … Bp y= (1 … yn) and X is an n x (p + 1) matrix with it row Til … Tip) ; i … • The table below lists the annual land-line phone cost per costumer: Year 2012 2013 2014 2015 2016 Cost (S) 692 610 580 495 434 a. Find a linear regression model for this data b. Interpret the slope… • Which method do I use to find the answers for this question? Thank you for the help.. 1 2 Exercise (7) Let A = – -1 1 1 . Compute det A. 0 1 3 • Practice and Homework Number Patterns Lesson 5.6 COMMON CORE STANDARD–4.OA.C.S jonesdis do’d walpre pattons. Haedhe rule to weiin the first meive numbers in the pattern. Describe another pattern in t… • Please explain and the questions.Â Thank you!. 18. (three marks) The point (6, 18) is on the graph of y = ax?. What is the value of a? 19. (three marks) The point (-2, 12) is on the graph of y = ax?…. • Please answer and explain questions 13-17. Thank you!. 13. (two marks) What is the axis of symmetry for the graph? 14. (two marks) What is the equation of the function y = 2(x – 1)2 -3 in the form y =… • Please explain the answers 9-12.. 9. (two marks) The table represents a quadratic function. What is the missing value? 0 6 N 14 3 38 5 56 10. (two marks) What are the second differences of the quadrat… • Which equation is a quadratic function? Explain how you know. Â A) y^2 = 3(x-2)^2 + 2 B) 10-3y = 2x C) x^2 + 2y = 36 D) 3x = 2y^2 – 4 • Does the equation y=4 Produce a horizontal line with all x-values of 4? Explain. • Which of these points cannot be generated by a linear function machine using the function y= x-4? a. (0, -4) Â b. (-1, -5) Â c. (2, 2) Â d. (5, 1) Please show work. • Consider the vector c = T2 of three observations and let 21 = and 22 = (1.1 01.2 are vectors of unknown constants. Also define, where of = (12.1 and 02 = 012.2 (3.1/ (3.2 21 Ti = (B1,1 B2,1) and T2 = … • Give two possible combinations of bags of chips and pretzels that Clark can buy.. X -2 -5 -4 -3 -2 -1 0 7. Clark is having a party at his house. His father has allowed him to spend at most 20 on snac… • Julia is a freelance bakers. She receives a few cake orders, for this Saturday. Since she got quite a number of orders on the same day, so she needs to deliver the cakes in an optimal way . She has st… • Which of the following vertex sequence describe paths in the graph given above? • Pls give full explanation, thank you! Â . (d) For any n 2 2, find a non-recursive formula for , in terms of 4, and 60. Hint: & can be found as the top component of the vector In or the bottom comp… • Pls give full explanation, thank you! Â . 67. Let A, B. P, X be 3 x 3 matrices for which det(A) = 4, det(B) = -2, det(P) = { for ( / 0. Compute the following: (a) det(ABA). (b) det(-2A"). (c) det… • Pls give full explanation, thank you! Â . 62. Let A be the matrix 1 2 A := 2 5 5 (a) Find a basis to the subspaces Null(A) and Col(A), i.e., find a basis to each of the null space and column space of … • Pls give full explanation, thank you! Â . (c) Find an algebraic identity / expression for a matrix A for which T(v) = Av for ve R. You do not need to calculate this all the way (just a product is suff… • Pls give full explanation, thank you! Â . (b) Let Prop : R" > R" be the linear transformation which takes a vector in R" and projects it on to the plane P. Find a matrix A for whi… • Given the points (-4,5) and (2,-1) a. Find the slope of the line that passes through the two points: b. Find the equation of the line that passes through the two points: • Module 11 – 2021S… Relationship Paper… PQ Is Genetically Modif Microsoft Word – K… PQ NAME: Written Homework – Introduction to Linear Functions 1. Suppose that you want to buy rolls of paper tow… • Let V be vector space of all functions S from R into R. W={f/f(3)=1+f(-5)} Check whether W is a subspace of V or not. • Find a linear transformation T:R^2 â†’ R^2 such that T(1,0)=(1,1) and T(0,1)=(-1,2). Determine the image of right-angled triangle with vertices {(0,0), (1,0), (1,1)} under T. • Find a linear transformation T:R? â†’ Ro such that T(1,0)=(1,1) and T(0,1)=(-1,2). Determine the image of right-angled triangle with vertices {(0,0), (1,0), (1,1)} under T. • If the matrix of the linear transformation T on V2(R) with respect to ordered basis B = {(1,0),(0,1)} is [1 1 1 1] What is the matrix of T relative to ordered basis B’= {(1,1),(1,-1)}. Also, find the … • -5 1-2 -5 -4 -2 7. Clark is having a party at his house. His father has allowed him to spend at most 20)on snack food. He’d like to buy chips that cost 4 per bag, and pretzels that cost 2 per bag. … • Clark is having a party at his house. His father has allowed him to spend at most 20 on snack food. He’d like to buy chips that cost 4 per bag, and pretzels that cost 2 per bag. • No ad. info. (30 points) Find the solution of the following system of difference equations using the Z-transform method: x n] 2 x[n – 1] y[n] 1 2 0 y[n – 1] + where yl-1] z n 3 z[n – 1] z • I know it is blurry, but it can be readable to some extent, can you please solve it because I have the photo of the question like that.. (15+6 pbs) For a given Wich 1, let (a) Determin (b) For a plwen… • No ad info. (35 points) Consider a real matrix A3x3 such that A"A = BI where B E R+. A has an eigenvalue of 1 E R with multiplicity k = 3 and a respective set of L.I. eigenvectors e1, e2, e3 E R3… • Let A and B be two nodes in a wireless network with access point X. Assume that A and B cannot hear each others’ transmissions, but can hear X. At time 0, X is sending a packet to some other node and … • With the start of school approaching, a store is planning on having a sale on school materials. They have 600 notebooks, 500 folders and 400 pens in stock, and they plan on packing it in two different… • Slope formula: m = 12 – V1 X2 – X1 1) For each line segment, determine its slope. y a) b) 4 2 A 4 X 4 X A • help with my math. actice Problem Af fixed costs are 25, variable costs per unit are 2 and the demand function is P = 20 – Q obtain an expression for a in terms of O and hence sketch its graph. (a) Fi… • X 70. 108 f_ 3.6 fi 15 ft 0 02 Save All Answers Save and Submit prt se delete backspace home • fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Once this final variable is determined… • Determine whether b is in the column space of A. If it is, then write I: as a linear combination of the column vectors of A. (Use v1, v2, and v3, respectively, for the three columns. If not possible, … • I need help with both a)&b). Thank you for assistance.. 25. A Matrix Equation. (a) Show that if a square matrix A satisfies A2 – 4A + 31 = 0 then A-1 (41 – A). (b) Verify these relations in th… • Could someone show me how to do Q18? Thank you.. 18. Compatibility of Matrix Products. (a) Show that if the matrix products AB and BA are both defined, then AB and BA are square matrices. (b) S… • The lines 4ax=3y+12 and 7bx-2=3y are parallel. Find the ratio a:b • which function is being represented by the graph. Which function is represented by the graph bel 10- HANWA MO NOO 10 -4 -3 -2 -114 1 (x) = log x O f(x) = e . x Of(x) =ex • iv already found one difference and one similarities:Â but i need help finding the othersÂ Â Â Â differences: One is linear and one is quadratic, Â Â Â similarities: they both create line segm… • A manufacturer has three machines I, II and III installed in his factory. Machines I and II are capable of being operated for at most 12 hours whereas machine III must be operated for at least 5 hours… • FarmProd produces two types of cattle feed, both consisting totally of wheat and barley. Feed 1 must contain at least 80% wheat, and feed 2 must contain at least 60% barley. Feed 1 sells for Â£1.5 per… • Calculate the final angular velocity of the drum. b) Calculate the angular acceleration of the drum. you should clearly state the formulae used and apply dimensional analysis techniques to show tha… • Do you prefer the corner point method or the isoprofit, isocost method? Why? • Description Students will produce a successful rollercoaster model.Â Â Objective Students will create a graph with the following restrictions below. You will need to find the equations for each func… • I need a detailed explanation of how each one is, or is not a vector space. None of the tutors provide a plain english response.. 4. Verify all vector space axioms and determine, if the set of all tri… • Verify all vector space axioms and determine, if the set of all triples of real numbers (ð‘¥, ð‘¦, ð‘§) is a vector space under the given operations (ð‘¥,ð‘¦,ð‘§)+(ð‘¥â€²,ð‘¦â€²,ð‘§â€²)=(?… • Determine if the function (u, v) = u,V2 tuz VI, where u = (uj, u2) and v = (v],v2 ) defines an inner product on R . Check each axiom to determine if it holds or fails. Show all work. • Could someone show me how to do 17 d) & Q19? Thanks.. oklet2022.pdf Open with Google Docs (c) A(B + C) = AB + AC 17. Two by Two Examples. Use trial and error to find 2 by 2 examples of the … • I could not solve for 16a) c). Could someone show me the steps? Thanks. 16. Matrix Properties. Consider the following matrices: 2 5 1 1 0 HO NOIN A = 3 1 -1 B = 2 C = 2 -1 2 1 2 4 Verify that (… • show me the steps for B and C,. QUESTION 5. Consider the basis B : {(1,1,1),(1,1, 0), (1, U, -â€”1)} and let S be the standard basis for R3. (a) Find the vector [VS] of [VB] 2 (1, -1, 1). (b) Find the… • QUESTION 2. Consider the following matrix Adi? ll loo 7] (a) What is the determinant of the matrix A? (b) Find the eigenvalues and eigenvectors of A. (C) Is the matrix A diagonalisable? If it is, writ… • please answer it on page by handwriting Â clear handwriting please don’t copy from other websitesÂ .. .. …… Task One of the basic operations used in computer graphics (for example, clipping) is … • EXERCISE 3 (6/32). (a) (2 points) Find the parametric equation of the plane passing through the points P = (1, 2, 3), Q = (2, 2,3) and S = (3, 1, 2). (b) (4 points) Consider the following parametric s… • QUESTION 6. Is it possible to construct a matrix whose column space contains the vectors (1,1,0) and (0,1,1) and Whose nullspace contains the vectors (1,0,1) and (0,0,1)? If it is possible, then write… • QUESTION 5. Consider the basis B : {(1,1,1),(1,1, 0), (1, U, -â€”1)} and let S be the standard basis for R3. (a) Find the vector [VS] of [VB] 2 (1, -1, 1). (b) Find the change of basis matrix PBS. (c)… • QUESTION 3. Consider the following complex numbers 21 :anâ€”rli, and 22:2â€”lâ€”bi Where a and b are both real numbers. (a) If a = 3, and 2:1 is one of the solution to the quadratic equation 22 – 62 +… • QUESTION 1. Consider the vectors v1 = (1, â€”2, â€”4), v2 = [â€”3,1,1), and V3 2 (2,1,k) (a) For what value of k is this set of vectors linearly dependent? (b) Does the vector 11 = (5, O: 2) an elemen… • QUESTION 5. Consider the basis B : {(1,1,1),(1,1, 0), (1, 0, â€”1)} and let S be the standard basis for R3. (a) Find the vector [v3] of [VB] 2 (1, -1, 1). (b) Find the change of basis matrix PBS. (c) … • Can you please explain the steps taken along the way?. 3. Construct a 3X3 non-triangular matrix, which has 3 distinct real non- zero eigenvalues. . Determine, eigenvalues and the corresponding eigen v… • Can you also make some commentary explaining what going on here? Thank you.. 1. Construct an example of a parallelepiped in 3-space providing the coordinates of all vertices (refer to a figure below)…. • . 15.7 (a) Prove if (an) is a decreasing sequence of real numbers and if _ an converges, then lim nan = 0. Hint: Consider antitan+2+. . .+ an for suitable N. (b) Use (a) to give another proof that _… • . 15.4 Determine which of the following series converge. Justify your answers. 1 (b) En=2 log n (a) En=2 Vnlog n n log n (c) Lin=4 n(log n) (log log n) 1 (d) En=2 n2 • AÂ contractorÂ wasÂ contractedÂ toÂ buildÂ aÂ goodÂ classÂ bungalowÂ (GCB)atÂ aÂ lumpÂ sumÂ price ofÂ 36Â million in Singapore. The SIA Measurement Contract (9 th Edition) was used for the contract.?… • . 15.3 Show En = 2 1 n=2 n(logn)p converges if and only if p > 1. • Use Sirug Approach in solving this simplex method, Thank you! Â Â . Simplex Maximization Problem: Try This Out! Example. A poultry raiser plans to raise chicken, ducks and turkeys. He has room for on… • Is it possible for a given nonzero vector to belong to more than one of the four fundamental subspaces of a matrix A? If yes, give an example; if no, explain why not. • . 2.3 Matrix Multiplication: Problem 9 (1 point) Given the matrix ï¬nd A3. Write A3 as A3 2 [an 6112] “21 “22 input your answer below: 0-11 = 012 = â‚¬121 = “22 = • . No.5. Let C : {f(x) E C'(-oo, 00) : f(1) = 0} -> C(-oo, co) be defined by C(f)(x) f'(x) – 2f(x). (i) Determine Null(C) and Range(C). (ii) For g(x) E C(-oo, co), what can we say about solving th… • . No.4. Suppose I: : B4 â€”) IR3 is a linear function with dÃ©m(Range(Â£)) = 3. For b E R3 given, discuss the existence and uniqueness of solutions to Â£(x) = b (i.e., will a solution exist for all … • . 2 No.1. Let T = t1 = t2 (i) Show that T is a basis for IR2. (ii) Suppose that C : IR R is linear and satisfies 2 1 C(ti) = -1 , L(t2) 6 Determine an expression for C(x). (iii) Express [(x) found i… • Sketch a graph based on the description given below. State the type of function and any key features related to the graph. “Q ‘â€” I’â€”l- a. The Panono ball camera is thrown up in the air by a p… • Determine what type of relationship exists (linear, exponential, other) using trendlines. See the Section 2.6 video for additional information on determining relationships. Explain why you would use t… • . Which of the following sets are vector spaces? 1 0 1. The set of all 4 x 4 matrices A such that A g = g and 4 0 5 0 6 0 A 7 â€” 0 . 8 0 2. The set of all differentiable functions f (3:) such that … • . Let B be an ordered basis for R3 defined by B = {(1, 0, 1), (2, -2, 1), (0, 1, -1)} Find the coordinates of v = (2, 0,2) with respect to the basis B. O a. [uB = (0, 4, -1)T O b. [UB = (5, -1, 2) T… • Problems 1. Consider the following sets: A = [7], B = {re A\| > >> 4) and C = 12 -1\| r ( B). List the elements of each of the following sets. a.) B b.) C c.) BOC d.) BUC e.) A-B () B-C 🙂 C -B… • Review \| Constants \| Pengdic Table The forces In (Figure 1) are acting on a 25 kg object. Part A What is a the I-component of the object’s acceleration? Express your answer with the appropriate units…. • . Given two m x m matrix X and Y, where XY = YX. Suppose Y is invertible and Yu is an eigenvector of X. Show u is an eigen- vector of X. • . Give the matrices for the following linear transformations with respect to the given bases. (a) T: R2 â€”> R2 given by T(x) = AX Where A = ( (1,0), (1, 1) for R2. (b) T: Rte]? â€”> R[m]53 gi… • . (4) A 3 x 3 magic square with entries in R is a 3 x 3 matrix A = (ajj) E R3x3 such that the sums of the numbers in each row, each column, and both main diagonals are the same. Note that for us, ma… • . 9. Suppose that (5″) is a sequence such that 3,1 â€”> 0. If (in) is a bounded sequence (not necessarily convergent), show that sntn â€”) U. • kofi is four times as old as esi and kofi is 36 years older than esi • Question on linear algebra Â . 5. Decide if each of the following sets of vector spans R" for the appro- priate n. Justify your answer in each case. (a) • (10 marks) Let A = ï£« ï£¬ï£¬ï£ 0 3 3 1 1 0 âˆ’1 2 3 ï£¶ ï£·ï£·ï£¸ . Assume that AB = A + 2B. Find the matrix B. • What is wrong with this LP model? MAX X1 + X2 – X3 ST X1 ï‚£ 10 X2 ï‚£ 10 X3 ï‚£ 10 X1 – X2 ï‚³ 5 X1, X2, X3 ï‚³ 0 • Linear algebra. -2 N O -3 2 -4 (a) For which values of x is the following matrix invertible? 4 X 4 (b) Give a vector in R’ which is not in the span of \| -4 \| and Check • (10 MARKS) Let 0 3 3 A = 1 10 -1 2 3 Assume that AB = A + 2B. Find the matrix B. • 1)Let U be a subspace of R15. Suppose U has a set of 9 linearly dependent vectors and a set of 7 linearly independent vectors. Which of the following statements is always true? 2)Let U be a subspace o… • Please see that attached question 4.Â Â I have provided below it solutions to other sections of this homework as a reference.Â Â thank you!!!Â Â Â Refence examples below:Â Â Â Â Â Â . Prob… • Hi, please see the attached. I am looking for assistance with question 2 E. Â (Sample solutions for 2a – 2d are provided for reference. )Â thank you!!!Â Â Â Â . Problem 2 Question: 2. (5 pt) Consi… • . 1. (5 marks) Consider two firms with identical goods that must each choose a quantity q; to produce. The price of the good is p(q1, q2) = max {0, 100 – q1 – q2}. Firm i’s payoff function is mi(q1,… • . Suppose that (5,1) is a sequence such that en â€”} 0. If (in) is a bounded sequence (not necessarily convergent), show that sntn â€”) U. lGive an example to show that the boundedness assumption in… • . Prove directly from the deï¬nition of convergence that for any real number p > 0, limnamï¬‚/n”) = 0. Using the previous problem, prove that for any rational number I E Q, there is a. sequenc… • What is a spanning set? If S spans the Vector Space V , is it necessary that S be linearly independent? Why or why not? • Which ones are the correct answers?Â . Introducing Domain and Range X C https://student.desmos.com/activitybuilder/instance/620fdbf8434ad64967d7d47e/student/62190a885a7acaf0316dae77#screenld=5912… Q… • Find a basis for the nullspace of the matrix A = 3 What is the dimension of Nul A? • . 3. (10 pts) Consider the following three vectors 3 u = -1 V = HOP W= Please find the vector x that satisfies the following equation 3( u u+x + w=2x -V. Is x parallel to u, v or w? • . 2. (15 pts) Given the matrix A and vector b. 0 1 1 A=101 b = HOA 1 1 0 Please compute A100b. • . 1. (25 pts) Suppose A is any skew symmetric matrix. (a) (10 pts) Show that I + A and I â€” A are both invertible. (b) (5 pts) Show A = â€”1 is not an eigenvalue of (I â€” A)’1(I + A). (c) (10… • Need help with this linear algebra question Â . 9. Let T be the mapping described by T 1 – 12 T2 I2 Prove that T is a linear transformation by establishing that it satifies both of the conditions in t… • . Simplex Maximizetion Problem: Try This Out! Example. A poultry raiser plans to raise chicken, ducks and turkeys. He has room for only EDD birds and wishes to limit the number of turkeys to a maxim… • . 2 1 2 2. Let A = and B = 2 1 0 1 (a) Find (AB), BTAT and ATBT. (b) (AB)-], B-A- and AlB-1. • Kindly help me by completing all these tasks. These questions are from a module titled “Artificial Intelligence & Machine Learning”. The topics are based on observation & approximation, Depend… • Answer (3) or (5) choose one to replyÂ THANKS. 3 ) Let A = [ ! , S be the lineor transformation s.+. [S] . = A It is tricky to try to visualize this transformation because its range is the difficulty… • left with a bunch of linear equations Question 3: Find all vectors 7 of length 2 parallel to the line through P = (1, 2, 3) and Q = (-1, 2, 1). Question 4: For any v w E R? show at • Question 2: Consider the parametric equations: x1 = 5t – 2, X2 = 3t, X3 = 5, x4 =7-t which correspond to a line in R4. Find a matrix A and a vector b such that the solution set of Ax = b is this line…. • Consider the matrices 1 0 â€”1 0 1 A = â€”1 1 â€”1 0 0 2 2 â€”6 1 6 0 2 â€”4 1 4 Solve AX : B and write your answer in vector form. • Subject: Supply Chain and logistic Listed below is the actual sales data of car A and car B starting at year 2008. Year Sales Car A Sales Car B 2008 116 119 2009 105 120 2010 80 130 2011 130 48 2012 9… • 1. Let Q2Ã—2 denote the Q-vector space of rational (2 Ã— 2)-matrices and consider the linear operator T: Q2Ã—2 â†’Q2Ã—2 defined,forall(2Ã—2)-matricesA,byT(A):=AT. (i) Show that Â±1 are the only ei… • 3. The union of the zero vector and the set of all eigenvectors with an eigenvalue A is called the 2-eigenspace. The dimension of the 2-eigenspace, or equivalently the maximum number of linearly i… • 2. Let D: R<2 -> R<2 be defined by D[f] := 21 (t -1) f"(t) +if'(t) + f(t) +12 f'(0) where f’ and f" are the first and second derivatives of the polynomial f respectively. (i) Le… • What assumption is reasonable for the magnitude of the vectors? Explain. • A linear programming problem with 2 decision variables and 3 resource constraints is solved and some of the results are recorded in the table below Which of the resource constraints are binding in the… • Topic: Slope of a Line Â Topic: The Equation of a Line Â . Please please please answer it completely. Please Make sure your handwriting is legible. Please please please make it neat and understandabl… • see attached Â . 1) Determine whether or not the following functions are inner products on 3. 2 1 y1 Let x = JC 2 and y = y2 y3 (a) (x, y ) = 3×122+ 2y12 + 2×393 (b) (x, y) = 32191 + 2×292 – 20393 (c)… • Hi can someone show me the step by step answer to this problem? using Simplex Maximization Problem (Sirug Method) Â . Example. A poultry raiser plans to raise chicken, ducks and turkeys. He has room f… • Respected Tutor, Â I want a written answer and not a coding answer. 2. Given vectors a = [a1, a2]’ and b = [b1, b2]’, map them to 2 x 2 matrices A and B respectively such that AB = (albi tazb2) 12… • Respected Tutor, I want a written answer and not a Coding answer Â Â . 4. Projection Operators: A projection matrix is defined as A(ATA) A where A is m x n and of rank n. Use the singular value decom… • . Express each column vector of BB as a linear combination of the column vectors c1, c2, and c3 of B. 2 â€”1 5 B: 0 1 3 3 5 4 Enter ï¬rst column as a linear combination of columns of B in terms of … • . 1. Consider the homogeneous system of differential equations ‘9’ = (â€”05 :5) y. (a) [1 mark] Write down the eigenvalues and corresponding eigenvectors of the underlying matrix. (You may use MAT… • . Question 4 of 10 < View Policies Current Attempt in Progress Using the matrices compute the following. tr(2ET â€” D) = n eTextbook and Media Save for Later â€”4 0 â€”2 2,E= â€”1 3 t1′(2ET â€” D… • . Question 3 of 10 – /1 E View Policies Current Attempt in Progress Using the matrices 2 8 A = 3 5 I, B = 6 compute the following. AB i H AB = i i e Textbook and Media Save for Later Attempts: 0 of … • Check attached image for questions Â . b) Create a system of linear equation that is convergent and would have a unique solution. Discuss what makes your constructed system of linear equation converge… • 7) Let Il. II be any matrix norm that is compatible with the norm being used in R ? Let weRR" be such that lull= 1 and let velR". Here is the fact that we’ll use in the proof of Gastinel’s T… • Hi there, would you solve it real quick, that would be helpful and appreciated. Thank you. I will give a good feedback too.. Quethan: Let H be a nonzero subs pace of V, and suppose T is a one- to- one… • Use elimination. 6. 6x + 9 y = 12 2x + 3y = 4 6 7. xty=3 7. 2x +2y =4 • 19 Solve each systems word problem by setting up two equations and solving. Show your work! 18. A store sells two different kinds of rulers, plastic and wood. The receipts show that they have sold 35 … • please answer the question below. In Exercises 33 and 34, 7 is a linear transformation from R into R’. Show that T’ is invertible and find a formula for T-1. 33. T(x1, X2) = (-5×1 + 9×2, 4×1 – 7×2) • 6) Let A be a 44 matrix such that. A O A (: )- ( 2 ) . (: ) . A A Q a) Find the Jordan Canonical Form of A. b ) Find o ( A) and ro (A ). c) Fix Exo. Find a matrix norm N for which N ( A ) = 1 / 3 + E… • The eigenvalue have been found to be -5 and eigenvector associated with it is (1,0). 5 1. Consider the homogeneous system of differential equations y’ = y. (a) [1 mark Write down the eigenvalues and c… • Let R be a ring and f(x) = Chair, g(x) = Et, bix e R[x]. Then h(x) = f(x)g(2) = >ko 2n crack, where Ck = Zio dibb-i. (i) Prove that if R is commutative then for any T E R, h(r) = f (r)g(r). (ii)… • Let B be a nonsingular n Ã— n matrix and let {A1, A2, . . . , Amn} be a basis for the complex vector space CmÃ—n (the set of m Ã— n matrices with complex entries, com- plex scalars, and the normal mat… • The set 13′ = {1,1+ 2t,1 +2tâ€”lâ€” 3t2 } is a basis for [132. 1 (a) What polynomial (i.e. 1lrector} has coordinates 1 with 1 respect to B? (b) Find the coordinates of p(t) = 2 + 3t â€” 4? with res… • Previous Problem Problem List Next Problem (1 point) Find the angle a in radians between the vectors 2 and 4 a= • Previous Problem Problem List Next Problem (1 point) Find the angle a in radians between the vectors -1 and 1 5 a= Preview My Answers Submit Answers • (1 point) Find the length of the vector x = -4 2 = Preview My Answers Submit Answers • Previous Problem Problem List Next Problem (1 point) Perform the following operations on the vectors u = (4, 4, -3), U = (5, -1, 2), and w = (1, 2, 5). u . w = (u . D)u = ((w . w)u) . u = u . U + U . … • Previous Problem Problem List Next Problem w (1 point) Let x = and y = 0 . Find the vectors u = 7x, u = x + j, and w = 7x + j. U = u = W = Note: You can earn partial credit on this problem. Preview My… • Use an inverse matrix to find [x] for the given x and B. 8 2 B = 4 X = 9 2 N [X]B = ( Simplify your answer.) • y h(x) + +x -9 -8 -7 -6 -5 -4-3 -2 1 2 3 4 5 6 7 8 9 y = g(x) Given that h(a) = (x – 4)2 + 2, write an expression for g (a) in terms of x. g(x) = • Isaiah bought a total of 22 pieces of candy. The tootsie rolls cost 3 cents each and the jolly ranchers cost 5 cents each. If Isaiah spent a total of 90 cents, how many of each candy did he purch… • please asnwer quick Â . 1. Answer the following questions regarding linear transformations. a 4 marks Find the standard matrix of the linear transformation T : R3 â€”> R2 such that ( ) ( T as _ 3:5… • The augmented matrix is given for a system of equations. If the system is consistent, find the general solution. Otherwise state that there is no solution âŽ£ âŽ¢ âŽ¡ â€‹ 1 0 0 â€‹ 0 1 0 â€‹ 6 âˆ’ 2 0… • I started i an ii, I feel confident about those. I don’t know how to do the others. Plz help.. Problem 3 [29 points (9, 6, 4, 4, 6)]: Ecology Consider an ecological system of plants, rabbits and foxes… • Problem 1 [22 points (6, 6, 6, 4)]: Line of Best Fit The following scatter and table describes the time spent studying and test score of 5 students: 100 95 test score (b) 90 85 80 . 0.5 1.0 1.5 2.0 ho… • . 4. Find an example of three vectors v1, v2, and v3 (in a vector space of your choosing) which have the following properties. (a) The set {v1, v2, 13} is linearly dependent. (b) The sets {v1, v2}, … • Solve the system of equation using elementary row operations x 1 + x 2 + x 3 = 6 x 1 Â Â Â + Â x 3 Â = -2 Â Â Â x 2 + 3x 3 = 11 • Determine the reduced echelon form of the given matrixÂ âŽ£ âŽ¢ âŽ¡ â€‹ 1 2 âˆ’ 3 â€‹ 4 5 âˆ’ 9 â€‹ âˆ’ 5 âˆ’ 4 9 â€‹ 1 âˆ’ 1 2 â€‹ 2 4 1 0 â€‹ âŽ¦ âŽ¥ âŽ¤ â€‹ • Find an LU factorization of the matrices in Exercises 7-16 (with L unit lower triangular). Note that MATLAB will usually produce a permuted LU factorization because it uses partial pivoting for numeri… • Determine whether the matric is in echelon form, reduced echelon form, or neitherÂ âŽ£ âŽ¢ âŽ¡ â€‹ 1 2 0 â€‹ 0 1 5 â€‹ 0 0 1 â€‹ âˆ’ 7 âˆ’ 2 2 â€‹ âŽ¦ âŽ¥ âŽ¤ â€‹ • please do number 34. 33. Use the algorithm from this section to find the inverses of and Let A be the corresponding n x / matrix, and let B be its inverse. Guess the form of B, and then prove that AB … • Find the inverses of the matrices in Exercises 29-32, if they exist. Use the algorithm introduced in this section. • Define a linear transofrmation T: P2 -> R2 by T(p) = [ p(0) p(0)}. Find polynomials p1 and p2 that span the kernel of T, and describe the range of T.. 44. Define a linear transformation 7 : P2 – R-… • A wheel hasÂ 5 Â equally sized slices numbered fromÂ 1 Â toÂ 5 . Some are grey and some are white. The slice numberedÂ 1 Â is grey. The slices numberedÂ 2 ,Â 3 ,Â 4 ,Â andÂ 5 Â are white. Â T… • . 5. Determine which of the following sets of vectors are linearly independent in R2x2. O 2 ( 2) 0 0 • Question 4 Let P E Mnxn, be an invertible matrix. Define L : Mnxn -+ Mnxn by L(A) = P-AP. Prove that L is an isomorphism. • Question 3 Let T := V – V be a linear operator from the n-dimensional vector space V over F. Prove that T is invertible iff 0 is not an eigenvalue of T. • Question 2 (I) (a) Let T : V – V be a linear operator on the n-dimensional vector space V, n > 0. Let W be a k-dimensional invariant subspace of T, with 0 < k < n. (see W22-A4) Show that ther… • Question 2 Let T : C – V be a linear transformation from C to the vector space V. Let c E C, with c 0, satisfy T(c) = v, for some v E V. What is T(x), x E C? • Let T : V – V be a linear operator on the n-dimensional vector space V with n > 0. Let W be a k-dimensional subspace of V, with 0 < k < n. We say that W is T-invariant to mean that Vw E W, T(… • Let M be a 3×3 matrix such that. M 3 6 4 00 O O M 1-4 3 O MT-I 7 25 10 Express M= Q TQ where T is upper triangular and Q is orthogonal. ( As a check, if you approach this how It anticipated, the en… • Find the volume of the right circular cone with r = 24.5 cm and h = 5.96 cm. H V ~ cm 3 (Round to the nearest ten as needed.) • A new spiral notebook contains 30 more sheets of paper than a new memo book, the total of the sheet of paper in 3 new spiral notebooks, and 5 new memo books is 810. Which system of the equation can be… • Change the given angle to an equal angle expressed to the nearest minute. -7.48 -7.48 = • This question: 1 Convert the following radian measure to degree measure. 5.381 radians = 5.381 (Round to the nearest hundredth as needed.) • Quiz: Week 7 Quiz Question 8 of 12 > This quiz: 12 point(s) po This question: 1 point(s Find the measure of the least positive angle that is coterminal with the angle given. Then find the measure o… • Find the volume of the regular pyramid with a square base of side 93 in and h = 120 in. h side V = in3 (Round to the nearest thousand as needed.) • Find the volume of the sphere for r = 0.869 ft. V = At 3 (Round to two decimal places as needed.) • Find the total surface area of the right circular cylinder with r = 672 ft and h = 242 ft. h r A= f1 2 (Round to the nearest ten thousand as needed.) • . 1. Consider (R2, 0, O) where (a1, a2) @ (b1, b2) T = (a1 + 61, a2 + 62) a (a1, a2) = (aal, a2) T Is this a vector space? Justify your answer. • Please answer this question. Thank you.. O OH 4 5. Let A = -1 0 and let b = 4 -4 (a) Find the rref(A). (b) Find the rank of A. (c) Find a such that b is in the column space of A. • Please answer this question. Thank you.. 4. Prove or disprove that the following sets are subspaces of the space of 2 x 2 matrices. (a) SI is the set of 2 X 2 matrices A such that: 0112 = â€”a21 and a… • . 2 0 2. Let A = 2 1 and B (a) Find (AB) T, BT AT and ATBT. (b) (AB)-1, B-1A-1 and A-1B-1. • . 7 (T&B 18.2). [10%] Social scientists depend on the technique of _regression_, in which a vector of observations of some quantity is approximated in the least squares sense by a linear combina… • Please I need them done on a paper. Question 1: Write the line passing through P = (1, 1) and Q = (-2,3) in vector form, parameterized form, point-slope, and slope-intercept forms. Question 2: Conside… • Find the volume of the cube with e = 7.09 in. e V = in3 (Round to the nearest whole number as needed.) • Calculate the area of the cross section of the airplane wing, using the trapezoidal rule. 1.00 ft 0.63 ft 1.05 ft 1.15 ft 0.51 ft 0.00 ft 1.00 ft 0.75 ft The area of the cross section of the airplane … • Find the volume of the right circular cone with r = 23.3 cm and h = 5.82 cm. Ih H V ~ cm (Round to the nearest ten as needed.) • Question needs help. 6 (Inspired by T&B 16.1). Let Q and A be 11 X n floating point matrices (meaning Q = f1(Q) and A = fl(A)), and suppose Q is unitary. Define f(A) = QA in exact arithmetic, and … • Problem 5: Suppose you have this matrix: 0 – 2 HA X3 = WOr Column vectors a1 az a3 A X ENG EK 103: Spring 2022 : Problem set 4 (Thurs) Mar 3, 2022 (a) Consider the column vectors a1 , d2, d3, and a, i… • -2 -1 -1 G = – 2 Column vectors g1 92 93 ENG EK 103: Spring 2022 : Problem set 4 (Thurs) Mar 3, 2022 (a) Fundamentally, do the vectors g1 , 92 , and g3 reside in IR , IR? , IR, IR space, or something… • Part a and b. Problem 2: Consider the 2 matrices: A = and G = HON -3 2 (a) Using the 2×2 shortcut formula in Lay, Ch. 2.2, find the inverse of A (b) Find the inverse of G using row reductions: Gauss-J… • Problem 1 (part 1): Consider the matrix: 1/2 1/V2 1/2 0 V = 1/2 0 1/2 -1/V2 (a) Calculate VTV and VVT . If the product is impossible, indicate so I (b) Are V and VT inverses of each other ? Justify yo… • . 1. Let T be the linear transformation on C3 deï¬ned by Tel 2 62, T52 = 33, and T63 = 61. Here the e,- are, as usual, the standard basis vectors. (a) (b) (C) (11) Give the matrix representation [T… • . 5. Let F : R and suppose that dim(V) is odd. Show that every linear transformation T 011 V has a real eigenvalue and a real eigenvector. • . Problem 4 [7 points (7, +2 extra credit)]: Eigenvector Multiplication Let matrix A have eigenvectors vo, V1, v2 associated with eigenvalues: do = 4, 1 = 2, 12 = 0. Suppose some vector x has: x = 1… • . Problem 2 [20 points (9, 11)]: Computing Eigenvalues and Eigenvectors -2 0 0 i Find the eigenvalues of A = -110 18 126 -20 -3 21 -13 8 ii Find the eigenvalues of B = 28 17 Additionally, find the e… • . Problem 3 [29 points (9, 6, 4, 4, 6)]: Ecology Consider an ecological system of plants, rabbits and foxes. Let’s assume that when one group ‘eats" another it is able to produce an equal numbe… • < Question 1, 2.5.7 E Homework: Section 2.5 Homework The widths of a kidney-shaped swimming pool were measured at 1.0-m intervals, as shown in the figure 7.2 m 6.3 m 5.2m 4.9m on the right. Calcula… • Apply the Gauss-Seidel method to exercise 1. 5. Apply the Gauss-Seidel method to exercise 3. Â . In Exercises 1â€” 4, apply the Jacobi method to the given system of linear equations, using the init… • Calculate the area of the cross section of the airplane wing, using the trapezoidal rule. 1.05 ft 1.25 ft 1.00 ft 0.63 ft 0.55 ft 0.00 ft 1.00 ft 0.85 ft The area of the cross section of the airplane … • Homework: Section 2.5 Homework Question 3, 2.5.9 HW Score: 1 < Points: 0 The fuel tanks for airplanes are in the wings, cross section below. The tank must hold 5400 lb of fuel with density 42 Ib/ft… • The widths of a kidney-shaped swimming pool were measured at 1.0-m intervals, as shown in the figure 7.3 m 6.2m 5.3 m 49m on the right. Calculate the surface area of the pool, using the trapezoidal ru… • Simplify Â ( provide the detailed solution ) Â 1) Â 50 – 10 / 12 + 8 Â 2) Â 30 + 8 [ 6^2 – 4 ( 3 – 1) ] / 4 – 6 Â 3) Â Â 268 / 4400 * 156 / 366 Â 4) Â 8600 ( 1 – 0.27 * 226 / 360 Â 5) Â ?… • . 6 (Inspired by T&B 16.1). Let Q and A be :1 X n floating point matrices (meaning Q = fl(Q) and A = HUD), and suppose Q is unitary. Define f (A) = QA in exact arithmetic, and let f(Q) = f1(QA) … • . 5 bonus points. Determine a rational function nEm nn(Em) = an + BnEm where an and B, can be any functions of n, such that nn(Em) Z en(Em), whenever an + Bnem < 1, and such that n(Em) – en(Em) =… • Please help to solve question 7. The previous question (Question 6) is also attached here. Â . 7. Keep the notation as in the previous question. (a) Show that Z[m]x = {:\|:1, :\|:w , :\|::.u2 }. (b) Show… • . 3. Let k be a field, and consider the polynomial ring k[x, y] in two variables x, y over k. Let R : = kx, xy, …, any"-1 , … ] be the subring of k(x, y] generated by R and the monomials xy… • Calculate the area of the cross section of the airplane wing, using the trapezoidal rule. 0.61 ft 1.05 ft 1.25 ft 1.00 ft 0.57 ft 0.00 ft 1.00 ft 0.85 ft The area of the cross section of the airplane … • Hi! I have this question for my assignment and I don’t know where to startÂ . 6. Let A be an m x n matrix with m s 11. Suppose that there exists a square matrix B of order m such that the ï¬rst m … • I don’t know how to turn this into a radical.. Find the characteristic polynomial and the eigenvalues of the matrix. The characteristic polynomial is 24 – 82 + 3 . (Type an expression using ) as the v… • please help this linear algebra Â questions with clear steps snd details, thank you so much • . (a) X = [-1, 1] and f (x) = 1/x; (b) X = [0, 2] and f (x) = min {x, 1} ; (c) X = [0, 2] and f (x) = x if x < 1 and = 2x otherwise; (d) X = [0, 1] and f (x) = 1 if x # 0.6 and = 2 if x = 0.6; (e… • Use Sirug or Victoriano Approach in solving the simplex method, Thank you! • Need help with some conceptual question in linear algebra • Working on some practice quizzes on linear algebra need help with these. • The question needs help, thanks! Â . 3 (T&B 14.2). (a) [6%] Show that (1 + O(Em))(1 + O(Em)) = 1 + O(Em). The precise meaning of this statement is that if f is a function satisfying f(Em) = (1 + O… • . Let X E RDXN be a data matrix whose columns are the vectors x1, …, IN E RP, i.e., X = [x12 … xx]. Let u; be the left singular vector of X associated with the ith singular value, where 01 2 02 … • . (2 pts) Let A E Rmxn with m > n have rank n. Show that On (A ) 1/A+ 1/2′ where on (A) is the smallest singular value of A and JA+], denotes the matrix 2-norm. • Show your COMPLETE solution following the STEP BY STEP procedure. Use excel to show your solution. Â Â Â Please provide an excel file to better understand. Please do follow the steps in the sent re… • I just need help finding c.. 5. Find bases for the kernel and image of the linear transformation L : R5 – R3 defined by L(r) := TA for 2 1 0 A := 2 -1 3 5 Find c so that (-4, -1, c) is in image(L). Th… • Can you solve this so I can see what you did?. 3. Suppose LE)E = CEAE-E with 7 9 15 AFE: -2 -2 -10 1 -1 1 For V the second basis in problem 2 above, if ry := (47, 25, 17), find IE, L()E, and L(E)v. • Please see attached. Let A be an m x n matrix with linearly independent columns. For { = 1. …, n, define ") as the solution of the constrained least squares problem minimize subject to er = -1…. • . 6. Compute the determinant of each of the following matrices and indicate the method being used for each. -3 OO O 0 a. -3 2 2 -2 -19 1 104 too sill ban A mot soulsmogis sidi bydil O 3 A pulushe of… • How do I solve this?Â Â . 1. Which of the following matrices induce isomorphisms of R"? (a) OHH NNO WOW (P) (3 8]. 1. Let S, T, and U be linear operators on R2 defined by Calculate the matrices … • How do I solve this?Â Â . 1. Let S, T, and U be linear operators on R2 defined by Calculate the matrices associated with the following linear transformations. • How do I solve this?Â Â Â . 8. Let 01.03.”..0″ be :3 real numbers not all of which are zero. Show that the function x1 :2 f : " 013:! + (13;: + " ‘ +6!ch II x? is linear from R" t… • How do I solve this?Â Â Â . 3. A furniture manufacturer produces two types of bookcases. The first type requires 10 square feet of wood, 40 nails, and 2 hours of labor. The second type requires 30 s… • At Aston University, four instructors teach Foundations of Business Analytics – a large compulsory module for more than 600 first year undergraduate students. The Aston timetabling team have split the… • Avalon Computer Depot Downsizing Problem Product Code Product Type Brand Cost to Liquidate (Â£ 1000s) Cost to Restock (Â£ 1000s) Space Required (m 2 ) Avalon Computer Depot is currently in the process… • No online solvers will give a good rating please, and Thank you. • Suppose that matrix A does not have a full column rank. How would you construct an orthogonal projector on the column space of A? • Dear Tutor: Please solve the following two vectors problem (they are related to each other). Explain your reasoning and show your work. Thank you. Â . Your quadcopter is on a special mission: inspecti… • Find the change-of-coordinates matrix from B to the standard basis in R". B= 6 – 9 . . . PB. â€” The set E = {1 â€” t2, â€” 2! + t2, 1 â€”tâ€” t2} is a basis for P2. Find the coordinate vector of… • Dear Tutor: Please solve this vector problem. You will need to use the answer from #4 (which I have included in the image) to solve this question. Explain your reasoning and show your work. Thank you…. • Please answer these linear algebra questions please show every working and details explaning how to solve the problems so I can learn how to solve them alsoÂ . 11. In each part, solve the linear syste… • Please answer these linear algebra questions also please show every little details and explain how to solve these problems so I can understand how to solve them also please and thank youÂ Â . In Exer… • Let u = (1, 2) and v = (-1, 3) . Design a projector P E R x2 onto the space S1 = (u) along the space S2 = (v). . Provide the entries of your projector . Show that your projector satisfies P2 = P. . Gi… • . Question 10 The function given in each graph below is f (x) = Ixl. Sketch the graph of the indicated new function and provide its algebraic formula. You MUST also show where the highlighted points… • . Problem 1 0 points possible (ungraded, results hidden) Using Cauchy-Riemann equations, show that the function f (2) = (2 + 4) is differentiable everywhere.. Problem 2 0 points possible (ungraded, … • . For the following linear programming problem: Max -0.5A + 4B s.t. -2A + 3B s 8 (Constraint 1) 6A – 2B – 5 (Constraint 2) A 2 1 (Constraint 3) A + Bs 5 (Constraint 4) A, B 2 0 (Note: IN ORDER TO GE… • Hi, in the below question, we are asked to guess the approximate valeus of U and D matrices, which are obtained by diagonalizing the covariance matrix of the dataset. There is no dataset for this ques… • . Given the following all-integer linear programming problem: Max 3361 + 10362 SI. 2951+ X3235 x1+ 63C25 9 161- Â£222 x1, x2 2 0 and integer a. Solve the problem graphically as a linear program. b. … • Graphical Solution Method The Graphical Solution Method cannot be used when…. Select one: a. the feasible region is unbounded. Â b. the decision variables must be integer. c.there are more than 2 de… • . 3. a) Let x is n-vector and p and q are two constant n-vector. f (x) = llax – pll– llax – all is f(x) linear, or affine or neither? Justify your answer. b) Here’s a system of 512 linear equations… • . 4. a) Suppose that a and b are any n-vectors. Find a scalar B so that (a – Bb) vector and the b vector are perpendicular b) Solve the following equations using gaussian elimination with row pivoti… • . 2. There are two vectors for given x. x is a n-vector. So, y, and yz are also n-size vectors. y1 = mx+ 10x x Y2 = max + 10 Find m, and m, so that the squared distance d between y1 and y2 would be … • . 1. Consider a 4-vector x = (X1, x2, x3, x4)T. Find the matrix A such that y=Ax where a) y=(x1, x1 +x2+x3, x2+ x3 +X4)T b) .V = (X1 + X2, X3,X1’X2)T C) y = {X1 + X2, X3 + X4, "xix: ‘ X1; X… • Hi, can someone please solve below as fast as possible. And show the necessary steps THANK YOU!Â Â Â . 2 Two period model Consider a consumer who lives for 2 periods. She is young in period 1 and ol… • A company manufactures three goods, X, Y and Z, each of which is made from three types of input, A,B and C. Each unit of X requires 1 unit of A, 7 unitsofBand3unitsofC.EachunitofY requires4unitsofA,3u… • Let n > 2022 be an even number, and define A E RX" and b E " so that aii =1 for i = 1, 2 or 4 <i<n-1, a33 = 4, an3 = 03n = 1, ann = 4, aij =0 otherwise, and b3 = bn =0, and bi =1 fo… • no reference information need iii> iv> and v> Â . Problem 3 [29 points [9, ï¬, 4, 4, 5}]: Ecology lE’tltnsider an ecological system of plants, rabbits and foxes. Let’s assume that when one… • no reference information need Â Â Â . Problem 3 [29 points [9, ï¬, 4, 4, 5}]: Ecology,-r lC’xtrrnsider an ecological system of plants, rabbits and foxes Let’s assume that when one group ‘eats?… • . Problem 4 [7 points (7, +2 extra credit)]: Eigenvector Multiplication Let matrix A have eigenvectors vo, v1, v2 associated with eigenvalues: 10 = 4, Al = 2, A2 = 0. Suppose some vector r has: r = … • . Problem 2 [20 points (9, 11)]: Computing Eigenvalues and Eigenvectors -2 0 0 i Find the eigenvalues of A = -110 -18 126 -20 -3 21 -13 ii Find the eigenvalues of B = 8 -28 17 .Additionally, find th… • . Problem 1 [22 points (6, 6, 6, 4)]: Line of Best Fit The following scatter and table describes the time spent studying and test score of 5 students: 100 95 test score (b) 90 85 0.5 1.0 1.5 2.0 hou… • Let A = {1, D, E), B – {2, 3, 4, A, C, D), and the Universal Set U = {1,2,3,4,A,B,C,D, E,F,G}. Determine: AnB • Linear Algebra coding (Julia) problem needs help, thanks! Â Â . c [3%] Create a plot showing eM and y" for em for 32-bit floating point numbers for n = O to 1 million. 5 bonus points. Determine … • Linear Algebra Â . Find the value of a for which 3 a U = 6 is in the set O 5 H = span -2 3 • The replacement elementary row operation is the key to producing these echelon forms. Explain why this replacement step is “replace a row by adding to it a multiple of a different row”, and not “repla… • Suppose you have two matrices A A and B B both of the same sizes. And suppose both A A and B B can be reduced to the same RREF, say R R using only elementary row operations. Explain why there is a seq… • Find the equation of the line that passes through the two points ( – 2, 8) and 3, 1 . Write your answer in standard form. Preview • Find the slope of a line a. parallel and b. perpendicular to the line – 4x – y = 7. a. Parallel: Preview b. Perpendicular: Preview • Solve for a if the line through the two given points has the given slope. ( – 3, 3a) and ( – 2, – a), m = 8 • CHALLENGE ACTIVITY 6.6.1: Complex eigenvalues and eigenvectors. 376940.2343920.qx3zqy7 Jump to level 1 V 1 V Complete the eigenvalues and eigenvectors for A = 2 Enter eigenvalues in the order a + bi, … • Please explain each answer in detail.. 5. Let T be the linear transformation T(x} = Ax, where 100 A=020. ODD a. (2 pts) Describe the transformation geometrically. That is, what action does A have on e… • Please answer these questionsÂ Also for these questions I would like for you to state every single step you do to complete these linear algebra questions please include every detail so I can understa… • (9) Decide whether or not each set of vectors in R3 is a. subspace of R3. Explain your answer. (a) The set of solutions to a. system of the form Ax = 0, where A is a 3 x 3 matrix. (b) The set of solut… • (10) For the matrix A, find conditions on a vector b to determine when b lies in the column space of A. That is, find equation(s) which must be satisfied by the coordinates b1, b2, b3. 1 1 A = 0 1 2 0 • . Which of the following functions are linear transformations? 1. The function T[f(x)] = f(x) + f(-x), defined on P4, the space of all quartic and lower-degree polynomials. 2. The function T[A = A A… • . Which of the following sets are vector spaces? 1 0 . 2 0 1. The set of all 4 x 4 matrices A such that A 3 = 0 and 4 0 5 0 6 0 A 7 _ 0 . 8 0 2. The set of all differentiable functions f(.’13) suc… • (6) Suppose A has the following LU-factorization: 3 A = 0 2 0 3 Use back-substitution twice to solve the system Ax = b, where b = NON • (5) Find the LU-factorization of the matrix A, i.e., factor A = LU where L is a lower triangular matrix with I’s on the diagonal and U is an upper triangular matrix: A = 09 No O • . Problem 5) Find the range and null spaces of the following matrices 1 2 A = -1 -2 2 -4. 1 2 37 B = -2 4 1 4 0 5 • 1 3. Suppose that A = 0 0 0 0 (a) Compute B = AT A and find B-1. (b) Explain why Ax = b is inconsistent, and write down the normal equations of the system. (c) Find the least squares solution to Ax = … • . Problems 3) In this problem, you will show that every 2 x 2 orthogonal matrix is either a rotation or a reflection. (a) Let Q = [a b be an orthogonal 2 x 2 matrix. Show that the following equation… • I only need help with part (d) and part (e). Please only answer if you are sure it is correct. I have attached the map that is needed for part (e) below.Â Â Â . Suppose T : V – W and U : W – Z are l… • . Let G be a group and H be a subgroup of G. (a) [3 points] For any g E G, prove that the set gHg 1 = {ghg 1 : he H} is a subgroup of G. (b) [3 points] For any g E G, prove that gig-1 ~ H. (c) [2 po… • Help me please, Kindly answer this with complete solution and clear explanation. Thank you. • write the equation of the line in slope-intercept form that is perpendicular to the graph of y=-2x+1 and goes through the point (2,-1) • Linear algebra question. will upvote if correct. thank you.. Let A and B be 3 x3 matrices, with det A = 5 and det B = 8. Use properties of determinants to complete parts (a) through (e) below. a. Comp… • Linear algebra question. will upvote if correct. thank you. Â Â . Let A and B be 3 x3 matrices, with det A = 5 and det B = 8. Use properties of determinants to complete parts (a) through (e) below. a… • Linear Algebra question. will upvote if correct. Thank you. Â . In P,, find the change-of-coordinates matrix from the basis B = (1 – 2t +1,4 – 7t+ 5t",2- 2t+ 5t~} to the standard basis C = (1,t,t… • i’m trying to solve this, but i’m stuck Â Â . The solution set of the linear system W + 3y – 14z = -10 -2 w + – 6y + 25 z = 17 may be written as {v1 + su, + tus : s, t ( R] for some vectors v1, 12, a… • The weight of a newborn isÂ 7.4 pounds. Consider the linear function that models the baby’s weight, WW, as a function of the age of the baby, in months, tt. The baby gained one-half pound a month for … • The weight of a newborn isÂ 7.47.4 pounds. Consider the linear function that models the baby’s weight, WW, as a function of the age of the baby, in months, tt. The baby gained one-half pound a month f… • I don’t know how to do C, D and E. Problem 3 (16 points) A n-dimensional permutation is a function over an n-dimensional vectors that reorders their elements in a fixed way. For instance, one 6-dimens… • I don’t know how to do C, D and E. I would appreciate if anyone could help me!. Problem 3 (16 points) A n-dimensional permutation is a function over an n-dimensional vectors that reorders their elemen… • Let V = Span{(1,2,3),(1,0,2)} and W = Span{(0,2,1),(1,1,1)}. Determine the relation between V and W, i.e. whether V subset of W, or W subset of V, or neither • Let V = Span{(1,2,3),(1,0,2)} and W = Span{(0,2,1),(1,1,1)}. Determine the relation between V and W, i.e. whether V subset of W, or W subset of V, or neither. • thx for help. The space of continuous complex functions defined on an interval [a, b] with the inner product (flg) = S f(x)g(x)dx (for any two functions f(x) and g(x) belonging to this space) is a Hi… • thx for help. Which of the following statements are true? Select one or more: O 1. An inner product needs to be defined before a complete space could be defined. U 2. An inner product needs to be defi… • thx for help. The solution u(x, t), -00 < x < co, of the wave equation utt = Uxx with the initial condition u(x, 0) = 0, u,(x, 0) = Uo cos(kx), where v_0 and k are some constants, can be found u… • thx for help. The solution of the 3D Laplace equation inside a sphere of radius R with the boundary condition (in spherical coordinates) u(r = R, 0, ) = sin(0) sin() can be expressed using only the fo… • thx for help. The orthonormal basis of eigenfunctions of the Laplace operator oz/Ox defined on the interval x E [0, 1] with the boundary conditions f(0) = f(1) is: Select one: O 1. em = V2cos[(x/2 + m… • thx for help. The wave equation u” = 0 2am for a function u(x, I) defined on the interval x E [0,1] and t 2 0 is to be solved with the boundary conditions u(0, t) = 0, 110′, t) = A sin(wt). Which … • thx for help. The first three orthonormal polynomials in the Hilbert space [2 [-co, co ] with the inner product (flg) = _of(x)g(x)w(x)dx, w(x) = e" ? are leo) = (2n)-1/4, le1) = (2n)-1/4x, \|e2) … • tthx for help. The first four Legendre polynomials leo >= , le1 >= \ 3x, le2 >= s 215 (3×2 – 1), Je3 >= 17(5×3 – 3x) are orthonormal on the interval [-1, 1] with the inner product (flg) = … • Decide whether the pair of lines is parallel, perpendicular, or neither. 12x – 8y = 15 8x + 12y = 19 O A. Neither O B. Perpendicular O C. Parallel • Please solve all of these questions. Thank you! Â 1. Given the matrix below, find A T B. Choose the correct answer here. ——————————————————————————… • 1 2 3. Suppose that A = 0 1 0 01 and b = 0 0 0 (a) Compute B = AT A and find B-1. (b) Explain why Ax = b is inconsistent, and write down the normal equations of the system. (c) Find the least squares … • . a) Show that the matrix A = is normal matrix and find matrix A unitary diagonalisation. • . For the following find (I, y) ER? sets and outline graphs of these sets: (a) (x tiy) = 1. (b) Re((1 + i)(x +iy)) <0. (c) 2<\|r+iy – 3 <5. • Show that the matrix N= 1 is normal. Find the unitary diagonalization OF the matrix N . 6) Calculate the singular value dispersion OF the matrix N. • . Find every vector r = (1, 12, 13, TA)ER that are perpendicular to the vectors: e, r, tER e = (1,1,1,1), r = (1,2,3,4), t = (1,4,9,16). Form a matrix and use Gaussian elimination. • . a) Show that the matrix A = is normal matrix and find matrix A unitary diagonalisation. 1 b) Calculate matrix A = SVD (A = UEV • . We are looking at two matrices y E R3x2 and A E R2x3 3 2 Why can’t YA = be true?’ • Math Linear System question Â Read attachment and solve correctly!. 2. Solve the linear system using the elimination method. Check your solution. 9x+1y=51 2 7x+1y=39 3 • Linear System Math QuestionÂ Read the attachment and solve. 3. Solve the linear system using any method. Check your solution. y=3x-27 X-1y=9 3 • Math word problem Read attachment below and ONLY choose one word problem.. 4_ Choose ONLY ONE WORD PROBLEM to complete. Circle the problem you are attempting. Option #1 (Level 3) Option #2 (Level 4) A… • Read attachment and solve quickly Â . 1. Solve the linear system using the substitution method. Check your solution. 3x+y=-13 -2x+5y=-54 • Let 9 42 -30 A = -4 -25 20 -4 -28 23 (a) Find an eigenvector for A corresponding to the eigenvalue 1. (b) Find a basis for the eigenspace of A corresponding to the eigen- value 3 • Direction: State whether the given problem is FUNCTION or NOT FUNCTION. Â Topic: Graphic of Functions. D. State whether the given problems is FUNCTION or NOT FUNCTION. 1. {(6, -3), (7, 4), (-7, -2), … • . The set C of all complex numbers, with the usual operations of addition and real scalar multiplication, forms a vector space. Elements of C can be written in the form x + iy, where i is the ima… • . Lecture attendance, laptops in lecture, and ï¬nal grade. A study collects data on a large number of students in a lecture course, with the goal of predicting the effect (or at least the associati… • A business has a (predicted) sequence of monthly liabilities (payments it must make) over 10 years, given by the 120-vector l. The business will purchase 20 different bonds, with quantities given by t… • can you help me with Introduction to Linear Algebra? 14) Â . 9) Given A= 2 Express A and 4" as a product of elementary matrices (EM): Solution: (Write out all EM ). Read the lecture notes) Answer… • can you help me with Introduction to Linear Algebra? Â . 5) Use Gauss-Jordan elimination to solve the system. Discuss the value of a and b such that the following system has a) no solution, b) Unique … • can you help me with Introduction to Linear Algebra? Â Â . 1) What single elementary row operation that will create a 2 in a lower left corner of given matrices and will not create any fractions in a… • . 12 O -2 2 -6 5 0 7) (5) For the matrix A= calculate tr(2A -4tr(-3A) x A) -1 1 -3 0 10 14 8 5 Solution: • . 5) (16) Use Gauss-Jordan elimination to solve the system. Discuss the value of a and b such that the following system has a) no solution, b) Unique solution or c) Infinitely many solutions. d) Sol… • . 2. Let G = (N , A) be an undirected graph. A node cover is a. set S Q N such that every arc in A has at least one endpoint in S. (a) Formulate an integer linear program to [incl a node cover for G… • . If A is a m x n matrix and Ax = b has a solution for each vector b in Rm then the linear transformation T : R" -> R" defined by T(x) = Ax is onto. O True O False • Hi, I have questions. I need your help. Thanks. Â Q1: Â Q2: Â . Is it true that every finite group is a quotient of some S”? La. if G is finite can one find an onto homomorphism S”L â€”> Gfor… • In this extra credit problem, you will derive a recurrence that counts the number of different ways to multiply n matricies. For example, for 3 matricies, it is easy to see that there are only two way… • Suppose that A = and b = -NAN 10 19 2 (a) What is the determinant of A? (b) If Ax = b, use Cramer’s rule to find r2. • Find a matrix A whose inverse is A- = 2. Suppose A is invertible. (a) Show or explain why A’A is also invertible for any matrix A. (b) Show that A-1 = (ATA)-1AT by computing the product of A and (A… • please answer the questions, thank you! Â 3.3 Â . 1. [41 Points] DETAILS HOLTLINALGZ 3.2.001. Perform the indicated computations when possible, using the matrices given below. (If an answer does not … • Hi!Â I was working in this homework, but I would like to double check it. Could you please solve it? Thanks!Â Â Â . Let B = {;]. . Letv = [v; be the vector whose coordinates in the basis B are [v]B… • Hi!Â I was working in this homework, but I would like to double check it. Could you please solve it? Thanks!Â Â . 2 â€”1 x Consider the matrix C=[3 1 0] The determinant of C is of the form detC = ax… • Hi, I was working on my homework, but would like to double check my answer. Could you please solve it? • Hi!Â I was working in this homework, but I would like to double check it. Could you please solve it? Thanks!Â Â . apply) Select which of the following collections of vectors are bases of R3 (Select … • Hi, I was working on my hwk, but would like to double check my answer. Could you please solve it? • Consider the system of linear equations.Â ,âˆ’4ð‘¥+6ð‘¦=4 2ð‘¥âˆ’3ð‘¦=ð‘˜Â Determine the constant ð‘˜ so that the system of linear equations has infinitely many solutions • . 4. (2 points) Give an example of a 2 x 4 matrix A such that im(A) is the line L = : 5x – 2y = 0 in R2. No justification is needed. • . 4. (2 points) Give an example of a 2 x 4 matrix A such that im (A) is the line L = : 5x – 2y = 0 in R2. No justification is needed. • . While we mainly think of Leibniz as one of the discoverers of calculus, he did some other pretty sophisticated mathematics that was in some sense well ahead of his time. Read Leibniz’ "On Det… • . Use the diagram to answer the question. Â© 2016 StrongMind. Created using GeoGebra. Which transformation maps polygon A BC D to itself? 0 a 90Â° counterclockwise rotation about its center a 90Â° c… • Please answer following TRUE/ FALSE question in detail with explanation. Thank You Â . 3.12 If A and B are 2 x 2 matrices, (AB)2 = A2B2.. 3.15 Assume that A and B are matrices such that AB is defined … • Please answer in detail. Thank You Â . 3.44 Find all 2 x 2 matrices B such that AB = BA, where N A = • Please answer in detail. Thank You Â . 3.38 Define a transformation 7 : 2 _> p2 by the following rule: T(X) is the result of first rotating X counterclockwise about the origin by – radian, then mul… • Please answer in detail. Thank You Â . 3.34 Define a transformation 7 : R2 _> p2 by the following rule: T(X) is the result of first rotating X counterclockwise by /6 radians and then multiplying by… • Please answer the following True/False questions in detail with explanation. Thank You Â . 3.2 It is impossible for a linear transformation from R2 into R2 to transform a parallelogram onto a pentagon… • Please answer in detail. Thank You. Figure 3.5 is attached. Â . 3.7 Find a 2 x 2 matrix A such that multiplication by A transforms the parallelogram on the left in Figure 3.5 onto the parallelogram on… • Please help me with a brief explanation. Â Thanks! Â . Suppose V is a finite dimensional inner product space. Let T’ be a self-adjoint operator on V. If T’ is a projection, then T’ is an orthogonal pr… • Dear Tutors, Your help and very brief explanation would be very much appreciated. Â Thanks! Q2 for my records.. Let 7 be a normal operator on a finite-dimensional complex inner product space. If all t… • A convenience store sells Gatorade at 2 per bottle and Powerade at 4 per bottle On a hot day they sell 412 worth of Gatorade and Powerade combined The number of Gatorades sold was 18 more than 2 times… • . Consider the 2D nonlinear system: (phase portrait given in the figure above) ic = x(1 – 92 – 22) – 3y y = y(1 – 932 – a2) + 520. 1 Let P1 = {(x, y) ER2, x2 + 942 5 2} P2 = {(x, y) ER2, NIK <x2 … • I need an explanation! Thank you. Week 7: For the following exploration, your matrix A needs to be relatively random looking. So, in particular it should have: . not too many 1’s or 0’s, Â«- no… • A Linear Algebra Question Needs help, Thanks! Â . 4. When analyzing a computation that is the result of a large numbers of floating point operations, we often end up with expressions like 3 = x(1+ 81)… • . Simplify the complex rational expression. a+1 – 1 a +3 • . 3. Divide the rational expressions. Simplify the quotient, if possible. (5 pts) y/2 -9 2y2 -y – 15 3y2 + 10y + 3 2y2 +y – 10 4. Perform the indicated operation and simplify the result, if possible… • 3. The union of the zero vector and the set of all eigenvectors with an eigenvalue 2 is called the 2-eigenspace. The dimension of the 2-eigenspace, or equivalently the maximum number of linearly i… • Hi can someone show my the step by step answer for this question? using Sirug Method Â . Try This Out! Simplex Maximization Problem Maximize: Z= 4X1 + 6X2 Subject to: -X, + X, 11 X1 + X2- 27 2×1 + 5… • Problem 4. (30%) Answer the following questions: (a) (10%) Let A c Remx3, b e Rmx]. Consider the following optimization problem which is not an LP in standard form: MAX ri + 212 + 4rs sub… • . ENTER FOR ENERGY PROGRAM The city of Chicago school system is considering launching a free lunch program for students in kindergarten through 5" Grade that would help teach healthy cating hab… • . EXERCISE 5.1 7. MENTAL MATH State the relation between the quantities involved. 1. {(3, 1), ( 4, 2), (5, 3) , ( 6,4 ) ,… } -8 -1 -6 -5 4 – 2. {(-2,2),(-1.5, 1.5), (-1,1), (0,0), (1,1),(2,2) }… • Advanced Engineering Mathematics By Alan Jeffrey Â Please solve all of the questions in the attached image. Thank you! Â . In Exercises 12 through 15 express the functions of u, v, and w in modulus-a… • Determine if each statement is True or False. You do not need to justify your answer. Just write T or F to the right of corresponding question label.. (i) The set of all solutions of a system of m hom… • Please help to solve the questions! Thank you! Â Â . 1. [-/1 Points] DETAILS HOLTLINALG2 3.3.001-004.WA.TUT. Determine if A is invertible and, if so, compute A . (If the answer does not exist, enter … • Please help to answer the questions! Thank you! Â Â . 1. [41 Points] DETAILS HOLTLINALGZ 3.2.001. Perform the indicated computations when possible, using the matrices given below. (If an answer does … • Compute in online OCTAVE if needed.. 5.) Solve the following system, if possible: x, +3x, – 2×3 + 2x, =0 2x, +6x, -5x, -2x, +4.5 -3×6 =-1 5x, +10x, +15x =5 2x +6.x, +8x, +4x +18x =6 6.) Solve the fol… • . Given an m by n by p tensor T, how do you decide if T has rank 1? • The four null spaces include the null space, the row space, the column space, and the left null space First calculate the RREF then find the bases. Find bases for the four subspaces corresponding to t… • . 1. In Lecture 8, we defined the criterion function for a 2-class Perceptron Learning problem (in augmented space) as: N J(w) n=1 Prove that J (w ) is convex. Hints: (i) Write J(w) replacing the in… • Please refer to the attached. Not much detail is required.. Question Part1 What is the rank of the linear transformation, 7 : R4 – R3, given by T(x) = Ar. For the matrix: A = 3 4 0 1 1 1 Select one or… • Once of the regression co-efficient of two perfectly correlated variables 0.5 , hence the other regression co-effecient is • Rewriting the equation V = t + a/ t – a in terms of V • A company that manufactures radios has determined that retailers are likely to purchase x radios at a price of x = 10 – x / 1000 per unit If the company’s fixed costs are R5 000 and their variable cos… • If the demand function for a certain product is P = 60 – 3,5Q where P and Q are the price and quantity respectively, determine the point price elasticity of demand if the price P equals R18. • I had some questions about basis and linear maps, so I have constructed these statements and please would like to know which is true and which is false.. Question Which of these statements are true an… • Find an article or other resource about the application of linear algebra concepts in the real world. For instance, search for the real-world applications of eigenvectors and eigenvalues, matrix algeb… • A company makes two products, A and B. Product A requires 12,5 hours of machining time and 30 minutes of finishing time per unit. Product B requires 10 hours of machining time and one hour of finishin… • The demand function for a commodity is Q = 6000 – 30P. Fixed costs are 72000 and the variable costs are 60 per unit produced. What are the equations for the Total Revenue and Total Cost in terms of … • 3 2. Jack drove at 50 km/h from Smithville to Dry Gulch. From Dry Gulch to Streetsville he drove at 80 km/h. The whole trip was 550 km and took 8 h. How far is it from Dry Gulch to Streetsville? • If Matlab does not work, please use GNU Octave.. 1.) Enter the following matrices into Matlab. 5 33 42 -7 A= 27 -9 and B = 10 4 3/4 5/6 2.) Compute AB and BA. 3.) Compute 2A-B’ and express your answer… • please help me!!!. ecore BIOL 1011K – LAB PACKET Create a line graph for each Data Table (1, 2, and 3) on this Combined Data Graph. When you finish, you will have 1 graph with three separate lines. Us… • Use the following diagram for exercises 1-5. G 10 mm H 6 mm 8 mm 1. How long is the side opposite angle G? 2. How long is the side adjacent to angle G? 3. How long is the hypotenuse? 4. What is the ta… • Please show work and solve all if possible.. Exercises 1.) Enter the following matrices into Matlab. 5 33 4 2 -7 A= 27 -9 and B = 0 4 3/4 5/6 2.) Compute AB and BA. 3.) Compute 2A – B’ and express … • A matrix is A is nilpotent if Ak is the 0 matrix for some k. A. Show that if an n Ã— n matrix A is nilpotent, then rank(A) < n. (Hint: Prove the contrapositive, “If rank(A)=n, then A is not nilpote… • Diagomiize the matrix A: 3 "1Â° . ([3) Explain how to quickly compute A”. (You don’t need to actually do it.) (c) Give the mlutiun to the differential equation x’ = Ax with x(0) = m • Problem 7. (10+10+10 points) Let 0 be a real number that is not an integer times 7. Let M be the 2 x 2 matrix representing clockwise rotation by the angle 0. (a) Explain geometrically why M cannot hav… • Determine the maximum value of the objective function subject to the set of inequalities above. In the graph below the following set of in equalities is represented with the feasable area in greg 10 0… • How do I find the 2nd and 3rd columns? ?Â . J 5. ’ 17 â€” 12 – 8 Let A = 378 240 176 . Find the second and third columns of Aâ€”1 without computing the ï¬rst column. 106 65 54 How can the second and… • The demand and supply functions for free-range meat are Pd= 50 – 0,6Q. and Ps = 20 + 0,4Q where P is price and Q is quantity. . If free-range meat are subsidised by R4, the equilibrium price and quant… • If the demand function is P = 80 – 0.5Q where P and Q are the price and quantity respectively, determine the expression for price elasticity of demand in terms of P only • Consider the market defined by the following demand and supply functions, Pd = 50 – 3Q. Ps = 14 + 1.5Q where P and Q are the price and quantity respectively. The producer and consumer surplus of the p… • Not yet answered Question 13 Marked out of 6.67 If the demand function of a commodity is \(Q=80- 2,5P,$$ where P and Q are price and quantity respectively, determine the price elasticity of demand whe… • Not yet answered Question 15 Marked out of 6.67 The demand and supply functions for free-range meat are Pa = 50 – 0, 6Q and Ps = 20 + 0, 4Q, where P is the price, and Q the quantity. The equilibrium p… • Suppose you are given an n Ã— n symmetric matrix of positive numbers, D with 0’s along the diagonal. Is it always possible to find a collection of n points (xi, yi), i = 1, . . . , n so that the Eucli… • Not yet answered Question 1 Marked out of 6.67 Solve the following system of linear equations: – 3y + 62 = 21 3x + 2y – 52 = -30 2x – 5y + 22 = -6 (30 The sum of values of x, y, and z of the solution … • can anyone solve number 36?. 34. Manufacturing A firm manufactures three products: tables, desks, and bookcases. To produce each table 36. Fishery A certain lake has three species of trout: requires 1… • can anyone solve number 34?. 34. Manufacturing A firm manufactures three products: tables, desks. and bookcases. To produce each table 36. Fishery A certain lake has three species of trout. requires 1… • Â determine the number of zeros (i.e., x-intercepts) of a quadratic function, using a variety of strategies (e.g., inspecting graphs; factoring; calculating the discriminant)Â Â Problem: Investig… • just finish Â b Â c Â d. Let Z denote the set of integers. If m is a positive integer, we write Zm for the system of "integers modulo m." Some authors write Z/mZ for that system. For comple… • Prove or disprove: If A is an uncountable set and B subset A is a countably infinite set, then the set difference A\B is uncountable. • 3.6 ab page 62 Find the distance between the pair of points given a) Pi(1,3,0), P2(4, 1,2); b) Pi(-1,2,-3), P2(4, -2, 1 ). c) P.(4,-2,-2), P2(3,2, 1 ). Ex.3.7, page 63 Let Pi(3,4,0) , P2(7,-8,5) be… • Find the midpoint of the segment AB, a) A = (4,3,5), B = (-2, -1,2) use the formula 2 2 2 5. Find the distance between each pair of points: a) Pi(1,3,0) and P2(4, 1,2); use the formula \| P1 P2\| = V… • Find the distance between the pair of points given a) P1(4,-2,1), P2(2,2,-3) use the formula \| P1P2l – V(x1 – X2)2 + (y1 – yz)2+ (21 – Z2)2. 2. Give in the component form a vector v that is represe… • Prove that the set {0, 1} x N is countably infinite. 2.) Prove or disprove: If A is an uncountable set and B âŠ† A is a countably infinite set, then the set difference A\B is uncountable. • . 6. Find the least-squares line that best fits the points (2, 3), (3, 2), (5, 1), and (6, 0). 7. Consider the data points (x1, y1), (x2, y2), . . . , (Xn, Un). Define the summations n n n n Ey = _y… • I don’t know how to solve B and C Â . (a) Find the reduced row echelon form of the matrix M. Your answers must be fractions (decimals are not allowed). 0 -7/3 -7/3 0 5/3 5/3 (b) Solve the system C1 – … • . Q4. Let Maxn (R) denote the n x n matrices with real entries. (a) Sketch a proof of the fact that Mnxn (R) is a vector space. (b) Prove that T(A) = A’ is a linear transformation from Maxn (R) to M… • I just need help with part B of the problem. Problem 4. (a) Find the inverse of the matrix A = 1 0 12 1 0 1 All = 100 100 4 0170-210 104 1 – 30 1 [100 /100 Q10 -210 O A = – 10 01 5-41 5 -4 (b) Use you… • . 2. Given two m x m matrix X and Y, where XY = YX. 1) Let u be an eigenvector of X. Show that either Yu is an eigenvector of X or Yu is a zero vector. (5 points) 2) Suppose Y is invertible and Yu i… • I’m trying to figure out the derivative of a Matrix, A, which is a function depending on t.Â Explicitly, how do I find dA/dt of (A(t))^2 ? The answer is not 2AdA/dt. The guide given with it is that … • . Problem 1: (a) Find det(B) and then find all the values of A for which det(B) = 0, where +5 -6 B = 0 1 – 7 -3 5 (b) Use a combination of elementary row operations and cofactor expansions to … • . 1. Given a matrix A with eigenvalues )1, . .. , An, calculate eigenvalues of the following matrices (show the procedure). (1) (21 + A2) -1 (5 points) (2) (3A + A-1 )2 (5 points) (3) ((3A + A-1)2 +… • Compute the determinants of the following two matrices. Say which row operation relates them, and deduce how performing that row operation changes the determinant. • . MATRIX ALGEBRA QUIZ 1.1-1.3 NAME Solutions SID # Instructions: Must provide complete answers to each problem including any procedures needed to receive partial/full credit, using methods covered u… • i need help. 3. Josie has 1.74 meters of pink ribbon and 2.35 meters of yellow ribbon. How much ribbon does she have? Show your work. Don’t forget to label your answer, or answer in a complete sentenc… • . Problem 1. (10 points) How many swaps does the Bubblesort algorithm shown in class perform on an input of the form A = [2, 1, 4, 3, 6, 5, …, 2, 2n – 1]. Express your bound in terms of an exact b… • How do you solve to find this?. find the Values Of ‘M’ aNd ‘N’. EXPLAIN YOUR ReaSONING. (N+12) M 114 • . 5. Find the least-squares solution to AT = b and the least-squares error associated with the solution for 1 1 0 -5 A = and b = 1 3 2 3 • . 3. Let {[1 = â€”21 and 1752 = _99 . Find an orthogonal basis for the subspace W = Span{u"1, u’Ã©}. â€”1 3 1 1 0 4.Letï¬1= 3 ,3: : ,andï¬‚’3= _21 â€”1 1 3 0 Let W = Span{u’i, â‚¬2,135}. F… • . 1. Let U be an m x n matrix with orthonormal columns. (a) Show that \| \|Uall = \|\|all for all a E R" so that the transformation defined by U preserves length. (b) Show that (Ux) . (Uy) = 0 if a… • . In this problem we will explore some general and practical implications of matrix rank. 1. Pick two different values 711, and n, each 3 or larger. Create two random matrices using numpy.random:… • I’m trying to solve this but i can’t get the right result • How do you show that a polynomial function has a real zero between two given numbers? Give an example to support your answer. • . Check here for instructional material to complete this problem. X 1 X2 Evaluate the following formula for p, – p2 = 0, p = 0.112878, q =1 -p, x1 = 60, X2 = 11, n, = 318, n2 = 311, p, =- -, and p2 … • A linear algebra question needs to be solved, thanks! Â . T&B Exercise 18.4. Explain why, as remarked after Theorem 18.1, the condition number of y with respect to perturbations in A becomes 0 in … • . Suppose a string is stretched with tension 1′ horizontally between two anchors at m = U and :r: = 1. At each of then â€” 1 equally spaced positions m; = k/ï¬‚, ls: = 1,. . . ,ï¬‚ â€” 1. we attach … • Two small planes start from the same point and fly in opposite directions. The first plane is flying 30 mph slower than the second plane. In 2 h, the planes are 540 mi apart. Find the rate of each pla… • (3 points) Suppose T: " – R" is a linear transformation. If, for any pair of orthogonal vectors v and w in R", the vectors T(v) and T(w) are also orthogonal vectors in R", then… • Determine the standard matrix for the linear transformation, T : R 2 â†’R 2 , that first reflects a vector about the horizontal axis (x-axis) and then scales it by a factor of 2. Â Â 2) For each … • can you help me with Introduction to Linear Algebra? Â Â . Linear Algebra 1 ) Use a calculator ( with matrix operation ) to solve the system for X, X,, X;, and x, 40 75 30 25 20 10 30 2) Find A given… • . Please please please answer it completely. Please Make sure your handwriting is legible. Please please please make it neat and understandable. THE EQUATION OF A LINE Comprehension Check State whet… • Please show the explanation how to get the result!Thanks. 3. Determine whether the following are subspaces of 2X2: 1. The set of all 2 X 2 diagonal matrices 2. The set of all 2 x 2 triangular matrices… • Please show the explanation how to get the result!Thanks. 1. Determine whether the following sets form subspaces of 2: 1. { (21, 202 ) 7’\| 21 + 202 = 0) 2. { (21, 202) 7 212 = 0} 3. { (21, 262) Tac1 =… • . 5. Determine whether the following are subspaces of P4 (be careful!): • Please show the explanation how to get the result!Thanks • . (7, 4, 1 ) [A]T = (9,8) (3, 2, 1) [A]T = (5, 4) (5, 3, 1) [A]T = (7,6) 4 9 3 2 [AIT = 3 9 8 -1 -1 2 10 8 6 2 10 8 -6R1 -2 – 3 1 7 3 -5R1 -1 -1 -1 0 0 0 8 16 8 1 2 6 12 6 0 All – A13 = -1 A12 + 2A1… • Homework 7: Standard Form This is a 2-page document! ections: Write each equation in slope-intercept form, then graph. Convert Graph x- y =3 X pe-Intercept Form: 5x + 3y = 6 • Question : Determine the state matrix, the [ð´] matrix, by linearizing problem #4 about equilibrium. You do not need to evaluate the terms numerically, and can use ð‘¥1ð‘’,ð‘¥2ð‘’, etc. instead…. • . Problem 7. (i) Find a linear map f : {0, 1 } – {-1,1} such that f (0) = -1, f (1) = 1. (ii) Find a linear map g : {-1, 1} – {0, 1} • Plot the three points on a graph X=0 Y=0 X=4 Y=2 X=9 Y=3 • Let T : R3 – R2 be the function T(x, y, z) = (2x ty – 3z, x ty – z). (a) Show that T is a linear transformation. (b) Compute a basis for the subspace ker T C R3. What is its dimension? (c) Compute… • . 1. Calculate the determinant for each matrix below. Clearly show all your steps in the calculations â€” numerical answers only will receive little to no credit. (Friendly note: if you make use of … • Roger’s class sold bottles of soda for$2.50 each and Mrs. Magoo’s class sold candy bars for $1.20 each. Together, the classes sold 63 items and earned$102.90 for their school.Â Â (a) write an… • Suppose that a matrix A has integer entries and an eigenvalue 0 < \|/\\| < 1. In this question, you will prove a big theorem: Any Aâ€”eigenvector of A has an irrational entry. (a) Suppose that… • Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation. (You’ll want to pick rows or columns with lots of zeroes if you can.)… • Compute the determinant of the following matrix by using cofactor expansion across the first row. Also compute it using cofactor expansion down the second column. (You should get the same answer eithe… • Solve the following linear system using elementary row operations on the equations or on the augmented matric: TI – 313 = 8 21 + 212 + 9.13 = 7 12 + 5.13 = -2 • For each of the following subspaces W of R", find the standard matrix of the linear trans- formation 7: R – R given by projection onto W. (Make sure you are working with an orthonormal basis o… • . We will continue to study the linear system in the above problem. Regardless of your work there, you should run Gaussâ€”Jordan elimination on this system in Python before starting this problem…. • How do I do this?Â Â . 1. Find a basis for the range space and the rank of the linear transformations induced by the following matrices.. 4. Calculate a basis for the kernal and the nullity of the li… • . Suppose that A is a 4 X 4 real matrix with characteristic polynomial det()\I â€” A) = A4 â€” 2A2 + 4. Suppose further that b is a 4 X 1 real vector such that the set {A4b, A31), A21), Ab} is linea… • EXERCISE 5 [25pt] (a) Implement the BFGS method by modifying the descentLineSearch.m function from tutorial 2. Make your implementation efficient i.e. avoid explicitly forming the inverse Hessian matr… • EXERCISE 4 (15pt] (a) Implement the Polar-Ribiere conjugate gradient method. Hint: Modify the descentLineSearch. m template from tutorial 2. Copy the relevant lines in your report. [2pt] (b) Apply Pol… • EXERCISE 2 [18pt] (a) Apply steepest descent with an appropriate line search to the function / in Ex.1 starting from 20 = (1, -1) and so = (-1,0), Plot the iterates over the function contours, State y… • EXERCISE 3 [15pt] (a) Implement the dogleg trust region method for strictly convex functions (with s.p.d. Hessian). Your implementation should return the Cauchy point whenever the gradient and Newton … • EXERCISE 1 [27pt] We are given a function f : R? – R f(x,y) = (y – cosx)? + (y -x). (a) Calculate the gradient Vf and the Hessian V2J. [2 pt] (b) Find the minimiser * of the function f and describe ho… • It is about elementary number theory.. 2. Find the remainder when 23612 is divided by 7. 3. Find the remainder when 23612 is divided by 9. 4. (a) Find an inverse of 18 modulo 29. (b) Solve the congrue… • 85 y 1 -5 15 -5 Y int = X int = Zeros = Max = Min = Domain = Range = End Behavior= • Please don’t copy from chegg and course heroÂ Â Â . Quiz.1. Solve the following recurrence by the method of backward subsitutions: A(n) = 2A(n – 1) + 2 for n > 1, with A(1) = 0. (30 Mar • . 4. Projection Operators: A projection matrix is defined as A(A’A) A’ where A is m x n and of rank n. Use the singular value decomposition of A = USV to rewrite P in terms of U alone. Next consider… • . Let T : V – V be a linear operator from the n-dimensional vector space V over F. Let (), v) be an eigenpair of T, and let p(t) = do + ait + . . . + at be a polynomial of order q > 0 with coeffi… • . Determine which of the following functions are real inner products on the indicated vector spaces. If they are inner products, then prove it, but if they are not, then provide a reason why. (A) Le… • In a learning experiment involving memory, subjects were given a certain amount of time to study the objects laid out on a table. Later, they were asked to recall as many items as they could. The prop… • A children’s medication prescribes 2.5 milligrams be administered to children aged two years old. As the child ages, the dosage increases by 0.5 milligrams every year. Â a. Find a linear function that… • Use elementary row or column operations to find the determinant. 2 -3 2 NWH 4 • Evaluate each determinant when a = 3, b = 1, and c = -4. 0 b (a) O X (b) C C -14 X • Use either elementary row or column operations, or cofactor expansion, to find the determinant by hand. Then use a software program or a graphing utility to verify your answer. HOM -1 Rectangular Snip… • How do I do this?Â Â . 9. In R2, let in = (cos ) i + (sin 0)j jo = -(sin0) i + (cos #) j Let To be the operator that carries in into jo and je into 0. (a) Calculate the matrix representation of To re… • How do I do this?Â Â . 7. Let T be a linear operator on R? such that T(i + j) = j and T(2i – j) = i + j. Find T(i) and T(j). 9. Two people agree to the following income-leveling scheme. The first per… • Answer all clearly please. Problem 1 (6 pts) Consider the vectors v1 Determine whether the set { v1, V2, V3} is linearly dependent or independent. If it is linearly dependent, find a linear dependence… • Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the stat… • Suppose that a matrix A has integer entries and an eigenvalue 0 < \|Î»\| < 1. In this question, you will prove a big theorem: Any Î»-eigenvector of A has an irrational entry. • Use Maple to solve the problem and show me the steps in Maple.. Solve the initial value problem 2y(5) – 934 – 14y3 + 61y" – 86y’ + 30y = et cos x, y(0) = 1, y'(0) = 0, y"(0) =2, y3(0) – -1, … • Please solve this question correctly. Make sure the formatting matches as expected in the directions. DIRECTIONS: Â Â QUESTION:Â Â . The next few questions may.r contain multiple choice or user-inp… • Solve this linear programming problem using the simplex method. (Suggestion: Use the row operations tool in decimal mode. Your answers probably will not come out as round numbers, but enter all answer… • . 3. Let A = 0 (a) Find the matrix B (RREF of A) and U (product of elementary matrices) such that B = UA. (b) Compute det (A). (c) Compute det (U) and det (US) (Hint: use properties of determinants … • Suppose that a company records the following quarterly revenue (in millions of dollars) over one year: Quarter \| Revenue Q1 20.2 Q2 16.5 Q3 15 Q4 17.1 20.2 16.5 This quarterly revenue data can also… • Let W be the subspace of R3 defined by the equation 2x – y + 32 = 0. (a) (2 points) Find a basis { v1, v2 } of W. (b) (1 point) Find a vector vy that is orthogonal to the plane W. (c) (3 points) Ex… • Which of the following are true? Justify your answer. 1. If A is singular, then adj(A) is singular. 2. If adj(A) is singular, then A is singular. 3. If A is not singular, then adj(A) is not singular. … • I have a module named “Artificial Intelligence & Machine Learning”. Kindly help me by doing all the Tasks given in the picture. Task 3 must be coded in Python programming language. The question. -… • . T 2. (a) Let X = y and -3 5 3 Find X. 2 (b) if A = 0 sin(a) 1 then find all the values of a in [0, 27] for which A is invertible 0 1/ V/2 1 (Hint: Compute first det(A)). • Consider the rotation matrix R(0) = cos 0 – sin 0 sin 0 cos 0 Show that a.) R(01) R(02) = R(01 + 02), V01, 02 ER HIPOA ‘T = (0-)(0) (q 12. Let us call a subset of matrices S commutative if Vr, y E… • Write the so-called belonging condition, I E A for the following sets, using quantifiers. a.) A = {x2 + 1 \| x E QA2x <1} b.) A = {3x + 1 \| x EZAx is a prime number} c.) A = {a3 + b3 + 3 \| a, beR… • . 3.6 (Dimension formula for preimages of linear maps) Let V and W be vector spaces over K such that dim V < co, U C W is a subspace of W and d : V -> W is linear. Show that dim -(U) = dim(U n… • . 2.10 Let V be an n-dimensional (n E N) vector space let U C V be a (proper) subspace. Show that there exists a basis B of V such that U n B = 0. • Let u = ( – 1, – 7, – 2) and v = ( – 4, 2, – 5). Find proj;(u). Enter your answer as a vector, using < and > as enclosing brackets. 0 x Invalid notation. • Can someone please help me set this up? Would Los Pollos Locos work 10 hrs each day? While Wild Duck works 8 hours a day? If someone could help me I truly appreciate it!! Â Â . SOCIAL MEDIA DIRECTOR … • Hi everyone, I am stuck on this linear algebra problem: Â Â Â Could someone kindly show me how to find Beta1 and Beta2 using the OLS estimator above?. Linear algebra 1. Partition matrices: Consider… • . 1. Let b A = Id e where a, b, c, d, e, f, g, h, i are some real numbers, if det( A)=5 answer the following questions: (a) Is A invertible? (Justify your answer). Find rank(A). (b) Let atd bte c+f … • Choose the correct answer * 1.If A and B matrices are of the same order and A+B=B+A, this law is known as. * * a. distributive law * b. commutative law * c. associative law * d. cramer’s law * 2. W… • . Find the LU-factorization of the matrix. (Your Z matrix must be unit diagonal.) 1 0 -4 1 LU = • . Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.) 4 1 2 31 7 18 6 14 10I 1 • . Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.) -7 4 33 -19 1 1 • . Find the transpose of the matrix. -3 4 D = 1 -1 -2 5 • . 2 0 : Perform the indicated operations, given c = -2, B = , and C = 3 -1 1 0 (CB) (C + () = • . Write the system of linear equations in the form Ax = b and solve this matrix equation for x. -2×1 – 3X2 = -15 6X1+ X2 = -11 X1 – 15 X2 -11 • . Find, if possible, AB and BA. (If not possible, enter IMPOSSIBLE in any single cell.) 1 0 -10 A = -4 4 B = 3 0 1 6 9 -1 71 (a) AB (b) BA • . Find x and y. x + 2 2 -7 2x + 6 2 -7 1 5 2y 2x 5 14 -8 1 -6 y+2 1 -6 9 (X, y) • . Show that if A is a 3 x 3 matrix and the sum of each column of A is 1, then the determinant of I – A is zero. You should not invoke the Invertible Matrix Theorem. • Hi I am seeking help with understanding the solution of part 2c below.Â I show the question and then the solution given, can you help me write out some of the intermediate steps..? or explain the sol… • Hi I am looking for assistance with question E below. I have provided my answer for Â a – d , i’m pretty sure they are correct, but i’m not sure how to do e? Can you help Â Â . 2. (5 pt) Consider the… • How to show that an upper triangular matrix is nilpotent if and only if all its diagonal elements are equal to zero by using induction? • . 3. (20 pts) Consider the following 3 x 3 matrix -3 O A = -2 6 ON 3 4 (a) (5 pts) Verify that x = NHO , is an eigenvector of A. What is its corresponding eigenvalue? (b) (15 pts) Find the other eig… • Hello! I have a question regarding Principal Component Analysis. Thank you in advance..! Â . In order to do PCA when the input space has more dimensions D than the number of training patterns n, a tri… • Use the Adjoint Method to find the inverse of the following matrix when , y, & are constants and y + 0: 0 y • Consider the following matrix: 0 0 40 – 2 3 -30 A = -1 3 -5 2 1 -5 0 2 0 – 2 0 0 2 ( 3 (b) [6 points] Consider the linear system C1 2 a 2 A X3 C4 Use Cramer’s rule to find a, ONLY. • Provided that the inverse of the matrix 4 -7 A = -11 20 8 do is -4 -22 9 -6 0 -18 -7 -5 A -1 5 14 5 4 3 Solve the following linear system by using the inversion method (6A) T • Let & be a real number, and let A be the matrix 2 -1 -2 A = 4 -1 1 8 -2 2+ 2x Determine for which values of a the matrix A is invertible. • Let A, B and C be the matrices 1 V2 0 0 V10 0 0 A = 0 7 8 B – -1 V2 0 0 1 10 0 2 24 0 1/ 3 V10 Calculate \| ABC). • Using the information from the answers obtained answer the following Â What is the number of mutually reachable dyads in the graph? Â What is the number of asymmetrically reachable dyads in the grap… • measurements from a scientific experiment. The variables a; are the experimental conditions and the bi correspond to the measured response in each condition. Suppose we wish to fit a degree n < m p… • Let W be the subspace of R* defined by W = span 3 3} For each of the following vectors v, determine if v is in W. For those vectors that are in W, find their B-coordinates. -3 (a) (2 points) v = OH… • (4 points) Use the Gram-Schmidt process to find an orthonormal basis for im( A), where 2 A = 2 -2 • Consider the following ellipse in the ry-plane: 3 (2, 2) (-1, 1) T. -3 -1 2 3 -1 (1, 1) (-2, -2) -3 Our goal in this problem is to find an equation for this ellipse in terms of r and y. (a) (1 poin… • Let W be the subspace of R3 defined by the equation 2x – y + 32 =0. (a) (2 points) Find a basis { v1, v2} of W. (b) (1 point) Find a vector vs that is orthogonal to the plane W. (c) (3 points) Expl… • Suppose that a company records the following quarterly revenue (in millions of dollars) over one year: Quarter Revenue Q1 20.2 Q2 16.5 Q3 15 Q4 17.1 20.2 16.5 This quarterly revenue data can also b… • Let v = , written with respect to the standard basis of R". For each of the following bases B of Ra, find the B-coordinates of v. (a) (2 points) B = 0 . i – ;} (b) (2 points) B – 1 2 . ; – } • Finding points with span/projection! Please help!. iii Give (or compute, if you must) the point in the span of { (2] . /} closest to iv Compute (here, you must) the point in the span of { } closest to… • . Problem 2 [24 points (6 pts each)]: Linear Independence Determine whether each set of vectors is linearly independent or linearly dependent by the method given: . Inspection: By observing and thin… • Finding inverse using RREF please!!!! [A\| Identity] –> [Identity \| A^-1] • This is linear algebra Â . 1. (5 pts) Suppose A, B, and C’ are invertible n x n matrices such that ABC = I where I is the n x n identity matrix. Prove det(B) = det(A) det(C).. 2. (25 pts) Consider the… • Let T: My, – RZ be defined by the equation (a) Find a basis for ran T (in R2). (b) Find a basis for ker T (in M22).. 23. If T: V – V and S: V – V are linear operators on a vector space V, show that… • Rotation in Hamilton’s sense using quaternions. Rotations of space Rotations are isometries, and they have a ï¬xed point. Again. we’ll assume thats the origin. because any other can be dealt with … • Exercise 9. Give the final formula we’ve all but arrived at for the mapping "rotation about v through angle " in the explicit form I’ = some formula involving r, y, z, v, and 0 y’ = some oth… • like Let’s say u = (2, 4) and v = (-3, 3). I know I said "unit vectors", but people don’t like dealing with unit vectors concretely. The idea still works, except the coordinates are a little… • Exercise 1. A mapping T from the plane R2 to itself is called additive if it has the property that T(ut w) =T(v) + T(w) for all v and w in R2. Show that rotation by an angle o is additive. Exercise 2…. • . A Zumba instructor remembers the data given in the following table, which shows the recommended maximum exercise heart rates for individuals of the given ages. Age (x years)Â Â Â Â Â Â Â Â ?… • Problem 5 [20 points (4, 5, 7, 4)]: Informative, Popular, Hip? (NYTimes, Reddit, TikTok?) We can describe a webpage on the internet as a 3 dimensional IPH vector: b = where: . i indicates how informat… • How do I test if it Span P2?. Let B be the standard basis of the space iii"2 of polynomials.Use coordinate vectors to test whether the following set of polynomials span P2. Justify your conclusio… • 1st picture: 2nd picture: Â Â Â Â . Linear algebra 1. Partition matrices: Consider using a partition matrix in OLS. That is, consider X = [X1, X2] and B1] B = B 2 where X1 is n X mi, X1 is n X r2, … • PROBLEMA 12 Una pequena compania aeroespacial planea ocho proyectos: N I Proyecto 1: Desarrolladona instalacion de pruebas automatizadas. Proyecto 2: Asignar un codigo de barras a todo el inventario y… • . Problem 5 ( 10 pts ) Solve the following system of linear equations 12 X, +11 x 2 + 10x, = 9 Show your work. 8 X 1 + 7 x 2 t b x 3 = S 4×1 +3 x 2 +2 x3 = 1 • . Problem 4 ( 10 pts) Let A be the following 3 x 3 matrix: – – 4 Determine whether A is invertible or not. A = 2 – 2 If A is invertible , compute the inverse of A 1 3 2 Show your work. • . Problem 3 ( 10 pts ) By definition , an nan matrix A is unipotent if I- A is bilpotent. Prove the following: If A is unipotent, then A is nonsingular. • . Problem 2 ( 10 pts ) By definition , an nan matrix T = ( tip ) is upper triangular if tip =0 whenever is j . Prove the following : If I is upper triangular and nilpotent, then ti; =0 for all i. • . Probleml ( 10 pts ) By definition , an nan matrix A is nilpotent if there is an integer kxo such that A"= 0, the zero matrix. Prove the following : If A is nilpotent , then ItA is nonsingular… • Find the coordinate vector [x] of x relative to the given basis B = {b,, b2, b3} [x]B = (Simplify your answer.) • Please solve all of the given questions. Thank you! subject; advanced engineering math Â . In Exercises 1 through 8 find the square roots of the given complex number by using result (10), and then con… • Hi can some one show the step by step answer on this problem? If it has graph then can you please include? if not it’s fine Â . 3. In order to ensure optimal health (and thus accurate test results), a… • Hi can some one show the step by step answer on this problem? If it has graph then can you please include? if not it’s fine Â . 2. A calculator company produces a scientific calculator and a graphing … • Hi can some one show the step by step answer on this problem? If it has graph then can you please include? if not it’s fine • Problem 3 (10 pts) By definition, an nah matrix A is unipotent if I- A is bilpotent. Prove the following: If A is unipotent, then A is nonsingular. • No additional infromation. Problem, 2 (10 pts) By definition, an non matrix T= (t;; ) is upper triangular if tir =0 whenever isj. Prove the following: If T is upper triangular and nilpotent, then t;; … • Problem! ( 10 pts ) By definition , an nan matrix A is nilpotent if there is an integer k >o such that A"= O, the zero matrix. Prove the following : If A is nilpotent , then ItA is nonsingular… • Hi can someone show the step by step answer for this questions? and if it has graph please include if not then it’s fine Â . 1. Maribel is selling regular and special juices. To make a bottle of regul… • Pls answer all 4 questions with deatils Â and each every step.Â . (1) How to divide $100 among four people according to given proportions: 1 : 2 : 4 : 8? (Use ractions only) (2) How to divide$100 amo… • Find the circumference of the circle with the given diameter. Use 3.1416 for It. d = 22.5 mm The circumference is mm. (Simplify your answer. Type an integer or a decimal. Round to the nearest tenth as… • Hi can someone show the step by step answer from this problem? can you include the graph? if the question has a graph • Hi can someone show the step by step answer to this? if the question needs to graph then please include if not then it’s fine Â . 1. Maribel is selling regular and special juices. To make a bottle of … • Hi can you show step by step answer of this question? if the question needed to be graph then please include if not then it’s fine • In the right triangle, find the length of the side not given. C a b = 0.496 in., c =0.787 in. b What is the value of a? a = in. (Round to three decimal places as needed.) • Convert the angle in degrees to radians. Express your answer in decimal form. 260 26 = radians (Round to two decimal places!) • Find LA in the figure to the right. A 8 740 B C LA = o • Use the fomulas developed in this section to find the area of the figure. 19 m 9.7 m 8.8 m 8 m 28 m A = m 2 • Hi can you solve this problem in step by step? if the question state that it needs to graph please show me the graph one if not then it’s fine Â . 1. Maribel is selling regular and special juices. To … • Quiz: Week 6 Quiz Question 10 of 12 > Thi Thi In the given figure AB is a diameter, TB is a tangent line at B and ZABC = 60. Determine ZCBT. O B T LCBT = • . 1. Maribel is selling regular and special juices. To make a bottle of regular juice, she needs 2 pounds of mango and 3 pounds of pineapple. On the other hand, 4 pounds of mango and 2 pounds of pin… • . 3. Kronecker Tensor Product: You are given two sets of vector space bases {x/} _ with x; C RAI and { vij = with y; ERA2. From these you construct a new set of vectors with components Zij,ab = Xiay… • Suppose you have a linear optimization model where you are trying to decide which products to produce to maximize profit.Â (a) What does the additive assumption imply about the profit objective?Â (b… • Consider an optimization model with a number of resource constraints. Each indicates that the amount of the resource used cannot exceed the amount available. Why is the shadow price of such a resource… • A nurse hangs aÂ 10001000-milliliter IV bag which is set to drip atÂ 125125Â milliliters per hour. Create a model of this situation to represent the amount of IV solution left in the bag afterÂ x hour… • Find, in terms of n, the number of n x n orthogonal matrices with integer entries. • (1 point) Solve the system using Cramer’s Rule 4 12 y – 41 x -40 7y – 15 z CT CO P 30 + 9y + 30z det = D = U = Z= • State the corresponding corner point for the given simplex tableau. P constant X y $1 52 0.25 71 3 1 0 0 HOOW 9 1 0 16 6 62 8 0.75 -2 -9 0 0 01 0 The corresponding corner point is (x, y) = • Use the given constraints and optimal solution to state the value of each slack variable. subject to 3x + 5y S 1,375 38x + 34y$ 1,140 x 20, y 20 optimal solution (x, y) = (30, 0) The slack variables … • Solve correctly on a sheet of paper. Â . 4. Use Cramer’s Rule to solve the system 2×1 + 612 + 813 16 4x, + 15×2 + 1913 38 271 + 313 6 • Week Quiz Nam Bernard Nkansah SHORT ANSWER. Answer each question. Make sure you show all work and use the method in directions. Find the requested measurement. Use it =3.14 and round your answer to th… • . Write the following linear program in standard form. maximize 4m + 562 â€” 1:3 such that $1 + 3×3 3561 ‘l’ 562 + 3333 IV \|/\ .561 2 0,562 2 0, 563 unrestricted • . Write the following linear program in standard form. maximize 4×1 + 2 -23 such that 1 + 3×3 – 6 3×1+ x2 + 3×3 2 9 x1 2 0, x2 2 0, X3 unrestricted • Juan owns majority stake in the Moncada Energy Group, a production company that sells two types of gasoline, G1 and G2, from two types of crude oil, C1 and C2. Moncada prides itself in producing signa… • How to find the simple Â linear equations of basis of W??. Consider the following matrices: 4 6 2 2 -2 2 12 2 2 01 = 2 3 , U2 = 1 -1 , 13 = 6 1 1 6 9 3 3 -3 3 18 3 3 -5 3 -3 4 6 W1 = 3 3 , W2 = 2 2 , … • Please show all the steps Â . Find the angle o in radians between the vectors 1 and. Find the angle o in radians between the vectors 4 and -1 4 • Eiji has just inherited the Hokama Industries and is excited to run the business. In order to rein in costs, he has negotiated with the employee’s union to let some employees retire early and others t… • . Let (-w) + dix, + d2x2 + . .. + dmxm = 0 () be the phase I objective function when phase I terminates for maximizing w. Discuss the following two procedures for making the phase I to II transitio… • An automobile company produces cars in Los Angeles and Detroit and has a warehouse in Atlanta. The company supplies cars to customers in Houston and Tampa. The costs of shipping a car between various … • A company manufactures two types of trucks. Each truck must go through the painting shop and the assembly shop. If the painting shop were completely devoted to painting type 1 trucks, 800 per day coul… • A = [a, 2, 8, 24 45] = 5 2 2 4 1 2 1 which row is equivalent to the 3 atrix 3 B = O O o ce 0 0 1. Find a basis for Null A 2 . Find a basis for CIA 3. Show how write one of the columns of A as a linear… • Find and post a real-life application for: Polynomials (most formulas are polynomials), or Quadratics, or rational equations. • explain what is meant by the term extraneous solution. Is it a solution or now? Explain why. • please provide answers neatly Â . Let { u1 (a) = 12, u2 (x) = 12x, u3 (2) = -12×2) be a basis for a subspace of P2. Use the Gram-Schmidt process to find an orthogonal basis under the integration inner… • Need help with question 6. estandard basis, show that its matrix is E 4 3 6. Find a basis of R4 with respect to which the linear map ‘reflect across the line x + 3y = 0’ has a diagonal matrix. Now fin… • Let R be the triangle with vertices at (x1, y1), (x2, y2), and (x3, y3). Show that {area of triangle} = 1/2 * det[ x1 y1 1 x2 y2 1 x3 y3 1 } • I need help for part b only. this is math 290 linear algebra • A county landfill has a maximum capacity of about 131,000,000 tons of trash. The amount in the landfill on a given day since 1996 is modeled by the function T ( x ) = 32,000 + 4 x . Here x is the numb… • . Problem 4 [18 points (6 cach)]: Inverse Existence For each of the matrices below: 2 . If the matrix has an inverse, compute it by row reduction. You do not need to document your row reductions, bu… • . Problem 3 [20 points (5 pts each]: Computing Projections Compute each vector below. Be sure to show your computation or justification in all but the final sub-problem. i Compute projection of onto… • . Problem 1 [18 points (6 pts each)]: Span Illustration Illustrate the span of each of the following sets of vectors by providing coordinate axes and shading the appropriate vector space. (If you’re… • Please explain what is involution in the following:. An n x n matrix A is said to be an involution if A2 = 1. Show that if G is any matrix of the form G cos 0 sin 0 sin 0 – cos 0 then G is an involuti… • please provide answers neatly Â . Let { ul (a) = -9, u2 (x) = -18x, u3 (a) = 12×2 be a basis for a subspace of P2. Use the Gram- Schmidt process to find an orthogonal basis under the integration inner… • Please provide answers clearly Â . 0 Let U1 12 2, 02 = -18 18 ],03 – 18 3 be a basis for a subspace of R2x2. Use the Gram-Schmidt process to find an orthogonal basis under the Frobenius inner product…. • Find the x coordinate of the point where the graph of the equationÂ y=(xâˆ’6+10/x)(1/2x)Â has a horizontal tangent line. • please provide answers to question neatly Â . 2 Let be a basis for R . Use the Gram-Schmidt process to find an orthogonal basis under the Euclidean inner product. Orthogonal basis: V1 V2 = a V3 b a = … • please provide answers for this question, they should be in decimals Â . Let U1 be a basis for R . Use the Gram-Schmidt process to find an orthogonal basis under the Euclidean inner product. Orthogona… • Hi , may I know how to solve the above 2 questions ? Preferably with step-by-step solution .. 1. A is 3×3 matrix with three distinct eigenvalues and P = is a matrix that diagonalizes A. Which of the f… • Solve for x. (Enter your answers as a comma-separated list.) X – 8 9 =0 2x-1 X = • 1 0 -3 -1 0 2 Let A and B be A = 2 1 0 B = 21 0 -1 0 2 1 0 -3 Find an elementary matrix E such that EA = B. E = 10 T • Express answers using mathematical vocabulary. The store “Plumas al aire” is dedicated to the sale of food for animals. They sell two types of seeds for bird food. The first type of seeds costs$11.75… • buenas noches, me podrian ayudar con este problema? gracias. • . Determine a potential function for F = 3×2(y2 – 4y)i + (2x y – 4×3)j and use it to evaluate fo F . dR for the points Q = (-1, 1), P = (2, 3). By potential function, find a function f(x, y) so that… • . Use Green’s theorem to evaluate fo F . dR (counter-clockwise orienta- tion), where F = e" cos(y)i – ex sin(y)j, and C is any closed path in the plane. • . Given the 4 data points in the (x; y) plane: (0; 3); (1; 2); (2; 4); (3; 4); ï¬nd the best quadratic least squares ï¬t: y : co + c111: + c2352 (a) Set the problem up as an inhomogeneous matrix e… • need step and explaination. Suppose that in an instance of LP we have 72 variables that are uconstrained in sign. Show how they can be replaced by n + 1 variables that are constrained to be nonnegativ… • Solve for a and b. 3 points Find the values of a and b such that the system 2x + 3ay = 5 x + 4y = 2b has infinitely many solutions. The unknowns are r and y. • . 0 2 N 0 -2 -1 Let A and B be A = 1 0 2 B = 1 0 2 0 -2 -1 0 2 3 Find an elementary matrix E such that EA = B. E = • . Solve for X. (Enter your answers as a comma-separated list.) x – 2 3 = 0 2x-1 X = • . Find the values of A for which the determinant is zero. (Enter your answers as a commaâ€”separated list.) )L 1 O 0 1+1 1 O 2 )L • A vector space, V, with the vectors, X, Y, and Z satisfies the following 10 laws, where r and s are real number scalars: :Law 3 commutative law X+Y =Y+X • . Find the values of a and b such that the system [2x + 3ay = 5 r + 4y = 2b has infinitely many solutions. The unknowns are c and y. • please answer question, I provided an example of the same type of question already answered below to show the process of how the answer was fouhnd Â . Find the orthogonal projection y of y = \|-1 onto … • Hi , may I know if my answers are correct . Kindly go through the above 3 questions with step-by-step solution.. 1. A is 3×3 matrix with three distinct eigenvalues and P = is a matrix that diagonalize… • Find the inverse of the given matrix using gauss jordan elimination method • . 2 1 3. Which of the following is a fundamental set of solutions for the SDE 1 1 y’= 2 NHK y ? 1 1 (1 mark) O eat e2t NH HAN eat est NH H O e OOH et OH O HOO est et pt HOO O O est et HOO • . 1. 1 A is 3×3 matrix with three distinct eigenvalues and P = 1 oo is a matrix that diagonalizes A. 1 3 Which of the following matrices must also diagonalize A? (1 mark) HO O NHO 3 O OOH O… • please provide answers clearly for what xr equals to Â . -2 -2 2 Consider the system Ax = b where A = and b = . A -6 -6 basis for the null space of A is given by and x = is a particular solution. Use … • . Let -1 A = 3 2 -2 4 Compute the following. A + AT = A – A = For any square matrix A, is the matrix A + A lower triangular, upper triangular, symmetric, skew-symmetric, or none of these? Choose For… • Use the data in Problem 4-27 to develop a multiple linear regression model. How does this compare with each of the models in Problem 4-27? • . 4 3 -2 4 2 If A : 2 1 and B = 4 -2 then -1 -1 0 4 O 3A – 4B = and AT = • Need helps with steps. A call center needs to plan its workforce. The following table shows the average number of calls handled by the call center per day for each day of the week. Mon Tue Wed Thu … • Previous Problem Problem List Next Problem 6×2 +2x + 3 (1 point) Let f(x) = Find the following: Vx f’ (x ) = (18x^2+2x-3)/(2x^(3/2)) f’ (25) = • Previous Problem Problem List Next Problem (1 point) If f(x) = 5x(x3 – 7,x+4), find f’ (16). f’ (16) = Preview My Answers Submit Answers • The answer above is NOT correct. 3×3- 6×2 (1 point) Let f(x) = Evaluate f’ (x) at x = 1. f’ (1 ) = Preview My Answers Submit Answers • please show me how to solve the following matrices using the Gauss-Jordan Elimination method. Find the inverse of the given matrix, if it exists. (a) CT O0 (b) HO (c) -3 (d) • please answer question and provide ANSWER CLEARLY Â . Find the orthogonal projection y of y = onto the subspace 1 W = Span u1 = -3 5 Ex: 1.23 = • A radioactive substance decays from 16 g to 12.5 g in 5 years. What is the half life of this substance? Round your answer to the nearest whole number. • De Metal S.A., produces three types of products: dining tables, desks and classic chairs. Each product is manufactured in three stages: cutting, construction and finishing. A dining table takes five h… • use the cross product to find the equation of the plane that passes through the points (-1,2,1), (0,1,1) and (2,1,2) • . Q 3. Perform one iteration of (a) Jacobi Iteration and (b) Gauss-Siedel iteration by using a cal- culator for solving the system Ar = b, where A is given below and b = [3, 5, -3]. Use zeros for in… • Please help on this linear algebra problem. Thank you. 3. Let V be a vector space. If P e End(V ) satisfies P2 = P, we call P a projection. Prove that for every projection P e End(V), there exist subs… • Exercises 19-23 telate to inhabitants of the island of knights and knaves created by Smullyan, where knights always tell the truth and knaves always lie. You encounter two people, A and B. Determine, … • Find a linear dependence relation among P 1 , P 2 , and P 3 . Then find a basis for Span{P 1 , P 2 , P 3 }. Â 2) Find a basis for H. Â 3) Find bases for Nul A, Col A, and Row A.. Consider the pol… • Write it on paper please. 6. Provide an example of a data set that would represent a quartic function when finite difference method is applied to it. C 7) The graph on the left was obtained by transfo… • Determine the amount of each current for the fol- lowing networks: (a) 16 volts 22 ohms 12 2 ohms A B 3 ohms 13 W (b) 2 ohms N. W 20 volts 4 ohms i2 A B 13 2 ohms (c 8 volts IN. 4 ohms 2 ohms 12 A B 1… • please solve it step by step. Suppose that in an instance of LP we have n variables that are ueonstrained in sign. Show how they can be replaced by n + 1 variables that are constrained to be nonnegati… • Solve each Systems of Linear Equations by Gauss Elimination X – 2x 2 + x1 =0 a) – x1 +3×3 =-4 2×1 – 5×2 – X3 = 3 2×1 – X3 – X1= -2 b) -x1 +3×2 -X3 + =1 (x1 + 3×2 -2×3 = 0 2x – 2x, + x, = 3 c) 3×1 + 1… • Solve each Systems of Linear Equations by Gauss Elimination 2x – 4x, + x3 =-4 d) 4×1 – 8×2+7×3 = 2 -2x, +4x, – 3×3 = 5 3x, + 6x, -9; =15 e) 2x 4 4×2 6×3 = 10 – 2x -3×2 + 4×3 =-6 x + 2×2 + 4×3 + x X -… • Find a unit vector in the same direction as: a) v = 31 – 4j, 3.6. Find the distance between each pair of points: a) (1,3) and (4,1); b) (-1,2) and (4,-2); • Solve System of Linear Equations using 1. Matrix Method 2. Cramer’s Rule 3. Gauss Elimination 4. Gauss Jordan Elimination 3x, + 2X2 – X3 = 4 2×1 – x2 + 3×3 – 5 1 + 3×2 + 2×3 – -1 • Solve each Systems of Linear Equations by Gauss Jordan Elimination method 3.x, + 2.×2 – 13= 4 a) {2×1 – x2 + 3×3 = 5 (x, +3.×2 + 2×3 =-1 * + 2.×2 + 3.13 =0 H b) x1 – 2×3 = 3 (X2 +x3 =1 (4×1 – 2x, + 3…. • Given three vectors v = (2.1), w =(-1,5) and u = (2,5) ). Find: b ) – 20 , C) BTW () v – 2W, 3.3. If v = 2 i – 3j and w =- i+27, find c) 1-3 7 + 2w/, 3.4. Find a unit vector in the same direction… • Write it on paper please. 1. Write the equation and sketch an example of a quartic function with the zeros at -4, 2 (order 3), if f(0) = 4. What further information about the polynomial function is ne… • . pe M My X Messag > Bb Colle DE CamSca Inbox (3\| (no sub ) New tal \| The https://www.myopenmath.com/assess2/?cid=131963&aid=9436743#/skip/6 a) Find the slope (m). m b) Find the y-intercept (… • Find the first derivative of f(x) and then evaluate it for x=2. f(x)=29x 5 -7x -1 +4x -26 f'(2)= If needed, round to 2 d.p. 12 Solve the following power equation for the appropriate variable . Â·In… • . Plot the point P(0, 0) -8 -7 -6 -5 -4 -3 -2 – 8 Clear All Draw: Dot Check Answer here to search O • This question has five parts. Â 2) This question has two parts. Â 3) This question has five parts. Â 4) This question has two parts. Â 5) This question has two parts.. Consider the following tw… • Use matrix inverse metod to solve each of the following systems: 3x + 2y + 2z=0 a ) 2x + 2y + 2z=5 -4x + 4y + 3z=2 Solve system of equations by Cramer’s rule 3X1 + 2X2 + X3 – 8 b) 2X1 -3X2 + 2X3 16 X1… • . M My X Blackbo > Bb Colle EDE CamSca Inbox (3\| i (no sub ) New tat https://www.myopenmath.com/assess2/?cid=131963&aid=9436743#/skip/2 a) Find the slope (m). m b) Find the y-intercept (b). b… • . Graph the given equation: y = 2x – 5. -6 -5 -4 -3 -1 5 16 Clear All Draw: • Â 2) Â 3) Â 4) Â 5) Â 6) Â 7) Â 8) This question has three parts Â 9) This question has four parts Â 10) This question has three parts. Let W be the set of all vectors of the form shown on… • Consider the following optimization problem: for some given parameters, 61,0}, 83- (a) how many extreme points are there? (b) 11ch many basic feasible solutions are there? 2 (c) Is (0.5,U.5,U.5) an… • Maximize: P = 20x +15y Subject to:x + y â‰¤ 10: 4x +3y â‰¤ 24; xy 20Â 4. Minimize: C = 65x +55y Subject to:9x +17y â‰¥ 210: 55x +40y 2 24: 2x +2y â‰¥ 40: xy â‰¥0 Â Â Â Â Â Â Â Â Â Â Â ?… • CP and Tucker Decomposition Â (proofs) Â Prove the following theorem, given the information in attachment: When U ji , V i , and R j are known, a reshape form of the core tensor C j can be estimated… • Elementary Linear Algebra. Find the inverse of the given matrix, if it exists. (a) (b ) -1 09 O (c) .3 (d) • . 2. (8 pts) Suppose that K and E are two square n x n matrices. Assume > is symmetric, but K is not symmetric. Both matrices have rank n. a) (2 pts) Is the matrix K"EK always symmetric? (Pr… • . 1. (2? pie) Censider a m x a full rank matrix X and a veryr useful matrix in menometrits: M = I â€” X {X ‘X }’1X {where I is a conformable identity matrix]. a) (2 pts] ls M symmetric? (Preve it … • Subject: Linear Programming Â You’re on a special diet and know that your daily requirement of five nutrients is 60 milligrams of vitamin C, 1,000 milligrams of calcium, 18 milligrams of iron, 20 mil… • . M MyOpE X ODE Course C https://www.myopenmath.com/assess2/?cid=131963&aid=9436743#/skip/4 a) Find the slope (m). m b) Find the y-intercept (b). b – c) Write the equation of the line in slope-i… • Graph the given equation: y=2xâˆ’1y=2x-1. Â Plot the point P(âˆ’6,âˆ’6)P(-6,-6)Â . • Elementary Linear Algebra. Find the inverse of the given matrix, if it exists. (a) (b ) HO (c ) -3 2 -3 (d) • Elementary Linear Algebra. Use the Invertible Matrix Theorem to answer the following questions. (a) Can a square matrix with two identical columns be invertible? Why or why not? (b) If an n x n matrix… • Elementary Linear Algebra. if a matrix A is 5 x 3 and AB is a 5 x 7 matrix, what is the size of the matrix B? Explain. • Elementary Linear Algebra. Let A = – 3.c= 3 . Verify that AB = AC and yet B # C. • Elementary Linear Algebra. Use the given matrices to compute each matrix sum or product if it is defined. If it is undefined explain why. 4 = 4 -5 2 B = 1 4 3] C = -2 – N D = 3 5 (a) Find -24. (b) Fin… • . Read the following, and then answer the questions below. Suppose that A is a matrix with complex number entries ajj. We define the complex conjugate of A, written A, to be the matrix whose entries… • With the complex numbers given, showing the complete procedure, you must perform the following operations: Note: For operations in polar form, in addition to expressing the result in polar form, you m… • how to make two matrices (A and B) multiplied into vector v in matlab? Â I need to showÂ AB = BA, Â ABâ‰ BA matrix…and the graph • Is the polynomial function model a reasonable model in predicting future net profits? Explain. • Due in 4 hours, 11 minutes. Due Fri 02/18/2022 11:59 pm The following problem refers to an arithmetic sequence. If the fourth term is 4 and the eighth term is 8, find the term a1, the common differenc… • 21 / 21 \| – 191% + 3. For each matrix A and basis B, find [LA], and an invertible matrix Q such that [LA]B = Q-1AQ. (a) A = ( 7) and s = {( ) . (1 ) ( b) A = 1 13 4 4 4 10 and B = { ( 12 ) . ( 1 ) . (… • . Let f: IR3 – R3 be the linear transformation that First rotates the xz-plane by – radians counterclockwise about the y-axis, Then reflects everything about the yz-plane (i.e., switching the sign o… • . What is the projection of b = (2,1,2) onto the plane spanned by (0,1,0) and (0,1,1)? • . For the graph G a. What is the incidence matrix A-? (Orient the edges whichever way you like.) b. What is AcAt? c. For a general graph G, compute the n x n matrix A, A,. There should be 3 cases: 1… • could you please help me with this question? Â . MATH210 Assignment 2D- LA winter 2022 2. For the following system of linear equations: â€”2y+3_- = 2xâ€” y+ _- = â€”2xâ€” y+6_- = â€”4 a. Find A’1 usin… • . 4. In one of the core exercises you found a distance between two vectors using the Pythagorean theorem. This problem is about a generalization of this statement and the Law of Cosines. The law of … • Problem 6: Consider the vector space P3 consisting of polynomials of degrees up to 3 over the reals, equipped with the positive-definite scalar product (a) For u(t) = 1 – 2t and v(t) = 2 + 4t – 3t2, f… • Here is the answer to problem #6 above:Â Please explain how to get the solution to question 6.Â . 561 – 263 6 (a) Solvable if b2 = 2b1 and 3b1 – 3b3 + b4 = 0. Then x = = Cp b3 – 261 561 – 263 (… • Here is the solution please explain how to get this answer to question 4 above. Â . Find the complete solution (also called the general solution) to 4 2 H y = 3 6 4 Z ON 2. complete = Cp + n … • . 5 15 (1 point) Are the vectors -5 and -7 linearly independent? 5 2 Choose If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are linea… • . (1 point) Which of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.) C A. B. C. D. C 9 6 E. 5 -4 8 9 -5 F. • . Previous Problem Problem List Next Problem 5 O 15 (1 point) Are the vectors 5 and -7 linearly independent? 5 2 Choose If they are linearly dependent, find scalars that are not all zero such that t… • Find the y-intercept and slope of the shown graph. Answers may be given in fraction or decimal CO rounded to two decimal places. X M 1.png . .. AY Slope: Blank 1 y-intercept: Blank 2 Last saved 6:38:1… • Which two points are on the line represented by 5x – 2y = 10? • For the following matrix in the image below complete 1a-1d.Â 1a. Compute its rref 1b. Find its “special solutions” 1c. Write down an expression for a generic vector in its null space 1d. If there are… • Using algebra, find the slope and y-intercept of the line represented by the following equations. (a) 9x + 3y = 6; m = Blank 1, b = Blank 2 b) 2.5y – 12.5 = 0; m = Blank 3, b = Blank 4 Blank 1 Add you… • (1 point) Determine the value(5) of a: such that 6 â€”1 â€”6 :r: [m 2 1] â€”1 3 â€”3 â€”1 =[0] â€”ï¬ 25 â€”1 â€”4 3:: Note: If there is more than one value separate them by commas.. (1 point) Let z =… • 0 10 (1 point) Are the vectors 0 and 5 linearly independent? linearly dependent If they are linearly dependent, find scalars that are not all zero such that the equation below is true. If they are lin… • plz answer. (1 point) Which of the following sets of vectors are linearly independent? (Check the boxes for linearly independent sets.) • . Prove convexity of the following:. the Lagrangian function of the nonlinear optimization problem max f(x) s.t. gi(x) < bi, i = 1, …, m, x EP, where P is a polyhedron. • . Prove convexity of the following:. the risk function of portfolio m Risk(:r:) :2 E [R(m)] + 6E [‘R($) â€” E [R(:L’)] ‘] where R(m) is the portfolio return and 6 is any positive scalar. • . Denote three scalars A = e E-le, B = u E-le and C = M E-Ip. 1. Prove that AC – B2 > 0. • . Let a be a positive number. The following matrix is the adjacenty matri 0120 1010 210 0 0 0 a lain your answers in detail without drawing the graph. Is there an isolated vertex? (2 marks) Find the… • . Let 11′, if E R”. Consider the following identity. IIi.-.’+f:’l\|2 + "13â€” {I’ll2 = QUIT"? + 2ll’ii’ll2 Explain in words how the above identity can be interpreted geoemtrically, by… • . Every day, Jonathan runs in a straight line in the direction 7 = (1, -1, 2). His running route goes through the point A(8, 3, -5). His favorite tree is at point T(0, 1, 4). a) Find the point P alo… • . Lel N we d LWU-uilleTISIonal Subspace UI I . Let P be the projection matrix that projects vectors in R onto S. (a) Determine the eigenvalues of P. (b) For each eigenvalue of P, determine the… • . â€”1 1 1 5 Let A â€” _1 1 1 5 Factor A into a product A = QR, where Q is a matrix with orthonormal columns and R is an upper triangular square matrix. Be sure to clearly show the steps you used to… • . Let S be a two-dimensional subspace of IR3. Let P be the projection matrix that projects vectors in R onto S’. (a) Determine the eigenvalues of P. (b) For each eigenvalue of P, determine the dimen… • . Decompose the vector p = (6.5) with respect to the orthonormal base a, b below. 3 2 b = 2 2 • 2 Define T: P(R) > Pi(R) by T(f(I)) = 2f'(1) – f(I). (a) Compute the matrix of 7"with respect to the standard basis c. (b) Determine whether T is invertible, and compute T if it exists. (c) Le… • [4 pts] 1 For each of the following matrices, compute the rank and the inverse if it exists. OH HHO • . B = {131: b23b31b4} ={(3 3H3 W? UH 3)} is a basis for M23, the vector space of 2 X 2 real matrices. A+AT 2 Let L be the linear operator on M23 given by L(A) = . (Here AT denotes the transpose o… • Consider the linear transformation / : R* – R* given by the rule atbtetd 2a +6+ 2c+ d = C 2a +5+ c+ 20 a-bte-d Find a nonzero vector in the kernel of f. Explain your answer. • (12 points) For each statement below, determine whether it is True or False, and circle the appropriate answer. If it is true, brieï¬‚y explain why it is true. If it is false, give a speciï¬c coun… • Consider the following system of linear equations. cty+2+1=0 2x+ + 2+ 1= 0 2xty+ z+ 21 =0 8-1+2-1=0 Isc =y =1,z = = -1 a solution to this system? Write Yes or No. Explain with 1 sentence why you sa… • What is the solution?. 2. Cold Start Electric produces three batteries from lead, acid and plastic as shown in the table: 24-month 36-month 48-month Lead (pounds) 4 6 8 Acid (ounces) 12 14 16 Plastic … • What are the constraints? What is the solution?. 1. One day, Gillian the magician summoned the wisest of her women. "Devoted sisters of the Coven," she began, "I have a quandary: As you… • answer the following question. 1. List the number of digits for the following numbering systems: Decimal Binary Octal . Decimal 101 . Binary 11110011 . Octal . Hexadecimal 2. Write a simple chart to s… • Projection Suppose v = (1, 3, 5, 7). Find the projection of u onto W or, in other words, find w E W that minimizes \|lu – wll, where W is the subspace of R4 spanned by (a) u1 = (1, 1, 1, 1) and u2 = (1… • Eigendecomposition 2 X 2 Matrix (a) Find all eigenvalues and corresponding eigenvectors. (b) Find matrices P and D such that P is nonsingular and D = p- 1AP is diagonal. • 2/18/22, 4:02 AM Quiz: Eigenvectors, Eigenvalues, Diagonalization, Orthogonal Basis and Complements Orthogonal Complement: Basis . Let w = (1, 2, 3, 1) be a vector in R4. Find an orthogonal basis for … • Diagonalization Symmetric Matrix 1] â€”8 4 . Let B = [â€”8 â€”1 4} (a) Find all eigenvalues of B. 4 â€”2 â€”4 (b) Find a maximal set S of nonzero orthogonal eigen- vectors of B. (c) Find an orthogonal… • solve the following pleaseÂ Â . -1 Find a basis for the null space of the matrix A = 2 -1 -1 3 3 Enter your basis column vectors as a list separated by commas. sin (a). 1 1 3 1 Find a basis for the r… • solve the following pleaseÂ Â . -1 3 The set spans R 4 – 5 1 -1 O True O False. The set 4 is linearly dependent. 4 O True O False • Please help me just for question 3.Â Â Matlab: x=[0 0]; y=[0 1]; plot(x,y) draws a line from (0,0) to (0,1). Matlab: x=[0 cos(pi/4)]; y=[0 sin(pi/4)]; plot(x,y) draws a line from (0,0) to (cos(pi/4)… • . Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem min Ex Ex s.t. ex=1, A x= R where the target return must be achieved exactly, short-selling is allo… • Prove convexity of the following: Â . . an ellipsoid in R”. . the risk function of portfolio m Risk(m) :2 E [R(m)] + (SE [R(x) â€” E [R(m)] H where R(:L’) is the portfolio return and (5 is any posit… • Please help me solve the questions below Â Show all the necessary steps. Thank you!Â Â Â . 4 Investment model 2 Consider that the adjustment cost of investment is equal to: C(I) = 12 (7) 1. Find th… • Hi there if you could solve in details, that would be very much helpful and appreciated. Thank you. I will give you a good feedback too.. Consider the polynomials pact) = It t Pelt ) = 1 – t and Polt … • Hi there if you could solve in details, that would be very much helpful and appreciated. Thank you. I will give you a good feedback too.. Suppose that 7 is a one-to-one transformation, so that an equa… • In the vector space of all real-valued functions, find a basis for the subspace spanned by { Sin(t) , Sin(2t) , Sin(t) Cos(t) }. If you can solve in details that would be helpful and appreciated. I wi… • Hi there if you could solve in details, that would be very much helpful and appreciated. Thank you. I will give you a good feedback too.. 45. Let Max2 be the vector space of all 2 x 2 matrices, T 52 a… • Hi there if you could solve in details, that would be very much helpful and appreciated. Thank you. I will give you a good feedback too.. Define a dinear transformation To Py z R 2 by TCP ) = ( plo) …. • Hi there if you could solve the problem in details, that would be very much helpful and appreciated. Thank you. I will give you a good feedback too. If you get confused on the figure, please let me kn… • Hi there, if you could solve in details that would be very much helpful and appreciated. Thank you. I will give you a good feedback too.. Let F be a fixed 3 x 2 matrix, and let // be the set of all ma… • . or too) Consider the matrix 1 -8 -6 10 10 -2 9 -8 -1 (a) On the matrix above, perform the row operation 7R3 + R1 -> R1. The new matrix is: (b) On the original matrix, perform the row operation … • all the nurses in Belvedere hospital are women, so women are better qualified for medical jobs • . w Describe the set of all matrices that are [OW equivalent to 1 ï¬‚ 1 U U D I Explain 1liour answer using complete sentences 1.Iirith correct grammar, spelling, and punctuation. • PLEASE DO PART C (and other parts if you want to help) Â Please note that the problem says to use the QR factorization and give algorithm complexity for solving it (in flops). The previous answers I’… • PLEASE DO PART B (and other parts if you want to help) Â Please note that the problem says to use the QR factorization and give algorithm complexity for solving it (in flops). The previous answers I’… • PLEASE DO PART A (and other parts if you want to help) Â Please note that the problem says to use the QR factorization and give algorithm complexity for solving it (in flops). The previous answers I’… • Please note that the problem says to use the QR factorization and give algorithm complexity for solving it (in flops). The previous answers I’ve gotten didn’t do either of these. Also, many tutors sai… • Find the values of 1 for which the determinant is zero. (Enter your answers as a comma-separated list.) 9 0 0 1+2 3 0 1 1 = X • Use expansion by cofactors to find the determinant of the matrix. 240 05 0 1 3 5 6 0 0 424 0 0 3 6 3 0 0 0 02 • For each of the following matrices in the image below complete 1a-1d.Â 1a. Compute its rref 1b. Find its “special solutions” 1c. Write down an expression for a generic vector in its null space 1d. If… • For each of the following matrices 1a. Compute its rref 1b. Find its “special solutions” 1c. Write down an expression for a generic vector in its null space 1d. If there are any free columns, write th… • . Q4 6 Points 12 Let A = NHN and b = 6 18 (a) Factor A into a product A = Q R, where Q is a matrix with orthonormal columns and R is an upper trianglular square matrix. Be sure to clearly show th… • Linear equations, parameters, row echelon form. Question 1 (8 marks) (a) A bag of coins has total value$8.15, and consists entirely of five, ten and fifty cent pieces. There are 37 coins in total. Al… • Find all minors and cofactors of the matrix. -2 3 1 6 4 5 2 -1 3 (a) Find all minors of the matrix. M11 = M12 = M13 = M21 = M22 = M23 = M31 = M32 = M33 = (b) Find all cofactors of the matrix. C11 = C1… • Use a software program or a graphing utility with matrix capabilities to find the determinant of the matrix. 5432 21 16 2 5 43 3 542 • Line of best fit, method of least sqaures. Question 11 (6 marks) A person recovering from an illness keeps a record of the value Y of how they are feeling, scored out of 10, after X days into their re… • Matrix, eigenvectors, eigenvalues. Question 10 (7 marks) (a) For the matrix 0 1 â€”4 M = 1 1 1 1 0 5 it is known that the three eigenvectors are â€”1 â€”3 â€”5 0 , i2 , 4 1 1 1 (i) Specify the corresp… • orthonormal basis with respect to the dot product. Iâ€”l Question 9 (6 marks) Consider the basis for R2 given by B = {b1, b2}, where bl = %(1, J3), b2 = %(â€”\/3, 1). (a Is 3 an orthonormal basis with… • . Problem 4 Consider the following LP: maximize Z = 6.7121 + 5:192 + 4.7123 + 5334 + 6.7125 $1+$2+$3+$4+$5 5331 + 4552 + 3333 + 2504 + :05 14 \|/\ \|/\ :0, 2 0,for i: 1,2,3,4,5 Solve the problem using… • . PDFfiller – wes form. \| LAB 1 FOR C HW5: Problem 16 Previous Problem Problem List Next Problem (1 point) Let V = (-5, 00). For u, v E V and a E R define vector addition by u flu :- uv + 5(u + v) +… • . 8.19 Let A be an m x n matrix with linearly independent columns. Explain how you can solve each of the following problems using the QR factorization of A. Give the complexity of your algorithm, in… • . Problem 3 Show that the dual of the linear program minx Ax 2 2 2 0 is the linear program max, bry ATy < C y 2 0 • Solve the system 401 -512 +413 +404- 3 +2 +213 +204 = 2 -4×2 +613 +614= 5 -201 +212 +403 +404= 4 14 14 13 12 12 11 +t • Please Show all work as well why you did a particular step. Thank you Â . 4 3 LADW # 4.2.3: Compute 11mm, where A = [I 2 . Write your answer as the product. of that: matrices. • Please show all work as well as why you chose to do a step. Thank you.. 4 3 LADW # 4.2.3: Compute 11mm, where A = [I 2 . Write your answer as the product. of that: matrices. • Please show all work and why you do a particular step. Thank you.Â . 1. Let 4 2 4 A = 2 1 2 2 (a) Find bases for each of the cigenspaces (called eigenbases). Note: There are two distinct eigenvalues, … • (4) Using the geometric series 0Â° 1 2:5″ : for \|33\| <1 7320 1â€”11: write the following functions as a series. (a) SEQ/(1 – II?) (b) 1/(1+ 9:52) • Linear transformation. Question 8 (6 marks) (a) Let T : “Pg â€”) ‘Pz be the linear transformation with standard matrix 1 l 0 [T]: 1 0 â€”1 . 0 1 l (i) Calculate T(mâ€” 2922). (ii) Find a basis for t… • 313 Use the Gauss-Jordan reduction to solve the following linear system: 412 813 11 11 – 1203 311 T1 O C3 • Matrix question. lâ€”l Question 6 (8 marks) Let A be a 5 X 5 matrix and let V1, V2, V3, V4 be column vectors with ï¬ve entries. Suppose that 1 1 0 0 0 â€”1 0 â€”1 0 0 0 1 â€”1 0 2 0 1 1 [AIV1 V2 V3 V… • Find all values of c, if any, for which the below matrix is invertible (try to reduce matrix using row echelon method to and prove that way based on the values that c cannot be please) .Â Â . C C • Find all values of c for which the given matrix is invertible. (attached matrix picture below) Â . C • Question about vector addition, span. Question 5 (7 marks) (a) Let H C R2 be speciï¬ed by H = {(ac,y) e R2: at 2 0, y 2 U}. (i) Sketch H. (ii) Show that H is closed under vector addition. (iii) Show … • Vectors in R^3. Question 4 (5 marks) Consider the vectors in R3 x=(1,1,1), u=(â€”l,2,2). (a) Calculate the cosine of the angle between x and u. (b) Find the vectors a and b such that x=a+h Where a is … • . Consider the basis for R2 given by B 2 {b1, b2}, where bl = %(1, x/g), b2 = %(â€”\/Â§, 1). the origin in the direction of b1. Specify [T] 3. (e) With T as in (d), calculate [T]3. • . 1. Let A be an m x n matrix. (a) Suppose that for each j = 1 . . . k the system of equations, AX = B; is consistent. If B is a linear combination of the vectors Bj, j = 1 . . . k, is the system AX… • . Find a set of basic vectors {11,55} in R4 so that every vector in the solution set of the equations can be written as a linear combination of the basic vectors. wâ€”zcâ€”z _ 0:. 5w+2mâ€”y+2z : 0. … • . Is Z7 a group under multiplication? What about if you remove [0], from Z7 will it be a group under multiplication? If so complete the group table for Z7 – {,} and find the inverses of every elemen… • . 1. Let A be an m X a. matrix. (a) Suppose that for each j = 1 – – – k: the system of equations, AX = B, is consistent. If B is a linear combination of the vectors B_,.-, j = 1 – – – k, is the syst… • Let W be the subspace of R3 defined by the equation 2x – y + 3z = 0. (a) (2 points) Find a basis {v1, v2} of W. (b) (1 point) Find a vector v3 that is orthogonal to the plane W. (c) (3 points) Expl… • The answer above is NOT correct. Solve the equation -1x + 2y – 3z = 0 -2 • Need help solving for x and y. Solve the system by finding the reduced row-echelon form of the augmented matrix. x – 2y – 2z = 3 -2x+ y + 7z -9 -2x + 5y + 3z reduced row-echelon form: T O O How many s… • Suppose that A, B, and A + B are invertible n X n matrices. a) Simplify the expression 111(14â€”l + B_I)B(A + B)_l. b) What does a) tell you about the matrix Aâ€”1 + B". • Consider the matrices: 2 O 0 4 N O W -4 6 -6 O N -1 4 0 0 2 0 D = OUINO 8 A = 1 2 0 – 1 B = C = 5 0 7 -3 4 O 7 OO O 4 -2 -2 Compute the following matrices, if they exist: AB, BA, (AB – 2DT)T, (CAB)T, … • How do I do this?Â ** Example 9:Â Â . 9. In Â£1,113! r’,-=(cosï¬‚)r’+(sinâ‚¬)j 1′,- â€”(sin9)r’+(cosâ‚¬)j Let T, be the operator that carries 1′, into j, and j, into 0. (a) Calculate the matrix … • How do I do this?Â Here is example 1:Â Â . 5. Prove that the function T from R’ into R2 defined by the equation is linear, by finding a suitable matrix A such that 7 = L, and applying Example 1 of t… • . CHALLENGE ACTIVITY 7.4.1: Finding an orthogonal basis using the Gram-Schmidt process. 377796.1678420.qx3zay7 v Jump to level 1 1 v Let {ul (x) = 9, u2 (x) = -18x, u3 (x) = -12×2} be a basis for a … • I need help finding which two of the equations is not vector Space in R^2, I need at least on axiom that is violated. S =(x, y) Â . utv = (uj, uz)(v1, V2) = (uj + v,, In uz + Invz) • im going to start sobbing pls help me. 1. What is the volume of this object? 3 ft 5 ft 7 ft 2 ft 3 m 7 ft 5 ft esc 20 988 #$ 2 3 4 Q W E R T U tab A S D F G H caps lock Z C B hift X N control optio… • Consider the following system of linear equations: 2x – 2y +z + 2w = 3 5x – 3y + 3z + 5w = 7 2x + 6y + 3z + 2w = 1 Â Write the system as an augmented matrix, find its RREF, and solve the system. • . 1. [Strong 2.1.3) When two equations in a linear system are exchanged, which of the following are changed: (a) the planes that make up the row picture; (b) the vectors in the column picture; (c) t… • a = 0 3 4 Â Â Â 0 3 3Â Â Â Â 1 -2 3Â b = 0 Â Â Â -3 Â Â Â 18 Let A be the matrix below and define a transformation T :R^3â†’R^3 by T (u)Â =Â Au . For the vector b below, find a vector u… • T is a transformation from r^2 to r^2. find matrix A that induces T if T is rotation by 7/4pi • t: r^2->r^3 this is a linear transformation find t(w • I’d appreciate help on this. Thank you so much! Â . Which of the following statements are always true? (i) Suppose that 1 = 9 is an eigenvalue of a matrix A. Then the equation Ax = 9x has exactly one … • I really appreciate any help on this question, thanks! Â . Let 4 27 6 A = 0 1 0 Find a basis for the eigenspace corresponding to 1 = 1. • I would appreciate help on this. Thank you! Â . Consider the following matrix A (whose 2nd and 3rd rows are not given), and vector x. a11 2 10 A = 2 Given that x is an eigenvector of the matrix A with… • I need help with this question. I appreciate it! • I would appreciate help with this question, thank you so much. • I need help on this please. Thanks! Â . Let 2 = 4 NID 0 Find det(-354 ) • Hello, I’d appreciate help on this question. Â . Find the determinant of the following huge matrix. O HOOO -5 10 10 6 5 -1 -8 10 3 224 1 222 -3 -1 -2 10 6 -5 -9 -9 -8 4 10 -7 • I’d appreciate help on this question, thanks! Â . Let a a2 a3 A= b by by C1 62 C3 Suppose that det(4) = 5. Find the determinant of the following matrix. a +4bj c1 301+ by an +4by c2 3an+ by a3 + 4b3 6… • I appreciate some help with this question.Â Â . Which of the following statements are always true? (i) If A and B are invertible matrices then then (BA) x = 0 has only the trivial solution. (1i) Supp… • I appreciate help with this question. Thanks a lot! Â . Consider the following two elementary matrices. 0 E1= E2 = Suppose that A is a 2 x 2 matrix such that Ez E1 A = 12 – Find the matrix A. What is … • Let A be an m X n matrix, B be an m x 1′ matrix, and let EA denote the RREF of A. Note that A and B have the same number of rows. (a) What is the RREF of the matrix (A\|B)? Prove your answer. (b) Wh… • I need help with this question. Thank you so much! Â . A matrix A is called self-inverse if A" = A. Which of the following statements are always true? (i) If A and B are both self-inverse 72 x 72… • I need help with this please. Thank you. Â . Suppose that A is p x 2, Bis n x m. C is 7 x q, and CBA is 5 x 3. What is the value of m? • Hi can I get help on this please, thanks! Â . Suppose that you were to try to find a function of the form y = ax + by + ex that passes through the (x, y) pairs (3. 1). (4, 8), and (7, 3). To obtain th… • Rex Things throws his mother’s crystal vase vertically upwards with a mass of 50.7 kg, compute for the velocity prior on throwing the crystal. Drag coefficient is 20.5 kg/s. 1. Plot its exact graph to… • Please help meÂ Â . The vector ï¬eld F(a:, y,z) = (2273; + 25,332,12) is conservative. Use the Fundamental Theorem of Conservative Field to evaluate / F -dr, where C C is an unknown curve from (1, ?… • Please help me with the following questionÂ 1.Â Â . Compute F . dr, where F(x, y) = (x2, -xy) and C is the part of the circle x2 +y? = 9 with x < 0 and y 2 0, oriented counterclockwise. ONone of … • . Question 6 0 pts (10 points) If T is a linear transformation with T(2u + 30) = ( 2 ) and T(a) = (3) then compute T(v – u). O O O O • . Question 5 0 pts (10 points) If T : R" – Rmis a linear transformation that is one-to-one. What can you say about m and n? What about if T maps R." onto R." what can you say about m … • Please help me with the following questions 1. 2.Â 3.Â Â . Use the Chain Rule to evaluate ag as at s = 4, where g(x, y) = x2 – y2, x= s’ +1, y=1 -2s. O 132 O -28 O 200 O 272 O -300 ONone of these … • . 8. A quadratic form Q(x) = xAx and its corresponding symmetric matrix A are called positive definite if Q(x) > 0 for all nonzero vectors x. (a) Prove that Q is positive definite if and only if… • . CHALLENGE ACTIVITY 7.4.1: Finding an orthogonal basis using the Gram-Schmidt process. 377796.1678420.qx3zqy7 v Jump to level 1 1 V Let { U1 = 22 8],U2 – [8 16].03=[0 71} 2 be a basis for a subspac… • Finding commutative matrices. Â My assignment is asking me to play around with shear, stretch, and stuff to find matrices A and B where the graph of ABv = BAv for all of the unit vectors v. Also, I h… • develop an equation that can be used to determine the value of each term pattern e.g 1.5 • . Question 2 0 pts Let 1 2 U 7 _. â€”1 _, 3 _, 1 _, 2 v = , v â€” , v â€” , v â€” 1 â€”1 2 5 3 4 4 3 1 1 U 5 1 0 U 3 The reducedechelonform ofA[Â§1,32,ï¬3,E34]= l] 1 0 2 0 0 1 â€”1 l] U U U (10 poi… • help Â . Let T be a linear operator on a complex finite dimensional inner product space V. If T is self-adjoint, then (T(x), z) E R for all x in V. O True O False • Help me with this problem and I would appreciate it as fast as possible. Â Keep explanations brief. Thanks! Â . Let T be a unitary operator on a finite dimensional complex inner product space V. Then … • If a nonhomogenous system has two linear equations in three variables. How many solutions might there be? Explain this geometrically. • . Question 2 (a) Define the column space CS(M), the row space RS(M) and the null space N(M) of a matrix M. (b) A linear transformation 7 : RS – R is defined by T(x) = Ax, where -2 UI N K X -3 X … • . Theset V={f:Râ€”>R\| l"(x)=a+bx+r:x2 wherea,b.cER} is a vector space where the operations of vector addition and scalar multiplication are defined as follows: For all f, g E V and all A E R… • (1 point) Given the matrix 4 4 6 a find all values of a that make \| A\| = 0. Give your answer as a comma-separated list. Values of a: • [9 pts] Let u be a solution to the homogeneous linear system Ax = 0 and 1 be a solution to the linear system Ax = b. Prove that v-9u is a solution to Ax = b. • 4 4. [12 pts] Factor A = as a product of elementary matrices. 1 0 • [12 pts] Solve the following LS by first obtaining the RREF of its augmented matrix. – X, + x, =0 X -X, +x, +3x, = 2 -2x] + 2×2 -3×3 -8X4 = 1 • Let Mï¬‚xï¬‚(R) denote the n X 5; matrices with real entries. (at) Sketch a proof of the fact that Mï¬‚Xï¬‚OR) is a vector space. (1)) Prove that T(A) 2 AT is a linear transformation from MHXAR) to M… • Given the system Â x1+rx2=3×1+rx2=3 x1+x2=sx1+x2=s Â Â For which values of r and s is there a single solution Â For which values of r and s are there no solution Â For which values of r and s are… • . If C is a 6 x 6 matrix and the equation Cc = v is consistent for every v in Ro, then Ca = w might have infinitely many solutions for some w in R6 Select one: O True O False If A is a 4 x 6 matrix,… • Given the system Â x1+rx2=3×1+rx2=3×1+x2=sx1+x2=s Â Â For which values of r and s is there a single solution Â For which values of r and s are there no solution Â For which values of r and s are … • . Question 1 [2x3pts] a. Let A and B be nonsingular N x N matrices. Show that (AB)-1= B-1A-1. b. Let A and B be Nx N matrices; let C=AB. Starting from the definition of a matrix multiplication, Ci =… • . 7 P 2 7 PK – 1 1xo – * PK Write the formula and find the limit. x x6 – x atb • . Question 7 If A has an inverse, then the system Ax = b in inconsistent for all b in RT Not yet answered Select one: Marked out of O True 1.00 P Flag question O False Question 8 If a set of v… • . The nullity of a matrix equals the number of pivot columns in the matrix Select one: O True O False. If A is a 4 x 6 matrix, and v is in R, and if Ax = v is consistent, then it must have infinitel… • . Solve using Gaussian elimination -201 + 12 + 213 = -11 4×1 – 212 + 213 = -14 211 – 312 – 213 = -13 = T2 = T3 =. If A has an inverse, then the system Ax = b is consistent for any bin R Select one: … • If AA is a 6Ã—86Ã—8 matrix with 5 pivot columns, then: The dimension of the column space is Â The dimension of the null space is • QUESTION 10 9. Consider the system of linear equations X1 – 2×2 + X3= 4 4×1 – 8X2+ 12×3 =16 (a) Find the reduced row echoleon form of the augmented matrix of the linear system. (b) Find all the soluti… • . DDSB Mob X Netflix X @ Homepage x 2 Thursday, x Meet O X Attachmer X Homeworl x It Find the ni X G vowels – G X + X CO orbit.texthelp.com/?file=https://drive.google.com/uc?id=1pZ8RwHMtIWNN4fhlqzwl… • If you subtract 3 from 2 times a number x, you get 6 • QUESTION 1 Let A be a 3 x 3 nonzero matrix. Suppose Ax = 0 has a unique solution x = 0. For any b in 3 the number of solutions of Ax = b is O A. 1 O B. infinite O C. not decided based on the informati… • Suppose columns of A are linearly independent and we know that Ax = b is consistent for some b. What can you conclude about the solution set of this system? Explain. (is it unique/infinitely many?) • The point(1,1) is on the graph of y(x)=x^5 . Find the corresponding point on the graph of y=2f(-1/4(x+2))-1. T / 4 Your answer 6. Provide an example of a quartic function in each of the following s… • can you give a step-by-step solution on how they reached the answer for question 3. • . NAME GRADE & SECTION Represent this function in the following forms DATE SCORE (For items 8 to 10): 8. Table of Values (Complete the table) MATH 8 Q2 PERFORMANCE TASK 2 Solving a Linear Functi… • One of the best all-wheel-drive sports utility vehicles was developed by KIA Motors, Ltd. (KML) during the Covid-19 pandemic era in 2020. As part of the marketing campaign, KML produced a video tape s… • Complete part B please. 8.19 Let A be an m x n matrix with linearly independent columns. Explain how you can solve each of the following problems using the QR factorization of A. Give the complexity o… • Complete part A please. 8.19 Let A be an m x n matrix with linearly independent columns. Explain how you can solve each of the following problems using the QR factorization of A. Give the complexity o… • If A is invertible, then the system of linear equations Ax = b has a unique solution x = A âˆ’1 b. Consider a non-invertible square matrix C and the homogeneous system corresponding to this matrix Cx … • Let S = {v 1 , v 2 , … v n } be a linearly independent set and let b âˆˆ Span(S). Suppose we can write b as a linear combination of these vectors in two ways, b = c 1 v 1 + c 2 v 2 + … c n v n and… • . 2. Keith and Michelle went out to dinner. The total cost of the meal, including the tip, came to $53.70. If the combined tip came out to$9.60, and each friend spent an equal amount, how much did … • Let c = 273x + 23 (mod 1155). Find two different values1 of x that yield the same value of c. Â So I’m stuck on question 1(b), and I have no idea how to solve this. Â I just need an explanation on… • Mohammed invested $875 at 5.2% per year. When the investment matured, mohammed received$22.44. Determine the term of mohammeds investment.Â Â 6. suppose the term of an investment at simple interest… • Let Î² = {v 1 , v 2 , …, v n } be a linearly independent subset of Rn for some n â‰¥ 2. Further, suppose we have that Span(Î²) = Rn. Consider the following sets: (a) A = Î² âˆª {z} where z (not belo… • Suppose that E 2Ã—3 and Eâ€² 2Ã—3 are row-reduced echelon forms of a matrix A for the system Ax = 0. (i) Do Ex = 0 and Eâ€²x = 0 have the same solutions? (ii) Can we say that E = Eâ€² ? Prove your ans… • Let {u, v, w} be a subset of distinct vectors in Rn for some n â‰¥ 2. Prove that {u, v, w} is linearly independent if and only if the set {u+v, v+w, u+w} is linearly independent. • linear algebra questions, in the first question you just have to determine if T it is linear justifying the answer in the second is to verify the two transformations through demonstrations and graphs … • Polynomial fitting. Suppose we observe pairs of points (ai, b;), i = 1, …, m representing measurements from a scientific experiment. The variables a, are the experimental conditions and the b; co… • Least Squares Approximation of Matrices. a) Derive the solution to least-squares problem min,, x – Tw\|; when T is an n- by-r matrix of orthonormal columns. Your solution should not involve a matrix… • . Suppose that A, B, and A + B are invertible n X n matrices. a) Simplify the expression A(A~] + B-1) B(A + B)-1. b) What does a) tell you about the matrix A~ + B-1 • 3-39. Dennis drives for 1 hour at average speed v. Then he drives for another hour at average speed 4v. (a) Find the overall average speed. (Recall that the definition of average speed is T – total di… • Suppose you are in charge of a grain store and you have the problem of managing the storage.Â Further, you know that for today: – Grain can be sold for $2.00/bushel, and you believe that the price wi… • . (10 marks) Let A, B, and C be square matrices of the same size. Using the definition of invertible matrix prove the following tw statements (proof that uses determinants will be given zero marks)…. • . UNIT# 4 The job is to develop it at home and then upload the attached PDF document the answers obtained are based on what was learned and developed in the compendium, asynchronous classes, synchro… • You are given the (2, 2)-matrices -7 -8 2 -15 A = B = -27 -31 -15 14 / C = ] -1 Solve the matrix equation below for X. (A-‘XA~ + B)A2 = (B + BT ) (C – CT) -25: -131 X = -971 -517 • . 5 12 8 2 10 2 -5 Let b = and let A be the matrix Is b in the range of the linear transformation x:Ax? Why or why not? – 2 0 1 3 4 -40 -8 4 Is b in the range of the linear transformation? Why or wh… • I need to find the point C, in its mathematical form Specify the surface f (x, y, z) on which \|S\| = 1. Nothing in Matlab. D1.2. A vector field S is expressed in rectangular coordinates as S = (125/ [(… • Please help to solve the questions. Thank you! Â Â . 12. [-11 Points] DETAILS HOLTLINALGZ 3.1.037. Find an example that meets the given speciï¬cations. A linear transformation T: R2 â€”> R2 such … • https://drive.google.com/file/d/1r2rrGqlYJMaDqjA7UExQBPMFdZkQKPDw/view?usp=drivesdk Answer question 1a,1b, and 2 clearly with fullworfs and explanations. Paragraph format and mathematical reasoning wi… • 3 to 4 sided objects to make your garden look beautiful • Linear Algebra Assignment 3 Due Sunday at 11:59 PM 1 Directions Complete the following problems showing all your work. You may use a calculator to check your work, but should write out (or TeX up) all… • Please help! Don’t know how to prove this.. Suppose that a matrix A has integer entries and an eigenvalue 0 < IAI < 1. In this question, you will prove a big theorem: Any Aâ€”eigenvector of A ha… • . 1. Let A E M3x3 and x, y, Z E R’ . Suppose that we have: Ax 0 Ay = 1 AZ = Find the value of the determinant of the matrix A. 2. Let O O 2 0 A = 4 5 B = 0 3 0 O O 4 Find the value of det(A2 B-1 A-2… • . Suppose that A B is invertible. Show that A and B are both invertible. • . Suppose that a matrix A has integer entries and an eigenvalue 0 < \|1\| < 1. In this question, you will prove a big theorem: Any 1-eigenvector of A has an irrational entry. 1. Suppose that x h… • . CHALLENGE ACTIVITY 7.4.1: Finding an orthogonal basis using the Gram-Schmidt process. 377796.1678420.qx3zay7 v Jump to level 1 1 V 2 Let {us – [7 2], 02 – [8 82 ],03 = [8 31} be a basis for a subs… • please find the slope of these 3 questions in the attached document. 5.5 Practice For use after Lesson 5.5 Find the slope of the line. (1, 3) ( 2, 4 ) (4, 4) in m N – O -3 O O (1, 1) m ( -3, – 2) M- … • Perform the following operations in polar form and convert the result to binomial form: Â . E) (240)3 a) 3450 . 2150 b) 3150* * 445 h) 1332 – 3410 216" 3120 () 133* * 216 341" i 21" d) … • Could somebody explain this task to me in detail and show all steps? Â Â Â Â The total costs of a company can be described by the function C(x) = -0.002xÂ² + 0.63x – 1 and sales revenue by the fun… • Could somebody explain this task to me in detail and show all the steps? Â Calculate the market equilibrium point forÂ Supply: p=0.1q+11 and Demand: p= -0.02qÂ² + 0.2q + 12 • Could somebody explain to me this task and show all steps? Â Calculate the market equilibrium point for theÂ Supply: p=0.1q+11 and Â Â Â Demand: p= -0.02qÂ² + 0.2q + 12 • Could somebody explain to me this task and show all the steps? Â Calculate the market equilibrium point for theÂ Supply: p=0.1q+11 and Demand: p= -0.02qÂ² + 0.2q + 12 • Could somebody explain this task to me in detail and show all steps. Â The number of clients from a company was 23’840 in the year 2003 and 20’504 in the year 2009.Â Â a) Find a linear function tha… • Write out the augmented matrix and reduce it to row echelon form using row operations in Octave, without using row interchanges. Write the reduced matrix such that there are 1s on the diagonal. Enter … • . Problem 3 Show that the dual of the linear program minx c’x Ax 6 0 is the linear program max, by ATy KC 2 • [5pt] Question 2. [continued from Question 1.] The following data are HDL cholesterol levels (mg/dL) of the first five study participants: Participant ID HDL cholesterol levels (mg/dL) 52 55 47 42 43 … • . 6. Let u1 -3 , U2 = Show that {ul, u2, u3} is an orthogonal basis for R3. 0 6 Find y = 2 as a linear combination of the basis. • . 4. For k = 1, 8, 12, define 1 1 Ak = 2/3 – 10-k 2/3 + 10-k Ck = AL Ak 1/3 + 2.0 * 10- 1/3 – 2.0 * 10-k (a) Use matlab or python to compute the singular values of Ak. Call them oik for i = 1 and i … • what is the best website to study drivers ed chapter 9 review its page’s 165-178 i need to figure it out cause i want to start driving • . MATH 111 MT3 2021-2 page 5 of 5 pages Q4. [9 marks] The diagram at the right displays three vectors in the plane. They are all drawn at the origin. I have omitted displaying the coordinate axes to… • I need an explanation. and Than please don’t copy the answer from Chegg. Thank you.. For 9/27: Step 1: Pick a matrix and find nul(A). Pick a matrix A of size no smaller than 3 x 5 (to get a good feel … • . 5. Evaluate the following determinants. 10 7 -9 N 3 (5a) 7 -7 7 7 v . (5b) -1 3 -8; 4 (5c) 2 -2 6 2 2 -5 2 -3 -3 4 I a a 2 a3 1 b 62 63 (5d) 11 C C2 3 l d d2 d3 [2+2+3+3 marks] • . 3. For each matrix, find the inverse or show that the matrix is not invertible 2 1 3 2 4 (3a) (3b) " . (3c) -3 4 6 3 1 2 3 3 1 (3d) 5 6 (3e) 1 2 1 ; 7 9 2 1 1 O -1 -1 0 -1 2 0 2 -3 1 -1 1 -2 … • The capacities at which U.S. nuclear power plants are working are shown in table for various years. Â Year Percent 1975 56 1980 59 1985 58 1990 70 1995 76 2000 88 2004 89 Let f(t) be the capacity (in… • Can you please explain these? Also how do i check span?Â . Consider the vector space Ro equipped with normal vector addition and scalar multiplication. Determine if the following subsets are subspaces… • Could you Please provide an explanation for these. Thank you so much. Consider the vector space R3 equipped with normal vector addition and scalar multiplication. Determine if the following subsets ar… • Given that Ve R with basis vectors v1 = [1 1] and v2 = [2 1]". Answer the following questions given their coordinates in two subspaces U and Ware: . [Vi]w = [7 1], [v2]w = [13 3] . [vily = [-125]… • Please provide detailed working for the answer. Thank you.Â . Find the following limits algebraically. Show all steps neatly and clearly. l l 1 2 • . 3. Let u, U ER". Show that \| \|u + 3\|12 + \| \|u – 31/2 = 21/4112 + 2118112. 4. Let W be a subspace of R". Let WJ be the orthogonal complement of W. Show that WA is a subspace of n. 5. Let … • . 7. Let U be an n x n orthogonal matrix. By definition, U has orthonormal columns. Show that since U is square, it also has orthonormal rows. • . 2. Pythagorean Theorem states that u, U E R" are orthogonal if and only if \|lu + \|12 = \|141/2 + 11312. (a) Prove the Pythagorean Theorem 5 2 (b) Let u = and v = 1 Compute u . 7, llu\|\|2, lv\|12… • Can you please provide the reasoning for each of these questions please. Â The answers are: T, F,t,t,t,f Â . 1. Indicate if each of the following statement is true or false. (a) If a linear system Ax… • . 1. Let u = -3 and 1 = 2 (a) Are the vectors orthogonal? Why? (b) Find the length of the vectors. (c) Find the distance between the vectors. • can anyone help?. 1. (2 points) Find the feasible region for the system of linear inequalities shown below. You may shade either only the true region or only the false region, but be sure to mark the … • can someone help?. A pet store sells large angelfish for$20 each and silver dollarfish for $16 each. Each angelfish requires 5 hours of care and 4 ounces of flake food to reach full size, and each si… • can someone help please?. Completely set up the following linear programming problems. DO NOT SOLVE THEM. 8. Manufacturing Furniture A firm manufactures tables and desks. To produce each table require… • Under topic rank of a matrix, question 10 only please.. 9 Show that the rank of AA is less than or equal to the rank of A. 10 Let A and B be matrices, each with p rows (but perhaps different numbers o… • Question 9 only please. Under topic, rank of a matrix.. the product is singular if and only if the other of the matrices is singular. 9 Show that the rank of AA’ is less than or equal to the rank of A… • Under topic, rank of a matrix.. 8 (a) Suppose that A and B are p x p and that AB is nonsingular; prove that A and B both are nonsingular. (b) Suppose that A and B are p x p and that one of them is non… • 4 only please.. O 4 * 4 Prove that x = 0 is the only solution to Ax = 0 if and only if the rank of the p x q matrix A equals q, the number of unknowns. 5 Suppose that the system Ax = b of p equations … • Question 2 only please.. [0 0 0] 2 Suppose that A is p x q with rank p and B is p X r; show that A B has rank p. 3 Each of the following matrices is the augmented matrix for a system of linear equa- • . File Home COllIllellt Stan EZI ((0 IE) Â§s@% View Form Assignmentï¬o’ljnll ndf , Form PDF Reader Protect Sllare Help 0 Tell me… AssignmenmLSpZZpdf x u Linear equations – Related problems 1. M… • Please prove convexity of the following:. the Lagrangian function of the nonlinear optimization problem max f(x) s.t. gi(x) < bi, i = 1, …, m, x EP, where P is a polyhedron. • Please prove question (2). Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem min -x Ex s.t. ex=1, u x= R ac 2 where the target return must be achieved exact… • . 1. AtA= -2 2- -30- 18 7 – Solve the following (if possible): a) AC b) CA C) AB d) BT A e) (AC)B • . Q4. [9 marks] The diagram at the right displays three vectors in the plane. They are all drawn at the origin. 1 have omitted displaying the coordinate axes to keep the diagram simple. They are lin… • . Problem 1. (a) Let V be a C-vector space, where dim(V) = n, and let Te Z(V). Show that, for k = 1, 2, …, n, there is a subspace Wk C V that is invariant under T, and has dim(Wk) = k. (b) Give an… • Hi can someone solve this problem step by step and explain it? Â Â Â Â . Solve the following LP models using the graphical method: 1. Maximize : Z= 4X, + 5X: Subject to: 20x, + 25x,$ 60 10×1 + 5x:… • . Solve the following LP models using the graphical method: 1. Maximize : Z = 4x, + 5X2 Subject to: 20x, + 25x, $60 10xi + 5x:$ 20 and x; 20, j = 1, 2 2. Maximize : z = 3x, +7×2 Subject to: 10xi +… • You are given the (2, 2)-matrices Solve the matrix equation below for X. (AX) = B-2 . C X = Check Last saved at 1:19:21 • My answer is wrong. Question 7 You are given the (2, 2)-matrices Tries remaining: 2 A = 15 10. B – [13 3]. C- [4 3’5] Marked out of 10.00 Solve the matrix equation below for X. Flag question (ATC – X)… • Reflect on the concept of polynomial and rational functions.Â Â 1. What concepts (only the names) did you need to accommodate these concepts in your mind?Â 2. What are the simplest polynomial and r… • For the following exercises, use the given rational function to answer the question. Â 82. The concentration C of a drug in a patient’s bloodstream t hours after injection is given by C ( t )=2 t 3+ … • ur answer is correct. Ise the diagram to estimate the perimeter, rounding each number to the nearest housand. 8,167 om 39,412 cm 43,812 cm • Can you provide me a step by step answer? Â Formulate the Linear Programming Model • solve the following pleaseÂ Â Â . Question 1 …………………………. 5 Is the vector @ = -1/5 1 in the nullspace of A = – 15 0 O @ is in the nullspace of A O * is not in the nullspace of A T… • . 211 What is the solution to the following system of equations? :1.- + 23; + z = 3 1.- + 33; – 2:: = 4 :I! T 1 .1: 1 â€”T {a} y = 3 + 1 e {e} y = 1 + 3 e 3 ï¬‚ 1 z G 1 I _1 _T [121} None of the abo… • . (1) How to divide $100 among four people according to given proportions: 1 : 2 : 4 : 8? (Use fractions only) (2) How to divide$100 among four people according according to given proportions: 1: ?… • Problem 2 Let M âˆˆ Mn(K) be any matrix (not necessarily symmetric), and define: qM : V â†’ K qM(v) = v tMv â€¢ Prove that qM is a quadratic form. â€¢ Assume BM is an alternating, bilinear form. Prove… • . 5. The vectors span R* because (choose all that apply) (a) the vectors (b) every vector in R is a linear combination of the vectors (c) given a vector in Ro, there are real numbers a, b, and c suc… • . 4. Use the accompanying figure to write the vectors a, b, and c as a linear combination of u and v. 2v V -V -2v -U C • Suppose that there are 50 scissors needed to be replenished at â‚±25 each, and that each glue costs â‚±16. If a budget of â‚±1890 is given for scissors and glues, how many glues may be bought with thi… • Geometrically, why does a homogenous system of two linear equations in three variables have infinitely many solutions? • Some airline tickets have a 15-digit identification number … where Â is a check digit that equals … mod 7. Find the check digit that follows each of these initial 14 digits of an airline ticket id… • Suppose A and B are m Ã— n matrices. Prove or disprove the following statements. To prove a statement is true, you should show it works for two arbitrary matrices. To show it is false, you just need t… • Consider the matrices If it is possible, compute each of the following. If it is not possible, explain why. Show all work necessary to justify your answer, which may include citation of previous compu… • Â Â Â i. Â What do we know about the pivots of A?Â Â Â Â ii. What do we know about the span of the columns of A? Â b) Â Â Â i. Â What do we know about the pivots of A?Â Â Â Â ii. Are… • . CHALLENGE ACTIVITY 7.4.1: Finding an orthogonal basis using the Gram-Schmidt process. 377796.1678420.qx3zay7 v Jump to level 1 1 -17 2 2 1 , U2 = 51 , Ug -1 be a basis for IR . Use the Gram-Schmid… • What if we want to find a linear transformation that does something specific? a) Suppose T is a linear transformation from R3 to R3 such that b) Suppose A is a 3 Ã— 2 matrix such that the equation hol… • Question 3 (5 marks) (3) Consider the matrix A below 3 â€”2 0 A: â€”2 3 0 0 0 5 (i) Show that A = 1 and A = 5 are eigenvalues of the matrix A. • . For parts a) to c) below, please assume the following: Let X = (X1 \| … [Xp) be an n X p random matrix such that Var((X );) = EVi, i.e. the _ is the covariance matrix for row i of X (the ith colu… • Let A be an m X 11 matrix such that rank(A) = 1′. Show that there exists matices, B an m x 7′ matrix, and C an r x n matrix, such that A = BC. • . Let V and W be ï¬nite-dimensional vector spaces, V1 be a subspace of V, and T : V â€”> W be linear. Show that T (V1) is a subspace of W. Apply the Rank-Nullity theorem to T\|V1 to conclude dim … • . Let V = M2(R). (a) Give an example of a linear map T : V â€”> V such that N (T) = R(T). (b) Give examples of distinct linear maps T : V â€”> V and U : V â€”> V such that N (T) = N (U) and… • are linear algebra exercises. The job is to develop it at home and then upload the attached PDF document the answers obtained are based on what was learned and developed in the compendium, asynchronou… • If A is an n x n invertible matix, then the equation Ax = b has a unique solution for any n x 1 column matrix b. Is this statement true or false? And why do you believe that? • Let A be an m x 13 matrix. Denote by (33- the vector in R” with all components zero except for a 1 in the jâ€”th position. Suppose that the system of equations AX = 83′ has a solution for each j … • . 1. Graph m 2 -3 on the 2. Without graphing, determine which of the following number line below. inequalities have (1, 4) as a solution. i. 3x + 4y > 19 ii. 7x – 2y 2 -1 3. Graph the inequalitie… • . 1 0 4 0 3 3. (a) Let A = 0 5 8 7 0 , and denote its columns by v1, . . ., Vs. Let b = 0 5 8 43 Using the row reduction 1 0 403 1 0 4 0 3 0 8 7 0 1 N 0 1 O – 0 5 8 4 3 1 O 0 -1 or otherwise, do the… • . 4. (a) A social club has a lucky draw every weekend in which the chance of winning a prize is 25%. Suppose that 0 Anyone who won a prize last week is not eligible for this week’s draw; they ente… • Â Based upon the data, is it reasonable to assume Bigfoot was heading inaÂ relatively straight line? Â If so, are there anyÂ reportedÂ locations that seem out of place? Â What possible reaso… • Fill in the table using this function rule. y = – 10x – 2 • Suppose A . B = 0. Show that this implies that C(B) is contained in Null(A). • Let A be an m x 71 matrix. (a) Suppose that for each 3′ = 1 ~ – . k the system of equations, AX = Bj is consistent. If B is a linear combination of the vectors Bi: 3′ = 1 ~ – . k, is the system AX … • . Problem 1. (25% + 10% bonus) Consider the following set X = {x, y ER : x2 – y> > 0, x > 0} CR2 and the following function f : X – R as f ( x, y) = Tr (2 ")"’). note that f(x,… • CT 0 1 2 -1 1 1 -2 2 1 4 1 6. Find the reduced row echelon form of the following matrices: 0 2 10 0 0 0 3 3 C = N -5 A = 2 1 -9 B = • Suppose a point x is drawn at random uniformly from the square [-1,1] x [-1,1]. • (a) Give an example of a 3 x 3 matrix A with C(A) = 3. (b) Give an example of a 3 x 3 matrix A with C(A) a line. 3. Show that for two matrices A and B one has that Null(B) is contained in Null(A . … • Consider the matrix and vector 1 X = 1 0 and b = 2 Note that X is defined identically in the preceding problems. a) Make a sketch of the orthonormal bases U and the columns of X in three dimen- sio… • Please prove convexity Â Â . the risk function of portfolio m Risk(m) z: E [R(m)] + (SE [Rm â€”E [R(m)] H Where R(w) is the portfolio return and 5 is any positive scalar. • . Question 1 (Markowtiz Efficient Frontier, (15 marks)). The minimum variance portfolio problem min x Ex s.t. ex = 1, ux= R ac 2 where the target return must be achieved exactly, short-selling is al… • . 5. In audio processing, one often wants to ï¬nd tonal sounds in segments of the recordings. This can be formulated as follows: We are given samples of an audio segment, 33k, k = O, . . . ,N â€” 1… • Linear algebra. Work Examples Q Solve the following 5 1) If A = -5 6. and B = [2 2 31 then find 2A+3B, 2B+3A, 3A-2B and 3B-2A? 2 2) If A = 3 -5 and B = 4 2 1 -3] then find 4A+5B, 4B+5A, 5A-4B and 5B-4… • Find the 22nd term in the following arithmetic sequence : 1, 4, 7,’ 10 Hint: Write a formula to help you. 1st term + common difference (desired term – 1) 1 + ? – 1) • HELP!. Find the 15th term in the following arithmetic sequence : 22, 19, 16, 13. Hint: Write a formula to help you. 1st term + common difference (desired term – 1) 22 + [? ]0 -1) • Solve asap. QUESTION III Gaussian Elimination/Gauss Jordan Reduction Method of Elimination and LU Decomposition (30 marks) Use the Gauss Jordan/Gaussian elimination to solve for the values of x. y and… • 1. Consider the three complex (2 x 2)-matrices X := 0 0 H := and O Y : 1 0 Problem C16.2 shows that B := (X, H, Y) is a basis for the linear subspace sI(2, C) of traceless complex (2 x 2)-matrices… • . (b) Let A be a 2×2 matrix that has: eigenvector u with eigenvalue 1 = 0.8 eigenvector v with eigenvalue a = 1.5 W Write the vector Aw as a linear combination of u and v and construct the linear co… • . Q4. [9 marks] The diagram at the right displays three vectors in the plane. They are all drawn at the origin. I have omitted displaying the coordinate axes to keep the diagram simple. They are lin… • . (Q3) (10+2+5 Points) Linear Recurrences. Consider a sequence ao, a1, a2, .. . that satisfies a linear recurrence of the form d an = C+ 61an-1 + 62an-2+ . .. +bdan-d =c+> bian-i (1) i=1 for all … • Quickly please. True or false: {-3×2 + 3x – 2, -2×2 – 7x – 1, 9×2 – 9 x + 6) is a basis for P2 (a) . • Quickly please. Let T : P1 (IR) – P1 (R) be given by T(s + tax) = (-s+t) +(-3s -3t)x a) Find [T]s where S = {1, x} : • Quickly please. Let B be an ordered basis for M2x2 () , the vector space of real 2 by 2 matrices. Find the coordinate vector [v]B, where v = 2 3 – VB = • Quickly please. True or false: {â€”73 + 10, â€”a: â€” 1} is a basis for P1 (3:), the vector space of real linear polynomials. • Quickly please. Don’t have to be specific. Let V = P2 (R) and W = R3. Define the linear transformation T : V -> W by a+2b+3c T(atbatca2)= 2 a+46+6 c -a-2b-3c (a):What is the Rank, Nullity and stand… • Quickly please. Don’t have to be specific. Â . Let B be an ordered basis for M2x2 (R), the vector space of real 2 by 2 matrices. Find the coordinate vector [v], where v = 3 3 4 • Quickly please. don’t have to be specific. Let V = P2 (R) and W = P2 (R). Define the linear transformation T : V -> W by T(atbatcx2)=2a-6b-6c+(-a-3b -4c)x+(a-b)x2. (a): Let A be the standard matrix… • Quickly please. Let V = P2(R) and W = nggï¬‚R). Define the linear transformation T : V â€”> W by aâ€”b+c 0 4aâ€”31bâ€”5c 4aâ€”25bâ€”3c)’ (a):What is the Rank, Nullity and standard matrix representa… • The world famous book: "How to break a belly down" that sells for $18.00 is sold 4000 times per week. However, when the price increases by$1, sales drop by 125. How many books should the bo… • The following table represents total cumulative sales (in millions) for The JoFe Group, a leading mathematical consultant group in the Caribbean. of Year X O 2 4 6 CO Sales 5.6 6 6.6 6.8 7 If x repres… • Given the following linear cost and revenue functions: C(x) = 38x + 152 R(x) = 53x it of Find the necessary number of items x to break even. • UB Math Club is going to organize a T-shirt drive for the upcoming Pi-Day on March 14 (3.14). The local T-shirt company is going to charge $125 for the T-shirt screen and$20 for each T-shirt produced… • Time left 0:47 If demand for a product is defined by q = – 20p + 45 and supply for the same product is defined by q =25p – 75, at what price should the product be sold for to ensure neither a shortage… • Students at University of Wakanda will pay $1835 for 9 credits and$2543 for 15 credits. If the cost function is linear, find the associated fixed cost. ROUND YOUR ANSWER OFF TO THE NEAREST WHOLE NUMB… • 2 Given 4x + 11, -10 <x<-2 of f (x) = x2 – 1, -2 <x<2 X +1, 2 <x < 10 Determine (if possible) f(11) O a. 12 Ob. 11 O c. None of these options O d. Undefined O e. 131 • The following table represents total cumulative sales (in millions) for The JoFe Group, a leading mathematical consultant group in the Caribbean. Year x 2 4 6 CO Sales 5.6 6 6.6 6.8 If x represents th… • Solve asap. Question 4 [10 points] Given the following matrices A and B, find elementary matrices E and F such that B – FEA You can resize a matrix (when appropriate) by clicking and dragging the bott… • . 9. Solve the following systems: 2×1 + 3×2 + 2×3 + 6×4 = 10 a) { X2 + 2×3 + X4 = 2 3×1 + -3×3 + 6X4 = 9 w+x – y – 9z = -3 b) (2w + 3x + 2y + 15z = 12}. 2w + x+ 2y+5z = 8 10. Determine A-1 if A i… • can you answer this problem please . and show the solution pleasse. Directions: Given the following system of linear equation: a) Solve the given linear system by the method of elimination. b) Write t… • can you help me answring this question please…and show the solution plsss. WEEK 2 Activity 2.1 Directions: Find the transpose of the given matrix. A= [6 8 9) c=\| Activity 2.2 Find the matrices obtai… • Quick answer needed. Question 5 (4 marks) Let V, W, X be finite-dimensional vector spaces over the same field. Let Ti be a linear transformation from V to W that is one-to-one. Let T2 be a linear tran… • Detail please!. Prove the following statement a C atc+2 V = R’, F = R, D b d b + d – 3 a aa a O Va, b, c, d, a E R, is not a vector space ab • Quick answer please!!. Show that the linear transformation T : P2 (R) – P2 (R) is diagonalizable, and find a basis B for P2(R) such that [T] B=D, a diagonal matrix, and state D. T(a + bac + ca?) = (2a… • Quick answer please!!!!. This question is continued on the next page. Let V be a real n-dimensional vector space and let W be a real m-dimensional vector space. a) 1 Mark Write down what it means for … • Quick answer please!!. Let U,V and W be vector spaces over the same field. a) 1 Marks Define what it means for the function L : U – V to be an isomorphism. b) 1 Mark Define what it means for the vecto… • Hi can someone explain and answer this difficult question? Formulate the Linear Programming Model Â . 7. An animal feed producer mixes two types of grain: A and B. Each unit of grain A costs P80.00 an… • Hi please help me this to showing step by step answer? Formulate the Linear Programming Model Â Â . 5. The Victoria Milling Company manufactures bags of concrete mix from beach sand and river sand. E… • Can you provide me a step by step answer? Â Formulate the Linear Programming Model Â Â . 3. A small generator burns two types of fuel: LOW SULFUR and HIGH SULFUR, to produce electricity. For one hour… • Hi can someone explain and answer this difficult question? Formulate the Linear Programming Model Â . 1. The Philippine Packing Corporation produces two pineapple products, which it sells in the marke… • . Exercise 17.5. Let F : R’ -> R be a function. (a) Consider a level curve F(x, y) = c defining y implicitly as a function y(x) of x. By using the Chain Rule for F(x, y(x)) = (Foh)(x) with h(x) =… • Hi can someone explain and answer this difficult question? Formulate the Linear Programming Model Â Â Â Â Â Â Â . 2. A local manufacturing industry in Iligan City produces three different … • Data from the lumber industry shows that the older the tree (a), the greater the number of board feet (n) that can be created from the tree. For a tree that is 160 years old, 18,200 board feet can be … • . Exercise 17.3. Let f : R2 -> R2 be the function f(x, y) = (x3y2 – y, xy3 – x), so f (1, 1) = (0, 0). By using the Chain Rule to compute D(f o f) (1, 1), give the linear approximation to (fo f) … • PP the minimum number of moves required to defeat the Tower of Hanoi game, based on the number of disks you must move. Think about the process of the game; and describe how your equation applies to it… • Thank you very much!! Â —– The mentioned Fibonacci numbers’ related information: Consider the Fibonacci recurrence F(n) = F(nâˆ’1)+F(nâˆ’2). Let the two solutions of the polynomial equation x^2… • . Exercise 17.2. Assume that f : R3 -> R3 is a function with f (1, 1, 2) = HNN and (Df) (1, 1, 2) = [1 3 1] 2 4 3 Let g(x, y, z) = Vx2 + y2 + 22. Calculate (go f)y(1, 1, 2). (Hint: this is an ent… • . 11.6 Show every subsequence of a subsequence of a given sequence is itself a subsequence of the given sequence. Hint: Define subsequences as in (3) of Definition 11.1. • Use the definition inside 10.6 and 5.4 to prove 11.8.. 11.8 8 Use Definition 10.6 and Exercise 5.4 to prove lim inf sn = – lim sup(-Sn) for every sequence (Sn).. 10.6 (a) Let (Sn) be a sequence such t… • En un comedor industrial se estÃ¡ planeando preparar el desayuno para el dÃa siguiente, teniendo dos alimentos disponibles A y B. El desayuno debe de contener los siguientes ingredientes nutritivos. … • Setting up an LP from a practical problem. In the fine tradition of bad puns on mathematical assignments, my colleagues and I are starting the soon-to-be-famous Opple Rubber Company. Our company ma… • Hi can someone solve and explain this problem? Formulate Linear Programming Model Â Â Â Â Â . 1. The Philippine Packing Corporation produces two pineapple products, which it sells in the market. P… • . Find the inverse of the matrix using elementary matrices. LE – 0 1 • . 0 3 0 -3 1 Let A and B be A = 1 3 8 = 1 3 0 -3 -1 3 2 Find an elementary matrix E such that EA = B. E = 10 1I • Answer all questions please. Problem 1 (8 pts) Let A, B, and C be the matrices given below. O 1 CO A = 2 0 -1 1 C = NN 2 3 B = O 0 0 2 Compute the matrices A – C, AT + B, AB, BA, AC, CA, BC, CB whenev… • Karl has completed 75% of his sticker album. Julia gives him the 10 missing stickers. How many stickers does the album contain in total? • . (i) if a vector space is the set of real valued continuous functions over R , then show that the set W of differential equation 2 gay -+2y =0 is a subspace ofV . dx2 dx (ii) Determine whether the … • Linear Recurrence Question: from Recurrence Relation on wikipedia (https://en.wikipedia.org/wiki/Recurrence_relation) Â Question: How to show that for any initial values a 0 â€‹ , . . . ,Â a d âˆ’ 1… • Q4 1 Point Let A be a square n X n matrix. If A is an invertible matrix then O The equation Ax = b has no solutions for each b in ". O A has less than n pivot positions. O The columns of A sp… • Question 3 x1=number of necklaces to be made X2= number of bracelets to be made X3= number of ear-studs to be made Maximize Profit (Z) = 75 x1 + 80 x2 + 85 x3 Subject to 5×1 + 2×2 + 2×3 – 80 gold stoc… • . Show that the only n x n matrix A that is invertible and also satisfies A2 = A is the identity matrix. • 5 marks Let m, n be two natural numbers. Let aij, yi, wj, Ci E R, be given for i = 1, . . . , m, j = 1, . . . , n. Compute the following. m n n max Eyi aikack + ci + wix; C1, ". . D’nER i=1 k=… • Could you please help me to solve this problem? Thank you so much!. 1. 5 marks Give an example of a dictionary (with more than one decision variables) for which the current basic feasible solution is … • . A. Transform each equation into standard B. Transform each equation into general form. form and determine the center and radius. 1. (x – 2)’ + (y – 3) = 25 1. 9x’ + 9y’ +6x =-5 2. (x – 5)’ + (y +7… • . Use at least three parts of the Invertible Matrix Theorem (Theorem 8 on p. 114 of our textbook) to give three independent proofs that the matrix A = (1) a is invertible if and only if b * ta. • I asked this question about getting vectors from a basic solution, <-3t+2s,t,s>, however I don’t know where the 1 and 0 are coming from or why they’re in that order. Can it be vice versa? Where … • Instruction: Formulate the dual problems for X2 the following linear programming model No need to compute/doing a tableau 1. Maximize Z = 2X 1 + X 2 Â Â Subject to: Â Â Â Â Â Â Â Â Â Â Â … • In the diagram shown below, Ilull = V40, llvil = V50, and proj,v = 1 . Find the length of d. V – – – – – – – u 1 0 5 0 -6 9. The matrix 01 3 0 1 W G7 00 represents the augmented matrix of a system … • The given homogeneous system has nontrivial solutions. Find the Basic Solution and vectors. Â I understand that the basic solution is Â <-2y+2z,y,z>, but how do I find the linear combination an… • (10 points) Find the inverse of the linear transformation 5×1 -10×2 +14×3 2×1 -3×2 +3×3 V2 +3×3 V3 X1 -2×2 VI + 32 + V3 . V3 : X= 32 + V3 . X3 = • (20 points) Suppose that you are in the garden supply business. Naturally, one of the things that you sell is fertilizer. You have three brands available: Vigoro, Parker’s, and Bleyer’s. The amount of… • I understand that the basic solution is <-3t+2s,t,s> but how do I get the two vectors from this? • 0 2 4 0 105 1. Let A = ONWH 0 0 0 3 0 0 0 0 Circle the correct answer to these questions: (a) Is A in echelon form? Yes No (b) Is A in reduced echelon form? Yes No (c) If A is an augmented matrix, doe… • SENSITIVITY ANALYSIS Â A tailor has the following materials available: 18 sqm cotton, 20 sqm silk, 5 sqm wool. A gown requires the following: 3 sqm cotton, 2 sqm silk and 1 sqm wool. A suit requires … • (12 points) Let i = 1 -1 , A = , B = 7 0 and b = 3 Whenever it is possible, find the following expressions. Explain why if you claim the operation is not possible. (a) Ab (b) 5 (AT) – 3A (C) BA (d)… • (2 points) Suppose that: 4 A = 3 2 Given the following descriptions, determine the following elementary matrices and their inverses. h. The elementary matrix Eg adds 3 times the third row of B to the … • Give explicit proofs of Corollary 2.32 parts 1 & 5. Corollary 2.32. Let A, B E Mmxn (IF). 1. If B, y are the standard bases, then [LA] = A 2. LA = LB < A =B 3. LA+B = LA + LB and LAA = ALA f… • Thank you!. Consider a sequence ao, a1, a2, . . . that satisfies a linear recurrence of the form d an = C+blan-1 + 62an-2+ . . . +bdan-d =c+> bian-i (1) i=1 for all n 2 d. Here d, c, bi are constan… • . rsonal Math Trainer 1.4 Reasoning and Proof R – Concepts 1 2 3 4 5 6 7 8 9 Question Example Step by Step Use a definition, postulate, or theorem to find the value of x in the figure described. ZAB… • need help please. 2 4 Question 3: Use Cramer’s rule to solve -1 2 I = 0 -1 2 Question 4: Find the determinant of 2 -1 -4 1 A = 1 -1 4 -2 2 -3 • need help my teacher also said this… In question 1 of assignment 5 I want you to find the determinants of all six items listed. I do not want to know what C^10 is for instance, just its determinant…. • Please answer like now. Time Below are linear functions, that represents either a linear demand function or a linear supply function. w (p)= – 10p+250 $(p) = 15p-75 B(p) = 35p +400 Determine which is … • Consider the Fibonacci recurrence F(n) = F(nâˆ’1)+F(nâˆ’2). Let the two solutions of the polynomial equation x^2 âˆ’ x âˆ’ 1 = 0 beÂ Î¦ = 2 1 + 5 â€‹ â€‹ â‰ˆ âˆ’1.618 and Î¦ ^ = 2 1 âˆ’ 5 â€‹ â€‹ â‰ˆ… • Question 1: Find the determinants of 3 1 3 4 – 2 1 . B – 2 C – OLOL NNON and then ATA, B + 212 and C10. L Question 2: Use the adjugate inverse formula to find A" ‘ when A = NO Question 3: Use Cra… • Elementary Linear Algebra. When it comes to linear algebra, we can even think of the polynomials as "vectors". Consider the transformation which takes real-valued polynomials of at most orde… • The question is The line segment joining points A(-8,-4) and B(-5,1) is perpendicular to the line Segment joining points CD. What is the slope of CD? • . 9. In 12′, let t’,=(cosï¬‚)i+(sin0)j j,= -(sin0)i+ (cosï¬‚)j Let 7; be the operator that carries 1′, into j, and j, into 0. (:1) Calculate tlte_ matrix representation of 1; relative to the basis… • . Q F: Assume that the solution set for the matrix A, which & A be a linear system in Xiy and S m 2m – 1 I MEIR 3 m+5 a ) find two distinct solution of the matrix ) Can ( 5 ) be a solution of th… • linear equation. 21. [-/1 Points] DETAILS HARMATHAP11 1.6.019. MY NOTES ASK YOUR TEACHER A manufacturer sells belts for$11 per unit. The fixed costs are $1800 per month, and the variable cost per uni… • linear equation and graph. MY NOTES ASK YOUR TEACHER 21. [-/1 Points] DETAILS HARMATHAP11 1.6,019. MY NOTES ASK YOUR TEACHER Decide whether the system of equations has one solution, no solution, or an… • Prove that f(x) = Ixl, + \|x, is also a norm on Rn (b) The "norm ball" is defined as the set of x for which an (arbitrary) norm f(x) = 1. Sketch the norm ball in R2 for the norm f (x) = I… • T is a linear operator on R’. Find a matrix A such that T(v) = Ao if (a) T(i) = i+ j, T(j) = i-j (b) T(it j) = i, T(i- j) = j.. 3. If e1, 93,. . . , e" is the standard basis for R”, obtain t… • 0 -31 -2 Let A and B be A = 2 3 1 B = 2 3 1 -31 -2 3 1 0 Find an elementary matrix E such that EA = B. E = • maximize his revenue. He has the following bottles available: 1 quart (32 oz.) Old Cambridge (a fine whiskey-cost$8/quart) 1 quart Joy Juice (another fine whiskey-cost $10/quart) 1 quart Ma’s Wicked … • . Give detailed answers Question 1: Find the determinants of 4- 3 5 8- 4 0- N 90 NONON and then ATA, B + 212 and C10 Question 2: Use the adjugate inverse formula to find A i when a – Question 3: Use… • . (1 point) Let f : RZ – R’ be the linear transformation determined by f(x) = Ax where 6 A = -3 -3 -2 a. Find bases for the kernel and image of f. vector A basis for Kernel(f) is -5, 1, -1 A basis f… • . (1 point) The matrix 2 4 -2 A =\-2.5 -5 -10 2.5 3 6 12 -1 is a matrix of a linear transformation T : Rk – R". a. Find k and n. k = ,n = b. Find the dimension of the kernel (or null space) and… • please use row reduction method. Determine if these vectors are linearly independent, 1 , 13 = 05 05 09 • I don’t understand. In Ra, we can rotate any vector v by / degrees by transforming it with the matrix, Re = cos 0 – sin 0 sin 0 cos (a) Show that rotating v by any multiple of 27 will result in the sa… • I don’t understand. n Ra, we can rotate any vector v by 0 degrees by transforming it with the matrix, Re = cos 0 – sin 0 sin 0 cos 0 (a) Show that rotating v by any multiple of 27 will result in the s… • Attention exploration (22 points) Multi-headed self-attention is the core modeling component of Transformers. In this question, we’ll get some practice working with the self-attention equations, an… • Let Pn be the space of polynomials in the variable X of degree at most n with coefficients in a field F. Define T âˆˆ L(Pn, Pn) by T(p)(X) = p(X + 1). Â a. (5 points) For n = 3, compute [T]Î± where Î±… • Let V be a finite dimensional vector space of dimension n over a field F. Let v be a non-zero vector in V . Let X be the set of all T âˆˆ L(V, V ) such that T(v) = 0. a. (3 points) Show that X is an… • Thank you.. (2 points) Consider the following Gauss-Jordan reduction: 6 2 0 0 0 0 O 0 0 0 O -6 O 0 0 1 0 = I 0 0 0 0 0 0 0 0 0 A EA E2 El A E3 E2 EIA EA E3 E2 El A Find 1 0 E1 = 1 0 0 Write A as a p… • Elementary Linear Algebra. Given the set of vectors, T= 2 (a) find a third, nonzero vector z such that the set { c, y, 2} is linearly dependent. (b) find a third, nonzero vector w such that the set {r… • (1 point) Find two unit vectors orthogonal to and A+0.40824 First vector. -0.816497 -0.40824 -0.40824 Second vector. 0.816497 • 05 09 (1 point) Let a = and b = be vectors. 1. Find the component of b in the direction of a. comp, (b) = 2. Decompose the vector b into b = v + w where v is parallel to a and w is orthogonal to a. V … • 2 (1 point) Let a = and b = be vectors. 1. Find the component of b in the direction of a . compa (b) = 2. Decompose the vector b into b = v + w where v is parallel to a and w is orthogonal to a. V W = • . 6. Determine whether or not the three vectors in R* are linearly independent. If not, find a linear combination that equals 0. 16. Find four vectors in R’ such that each subset of two vectors is l… • Briefly describe why casino resorts are superior to the stand-alone operations. • I need help with part B. I am not sure what vectors to use. • . 7. Any cubic function in one variable looks like f (x) = c323 + c2x2 + cix + co for some co, C1, C2, C3 E R. (a) Write the equation (in co, C1, C2, C3) which determines when the point (3, 2) is on… • I need help with part B. I do not understand how to answer it.. Let W be the union of the first and third quadrants in the xy-plane. That is, let W= \| \|: xy 20. . Complete parts a and b below. a. If u… • Solve the system of equations. If there is no solution, state "no solution". x ty + z =4 -2x – y + 2z = 6 x + 2y+ 5z = 18 • Solve the system of equations. If there is no solution, state "no solution". -2x +3y = 10 2y – z= -3 4x + 3z=-11 • Solve the system of equations. If there is no solution, state "no solution". xty – 2z = 6 2x – 3y + z =12 -3x – 4y + 3z = 2 • Solve the system of equations. If there is no solution, state "no solution". x ty + 2z = 6 2x – 3y + 5z = 1 2x+ y + 3z =5 • please provide detailed solution quickly. (3) Let A. B, and C be square matrices of the same order. Which of the following statements is false? (1 mark) (a) If A is similar to B, then A and B have the… • please use row reduction method. When it comes to linear algebra, we can even think of the polynomials as "vectors". Consider the transformation which takes real-valued polynomials of at mos… • Could you please help me understand and solve the below question. It feels somewhat cryptic. Thanks in advance. • This question is related to orthonormal vectors. Could you please help me out with this ? Thanks in advance. Edit 1: Unfortunately, this is all the information. Question is cropped from the homewo… • see attached Â . Piazza Question: Let T : V – W be a linear transformation, where dim(V) = n and dim(W) = m. Let B be a basis of V and let C be a basis of W. The matrix of the linear transformation c[… • see attached Â . 1) Let A = = 1 \| and define the linear operator T : M2(R) – M2(R) such that T(X) = AX – XA. a) Let = = {[: : ]. [8 : ]. [8 8 . 8 1 denote the standard basis of MILK ) Find [TE. (b) Fi… • Hi can help me with these 2 questions with step-by-step explanation ? Thanks in advance !. 2_ Given a 3 x 3 matrix A with eigenvalues 1, -1, 2, and corresponding eigenvectors v1, v2 and v3. Which of t… • Topic: Complex Numbers Solve all of these, thank you!. Practice Exercise 97 Operations involving Cartesian complex numbers (Answers on page 869) 1. Evaluate (a) (3+j2)+(5-j) and (b) (-2+j6)-(3-j2) and… • UNIT# 4 The job is to develop it at home and then upload the attached PDF document the answers obtained are based on what was learned and developed in the compendium, asynchronous classes, synchronous… • https://drive.google.com/file/d/1r2rrGqlYJMaDqjA7UExQBPMFdZkQKPDw/view?usp=drivesdk Answer questions 1a,1b and 2. Clearly with detailed work following all that is asked. Only original working and good… • https://drive.google.com/file/d/1r2rrGqlYJMaDqjA7UExQBPMFdZkQKPDw/view?usp=drivesdk Full questions here^ Answer all 3 parts. Questions 1a,1b and 2. Need full detailed answer with original working and … • Elementary Linear Algebra. (a) Determine if the homogeneous system BE = 0 has a nontrivial solution with, B = 3 -6 (b) What does your solution suggest about the columns of the matrix B? (c) Write your… • Elementary Linear Algebra. Find an example of the following: (a) A matrix transformation which is one-to-one but not onto. (b) A matrix transformation which is onto but not one-to-one. • Elementary Linear Algebra. Find the standard matrix of the transformation 7: R2 – R2 that is a vertical shear which maps e, to e1 – 2e2, but leaves ez unchanged. • . c(x) = ($ – 4)(x), $(.) is strictly increasing , not necessarily concave "(x) is strictly concave, For functions as defined in the previous part, consider the global maximizer of d(x), hereaf… • What matrix has the effect of rotating every vector through 90Â° and then projecting the result onto the x-axis? What matrix represents projection onto the x-axis followed by projection onto the y-axi… • linear equation. 10. [-/1 Points] DETAILS HARMATHAP12 1.R.053. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER In a certain course, grades are based on three tests worth 100 points each, three quizze… • equation and inequalities. 10. [-/1 Points) DETAILS HARMATHAP12 1.R.053. 0/100 Submissions Used MY NOTES ASK YOUR TEACHER In a certain course, grades are based on three tests worth 100 points each, th… • (1 consider the Following Statement : Every linear system having 37 and 62 and 12 -478 as a solution 2022 IF the above statement is true , prove it . If it is False , give an example of linear System … • Could Someone help me with these questions?! I need help!. 8. Let S be a linear system whose augmented matrix is 2 0 k – 2 0 0 k2 – 3k + 2 where k is some constant. The number of different values of k… • Exercise 10. Show that Hamilton’s scalar product is just the negative of the "dot product"; if v = bi + cj + dk and w = ei + fj + gk then S(vw) = -(be + of + dg) = -(b, c, d) . (e, f, g). Th… • Question 5 [10 points] Suppose T: P3->M2 2 is a linear transformation whose action on a basis for P3 is as follows: + T(3×3+3) = – 6 6 -12 T(x3 + x2+1 ) = -4 T( – 2x ) = T(3×2+3x+3) = 0 -3 Determin… • Elementary Linear Algebra. (a) Determine if the homogeneous system AT = 0 has a nontrivial solution with. A = (b) What does your solution suggest about the columns of the matrix A? • [1 0 0]. [6 -2 -3] [0 1 0]. X [8 10 6] [0 8 1] [-1 1 -1]. (3 points) In each part, find the matrix X solving the given equation. 18 91 x=1281x – b. 9 8 x =13 2. x = X = 4 13,024 FEB • Elementary Linear Algebra. please use row reduction method. Boron sulfide reacts violently with water to form boric acid and hydrogen sulfide gas (which smells like rotten eggs). The unbalanced equati… • Assignment 4: Problem 7 Previous Problem Problem List Next Problem (1 point) Enter a 3 X 3 skew-symmetric matrix A that has entries a21 = 5, a13 = 1, and a23 = 0. A = • Hi can we have help with this please. 3. Consider the following subset H of GL2(R) H= ((c d) : a,b. c, d EZ, ad – bc = +1 (a) Show that H is a subgroup of GL2(R) [3] (b) Let P = and 0 be the map that … • Elementary Linear Algebra. In Ra, we can rotate any vector v by & degrees by transforming it with the matrix, Re = cos 0 – sin 0 sin cos 0 (a) Show that rotating v by any multiple of 27 will resul… • Assignment 4: Problem 6 Previous Problem Problem List Next Problem (1 point) Enter a 3 X 3 symmetric matrix A that has entries all = 5, a22 = 2, a33 = 1, a12 = 4, a13 = 3, and a23 = 0. A = Preview My … • History Bookmarks Profiles Tab Window Help M Q 8 S – Yahoo C x C WeBWork : MATH_204_WINTE X + ordia.ca/webwork2/MATH_204_WINTER_2022/Assignment_4/4/?effectiveUser=s_ntoin&user=s_ntoin&key=2VO1… • (1 point) Write the system of linear equations 1x + 2y-1z = -8x + 6y+1z = -5 -3x – 3y + 2z = as a matrix equation. X E Note: You can earn partial credit on this problem. • Question 8, 2.3.31 Linear Functions & Slope Part 1 of 2 Use the given conditions to write an equation for the line in point-slope form and slope-intercept form. Passing through ( – 2, – 2) and (2,… • Elementary Linear Algebra. Determine if these vectors are linearly independent, 05 05 09 -2 , U2 1 • . 1 letm and n be nonzero constant. lets be the following system in the unknowns y and i nx tmy = n Which of the Following is true ?" ( a) S always has in Finitely many solutions. (b) S always … • Please solve this linear algebra problem. 1. Consider the subset S c Fo defined by S = { (21, 202, . . .) : an + En+1 – In+2 = 0, Vn > 1}. It is not hard to see that S is a subspace.’ Is S finite-d… • . Problem 3 [33 pts]: (5, 6, 9, 13) Penguin Perceptron Consider building a linear perception two distinguish two species of penguins’. We use the con- vention that Adelie penguins belong to group 0 … • Suppose that A, B are matrices and that AB = Im where Im is the m x m identity matrix. (a) A may not be a square matrix. How many rows and columns could A have? (b) Explain why the column space Col… • . 3 Simplex Method (score 20=7+7+6) Consider the LP problem: max 2 = 2.771 â€” 33:2 â€”â€” 373 â€” {104 st. â€”x1 â€”â€”$2 â€” 5103 S 4 .L’l â€” [Lg â€”â€” {Â£3 â€” {114 S 4 (P) â€”Â£E1 — 332 â€” … • . The bartender of your local pub has asked you to assist him in finding the combination of mixed drinks that will maximize his revenue. He has the following bottles available: 1 quart (32 oz.) Old … • 7) An insurance company is offering two insurance policies to your business. The monthly premium for Policy A is $50, and it gives$10,000 of theft insurance plus $180,000 in liability insurance. The … • . Consider the linear program Maximize z = x1, subject to: -x1 +x2 = 2, x1 +x2 5 8, -x1 + x2 2 -4, x1 20, X2 2 0. a) State the above in canonical form. b) Solve by the simplex method. c) Solve geome… • . 5. [5 points] Let A = =[3$1.B=[33 93], Dis a diagonal matrix such that D’ + A" = B. How many possible D matrices can be found? (A) 1 (B) 2 (C) 3 (D) 4 (E) 0 (F) More than 4 • Match the Device to its correct description. Group of answer choices Carries the Highest Amount of Traffic Â Â Â Â [ Choose ]Â Â Â Â Â Â Â Â Layer 1 SwitchesÂ Â Â Â Â Â Â Â Layer 3 Switches?… • I need 3 high quality comments on this page. They can be either detailed questions or comments. Need to show that I understand the content that is being demonstrated thanks • . 1 2 â€”3 I . A A . J A If I: ] Is the augmented matrix of a system of linear equations, then for what value of h, Is the system Inconsistent? 5 h 4 Oh=15 Oh=lï¬‚ Oh=5 Oh=ï¬‚ • . 1 ‘U’ Eonsider the matrices A = [I 1 [J I] 1 Which of the following statements is true? ï¬‚ 1] El 3 e. U: H Donâ€”L our: DIâ€”IID lâ€”ICID 0 Neither A nor B is in reduced echelon form. C) Both A a… • . 2 Let a1 = 3 , 82 = 2 and b = 1 . For what value(s) of h is b in the plane spanned by a1and a2? h Oh = -2 Oh =-1 Oh =-5 Oh = -10 • What point do you get if you rotate the point p = 3 âˆ’1 Â 5 T by 200â—¦ around the axis (through the origin) described by the vector Ï‰ = âˆ’1 2 âˆ’3T ? Â . What point do you get if you rotate the po… • . Which of the following sets of vectors is linearly independent? • . Oh=0 Ohaï¬ï¬‚ h, â€”1 D . For what value5{5) of h, are the vectors. V1 , V2, and V3 linearly independent? • . Which of the following defines a linear transformation from R2 to R3? O T = 2×1 – C2 3×2 + 5 O C1 T = C1+3 + 4 O T 1+ 2×2 = • . Suppose A is a matrix with four rows. Three of the following statement are equivalent. What statement is not equivalent to the others? O Every b in 14 is a linear combination fo the columns of A. … • . Let 7 : IR2 -> IR? be a linear transformation such that T and 7 ( _1) – [_3].F Find T’ O 2 CO O 12 – 09 O • . 1 1 1] Let 7 : 13 -> IR3 be a linear transformation whose standard matrix is A = 2 3 1 . Which of the following statements is true? 1 0 2 O T is neither one-to-one nor onto O T is one-to-one an… • . The selling prices of a number of houses in a particular section of the city overlooking the bay are given in the following table, along with the size of the lot and its elevation: Selling price L… • A laptop manufacturer ships products to either an electronics store in Los Angeles or Boston. The Boston store needs at least 32 more laptops for their inventory, while the Los Angeles store needs at … • A farming community grows both potatoes and carrots. They possess 500 acres total for the crops to be grown. At least 100 acres must be devoted to carrots, and at least 200 to potatoes. Potato storage… • plz help solving all needed fast. Extra Practice Linear Regression Example 1 1. SPORTING GOODS The table shows the revenue of a sporting goods company, in millions of dollars, each year since 2015. Le… • Consider the feasible set of a linear program in standard form, after the introduction of slack variables: D = {x : Ax = b and x; 2 0 for each j} where A is m x n with n > m and rank(A) = m. Suppos… • . 6. Every year, people move between cities A, B, and C with respect to the following approximate per- centages each year . A – B, B ., A, A , C, C .15, A, C 0.2, B, B .1, C. This is summarized i… • Exercise 19. What series do you get when you plug i0 in for a in the even-degree terms 2n n=0 (2n)! "It converges for all real numbers; it’s possible to discuss convergence for complex series, an… • . Minimize Z = (-1)(X1 – 2X2 + 3X3) Subjected to, X1 + X2 + X3 5 7 – X1 + X2- X3 5-2 3X1 – X2 -2X3 = -5 X1 , X2 20 • . Linear algebra 1. General Least Squares: Suppose we are estimating a linear model: y=X6+e However instead of the Var(e) = 0’21”, we know that Var(e) = 2. Derive the optimal OLS estimator given t… • Please let me know if you have some question . Thank you so much!. Problem 3 [33 pts]: (5, 6, 9, 13) Penguin Perceptron Consider building a linear perceptron two distinguish two species of penguins . … • What point do you get if you rotate the point p around the y axis for 140â—¦ ? Â . What point do you get if you rotate the point p = [-1 3 -4] T around the y axis for 140? • 5. Solve the following two versions of a similar problem: (a) Can you find a rotation matrix that would rotate the point q1 = 2 -6 3 into the point 92 = [0 7 0 ? Explain your answer. (b) Can you… • Show the full solutions. Do not get the same question. Activity 5 1. Sketch the solution sets of the following systems of linear inequalities. (a) 2x – y s 10 3x + ys 5 (b) 3x + 4y 2 14 X – yS5 2. Sol… • ——————– Â solve please with steps. each question in page separate from anotherÂ . Save Compute the value of a Gaussian pdf, N(m, S), at x1 = [0.2, 1.3] and x2 = [2.2, -1.3] , where m = [… • . Solve the system of equations using LU decomposition method given that coefficient matrix factors as (1X25 = 25 marks) 1 5 O 5 1 2 0 ON P = 14 O 4 9 x + 5y+z = 14 2x+y+ 3z = 13 3x + y+ 4z… • ALL PARTS. Is the given set of vectors a vector space? Give reasons. If your answer is yes, determine the dimension and find a basis. (v1, U2, . . . denote components.) 27. All vectors in R with v1 -… • Rob is 20 years older than Kate. In 5 years time, Rob will be twice as old as Kate will be then. Find Kate’s age. • Need clear explanation for a),b),c).Â . Let T : M2x2(R) – M2x2(IR) be given by r( 1) = (8-t-34 -+2u – stu a) Find [T’]s where S Your response Correct response -1 -3 0 0 2 0 1 0 0 0 0 Auto graded Grade… • . Question1: Score 0/3 Show all attempts There are 3 parts to this question, (a) , (b) and (c) Please click the "verify" button when you finish each part, otherwise your answer will NOT be… • . Problem 5 [20 pts]: Machine Learning Ethics Choose one Machine Learning application with ethical implications and write a few paragraphs (10 – 16 sentences total) which:. . Describe the applicatio… • iii and iv Â Â . Problem 3 [33 pts]: (5, 6, 9, 13) Penguin Perceptron Consider building a linear perception two distinguish two species of penguins’. We use the con- vention that Adelie penguins belo… • . Problem 5 [20 pts]: Machine Learning Ethics Choose one Machine Learning application with ethical implications and write a few paragraphs (10 – 16 sentences total) which: maybe it doesn’t need upda… • . Problem 3 [33 pta]: (5, 6, 0, 13) Penguin Perceptron Consider building a linear perception two distinguish two species of penguins’. We use the con- vention that Adelie penguins belong to group 0 … • . Problem 2 [13 pts]: [6 each] IMatrix Multiplication Geometry Provide an expression for the matrix A which. when multiplied by vector 1′ as Am, performs each of the transformations described in the… • . Problem 1 [18 pts]: (3 each) Matrix Vector Computation Solve for , which may be a vector or matrix, in each of the equations below. For each valid matrix- vector or vector-matrix multiplication, s… • What is true about all the frequent itemsets for the involved transactions? • Show your full solutions. Will give thumbs if satisfied with answers.. Activity 5 1. Sketch the solution sets of the following systems of linear inequalities. (a) 2x – y s 10 3x + ys 5 (b) 3x + 4y 2 1… • C Perform the indicated operations when A = 2 -1 A(I + A) = • . QUESTION 5 6 25 a) Let A = 6 0 0 O i) Find all eigenvalues of A. (2 marks) Find the eigenvalues of A6 and A-2. (4 marks) iii) Find the basis for the eigenspace corresponding to the smallest eigenv… • Given two objects represented by the tuples (21, 12, 3, 17, 48, 11, 82, 41, 35) and (34, 5, 13, 3, 57, 26, 69, 55, 27), calculate the Supremum distance between the two objects • Consider the following SLE’ x1+ x2+x3 +x4 =3 x1+2×2 + 3×3+4×4 = 2 X2 + 2×3+3×4 = 1 (a) Write the SLE as a matrix-vector equation. (b) Row reduce the augmented matrix for this SLE to a row- echelon … • A chemist wants an 84 milliliter acid solution with a concentration of 12%. She has concentrations of 8% and 14%. How much of each solution must she use to get the desired concentration? • Canonical form. (a) Consider the following LP (P1): max (0, -3, 0, -1,0)x + 4 subject to 1 10 0 2 0 0 6 1 20 * = CO OT CT 1 x 2 0. Transform this LP into the canonical form for the basis B1 := {2, 4, … • AnÂ individualÂ sphereÂ containsÂ aÂ higherÂ volumethanÂ aÂ cubeÂ ofÂ similarÂ dimension.Â WhatÂ evidence Â Â Â reading plus • . Find x such that the matrix is singular. 4 X A= -2 -3 X • . Question 1 (2 marks): Consider the following vector optimization problem: P: Max z1 = &1 + 1×2 Min z2 = x1 – 3×2 s.t. 4×1 + 15×2 < 60 7×1 + 1 1×2 <77+ i XI – 2×2 2 0 X1, X2 2 0. (a) Plot… • Describe the procedures for solving linear inequalities using a graphing method. What are some of the pitfalls you would watch out for? Are there some linear inequalities you cannot solve with the gra… • . -1 -2 2 0 -3 3 2) Given A = 0 5 -3 4 -2 2 , show that row rank and column rank of A are -3 5 -1 -3 2 1 equal. • Answer all three parts clearly with good working. Only original work and not copied thank you. Detailed steps and answers are required, thanks. • . Let 0 A = 0 4 1 6 a) (3 marks) Find ) for which the matrix A is non-invertible. b)(5 marks) Find the number k such that k . tr(C) . A-1 = 612 • Linear Algebra and Probability Take Home Midterm1 IdIaft P Pait February 2019 Instructions: 1. A function f : R" â€”> R is strictly convex if for all x,y E lR’l and all ‘ Open book takeâ€”ho… • No online solvers will give a good rating please, and Thank you. Â . Suppose that a. vector 11 in the mtg-plane has a. length of 7 units and points in a. direction that is Hull}ï¬‚ counterclockwise fr… • Please help me out with this problem. Â . Let V be a vector space with inner product (,}. Let T be a linear operator on V. Suppose W is a T invariant subspace. Let Tw be the restriction of T to W. Pro… • Please help me out. Â . Let V be a complex vector space with inner product (, ). Let T be a linear operator on V, with adjoint T. Prove that T = if and only if (T(v), v) E R for all v e V. • . 4. (3 pt) Consider the following problem: 3 = arg min \|\|0\|13 BERP subject to y = X0, where X E R"xP and p > n. Show that B = Xty. • . 2. (5 pt) Consider the following least squares problem: B = arg min y – XBl\|; (1) BERP (a) Show that if v E RP is a vector such that Xv = 0, then 8 + c.v is also a minimizer of the least squares p… • . 1. (3 pt) Let y E Rnxi, B E RPX] and X E Rnxp. We use T to indicate the transpose. The rows of X are designated with x, E RIXP for i = 1, … ,n and the columns of X are designated with o; E Rnx] … • Hi Dear Tutors; Please help me with this problem. Â Thanks! Â Â . Consider the Vandermonde matrix t1 th … M (t1, . . . , t ) = . . And let V(1 . .., tn) = det(M (t1, …, tn)). Prove by induction … • . 1) Suppose you throw a dart at a circular target of radius 10 inches. Assuming that you hit the target and that the coordinates of the outcomes are chosen at random (i.e. it is equally possibility… • Please answer all question 2 to 7. Â Please give me in detail if possible. Â Â . 2. Prove or disprove the following statements. (a) Let A E Mnxn (R). If AX = 0 for all & E R", then A = Onxn… • Solutions to all of these math of data model problems? I would appreciate if answered before tomorrow 🙁 Â Â Â . Problem 1 [18 pts]: (3 each) Matrix Vector Computation Solve for I, which may be a ve… • It’s about linear Algebra: Hamilton’s Quaternion. Exercise 4. As an application to a different area of mathematics, explain why this proves the following: if two positive integers are both sums of fou… • Given a m x n matrix X and a n x m matrix Y, suppose I + XY is invertible.. (2) Show that Y(I + XY) -! = (I+YX) Y. (20 points • Find a vector that goes from the point (2, 2) to the line y = 2x + 1 and meets that line at a right angle. (Hint: First find any vector w that goes from the point (2, 2) to the line. Then find the… • True or False Â The determinant of an orthogonal matrix has real values • . 2. Find all solutions to the equation 3 2 (2a) CI + C 2 (2b) [10+10 marks] • . 1. Find each of the following systems: (1) Write the coefficient matrix and the augmented matrix (ii) Find the rank of the coefficient matrix and the rank of the augmented matrix (iii) Determin… • 3. 4. 5.. What is the point of intersection of a – y < 0 and 3x + 4y – 28 < 0? O (3,2) O (2, 3) O (2, 5) O (5,2). The linear programming model of the maximum profit is shown below. What is… • solution please. Separation theorems and supporting hyperplanes 2.20 Strictly positive solution of linear equations. Suppose A E Rixn, b E R", with be R(A). Show that there exists an r satisfying… • while performing the simplex method, there are 2 positive entries in the bottom row left of the vertical line in the simplex table. the rest of the entries in the bottom row are zero how many iteratio… • What must be the value of P be in the linear system x ( P – 1 ) – y = -6 and 2 x + y = 3 if the system has no solution? • 2 Linear Algebra Questions Â . 5. Consider the 3d-volume given by 5pan((2,2,2,1,0), (1,0, â€”1,1,2), (02.4.2, â€”4)) in R5. a. Find all vectors in R5 that are orthogonal to this set. (Set up a system … • 2 Linear Algebra Questions Â . 3. In each case you are given the RREF arising from a certain system of equations. Write out the set of solutions for this system in vector form and then describe this s… • 2 Linear Algebra Questions Â . 1. Find the vector equation of the line (i.e. x = X, + tv) with the given properties. a. In R2, passing through the point (-1,2) with direction given by the vector (4, -… • Prove that if V and W are the three dimensional subspaces of R5 then V and W must have a nonzero vector in common. • hi I am doing lamberitian shading and I have a questionÂ if the formula is under picÂ what will be L like ? vector or just one number ? This is because when I calculate all shading after diffuse Spe… • how can I calculate intensity of the light of Diffuse shading? is there any formula for this ? • . Question1: Score 0/1 What is the change of basis matrix from N1 to N2 where: N1 is { -4×2 – 5x -5, -4×2 -3x -4, -5×2 -6x-6}, N2 is { -6 x + 10, 4×2 + 2 x + 6, 7×2 +3x+ 11} . Your response Correct … • . Question1: Score 0/1 What is the change of basis matrix from S to N where: Sis {1, x, x2 } and Nis (6×2 + 5 x + 5, -5×2 – 4x – 5, 5×2 + 4x+4). Notice: S is { 1, a, a2 } and N is in form of ax2 + b… • . Linear functions. For each description of y below, express it as y = Ax for some A. (You should specify A.) (a) y; is the difference between a; and the average value of 21, …, ;-1. (We take y1 =… • . QuestionT: Score 0/ 1 Let B – 13 x2 + 2x + 2, 4×2 + 5 x + 4, 2 x2 + x + 1 } be an ordered basis for P2 (a), the vector space of real quadratic polynomials. Find the coordinate vector [v] , where v… • . 1. Find each of the following systems: (i) Write the coefficient matrix and the augmented matrix Find the rank of the coefficient matrix and the rank of the augmented matrix (iii) Determine the so… • . (a) Consider the linear transformation f: R3 – R3 given by f (x, y,z) = (3y, -xty- 2z,x – z) Express f as a 3×3 matrix. (b) What is f- (x, y, z)? • (10 points) By setting up and solving an appropriate system of linear equations, determine whether there is a parabola that passes through the points (-1, 2), (0, -1), and (1, 3). • (20 points) Consider the following matrix. 612 144 -12 -1662 8 51 -12 2 141 -10 A = 102 24 -2 -277 -1 153 36 16 368 4 204 48 20 -494 -20 (a) Use software to find the RREF of A. Make sure to indicat… • Bryce invests $5900 in two different accounts. The first account paid 5%, the second account paid 6 % in interest. At the end of the first year he had earned$335 in interest. How much was in each acc… • . Vandermonde matrices. A Vandermonde matrix is an m x n matrix of the form t1 . . 1 t2 V = tm fn- m where 1, . .., tm are numbers. We will assume that these numbers are distinct, i.e., different fr… • . 2. Let C be the point on the line segment AB that is 3 times as far from B as it is from A. a) Show that OC = 20A + 10B. Include a properly labeled diagram of the 3 pts situation. b) Let 7 = (-1,2… • Please see attached image. Explain haw gun can solve the fellewing problems using the QR factorization. {a} Find the vector :1: that minimizes: IIAJ: â€” in”:1 + "A: â€” in”2 The pmbl- data a… • . 1. Let u = (7.6). a) Let u = (2.3), and w1 = (-1, 0). 2 pts Find a, be R such that 1= of + buj. Do this algebraically by starting with (7. 6) = 0(2.3) + b(-1, 0). b) Draw the vectors 71, w1, at, a… • . Show that Tis a linear transformation by finding a matrix that implements the mapping. Note that X1, X2. … are not vectors but are entries in vectors. T(X1,X2 X3) = (X1 -72 + 2X3. X2 – 8X3) • . Use a rectangular coordinate system to plot u = and their images under the given transformation T. Describe geometrically what T does to each vector x in R2. Which graph below shows u and its imag… • . Find all I. in R4 That are mapped into the zero vector by the transfennatien IIIâ€”3A.! for The given matrix A. 13133 1114â€”5 A: U134 â€”148 5 • . Assume that T is a linear transformation. Find the standard matrix of T. II T: Ilaâ€”412, rotates points {about the origin} through â€” E radians. • . 0 0 3 12 a Let A= 0 0 3 . and v = c . Define T: R-R* by T(x) = Ax. Find T(u) and T(v). – 15 h 0 W \| – • . 2. Let C be the point on the line segment AB that is 3 times as far from B as it is from A. a) Show that OC = 20A + 10B. Include a properly labeled diagram of the 3 pts situation. b) Let 7 = (-… • A company produces a niche hunting blind. Because of the nature of this product, demand varies greatly by season. In fact, demand exceeds production capacity during hunting season (fall,) but is very … • I need help with these questions, please? Â . Linear Functions Modeling Real-Life Data Lake Pamela is the large lake that you see on Valencia West Campus as you enter the school from Kirkman Road. Fro… • . 3. Let A be a 3×3 matrix and u, v, w be linearly independent vectors in R3 such that: Au = V, Av = u, AW = V. Which of the followings are true? (Select all correct answers). (1 mark) 0 u is an eig… • . 2. a 1 0 1 a 0 Which of the following vectors are eigenvectors of the matrix where a is non-zero? (Select all correct answers) 0 a 0 0 1 a (1 mark) O OOLH O -LOO. O -a O • . 1. Suppose T: R3 -> R3 is a linear transformation and we are given 1 1 1 2 T 2 = 1 and T 0 = 2 0 1 2 2 1 What is T 1 ? 1 (1 mark) 000005 \â€”/ \â€”/ \_/ < < H (1 < None of the above. O… • . 1. Suppose T: R3 â€”> R3 is a linear transformation and we are given 1 1 1 2 T 2 = 1 and T 0 = 2 0 1 2 2 1 What is T 1 7 1 (1 mark) (a) > (2) <1) (2) None ofthe above • . Q 1. Find the directional derivative of the function f(x 1,2) = 3x , – 2x 2 + 2x, – 5X2 + 4 at the point (1, – 1) in the direction of the vector v = (5, 12) . • . Q 2. Find the gradient and Hessian of the following cost-function that appears in the formulation of least square problems J(B) = 2mkXB – yk-2, where A e Rms, y ER". • use elementary row operations to transform each augmented coefficient matrix to echelon form. Then solve the system by back substitution.Â . 20. 2×1 + 4×2 – X3 – 2×4 + 2×5 = x1 + 3×2 + 2×3 – 7×4 + 3… • Steps in solving equations with two variables on both sides • Given the linear program Max Â Â 3 A + 4 B Â s.t âˆ’ 1 A + 2 B â‰¤ 8 1 A + 2 B â‰¤ 1 2 2 A + 1 B â‰¤ 1 6 A , B â‰¥ 0 • Let A be a 5×7 matrix such that the columns of A generate R5. Which part is false? Â -The columns of A are linearly independent. -Ax=b has an infinity of solutions -Matrix A has a pivot in each row. • . 1. (a) (Strang 1.1.9) If three corners of a parallelogram are (1, 1), (4, 2) and (1, 3), what are all possible locations of the fourth corner? Support each answer with a sketch of the parallelogra… • . Q1. (a) Let C be a line in R- with vector equation / = t ,te R. Prove that a point (a, b) is on C if and only if ap = bq. (b) Using (a), prove that if two lines in R- are not parallel, then they m… • . Q1. For each of the following systems of linear equations: i) Give the coefficient matrix and the augmented matrix. ii) Determine the reduced row echelon form of the augmented matrix. State the ra… • [5 points) Prove that it is impossible for a system An? = 5 of equations to have exactly two distinct solutions, e.g.1 if? = 11′ and f = if. Hint: You can show that if the system has two solutions,… • Please solve the the system of equations simultaneously for Y 1 (s) and Y 2 (s) using the matrix.Â Â Equation 1: Â (2s + 2)Y 1 (s) + 3Y 2 (s) = 2U 1 (s) Equation 2: Â -4Y 1 (s) + (s + 6)Y 2 (s) = 2… • 7) Practice: Organizing Information and Drawing Inferences Use the two similar triangles on page 12 to fill in the table below. Then use the table to infer a rule. Corresponding angles AABC A DEF ZA Z… • Can someone walk me through the steps to prove this? Please include reasoning/explanation so I can learn please.. It turns out that positive semidefinite matrices have the property that they do not sw… • Homework Help please Thank you. QUESTION 2 ( pts). Using the identity (3.11) from Olver & Shakiban (no need to prove this one), prove that the max norm on R has not associated inner product. • Homework Help please Thank you.. QUESTION 2 ( pts). Using the identity (3.11) from Olver & Shakiban (no need to prove this one), prove that the max norm on R has not associated inner product. • Question 3 and 4. 3. Consider an economy with three sectors, Chemicals and Metals, Fuels and Power, and Machinery. Chemicals sells 30% of its output to Fuels and 50% to Machinery and retains the rest…. • Questions 1 and 2. 1. Consider the matrix NO 1 2 A = 0 1 5 (a) For a general 4 x 3 matrix A, argue why it is reasonable to expect that the homogeneous equation has only the trivial solution. (b) Find … • . 12. Given the input-output matrix Industry Final Industry Health Oil Housing Demand Health 240 180 144 36 Oil 120 36 48 156 Housing 120 72 48 240 Other 120 72 240 Suppose final demand changes to 7… • Decide whether the lines given areâ€‹ parallel, perpendicular, or neither. Â The line through â€‹(4,âˆ’4â€‹) and (9,1â€‹). Â The line through â€‹(1,âˆ’3â€‹) and â€‹(âˆ’4,2â€‹). are these two line… • True or false? Give a reason if true and a counterexample or explanation if false. (a) (2 points) The vector – 3 2 0 0 is in Col 1 -5 1 0 2 0 0 3 1 (b) (2 points) The product of two rank 1 matrices… • This is a linear algebra. 2. (20 points) Find a row echelon form of the following matrix. Use your result to decide if the system of equations represented by this augmented matrix is inconsistent, con… • This is a linear algebra. 13 -3 3. (20 points) Is the vector w = -32 in the span of the vectors v1 = 5 12 12 2 2 7 5 V2 = and v3 = ? 22 0 13 • This is a linear algebra. 1. (20 points) Write the augmented matrix for the following system, find the reduced row echelon form and solve the system. 9 + 42 + 313 X 1 31 1513 6×1 + 2512 + -121 3×2 191… • Northeastern University DS5020 – Introduction to Linear Algebra and Probability for Data Science Fall 2017 Midterm Exam (Due: Thursday, Nov 16 2017, 11:59pm) Note: The midterm is 20% of the total grad… • . Find the characteristic equation and the eigenvalues (and a basis for each of the corresponding eigenspaces) of the matrix. D â€”3 9 â€”4 4 â€”18 O O 4 (a) the characteristic equation X (b) … • Solve the given linear system by any method X1+ 3X2 + XA 1 + 4×2 + 2×3 – 2×2 – 2×3 – X4 2x – 4×2 + X3 + X4 X – 2×2 – X3 + X4 • short and quick answer, please, i do not have much time Â . 3 Apply the Gram-Schmidt orthonormalization process to transform S given by S = {(1, 2, 1), (1, 1, 3), (2, 1, 1)} into an orthonormal basis…. • Suppose that you have a bucket that holds 57 cups, and one that holds 21 cups. How could you use them to measure out 6 cups of water? Â Â I need to use linear combination techniques such as the Eucli… • Consider the following data set with two features I] , $2 and two classes y E {â€”1,+1}. [1} Are the data linearly separable? Select “yes” or “no". ‘ [Select] V" [2} What set of feat… • Question 25 2 pts Consider the following data set with two features #1, @2 and two classes y E{-1, +1}. (1) Are the data linearly separable? Select "yes" or "no". [ Select ] (2) Wh… • Can anyone please solve #2 and #3 for me? I am running out of questions, so if you could answer both, I would highly highly appreciate it. Thanks. • Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE.) 72 -4 6 2 -4 6 2 -3 T • Use a software program or a graphing utility with matrix capabilities to solve the system of linear equations using an inverse matrix. X1 + 2×2 – X3 + 3X4 – X5 = -4 X1 – 3X2 + X3 + 2X4 – X5 = 1 2×1+ X… • Solve the system of equations. (5 pts) 2xty -2z = 6 -2x+ y+5z =1 2x+3y+z=13 7. Solve the system of equations. (5 pts) X – 3z = -3 3y + 4z = -5 3x – 2y = 6 • 1. Let U, V, W be three K-vector spaces such that both U and V have finite dimension. Consider the linear maps S: U – V and T : V – W. (i) Demonstrate that dim (Ker(TS) ) < dim (Ker(S) ) + dim(… • For this discussion, I would like to see some explorations of computational complexity using MATLAB. Please use MATLAB’s ability to represent sparse or dense matrices to experiment. For example, gathe… • a = [[-3,-3,0], Â Â Â Â [3, k, 0], Â Â Â Â [-3, 3, 5]] Â find what k needs to make the matrix invertible? if possible. • x1 + x2 – x3 = 4 -x1 +x2 – x3 = 10 3×1 + 2×2 – x3 = 0 Â A is coefficeint matrixÂ Â find the inverse of A Â then find x • Consider the following 3 x 3 system of linear equations: x + 3z = 2 3x + 2y + 7z = 6 k^2y – 4z = 1 Here, k is a real constant For which values of k does the system have a solution? • . 5. (5 points) Consider the problem min – 1 – C2 s.t. 21 – 22 <3 1 +22<6 T1, T2 2 0. Convert the problem into standard form and construct a basic feasible solution at which ($1, X2) = (0, 0)…. • Please help to solve the questions. Thank you! Â Â . 2. [0/1 Points] DETAILS PREVIOUS ANSWERS HOLTLINALG2 2.2.008. Determine if b is in the span of the other given vector. If so, express b as a linea… • How do I do this?Â Â . 4. Let A be the matrix: 2 8 0 2 1 4 3 4 0 O 1 l (3) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the row Space of A, co… • How do I do this?Â Â Â . 6. Determine whether or not the three vectors in R* are linearly independent. If not, find a linear combination that equals 0.. 16. Find four vectors in R3 such that each su… • How do I do this?Â Â . 4. Find a basis and the dimension of the space of solutions to each system of linear homogeneous equations.. (d) x1 + X2 + 2×3 – 4×4 = 0 +x – X – XA 0 2x, + 2×2 – X] – 3XA = 0… • Although I solved two problems, I need some help in understanding how to find the coordinates of all corner points.. Sketch the region that corresponds to the given inequalities. 3x – y $3 x + 2y s1 y… • . 6 (5 points, plus 2 BONUS points) Read the following carefully, and then answer the questions that follow. In the latest tutorial worksheet, you were introduced to the notions of the "thickne… • . 5 (3 points) Read the following definition carefully, and then answer the question below. A square matrix A is called bizarre if det (A) = i or det (A) = -i. Def (Note that "i" here is t… • please give me detailed explanation. 4 (9 points total – 3 points per part) Determine if the statements below are true or false. Make sure to justify your answers! You will receive no credit for simpl… • . 3 (4 points) Find all complex number solutions to the following equation: 25 =2-2V3i Put your answer(s) in polar form, e.g. re, where 0 E [0, 2x). Show all steps in your work. You do not need to s… • . 2 (4 points total) 2.1 (3 points) Determine all solution(s) to the following system of equations, by putting your solutions in parametric form and showing all steps in your work, or by explaining … • . (5 points total) 1.1 (2 points) Suppose that A and B are 3 x 3 matrices and suppose we know that det (A) = -3 and det (B) = 2. Compute det (A-1 . 3(Bt) 2). 1.2 (3 points) Consider the following ma… • [Solved] Question 1 Tries remaini X iCMA 53 (page 11 of 17) X Course Hero X [Solved] An object of mass 28 kg x + X > C A learn2.open.ac.uk/mod/quiz/attempt.php?attempt=5224493&cmid=1823062&am… • How do you use R to solve this? Thank you. Given the matrices X and Y X = A OS OT CO N Y = 23 25 1 17 evaluate (using R) the expression B = (XTX)-1XTY. What is the dimension of ? Also, plot the points… • Can someone help me with this? I don’t know where to start.. Prove that for any 2 x 2 matrix A A = all a12 a21 a22 that A-1 (if it exists) is given by A- 1 = a22 / D -a12/ D -a21 / D all / D where D =… • Can someone please help me solve this using R?. Given the matrices N 3 71 A = 6 B = 3 JNN 2 C = 4 7 5 3 – 1 calculate (by hand and with R) A + B, A – B, AC, CA, ABT, BTA. • I only need help with part 2 (ii). Can someone help me solve this using R?. Solve the following systems of linear equations (i) by hand (using back-substitution, Gaussian elimination, or what- ever me… • Decide whether each of the following is true or false. Explain your answers! (a) The matrices A and 2A have the same column space (b) Given a matrix A, the set of vectors b such that Ax = b has a s… • simplify the expression combining the terms: Â 2a raise to power 2 +8+8a raise power 2+5 • Look at the following graph for a bad company and how much they charge for trips can you answer all the questions. Modules O Announcements Account Grades 28 Syllabus Dashboard 9 Outcomes Cost (Dollars… • A person is the manager of a firm. He is considering manufacturing a newâ€‹ product, so he asks the accounting department for cost estimates and the sales department for sales estimates. After he rece… • what is the domain and range,. Home Use the graph to answer the following questions Modules O Announcements Account Grades Syllabus Dashboard Outcomes x Courses BrainPOP Class Notebook Calendar Eh Inb… • Look at the following expression: 30m2 kp-2 45m-6k Ip A. Solve and simplify (as much as possible) the following question showing all your work (mathematical reasoning)! 6 marks THERE CANNOT BE ANY NEG… • Prove the set of 2 x 2 diagonal matrices is a subspace of M 2 12 • Please help me with the following. Â Brief explanation please in order to get a prompt answer. Thanks! • Let Ti : R – R’ denote the reflection through the origin and T2(x, y) = (-x + y, 2y, x + 1). Suppose S is the transformation obtained from applying 71 followed by 72, namely you first apply 7, and the… • A line has a slope of â…— and crosses the y-axis at 2. A parallel line, to this first line, crosses the point (10,13). Find the equation of the parallel line as the following forms:. Linear vs Non-… • A bus company has sold Â£15 single tickets, leaving from Kingston at full capacity and with 115 passengers on board, among them there are 5 senior passengers {over 65 years old). The bus has stops in … • Consider 71 : R – R’ given by Ti(x, y) = (2x + 4, x – 1), and let T2 be the transformation that rotates every point in the plane about the origin through " and clockwise. Suppose A and B respecti… • Questions 2 and 3.. We can think of matrix multiplication in terms of matrix-vector multiplication. In turn, we can think of matrix-vector multiplication in terms of column operations, specifically as… • Make sense of the problem. (6 points: 2 points for each answer) What do you know? What do you want to find out? What kind of answer do you expect? • . 2. (10 points) A department store is attempting to decide on the types of advertising the store should use. The store manager has invited representatives from the local radio station, television s… • Graphing assignment Q12. Give the slope of the lines between the two points. Then circle A, B, C, or D A) Graph Rises left to right; B) Graph Falls left to right; C) Horizontal Line D) Vertical Line 1… • Prove whether u is a linear combination of vi, U2, 13. (a (5 points) vi – 2 . 3- 83-[2]–[81 (b) (5 points) vi = 3. (5 points) Consider the matrices Compute (if possible) AB, BA, AC, CA, BC, CB. If… • The graph below represents a section of past students’ overall class percentage plotted with total time spent studying for the semester. Â Create a linear model relating h hours and P percentage Â ?… • 7 Theorem: Suppose that A is a p x p nonsingular matrix. Then the cost of computing the LU-decomposition A = P LU of A is p3 /3 – p/3 multiplications/divisions, and p3 13 – p2 /2 + p/6 additions/sub… • 6 Show that the product of arbitrarily many upper (or lower)-triangular matrices is upper (or lower)-triangular. 7 Theorem: Suppose that A is a n p nonsingular matrix. Then the cost of computing • * 4 Show that a lower-triangular matrix L is nonsingular if and only if its main diagonal entries are all nonzero. 5 Show tha is lower (or • 1 Consider the matrix 1 A = -2 -4 2. 0 5 0 (a) Apply Gaussian elimination to reduce A to a unit upper-triangular matrix U. (b) Express U as a product of elementary matrices and A. (c) Compute the low… • Using Jupyter Notebook (language Python): Â . Consider the linear system Ax = b of size n = 600, where Solve by the conjugate gradient method, the method of steepest descent, and the BOP method with c… • Using Jupyter Notebook (language Python): explore spectral radius, norms, and convergence speed for various examples, make your own matrices with the given eigenvalues? • Suppose W is a subspace of V . Show there is a linear T : V â†’ W so that T(Î±) = Î± for all Î± âˆˆ W. • . 6 (1 point) Write X = 1 9 0 as a product X = E1 E2 E3 of elementary matrices. 2 0 0 -3 0.5 E1 = E2 E3 = 6 0.5 • A Linear Algebra Question needs help. Thanks! Â . 4. When analyzing a computation that is the result of a large numbers of floating point operations, we often end up with expressions like ac = x(1+ … • Counting 1.5 cu. M. Of coal to a tonneau, how many tones will a coal bin 6 m. Long 6ft. Wide and 9 ft deep contain when level full • . (1 point) Each picture below depicts a system of three linear equations in three unknowns. Determine whether each system has a solution and, if it does, the number of parameters. no solution 0 par… • . (1 point) Let V = R. For 11, v 6 V and a E R deï¬ne vector addition by u _l v â€” u l v 1 and scalar multiplication by a, E u = au â€” a + 1. It can be shown that V is a vector space under the op… • Answer the question with requirement like 1st must use GaussJordan Â elimination to solve.. 3. Given matrixB and its reduced row echelon form U 4 0 -11 1 -4 OOO B= DU = O O OW O can every vector in K&… • . (1 point)Asquare matrix A is idempotent if A2 = A. Let H be the set of all 2 x 2 idempotent matrices with real entries. Complete the following statements to determine if H is a subspace of M23. If… • . 2. Four guys ordered four custom burgers at a restaurant that has specific prices per topping. The bun and pattyr cost$2.44. – Mike’s burger cost $8.44 and had 2 avocados, 5 bacon strips, and 3… • . 6. Letr = (2,5, x) . For what value(s) of x is \|r\| = 6? (2 marks) • . (c) min 3×1 – 2×2 s.t. y1 + 92 – 1.5 y1 5 -21 y2 – 2 y2 – -12 0 < y1 <1, 0 <32 – 1, (d) min 3×1 – 2×2 s.t. x1 + 2 – 1.5 0<x151, 0<x2-1. (e) min 3×1 -2×2 s.t. 1 + 22 – 1.5$1 – 22 – … • need help with graphing the system of equations and find the solution. two points. Your Sets \| Quizlet Graph the system of equations and find the solution. y = -3+2 Line a Line b • also after solving need to two plot graph. PART A Graph the system of equations and find the solution. y = -x – 3 4 y = x + 2 Line a Line b • Let A = 2 00 In 09 09 3 12 (a) Determine the solution set of the homogeneous system with coefficient matrix A. (b) Prove that I = is a solution to the non-homogeneous system AT = 13 -11 (c) Use the… • Prove or disprove the following statements. (a) A system of 3 equations in 5 variables has infinitely many solutions. (b) A system of 5 equations in 3 variables cannot have infinitely many solutio… • For each of the ffg:Â Express the product of disjoint cycles? What is the order of each? Express as a product of transposition? Is it even or odd?. For each of the ï¬g: a. Express the product of dis… • In a cyclic group G generated by B of order 12, what is the order of ? Find the order of 3 in 24 Find the order of 32 in • Consider the . Determine the following :. Considerthe (241). Determine the ï¬g: If it has a subgroup of order 3? If it has exactly one subgroup of order 4? If it has exactly one subgroup of order 8?… • Use the Newton-Raphson Method to solve the linear system of equations [10 4 2 4 I 31 25 5 1 â€”2 4 1:2 _ 14 D 6 â€”2 2 I4 8 Comment on the behaviour of this method for a linear system of equations in … • a man buys cartons of canned mushrooms Each carton contains 12 cans and Each weight contains 825g 825 g825g How many cartons of canned mushrooms are needed to get a weight of 79.2kg? • . TI". Be the following lines in [IRE intersect? If 50, where? {3 marks] r = (1.! â€”?l2) + t(514l â€”3) I22â€”3 y=â€”1ï¬+23 3:3 • Please help me to solve this Â Â . Q1. Let {â€”2, 1: 1) and (4:2, 2) be two points on a line 15 in R3. (a) Find a vector equation of E. (b) Find parametric equations of E. (c) Determine whether the p… • . 5. Let 11,3 be 5 x 5 matrices such that: . a _{2,ifij==1â€” "1 â€” 1, otherwise =3â€”aï¬ ‘ ï¬‚; 1 Solve the matrix equation AB]: = (â€”2,0,5,l,3) for 17, correct to 4 decimal places {if poss… • . 2 3 -1 -2 1. Let Y = 4a 6a 2a -4a Show that rank(Y) = 1. (3 marks) 4 – 4a 6 -6a 2a -2 4a -4. • Please show ALL work.. Use an inverse matrix to solve each system of linear equations. (a) X1 + X2 – 4X3 = 0 X1 – 4X7 + X3 = 0 X1 – X2 – X3 : (X1, X2, 3) = (b) xit . X7 – 4X3 = -9 X1 – 4X2 + X3 = 6 X… • Please show ALL work.. Find the inverse of the matrix (if it exists). (If an answer does not exist, enter DNE. -2 -2 4 2 1-2 4 2 -3 10 1 1 • Please show ALL work.. Verify that (AB)T = BTAT. 2 0 -1 1 -8 A = and B = 8 8 0 1 1 -1 J Find (AB)T. Find BAT. • Please show ALL work.. 1 1 -1 -1 3 0 Perform the indicated operations, given A = B= , and C = 3 0 2 1 2 0 1 B(CA) E T • Please show ALL WORK.. Write the system of linear equations in the form Ax = b and solve this matrix equation for x. X1 – 2×2 + 3X3 = 24 -X1 + 3X2 – 3 = -11 2×1 – 5X2 + 5X3 = 42 24 X1 -11 X2 42 X3… • Please show ALL work.. Find, if possible, AB and BA. (If not possible, enter IMPOSSIBLE in any single cell. 5 2 1 -1 2 A = 2 3 3 B = -4 -2 -4 2 -1 -2 1 0 3 (a) AB (b) BA + • Please show ALL work.. Consider the Following points. (â€”1: 6}: (U, a}: (1! 1}: (4r 52} (a) Determine the polynomial function of least degree whose graph passes through the given points. pm = {b} … • Please show ALL work.. Determine the currents In, I2, and Is for the electrical network shown in the figure. (Assume V, = 10 V and V2 = 7 V.) I = A I, = A I= = A V1 R1 = 40 R, = 30 R3 = 10 V2 • Please show ALL work.. Find the solution set of the system of linear equations represented by the augmented matrix. (If there is no solution, enter NO SOLUTION. If the system has an infinite number of… • Please show ALL work.. Solve the system using either Gaussian elimination with back-substitution or Gauss-Jordan elimination. (If there is no solution, enter NO SOLUTION. If there are an infinite numb… • . 4 Let A = 2 1. Given p1 = 1 find a vector p2 with unit 2-norm, i.e., \|p2\|/2 = 1, and A-conjugate to P1. 2. Give two vectors p, and p2 with unit 2-norm such that they are A-conjugate and also ortho… • . CDnsider the function: ï¬‚an, 3:3) = I? + 171122 + $433 â€” 3:1 â€” 2.13:. At the point xi”) = (â€”1, 1], 1. give the search direction vector p for the steepest descent method. 2. Let c1 = 0.25,… • . Problem 4. Consider the set V of real polynomials f of degree at most 3 which satisfy f(1) = f(2). Define addition and scalar multiplication of such polynomials by (f +g)(x) = f(x) +g(x) and (af) … • Need ALL parts. n 3 Find all solutions of the system (E) of linear equations 15x + 45y + -87z = 1947 191 3 4x + 12y + -17z 389 out of 2x + 6y + -32 = 79 Answer, step 1: The RREF of the augmented coeff… • A car rental agency charges$250 per week plus $0.10 per mile to rent a car. (a). Write an equation that expresses the weekly cost to rent the car, y, in terms of the number of miles driven during the… • Te tiene los puntos o lineales y consecutivo A, B, C y D. Calcular BD, sÃ BC=6 AB/CD=2/3 y AB/BC= AD/CD. a) 12. Â b) Â 16 Â c) 18 d) 22 Â e) 24 • Solve system by elimination 1 – 3x +5 y = -19 – 4x + 8y – – 24 • Write the slope-intercept form of the equation of each line. 5) 6) Ay Ay 5 -4 -3 -2 – 2 3 4 5 x 5 4 -3 -2-1 x • Solve each system by elimination. 9) -7x – 5y =-17 5x + 9y=-15 • Problem 4. Consider the set V of real polynomials f of degree at most 3 which satisfy f(1) = f(2). Define addition and scalar multiplication of such polynomials by (f +g) (x) = f(x) +g(x) and (af) (x)… • Specify the RREF of the augmented matrix of the above system and specify which variable(s) are basic/free. 2. Write a vector equation that is equivalent to the above system of linear equations. 3. … • . 37. Determine the vector form of the general solution to the system of equations with the corresponding augmented matrix in row echelon form: 2 -3 3 (a) No answer gives the general solution in vec… • Problem 3. A matrix A = (v1, …, V6) with column vectors v1, …, V6 E R& has row reduced form 102010 B = 013020 000130 000 001 (a) Do the vectors V1, …, V6 span R4? (b) Are the vectors v1, …… • Rewrite the set G by listing its elements. Make sure to use the appropriate set notation. G= {z\|z is an integer and -2 • . 2. (Relations, 15pt) For each of the following binary relations ny, or R (x, y) if you prefer, state whether it is (i) reï¬‚exive, (ii) synnnetric andfor (iii) transitive (so a yesfno answer for e… • Rewrite the set G by listing its elements. Make sure to use the appropriate set notation. G= {z\| z is an integer and -2 <2 â‰¤ 0} G = • Let P3 = h1, x, x2 , x3 [4] i âŠ‚ C[âˆ’1, 1] be the space of polynomials of degree â‰¤ 3 on [âˆ’1, 1]. Use the Gram-Schmidt procedure to find an orthonormal basis of P3 with respect to the L 2 -norm o… • Determine the slope (m) and the -intercept (b) from each equation a) 2y = 2x+8 b) 3x-8y= 24. Parallel Lines 6. A line has a cl- – Finding Slope and Y-Int from an Equation MUST SHOW WORK where 4. Deter… • Please help to solve the question. Thank you! Â Â . 1. [â€”11 Points] HOLTLINALGZ 2.2.004. Find three oitterent vectors that are in the span of the given vectors. \|:\|\|:\|\|:\| \|:l\|:l\|:l \|:\|\|:\|\|:\| 2. [-1… • can you explain #1 please so I can teach my child how to do this?. Lesson 7.4 Place a point on the number line given for each of the following irrational numbers. 1. Point A: V2 2. Point B: V17 3. Poi… • . 2. Let f, gi : Rn -> R be smooth (continuously differentiable ) functions. Let a E R, K E R+ +, and suppose further that gi(x) < bi, Vi = 1, …, m f(x) 2 0. Suppose that you have a general … • . Three scents are combined to form three types of perfume. "Excellent" perfume is 10 percent scent A, 30 percent B, and 60 percent C. "Good" perfume is 20 percent A, 30 percent … • . QUESTION 1 (12 Marks) As a potential athlete representing Kolej Siswa J aya, you are consulting with the UTM’s Medical Center nutritionist about your sudden low energy level. The nutritionist ha… • The population of a country in 2015 was estimated to be 321.2 million people. This was an increase of 28% from the population in 1990. What was the population of the country in 1990? • Solve the following sums elaborately. Â . 4. Find characteristic equation, eigenvalues and corresponding eigenvectors of the matrix -2 -2 5 -2 -6 6 -3 • SUPPORT OF MATHEMATICAL MODELI SECTION 101 Â Â Â Â Â CURRENT OBJECTIVE Solve a formula for a specific variable Â Â Solve the formula P=2L+2W for W. • . – 0 1 1 Question 3: Suppose A = and b = First find A then 0 -1 0 1 0 solve Ar = b by calculating a = A- b. 1 2 – Question 4: Write the invertible matrix A = 1 1 ( as a product of elemen- 0 2 tary … • . Question 1: Compute AB, BA, 3A + BBQ, A – A’ + A – 21, for Question 2: Either find the inverse of the following matrix or determine that it is singular (not invertible): (2) 3 2 (b) 0 1 -2 0 1 -1 … • . Exercise 14.9. This exercise uses a linear transformation to relate the curve H+ defined by x2 – y? = +1 to the hyperbola defined by xy = 11/2 (i.e., y = +1/(2x)), with a single choice of sign (+)… • I need help solving these 2 questions . With detailed step-by-step solution ^. QUIZ 4 3. Let A be a 3×3 matrix and u, v, w be linearly independent vectors in R3 such that: Au = v, Av = U, Aw = V. Whic… • Determine if the columns of each matrix form a linearly independent set of vectors. 2) Find all value(s) of h for which the vectors are linearly dependent:. -8 3 -7 4 (a) -1 5 -4 1 -3 2 4 -3 (b) -2… • Please help me with the last 4 parts of the question: 5,6,7,8 Given an infinitely differentiable function f : Rn 7â†’ R, its Hessian matrix is given by. Problem 3 1. (5 %) Given an infinitely differen… • Please help me with the first 4 parts of this question: 1,2,3,4 Given an infinitely differentiable function f : Rn 7â†’ R, its Hessian matrix is given by. Problem 3 1. (5 %) Given an infinitely differ… • . 6. Find the major/minor axes and their lengths for the ellipse defined by 3×2 + 2xy+7y2 =1 • Determine if the following set of vectors are linearly independent or linearly dependent: ð‘†={(âˆ’4,âˆ’3,4),(1,âˆ’2,3),(6,0,0)} • . CO 1. Given vector a = find a Givens rotation G and a Householder matrix H such that Ga = Ha= D • A grade 9 class earns a profit of 534 for S each program they sell for the school play. 1600- Program Sales 1400 1200 1000- Income ($) 800 600 400 – 200 500 1000 1500 2000 2500 p Number of Program… • 1 = -6? 11. The table of values represents the drop in temperature after noon on a winter day. Time, t (h) 1:00 2:00 3:00 4:00 5:00 6:00 Temperature, 1 0 -1 -2-3 – 4 d (c ) a) Plot the data on a graph… • . c) (4 pts) If c = 0 and all other parameters are greater than 1, would you expect this model to oscillate? Justify your answer by referencing the specific features of the model that helped you dec… • . c) (3 pts) Keeping the notation and assumption from parts a) and b) of this exercise, now assume further that g(e]) = () Could g now be linear? Justify your answer. d) (3 pts) A set of vectors lik… • Photo World charges $3 for the first enlargement and$2 for each additional enlargement. a) Properly graph the given table of values. (Remember: all labels, scales, straight lines, etc) (7 marks) N… • 1 2 3 LetA= 1 â€”1 0 2 1 1 (a) Use three row operations E1, E2, E3 of type III to transform A into an upper triangular matrix U. (b) For each row operation used in part (a) write the corresponding ele… • Assignment 3: Problem 7 Previous Problem Problem List Next Problem (1 point) Let f(x) = 3×4 – 2. (a) Use the limit process to find the slope of the line tangent to the graph of f at x = 4. Slope at x … • The elementary matrix E6 multiplies the third row of B by -2. E6 = EG = g. The elementary matrix E7 switches the first and third rows of B. E1 = , ET = h. The elementary matrix Eg adds 3 times the … • Problems -2 1 -5 3 -4 A = and B = 4 4 LO Problem 1 -2 5 Problem 2 v Problem 3 Problem 4 Given the following descriptions, determine the following elementary matrices and their inverses. Problem 5 a. T… • Assignment 3: Problem 4 Previous Problem Problem List Next Problem (1 point) Solve for X. 13 21 x + 13 21 = 1821 x X = • Assignment 3: Problem 3 Previous Problem Problem List Next Problem (1 point) Find a 2 X 2 matrix such that = 4 O E Preview My Answers Submit Answers You have attempted this • For the next set of questions, let A and B be 3 x 3 matrices with det(A) = 2 and det(B) = -1. Compute (a) det(4A) (b) det(AB8) (c) det(AAT) (d) det(AA-1) (e) det(AE) where E is the elementary row oper… • . J.CELI_MATE_1ro A. By C_S_11-12_2Q (Vista protegida) – Word Daniela Fernanda Celi Castillo DF X Archivo Inicio Insertar Dibujar Diserio Disposicion Referencias Correspondencia Revisar Vista Ayuda … • The matrix of a certain linear functi0n f (relative to the standard basis) is 1 3 5 2 4 6 a) (5 pts) What is the domain of this function f? Justify your answer in terms of the standard basis vector… • a) (8 pts) For each of the following functions, state whether it is linear, is not linear, or, based on the given information, could be linear. Justify your answer. In your justification, you can u… • Describe the feasible region of x + y â‰¥ 10 and x + y â‰¤ 4 • . Perform the indicated operation and simplify. 28) x+3 x -5 x2 – 2x -8 x2 -12x +32 Simplify the complex fraction. WIX – 6 31) 10+ 15- XIN 32) Colx +4 16 33) * – 2 4 6 x+1 x Hint for # 34: Don’t for… • . Complete parts (a) and (b) for the matrix below. 0 3â€”9â€”4 144â€”7 A=â€”1â€”56â€”9 -1â€”8â€”3â€”1 8â€”438 (b) Find k such that Col{A) is a subspace of Rk. em • Find two linearly independent sets 81 and 32 in R4 such that 81 n 32 = Q and 31 U 32 is linearly dependent. • 7 marks We say that a function f : R" – R is convex if for any x, y e R" and A E [0, 1] (i.e. 0 < >< 1), we have f((1 – X)x + Ay) < (1 -A)f(x) + Af(y). (a) 2 marks Let n 2 3. … • 5 marks Use the two-phase method to solve the following LP problem: maximise z = 3×1 + 2×2 + 323 subject to 2×1 +12 +23 IV IA 3×1 +4×2 +2×3 and 21, $2, 23 2 0 This requires around 3 pivots. Remember t… • I understand how to prove it, I just need help with drawing the picture. • Need help with both parts. Â Please explain thoroughly because I want to make sure I understand it. • 5 marks Use the two-phase method to find the solution of the following LP problem: maximise z = 3×1 + 22 +2 2 subject to -201 + 12 IV IV IA 3×1 + 202 and 21,22 20 This requires around 3 pivots. Rem… • . (Problem 2 continued.) b) If this model is a linear function, write down the Leslie matrix that represents it. If this model is not a linear function, explain which term(s) fail(s) the properties … • 3 marks (a) Is there a pair (x, y), with x, y 2 0, that satisfy the following three inequalities simultaneoult? -x+2y<-2 2x+y21 – 3x ty 2 -4. Justify your answer. Remember to use Anstee’s rule: … • . Find the vector x determined by the given coordinate vector [x], and the given basis B. 6 5 2 B = , XB = 4 4 X = swer ) (Simplify your answer.) • . Complete parts (a) and (b) for the matrix below. 0 3 -9 -4 1 4 4 -7 A = -1 -5 6 -9 -1 -8 -3 -1 8 -4 3 8 (a) Find k such that Nul(A) is a subspace of RK. K= • . 4 3 – 2 – 2 Determine if w = is in Nul A, where A = 9 -5 -5 – 1 1 5 Is w in Nul A? Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) O A. … • Let TER22 be positive matrix, to which applies 72 = 7 Show that Tis form: cos(t) T = sin(t) [cos(t) sin(t) ] • R U E R2x2 Show, that all real( ) unitary matrices are of the form: cos(t) – sin U = sin(t) cos (t) • . 6t Let H be the set of all vectors of the form Find a vector v in R such that H = Span(v}. Why does this show that H is a subspace of R? 5t . . . H = Span{v} for v = • 2 Calculate matrix T = eigenpairs 5 2 (1, [x,y]] ) and ( 12, [z, w]] ) and give its vector cordinates ratio (x/y and z/w). • Let Tu = Au and Tv = uv, where Te CS, u,DEC3x1 and ME C so that all # 0 / land) # # Why can’t u + v be matrix Teigenvector? • Group Theory Â Find the unitary matrices of all irreducible representations of the symmetry group D 4 Â Recall that there are 5 irreducible representations for the D 4 group which are given as the r… • A boy finds$1.05 in dimes, nickels, and pennies. If there are 17 coins in all, how many coins of each type can he have? • (4 points) A poultry scientist oversees a farm where there are two breeds of chickens. The eggs produced by those two breeds differ in size. The scientist has found that eggs from both breeds have … • MatricesÂ Â . E. The matrix of a certain linear function I [relative to the standard basis) is 1 3 5 2 d E a] {5 pts] What is the domain of this function f? Justify your answer in terms of the standa… • Establishing linearityÂ Â . 5. a) (8 pts) For each of the following functions, state whether it is linear, is not linear, or, based on the given information, could be linear. Justify your answer. In … • Need help on the last question on line 9. I got half of the last question correct need the “eigV = ” Â . Script Save C Reset BE MATLAB Documentation 1 XEnter the matrix C. 2 C = [6 3 -8; 0 -2 0; 1 0 -… • . Find a basis for and the dimension of the solution space of the homogeneous system of linear equations 9×1 – 4X2 – 2×3 – 20X4 = 0 12×1 – 6X2 – 4×3 – 29X4 = 0 9X1 – 4X2 – 7×4 = 0 6X1 – 3X2… • Ty decided to invest $600 in an account and was able to take out$670 from the account after 8 years. Â The interest in the account was compounded continuously. What was the interest rate for his acco… • Construct orthonormal Basis using Gram-Schmidt orthogonalization processÂ from the set of vectors {(2, 2, 1), (âˆ’2, 1, 2), (18, 0, 0)}. • Problema de disminucion de inventario: Una empresa produce varios accesorios para bano, entre ellos accesorios decorativos para toallas y cortinas de bano. Cada uno de estos accesorios comprende un… • COuld you do the underlined part please. Lectu Do Homework – OQ4 – Brave X + mathxl.com/Student/PlayerHomework.aspx?homeworkld-621334847&questionld-3&flushed=false&cld-6839870&centerwi… • . the definitional properties of linear functions as well as any theorems pertaining to linearity. If you are using a theorem, you must be specific about the way in which the theorem applies to the … • Hi, I need help proving this. I tried to prove this two ways but now I am confused Â if I did it right. I only need to prove it one way. Also per my class it needs to be proved in this manner. Â Thank… • 2 -5 4 1 Solve for X in the equation, given A = -4 -3 and B = -2 2 0 1 1 4 (a) 3X + 2A = B X = (b) 2A – 58 = 3X X = (c) X – 3A + 2B = 0 X = (d) 6X – 4A – 3B = 0 X = • Let M2x2 be the vector space of all 2 x 2 matrices with real entries. Show that is the basis of M2x2. Also find at least three two-dimensional and three-dimensional subspaces of M2x2. • Problem 1 (14 pts) For each of the following three systems of equations, use Gaussian- Jordan elimination to obtain the augmented matrix in reduced row-echelon form. Indicate whether or not the system… • . 1 2 â€”1 Use the inverse of â€”4 â€”7 7 to ï¬nd the solutions to the linear system below. â€”1 â€”1 5 x1 + 2×2 â€” X3 = â€”3 â€”4×1 _ 7×2 __ 7×3 = 1 â€”x1 _ x2 __ 5×3 = _1 (x1 X2 X3) =( ) X Submi… • Let A = [ i s J b – [iz evaluate " ) At B b ) A A + 2 B C ) AB d ) BA 9 ) +C ( B BT ) ( ) for (BIB • . Assume that T is a linear transformation. Find the standard matrix of T. T: H- -R, T(@1 ) = (2, 1, 2, 1), and T(ez) = (- 4, 3, 0, 0), where e, = (1,0) and ez = (0,1). • . Question 2. (a) Let A, B be two 3 x 2 matrices such that for any vector 7 in R2 , AT = BU. Is it necessarily true then that A = B? (b) Let A, B be two m x n matrices such that for any vector 7 in … • . Question 2. (a) Let A, B be two 3 X 2 matrices such that for any vector 17 in R2 , A1? = B17. Is it necessarily true then that A = B? ] (b) Let A, B be two m X n matrices such that for any vector … • . 1 Question 1. Let A = 1 (a) Compute A2, AS, A (try it without MYLAB). (b) What is A"? How can you justify your answer? • Inner product "ï¬‚ue or False, two vectors, say ï¬t) and 90:), are orthogonal if their inner product is zero. Using your response to the above prove that ï¬t) = sinï¬mt} and 9155) = sin[mnt) a… • solve the question with steps or explanation. 1. Suppose {V,, V2) is a linearly independent set in R" . Prove that (v , v, +v, , is also linearly independent. 2. Suppose v , V2, v, are distinct p… • Solve the question with steps or explanation. 2. Prove that if A is an m xn matrix, u andv are vectorsin R’ and c is a scalar, then A(utv) = Au + Av. Justify each step.. 3. Given matrixB and its reduc… • Bases and frames of R2 Given the following sets of vectors: ‘1’1 ={s01,a, (91.1} = { [E] a [31]} (1) 2 {>2 = {Wm Wm 902,2, 992,3} = {[222] , fï¬] , [ï¬] , [%]} (2) l} ‘3) so {i) For each of t… • Page 2 Problems 1. Let Pn be the vector space of polynomials with real coefficients and degree at most n. For each of the following functions, determine if the function F is a linear transformation. I… • Let V be a vector space and X], X2, …, x,, be vectors in V. If x, # 0, x2 & sp( ( x, )), x} sp({ x] , X2 )), . . ., x,, # sp((x], . . ., x,_1)), show that the vectors X], 2, . .., x,, are l… • Show that B = {(1, 2, -1), (-2, 1, 1), (1, 2, 1) } is a basis of R3 by finding the unique representation of the vector (a, b, c) E R3 as a linear combination of the elements of B. • 4) Find the solution of the following linear-equation system by Cramer’s rule and by matrix inversion. [we discussed this in the lecture] 7×1 – X2 -X3 = 0 10×1 – 2×2 + X3 = 8 (6×1 + 3×2 – 2×3 = 7 • Find the solution of the following linear-equation system by Cramer’s rule and by matrix inversion. [we discussed this in the lecture] 7×1 – X2 -X3 = 0 10×1 – 2×2 + X3 = 8 (6×1+ 3×2 – 2×3= 7 • Solve the following systems of linear equations by matrix inversion. (4x + 3y = 28 4×1+ X2 – 5×3 = 8 (i) 2x + 5y = 42; (1i) -2×1 + 3×2 + X3 = 12 3×1 – X2 + 4×3 = 5 m by Crame • Compare and contrast three different ways to compute the determinant of a 3Ã—3 matrix. Be prepared to illustrate your discussion with examples of each method. • Problem 3 (12 points) Rank the following functions in order of their asymptotic growth. That is, find an order f1, f2, f3, … . f6, such that f1 = O(f2), f2 = O(f3), and so on. You do not need to pro… • . There has been lots of vocabulary in this course. It is important to understand the relationship between the definitions by combining words and constructing examples. Be able to give and explain m… • Verify the following properties of convex functions:. Problem 2 Verify the following properties of convex functions: 1. (10 % ) If f(x), g(x) are both convex, then of(x) + Bg(x) is also convex, if (, … • Find a function of x and y to describe the matrix above. X -5 -4 -3 -2 -1 O 2 3 4 5 +—– 8 1 w 1 3 H W V 3 4 a W O – 3 -2 wo war -9 – 7 – 5 11 OOO – On UT P W N WNHOHNWP -1 -15 -12 -9 – 6 w ww ww ww… • Let u and v be distinct vectors in R^n. prove that the set {u, v} is linearly independent if and only if the set {u + v, u – v} is linearly independent. • Let 7: R3 – R’ be the transformation given by orthogonal projection onto the line y = mr. where m is a constant. (a) (1 point) Use geometric reasoning to find im(T). (b) (1 point) Use geometric rea… • Let Wi and We be subspaces of R". Define Wi + Wa to be the subset of all vectors w1 + w2. where wj is in Wi and w2 is in W2. (a) (3 points) Is Wi + We always a subspace of R"? If yes, the… • Find an example of linearly independent subsets {u1, u2} and {v} of R^3 such that {u1, u2, v} is linearly independent. • . Perform the indicated operation and simplify. 3×2 6x 4x 6) 7) 5x 15 8) x -8 x-8 x+3 x+3 2x -1 2x -1 Find the LCM (least common multiple) of the polynomials. 12) 24×2 and 8×2 – 16x 13) x2 – 25, x, … • Perform the indicated operation listed on the paper for 6-8 perform the operation and simply for 12-14 find the lcm of the polynomials for 17-18 solve • u I b) Suppose you are given initial parameters for linear regression: we = 1,191 = 2,102 = 0. Compute the LMS error for this initial model. Compute the gradient for these parameters. You do not need … • 6) What is the dimension of the vector space spanned by the set of vectors { (001010), (101000), (001100), (100010), (011111)} over GF(2)? What is the dimension if the ground field is changed to GF(4)… • Answer the question in the attached image below. Show your work 🙂 Â . Problems: 3ritz+ 23 + =0 1. Solve the linear system 521 – 22 + 23 – 84 =0 Be sure to express your solution in point form, row for… • . (1 point) Which of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. JA. T(XO, X1… • Identify the graph of the linear equation x + 4 y + 2 z = 8 in three dimensional space. • numerical methods. For the following system of linear equations, find the number of digits required to use in the calculations to achieve a precision of 12 significant figures in the numerical solutio… • numerical methods. For the following system of linear equations, find the number of digits required to use in the calculations to achieve a precision of 10 significant figures in the numerical solutio… • . A country has four age groups: adult, teenagers, older children and younger children. Adults and teenagers require 3 doses of a vaccine, each one being 30 micrograms (ug). Older children require 2… • Problem 2. [33 points] The general form of Thwaites Method is: 0.45v x 02 [Ve (s)]5ds + 0% Ve (xo)) [Ve(x) 16 Jxo Ve (x) where the boundary layer momentum thickness at x = Xo is 60. Remember that 02 d… • Show that if A is an invertible matrix, then so is A-I and (A-l)-l = A. • see attached Â . 2) Let V be a vector space over R where dim(V) = n, and let B = {v1, v2, …, Un} and C = {w1, W2, …, Wn} be two distinct bases of V (so B C). For v E V, define the map L : VRn L(… • see attached Â . 1 In M2 (R), let B = 1 -1 2 -2 and consider the following set: U = {A E M2(R) \| AB = 02} . (a) Prove that U is a subspace of M2 (R). (b) Determine the dimension of U and give an expli… • . A country has four age groups: adult, teenagers, older children and younger children. Adults and teenagers require 3 doses of a vaccine, each one being 30 micrograms (pg). Older children require 2… • . 1. Suppose that the functions f, g are given by f(x) = x3 and g(x) = 2x + 3. Find formulae for the composite functions fg and gf. Explain why f and g have inverse functions, and find formulae for … • find using method of variation of parameters Â 1. Â Solve (D^2+2D+5)y=e^-x tanx • PgD [MAI 4.4] LINEAR REGRESSION PgUp 2. [Maximum mark: 6] Num Lock Consider the following data e x 2 3 4 y 2 3 7 8 The regression line for y on x is y = 2.2x – 0.5 (a) Solve the equation above for x t… • . QUESTION ES 20 marks (a) Let mezo>– : Find two vectors, x and y, such that span {x, y) = span {u, v} and x is orthogonal to y. Explain why your choice of x and y works. (4 marks) b) Determine … • I NEED HELP WITH ANSWERING THE SECOND PART OF THE QUESTION USING SIMPLEX DUAL METHOD , PLEASE DO NOT ANSWER THE GRAPHICAL PART. A brick manufacturing plants makes two types of blocks. Each Type 1 Bloc… • . Let A be the matrix (i) Find a basis for the nullspace of A. (ii) Create a basis for the row space of A. You need to justify your choice of creation. iii) Insert additional vectors into your creat… • . Let A be the matrix below, 2 2 1 2 2 Write out the characteristics polynomial of A and determine all eigenvalues of A. (ii) Determine a basis for each of the eigenvalues of A. Let C be a 2-by-2 ma… • . Find two vectors, x and y, such that span {x, y) = span {u, v; and x is orthogonal to y. Explain why your choice of x and y works. Determine the values for a, b and c such that the set below is an… • Please show detailed steps. Thank you.. An important feature of linear transformations T : Rm â€”) R” is that T65) = 6 (multiplying a matrix by a vector always gives back a zero vector). 931 992 pro… • Consider the following column vectors: b = âŽ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽœ âŽ› â€‹ 3 0 âˆ’ 2 1 4 âˆ’ 2 âˆ’ 3 âˆ’ 1 â€‹ âŽ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽŸ âŽž â€‹ ; c = âŽ… • Use the given conditions to write an equation for the line in point-slope form and slope-intercept form. Slope = – 5, passing through ( – 2, – 2) Type the point-slope form of the line. (Simplify your … • . 3. NCDOT looks at the following metrics when analyzing traffic data: . Crash Rate, which is defined to be the total crashes per 100 million vehicle miles (MVT). . Injury Rate, which is defined to … • . 2. The number of crashes, injuries, and fatalities that occurred on Durham County roads between the years of 2019-2021 are given in the table below: Durham County Vehicle Crashes (2017-2021) Year … • . 1. The number of crashes, injuries, and fatalities that occurred on Wake County roads between the years of 2018-2021 are given in the table below: Wake County Vehicle Crashes (2013-2021) Fatalitie… • . QUESTION 5 (9 MARKS) Aaron, Bob and Caleb are painting doors for houses. In a day’s work, the three of them could paint 30 doors combined. Find out how many doors did each of them paint if Bob and… • . a) Consider the system of linear equations: Cty= O yfz= -Z = ax + by + cz = 0 Find the values of a, b, c such that the system has a unique solution, no solution, infinite solution. • Suppose A = a b b a for some a, be R. (a) Is A necessarily invertible? Explain your reasoning. (b) For what values of a and b does A have two distinct eigenvalues? (c) Suppose A has two distinct ei… • I’m having some trouble with this question, can someone help. • . QUESTION 6 (12 MARKS) Solve the system of linear equations below using the LU-decomposition method 4x – 3y + 2z = 20 -2x – 4y + 3z = 3 xty- z = 2 3/7 • . QUESTION 9 (8 MARKS) Show whether the vector w is a linear combination of vectors v1, v2 and v3 where W = 2 v1 = v2 = WHO D3 = -1 2 • . QUESTION 10 (15 MARKS) A matrix K is given as below. 18 0 0 K = 6 6 11 1 0 1 1 Solve K4 by using the formula K4 = PD4p-1 • Use the graph to determine a. open intervals on which the function is increasing, if any. b. open intervals on which the function is decreasing, if any. c. open intervals on which the function is cons… • . QUESTION 8 (8 MARKS) While experimenting with 3 cups (A, B, C) and measuring their volumes in tablespoons. It is found that water from 1 cup of A and 4 cups of C gives the same volume as 2 cups of… • If a non-homogeneous linear system has exactly one solution, then the associated homogeneous system has no non-trivial solution. 1 (e) If 3 is a solution to a non-homogeneous linear system A and ?… • El Ing. Noriega compro un total de 200 sensores para el CEDIS en Xalostoc Estado de Mexico, entre ellos se encuentron: sensores de temperatura, de presion y de movimiento, invirtiendo un total de 7500… • (Valor. 1 de 10 puntos) Determina si el conjunto de vectores dado es o no una base para el espacio IR2. Justifica tu respuesta. {( -2 , 2) , (2,4)} Si pertenecen por tener cada uno contiene 2 coord… • (Valor. 1 de 10 puntos) Determinar si los vectores de R3 v1 = (1, -3,0), v2 = (3,0,4) y v3 = (11, -6,12) son linealmente independientes o dependientes. Como son componentes no proportionales entonc… • (Valor. 3 de 10 puntos) El conjunto de matrices M2x2 con terminos reales, &Es un espacio vectorial? Justifica tu respuesta a M2x2= : a, b, c, deR b • Un hospital particular suministra tres diferentes medicamentos a un piso que tiene a tres diferentes tipos de pacientes. Cada paciente del tipo 1 consume cada semana un promedio de 1 unidad del medica… • Q3 (1 point) Let ð¹ be a finite field of cardinality ð‘, and let ð‘‰ be a vector space containing a basis (ð‘’1,…,ð‘’ð‘›) of cardinality ð‘›. Show that ð‘‰ is a finite set, and calculate i… • . Given f (a, y) = fetty + xy. Find the determinant of the matrix V2 f (1, 1). a b Hint: The formula for calculating determinant of a 2 x 2 matrix is ad – bc. C d O 2e2 + 1 O 2(e2 + 1) O e2 + 1. -5 … • . Given the matrix A = 0 1 0 , which of the following values can a Ax attain (for 0 0 1 appropriate choices of @)? Select all that are correct. O 104.99 0 0 0 -8.7322 O 21.81 • #2 Q.#7 Q.#8 Â . A circular birthday cake is being adorned with candles, equally placed around in a circle on the top of the cake. There are fourteen candles in total to be placed, each having a di… • Assume that your mortgage payment is $1350/month. Calculate how many months it willÂ take to pay off each property (assuming 0% interest) in column I.Â 21. Create a column header in J1 – “Crisis”Â … • In this question, we revisit a theme from Assignment #1: using linear algebra to solve non-linear problems. On the previous Assignment, we found a parabola through three points. In this question, … • Next, introduce a parameter for this variable: Let z = t, where t is any real number. Rewrite the first and second equations from the system replacing z with t: X+ 29 13t =20_ and =y +4t = 50 Solve… • 134 2259 3.1415 4 x 109 1. Find the second row of the product CD where C = 1 1 1 1 -2 1 2 and D = 20 40 10 20 2. Suppose that 2 3 N A = 1 0 1 Determine whether A had an inverse and, if so, find it. 3…. • please solve. Let HN – Describe all solutions of Az – 0 +X4 • Please provide answer for this question clearly Â . Let T : P3 – R3 be defined by 3a + b+ 3c+ 2d T (ax’ + ba? + cox + d) = a + 2b – 2d Let u = x3 – 3x, B – {1, x, x2, x3 ), and 3a – c – 3d -3 3 2 Give… • . 6. let A be the matrix A = [v1 v2 . . . 15], and assume vi = 2v2 + 3v3 + 4v4 + 505. A. Determine a nontrivial solution to Ax = 0. B. Assuming the columns 2, 3, 4, and 5 are pivot columns of A , de… • Please provide answer for this question Â . * 1 2×1 + 2×2 Let T : R’ -> R’ be defined by T Let u -3×1 – 22 and c =1 51 , [3]} 1 -2 Given Pc = , use the Fundamental Theorem of Matrix Representations… • Please help me with the problem above.. Problem 1 Let V = Cc([0, 2x]) be the space of all continuous functions from [0, 27] to C (This is a vector space over C). Define .2n (f, g) = : f (t)g(t ) dt. 2… • . Exercise 2.5.14 While trying to invert A, [ A I ] is carried to [ P Q ] by row operations. Show that P 2 QA.. Then complete Exercise 2.5.14. Note: the matrices [P Q] and [A I] in this question are… • (ii) For each of the following column vectors, please state its dimension and compute its norm; also state whether the vector is a unit vector or not and give a reason for your conclusion. (a) u = [10… • State the value of x for the year of 2004 b. Find the amount of waste generated in 2004. c. Find the amount of waste recovered in 2004 d. Find the percent of waste recovered in 2004. 5. What is … • . 3. According to True Car (www.truecar.com), a 2018 Toyota Camry (conventional) (29/41 MPG city/hwy) sells for an average of$22,030, and a Camry Hybrid (51/53 MPG city/hwy) sells for an average of… • In this question, you will ï¬nd some 2 x 2 matrices that do not behave like real numbers. (a) Find two matrices A such that A2 = 0. (b) Find three matrices A such that A2 = I. (c) Find two matric… • There are three types of solutions to any system of two equations. What are the types? What do those solutions look like graphically? • The graph shows the prices of different numbers of bushels of corn at a store in the current year. The table shows the prices of different numbers of bushels of corn at the same store in the previous … • Since all the variables in the canonical form of a linear program must be non-negative, it is not possible to solve a linear program where one or more of the variables can take on negative as well as … • These two constraints are equivalent: j=1 n âˆ‘a ij .x ij < Â Â b iÂ Â for i = 1,2 .. MÂ Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â j=1 n âˆ‘(-a ij ).x ij > – b i for i = 1,2 .. M Â True or False? • can someone check this problem please?. Part II: Systems of Linear Equations with Parametric Solutions. a. The following augmented matrix represents a system of linear equations. Use your calculator t… • The equilibrium point for a leisure boat market is 13 5 4.89 , where x represents the quantity of boats (in thousands) and p(x) represents the price per boat (in hundreds of dollars). Write a sent… • . Convexity (15 pts.) Consider the theory consisting of strictly positive real-valued variables, real-valued constants, multiph- cation, and the binary predicates g, 2, and =. Ground terms in this t… • Please provide answers neatly. Let B = { u1 – [i]. = andc= V Find PB, the transition matrix from the standard basis of R to the basis B Ex: 5 PB = • . Determine whether the following set {v, v , v, } is orthonormal basis of R. V18 4 0 18 18 • A an B are solids which are mathematically similar. The length of A is 12cm and the length of b is 6cm. If the volume of A is 72cm cubed what is the volume of solid b? • Suppose A = a Â b Â Â for some a, b âˆˆ R Â Â Â Â Â Â Â Â Â Â Â Â Â b Â a Â (a) Is A necessarily invertible? Explain your reasoning.Â (b) For what values of a and b does A have two … • Suppose A = a Â b Â Â for some a, b âˆˆ R Â Â Â Â Â Â Â Â Â Â Â Â b Â a Â (a) Is A necessarily invertible? Explain your reasoning.Â (b) For what values of a and b does A have two dist… • Suppose A = a Â b Â Â for some a, b âˆˆ R Â Â Â Â Â Â Â Â Â Â Â Â Â b Â a Â (a) Is A necessarily invertible? Explain your reasoning.Â (b) For what values of a and b does A hav… • A matrix C is called idempotent if C 2 = C . Let A be an m x n matrix and B be a n x m matrix. Prove that if BA = I n , then AB is idempotent. • Remaining time: 101:05 1 2 3 4 Question 2 For a function f(x), the Newton divided differences table is given by f[Xx- 1.Xx] XO = f[Xol = A 0.0 X1 = f[X1] = B f[Xo, X1] = C 0.2 X2 = f[X2] = f[X 1, X2] … • Question 1 Find the Singular Value Decomposition of the matrix A = 3 2 Show all the steps and calculations in your answer. 2 3 Your swer. • Two mining companies, Red and Blue, bid for the right to drill a field. The possible bids are $15 Million,$25 Million, $35 Million,$45 Million and $50 Million. The winner is the company with the hig… • Please answer in detailed steps. Thank you.. 1 Suppose that A is a 2 x 3 matrix and E E R2. Suppose further that if = 2 and 113 = 3 5 â€”3 are both solutions to the equation A5 = b. 1 (a) How many sol… • Please answer in detailed steps. Thank you! Â . Let T : R2 – R3 and S : R3 – R2 are linear maps. Can So T : R’ – R be onto? Can it be 1-to-1? Give examples or counter-examples. The notation S o T mean… • Please answer in detailed steps. Thank you! Â . Consider the triangle A in R with corners (1, 0, 0), (0, 1, 0) and (0, 0, 1). a) Find the image of A under the linear transformation T : R3 -> R2 by … • Hi, I was working in my hwk, but would like to double check my answers. Could you please solve it? Â Â . Problem 5 0 solutions submitted (max: Unlimited) Write a function called Leibniz that will rec… • Please show detailed steps. Thank you. Â . â€” . v” nu… An important feature of linear transformations T : Rm â€”> R” is that Taj) = 6 (multiplying a matrix by a vector always gives back a ze… • Using Schur’s triangularization theorem, prove that for any matrix A âˆˆ M n with eigenvalues Î» 1 , Î» 2 , . . . , Î» n we have: det(A) = âˆ Î» i and tr(A)= âˆ‘ Î» i • . 5. Given an n x n matrix A, define a subset of R" by W = {VER" \| A . V = (VT . A) T }. Show that W is a subspace of Rn. H 0 Hint: To get a better handle on W, start with a matrix A, say … • lli’Jonsilï¬‚ertheplane râ€” y+z =[lin R3. {a} [2′ points} Write clown the matrix of a linear transformation T with kerlT] equal to this plane. {h} [2′ points} Write down the matrix of a linear tra… • A researcher test whether room temperature has a performance in college students by giving her different biology sections their first exam in either a cold warm or hot room • Q1a) Q1b). Let X be a set. and let F be a field. Let V be the set of functions from X to F , with addition and scalar multiplication defined as in class. Show that V is a vector space, by checking tha… • How do I do this?Â 1.Â 2. 3.Â Â . (i) Consider the system 12 – 13 = 13 + 14 = HNNON 13 – CA – 15 – 16 C4 – 15 + 16 = Determine whether it has a unique solution, no solutions, or infinitely many sol… • (40 points) Similar to the concept of lines and planes in R3, a hyperplane in R4 is a 3- dimensional subspace defined by a single linear equation, and a plane in R4 is a 2-dimensonal subspace defin… • MAX 25X1+30X2+15X3 (PROFIT) S.T. 1) 4X1+5X2+8X3<=1200 2) 9X1+15X2+3X3<=1500 OPTIMAL SOLUTION Objective Function Value = 4700.000 Variable Value Reduced Costs XI 140.000 0.000 X2 0.000 10.000 X3 … • Need part F) ONLY. 1. (30 points) Let a, b be nonzero vectors in R2. a) Suppose \|all = 2 and \| \|b\| \| = 3. Compute (a + b) . (a – b). [2] b) Use the dot product to show lla + b/12 + lla- b/12 = 21/a/12… • Need it. Midtown Delivery Service delivers packages which cost$1.70 per package to deliver. The fixed cost to run the delivery truck is $84 per day. If the company charges$3.70 per package, how many… • Please work need it now. Time left 0:1 Solve the problem. Let the supply and demand functions for raspberry-flavored licorice be given by p = S(q) = =q andp = D(q) = 60 – -q , where p is the price in … • . Let V, W, and X be vector spaces. Let T1 : V â€”> W and T2 : W â€”> X be linear transformations. Consider the composite linear transformation T2 0 T1 : V â€”> X, deï¬ned by (T2 0 T1)(V) … • 2 Cancellation in matrix equations is valid if approrpiate inverses exist. (a) Suppose that A has a left-inverse and that AB = AC; prove that B = C. (b) State and prove the analogous result for right-… • Suppose that D = diag(d1, .. ., dp). (a) If di + 0 for 1 < i < p, show that D is nonsingular and that D- = diag(1/d1, . . ., 1/dp). (b) If some di = 0, show that D is singular. • Please answer now. Solve the problem. Suppose that the price and supply for a certain model of graphing calculator are related by p = S(q) = 1.5q, where p is the price (in dollars) and q is the supply… • 5 Suppose that A is nonsingular. (a) Prove that A is symmetric if and only if A is symmetric. (b) Prove that A is hermitian if and only if A is hermitian. • Use Gauss elimination for augmented matrices to solve -X1 – X2 + 2×3 EX-5 -3×1 – X2 +7×3 = -22 X1 -32 – X3 = 10. • Please provide answer neatly Â . Let B =u = 72 – [MY ac- {v1 = [2],v2 = [2]} Find Pc, the transition matrix from the standard basis of R to the basis C. Ex: 5 Pc • 3 X1 – 3X2 – X3 = 10. 7 Find all 3 x 1 column matrices b for which there is at least one solution to Ax = b and find all solutions x associated with that b, where 4 -1 2 6 A = -1 5 -1 -3 3 4 1 3 8 Giv… • Please answer now I need it now. Write a cost function for the problem. Assume that the relationship is linear. A moving firm charges a flat fee of $35 plus$30 per hour. Let C(x) be the cost in dolla… • 3 4 1 3 8 Give an example of (a) a system with fewer equations than unknowns but no solu- tions; (b) a system with fewer equations than unknowns and infinitely many solu- tions. 9 Use Gauss eliminatio… • 9 Use Gauss elimination to reduce the augmented matrix H below to reduced form G. For each elementary row operation performed, find the corresponding elementary matrix. Find a nonsingular matri… • need like now please peruse and answer. Write a cost function for the problem. Assume that the relationship is linear. Marginal cost, $60; 40 items cost$3500 to produce Select one: O a. C(x) = 60x + … • 6 8 10 Let A be p x q and consider postmultiplication by q x q elementary matrices. Show that: (a) Postmultiplication by E;; interchanges the ith and jth columns of A. (b) Postmultiplication by E;(c) … • B =0. 13 (a) Suppose that A is p X q, B is q x q, and AB = 0. Prove that either B is singular or A = 0. (b) Suppose that A and B are p x p and AB = 0. Use (a) and 3.10.1 to prove that either A = 0 or … • Sorry bottom three lines of answer are cutoff. Can you please help. Original question: suppose that A is p * p, B is p * q, AB = 0. Prove that either A is singular or B = 0.. amugh1773 answered 2hr ag… • Please review answer need it now. Midtown Delivery Service delivers packages which cost $1.10 per package to deliver. The fixed cost to run the delivery truck is$364 per day. If the company charges $… • . 2. The table below gives, for each of three regions , the number of parks it contains, an estimate of the standard deviation of the number of rare trees growing within each park in that regi… • Please provide answer for this question neatly. Â . x1 x1 – X2 Let T : R’ – IR be defined by T C1 Let B { m = [2], 232 – } andc = { v1 = .2 = } Find [T’], the matrix representation of T’ with respect … • Please review I need it like now. A book publisher found that the cost to produce 1000 calculus textbooks is$25,200, while the cost to produce 2000 calculus textbooks is $52,000. Assume that the cost… • . 1. A pair of mean corrected and standardised variables X and Y are correlated with a negative Pearson’s correlation coefficient of -0.1923 . You are also given that the variances of X and Y are… • Please review and answer need it now. Given the supply and demand functions below, find the demand when p =$12. S(p) = 5p D(p) = 120 – 4p Select one: O a. 48 O b . 72 O c. 132 O d. 60 • . Problem 2 Consider the following models: In Y* = a1 + 02 In X* + ut In Ya = B1 + 32 In Xi tui where Y* = wiY; and X* = w2Xi, the w’s being constants. Establish the relationships between the two se… • U is a subspace of the finite-dimensional vector space V, how to prove(i)dimU=dimV (ii)U=V. • Please answer this question. Â This is an example on what the answer should look like: Â . 1 -3×1 Let T : R’ – R2 be defined by T 2 2 2 + 23 6 Let B = U1 = -3 ; U2 = 4 , 13 = What augmented matrix sh… • Please review and answer. Solve the problem. Ten students in a graduate program were randomly selected. Their grade point averages (GPAs) when they entered the program were between 3.5 and 4.0. The fo… • . function approximate_inv_x(} Approximate x” a5 a linear combination of expEx}, 5in(x), and gammaï¬x). # Retupn5 â€” ‘est’: estimate of the L2 norm II x" – {c_exp * exp(x) + c_5in * 5in… • Please answer need it asap. Find the equation of the least squares line. Two separate tests are designed to measure a student’s ability to solve problems. Several students are randomly selected to tak… • Please answer need it asap. C. y = bi.3+ d. y = 33.7 – 2.14x Find the equation of the least squares line. The paired data below consist of the costs of advertising (in thousands of dollars) and the nu… • Review and answer please need it like now. Time left 1:26:3 Find the equation of the least squares line. The paired data below consist of the test scores of 6 randomly selected students and the number… • Read view and answer. For some reason the quality of production decreases as the year progresses at a light bulb manufacturing plant. The following data represent the percentage of defective light bul… • Suppose that A is p * p, B is p * q and AB = 0. Prove that either A is singular or B = 0. • Suppose that A is p * p and nonsingular. If A’ is obtained from A by interchanging two rows of A, then A’ is nonsingular and its inverse can be obtained by interchanging the corresponding columns of A… • Suppose that A, B, and A + B are all nonsingular p * p matrices. Show that A^-1 + B^-1 is also nonsingular and that: image below and that this in turn equals B(A + B)^-1 A.. A(A + B) B • Prove that L is a left inverse for a matrix A if and only if L^T is a right inverse for A^T. Prove that R is a right inverse for a matrix A if and only if R^T is a left inverse for A^T. • Read the question and answer. Find the equation of the least squares line. In the table below, x represents the number of years since 2000 and y represents the population (in thousands) of the town Bo… • Read the question and answer please. Find the equation of the least squares line. In the table below, x represents the number of years since 2000 and y represents annual sales (in thousands of dollars… • Please help me out with this practice problem so I can get ready for the midterm. Â Thanks! Â . Problem 4 Let V be a vector space with inner product (, ). Let T be a linear operator on V. Let 7* be it… • Please help me out with this problem for practice. Â Thanks!. Problem 3 Let V be a vector space with inner product (,}. Let T be a linear operator on T. Prove that the following are equivalent: (a) Fo… • PLEASE SOLVE THE SECOND PART OF THE QUESTION USING SIMPLEX METHOD PLEASE DONT SOLVE USING THE GRAPHICAL. respectively. The company requires at least 240 kg of material X, 100 kg of material Y and 290 … • A company purchasing scrap material has two types of scarp materials available. The first type has 30% of material X, 20% of material Y and 50% of material Z by weight. The second type has 40% of mate… • I NEED HELP WITH THE SECOND PART OF THE QUESTION PLEASE USE THE SIMPLEX METHOD TO ANSWER THIS QUESTION . NOTE DONT ANSWER USING THE GRAPHICAL METHOD. Question 1. A brick manufacturing plants makes two… • PLEASE USE THE SIMPLEX METHOD TO ANSWER THIS QUESTION WITH CLEAR AND CONCISE EXPLANATIONS. ii. Maximize Z = 3×1 – 2X2 + X3 Subject to X – 2xz +X3 25 -Xi +3×2 + 4×3 5-10 2x, + 4×2 + 5×3 = 20 3×1 – X2 -… • . II. 1. Conjugate Functions. (4 pts) Find the conjugate functions of the following functions. (1) f(x) = x Ax + bix, where x E Rn, A E Rnn, b E Rn. Discuss your solution for the following three cas… • . II. 2. Compressed Sensing. (18 pts) For compressed sensing, we formulate the sensed signal yt with an equation y = Ax + b, where y E Rm, A E Rmn, x E R", b E Rm. We have vector y contains the… • Make three simple propositions out of things that matter to you most. What compound proposition involving logical operators shall represent your plan regarding these three propositions? Express it in … • Please use SAS coding. 1. Using the matrix A = 3 1 1 âˆ’1 3 1 Find (a) AAâ€² and (AAâ€²)âˆ’1. (b) The eigenvalues and eigenvectors of AAâ€². (c) The eigenvalues and eigenvectors of (AAâ€²)âˆ’1 Â Plea… • why is PV of lease payment calculated this way?. QUESTION 2 50,000 1 PV of the lease payment = $50,000 + (1 5% (1+5%) =$405,391 Amortization Table Lease Payment Interest Principal Payment Lease Payab… • please Help. Problem 6: Linear Functions The parts of this problem are independent of each other. Show any work necessary. A. (10 points) Based on the information given for each function, state if eac… • A sales-response curve relates the sale of a product to the effort expended to sell it, which in this case is measured by the number of calls made by the sales force. A suitable family of non-line… • I’m still not very familiar with bra-ket notation, so I am having trouble figuring out how to get started on this problem. • . 1. Frobenius method: (a) By looking for an infinite series solution, find one solution w1(z) of the ODE w" ( z ) + – w'(2) + 42 2w(2) = 0. Z Rearrange the form of the series to write your … • This question uses Bra-Ket notation. For the set to be orthonormal, the inner product, <psi_n\| psi_m> = 1 if m=n and 0 otherwise. It is assumed that { \|1>, \|2>, \|3> } is orthonormal. So… • . 3 -2 0 4 5 15 1 8. Let A = -22 13 12 and suppose A is can be row reduced to 4 12 -1 -14 6 1 6 8 NN 4 0 the matrix 0 O O a. Give a basis for Col (A) b. Give a basis for Nul(A). 9. Determine whether… • . 4 0 -3 11 Given the matrix 2 5 which statement below is correct: 4 0 a. The matrix is not invertible. If the given matrix is A, then the columns of A do not form a linearly independent set b. T… • . 1 3 6 Let v1 = 3 , 12 = -3 L,W= 3 . Determine if w is in the subspace of R 3 7 10 generated by v1 and v2. Show your work to get credit for this question. 7. Let b1 = _, b2 = 3 , x =. Given the … • . 5. An economy is based on three sectors – agriculture, manufacturing, and energy. For each unit of output, agriculture requires inputs of 0.30 unit from agriculture, 0.30 unit from manufacturing, … • . 4. Let A = 8 18 1. b1 = [of ]. 12 = [261. b3 = ["4] a. Find A-1 b. Use A ‘ to solve the equation Ax = bj . You must show your work to get value for this question. c. Use A to solve the equ… • . 4. Let A = 8 18 1. b1 = [of ]. 12 = [261. b3 = ["4] a. Find A-1 b. Use A ‘ to solve the equation Ax = bj . You must show your work to get value for this question. c. Use A to solve the equati… • . 3 3 N 4 -2 -21 1. Let A = -2 2 \|, B = -1 2 ,C= L5 -1 3 3 2 N Solve the following (if possible): a) AC b) CA C) AB d) BT A e) (AC)B • . 2. If a matrix A is 6 x 8 and the product AB is 6 x 2, what is the size of B? 3. Find the inverse of the matrix 5] • Please answer this question: Here’s an example of what the answer should look like from a previous question: Â . 1 Let T : R’ – R’ be defined by T 21 – $2 -203 2 -6 5 Let 3 = 11 1 * 42 = -4 -5 and C =… • (5) Let 2 := {(r, y) E RÂ² \| 1 < rÂ² + y < 4} be a punctured disc and let f: 0R be a continuous function which is “locally convex” (that is for every point r in the interior of N there is a smal… • Given Z1 = 2X +Y and Z2= X-2Y. Show that Cov(Z1,Z2) = 2Var(X) – 2Var(Y) – 3Cov(X, )). 2. If My(t) = e(2et-2) and My (t) = Get + 2)10 and X and Y are independent, find a) P(X+Y=2). b) P(XY=0). C) E(… • . (5) (20 points) (2′) Determine which of the following are SubSpaoes: (a) The set of all vectors in R5 of the form where at least two of the$1,:r2,:r3,:1:4,a:5 are 0. (b) The set of all vectors in… • Can you solve in detail with explanation. It will be very much helpful. Thank you. I will give a good feedback too.. Suppose : vectors bi,..:, b, span a subspace W, and let amay be any set in W contai… • Would you please solve in detail with explanation. That would be very helpful and much appreciated. Thank you. I will give you a good feedback too. Thanks again.. Suppose vectors b1,…, bp span a sub… • Would you please solve in detail with explanation. That would be helpful and much appreciated. Thank you. I will give you a good feedback too. Thanks again.. Let T : R" – R" be an invertible… • Would you please solve in detail with explanation. That would be helpful and much appreciated. Thank you. I will give you a good feedback too. Thanks again. • Consider two companies, Company A and Company B, producing the same model of iPhones. The demand for the iPhones produced by Company A is DA, and the demand for the iPhones produced by Company B is DB… • Please answer and explain the question above.. Find a unit vector perpendicular to v = (1, 1, 0, 1, 1) E R’. Find another. Find another. • Please answer and explain the question above.. Suppose S is a 3 x 3 symmetric matrix with pivots 2, 1, 1 (in that order) and multipliers (2,1 = 2, 43,1 = -2, and (3,2 = 0. What is S? • Please explain your answer to the question above. Â . Suppose matrices A2x3 and B3x2 are such that, for 1 < i < 3, 2 col; (A) row; (B) i 2i What is AB? (Note: you do not need to find A or B) • Are the following statement a-c TRUE OR FALSE? a) b) c) • Are the following statements a-c TRUE or FALSE? a) b) c) Â . If A and B are both n x n invertible matrices, then AB is also invertible.. The set of linear combinations of the vectors 4 2 6 U = 2 U=… • This is my little sisters homework it just doesn’t make sense? I need help. MASK IS A MUST COMPANY Mask is a Must Company charges PHP 15,000 (production cost) plus PhP 120 per KF94 mask Quantity 4 5 P… • . 5. (5 pts) Write b as a, linear combination of the vectors v1, V2,, and V3, where 2 â€”1 1 6 v1 = 3 , V2 = 0 , V3 = 2 , and b = 11 5 1 0 13 Show your work: set up and solve a. linear system. • . Distance versus angle nearest neighbor. Suppose =1. ….#m is a collection of n-vectors, and r is another n-vector. The vector , is the (distance) nearest neighbor of a (among the given vectors) i… • . Distance versus angle nearest neighbor. Suppose =1. ….#m is a collection of n-vectors, and r is another n-vector. The vector , is the (distance) nearest neighbor of a (among the given vectors… • Let A be any coefficient matrix of size mÃ—n (m rows, n columns). Part A) For the linear system Ax = b, how many equations are there? How many unknowns are there? Part B) Suppose m = 4 and n = 2. The … • what is a, b and c?. Let { u1 (a) = -12, u2 (x) = 12x, u3 (x) = -12×2} be a basis for a subspace of P2. Use the Gram- Schmidt process to find an orthogonal basis under the integration inner product (f… • Formulation Problem A forestry company has four sites on which they grow trees. They are considering four species of trees, the pines, spruces, walnuts and other hardwoods. Data on the problem is g… • Compute 2u, – v, u + v, and 2u – v. b. Use the triangle rule to sketch u + v . c. Use the triangle rule to sketch 2u – v . • For each matrix-matrix product AB , Compute the product or state that it is not defined:. b. (2 pts) a. (2 pts) NO [1 3 A = 2 5 1], B= 0 A = 2 7 c. (2 pts) d. (2 pts) A = 3 NO , B = 2 • For each of the following, does the set span R 2 ? Â If so, justify your answer. If not, sketch the span of the set of vectors.. {[3] . [-2]} • Seorang petani yang memiliki 8 ha tanah sedang memikirkan berapa ha tanah yang harus ditanami jagung dan berapa ha yang harus ditanami gandum. Dia mengetahui ahwa jika ditanami gandum, setiap ha tanah… • a) Solve for a and b in the equation 2 3 a + b 14 b) Use your answer to part (a) to give the coordinates of 9 14 with respect to the 2 basis 11. Give the coordinates of 5 with respect to the basis… • . 4×1 – 22 Let T : R3 -> R be defined by T = 3×3 2 3 2 Let B = < u1 -2 , U2 = 1 ; 13 = 7 What augmented matrix should be used to find [7], the matrix representation of T with respect to the ba… • . 1 Let T : R’ -> R’ be defined by T 20 2 1 + 2 Let u = [8 ] andc = { v = , va = } Find T(u), the image of u under T. Find [T(u)]c, the coordinatiation of T(u) with respect to the basis C. Ex: 5 … • 1 -17 3. Consider A = 1 4 2 (a) Denote the span of the column vectors of A as linear combinations of the column vectors. (b) Let b = , where s is a real number. Using elimination, determine the condit… • Include the formulation along with the original optimal solution and optimal value. Create and include the Sensitivity Report. Â Thank you.. 18. I&L Branstrator Inc grow grapes to sell to wine ma… • 2 1 1. Consider A = 6 3 4 2 (a) Bring A into the upper triangular matrix form U using elimination without interchanging or scaling rows. (b) Find the pivots of A. (c) Bring U to the reduced row echelo… • . Let Ti : R2 – R2 and T2 : IR2 – R2 be linear transformations defined as follows. -3×1 T1 -5×1 + 5×2] 2×1 T2 2 4×2 2 Ex: 42 (Tio T2) 2 Ex: 42 (T2 0 71 ) b • . In Exercises 37 and 38, describe the possible echelon forms of the standard matrix for a linear transformation 7. Use the notation of Example 1 in Section 1.2. 37. T : R3 – R4 is one-to-one. 38… • . Let Ti : R’ -> R’ and 72 : R – R be linear transformations defined as follows. – 5×1 1 T1 = 2×1 + 4×2 ] 4×1 T2 2 Ex: 42 (T2 0 1 ) • Plot the following set of points, (x, y) on a graph of y vs x, then sketch a line that fits the points as closely as possible. Estimate the equation for this line from your sketch.Â (1, 1),(2, 3),(3,… • . Solve the following using the graphical method: 1) X >= 125 2x + y >= 600 X + y >= 350 2) y > 2/3x – 7 X <-3 3) y < 2x – 3 y <= 1/2x + 1 • Let T : P3 â†’M2Ã—2 be the linear transformation defined by T (a + bx + cx2 + dx3) = [a + d b âˆ’c b + c a âˆ’d]. Â Let A= {1, 2x, 1 + x2, 1 âˆ’x + 2×3} and B = {[1 0 0 0] , [1 1 0 0] , [0 0 2 0] , [… • Suppose A is an n Ã— n nonsingular matrix. Prove that there exists a polynomial q(x) such that A -1 = q(A) • . QUESTION: Two water trucks have leaks and are each carrying 10,000 gallons of water when leaving a fill- station. The first truck is losing 95 gallons per hour, while the second truck is losing 12… • Does there exist a linear transformation T : R4 â†’R with null space ker(T ) = {(a_1, a_2, a_3, a_4) âˆˆR^4 : a_1 = 3a_2, a_3 = a_4}? If yes, give an example of such a transformation. If not, give pro… • . Let T : U â€”} V be a lineertranstormation. Use the rankâ€”nullity theorem to complete the information in the table below. • Help me with A, B, and C please.Â . An Earth satellite of unit mass can be modeled as a point mass moving in a plane about the origin under the inï¬‚uence of an inverse-square force law. Representing … • (Tucker 1.2.7) A furniture manufacturer makes tables, chairs and sofas. In one month, the company has available 300 units of wood, 225 units of upholstery and 350 units of labor. The different product… • . 5. The setting for this problem is RS. Suppose you have a plane P and two vectors a and b in P. The task is about the general question, "If you add two vectors in a plane, is the result still… • Suppose A : X â†’ Y is linear and V is a subspace of X. Denote by AV the subspace {Av : v âˆˆV } of Y . Prove the following: (i) AV is a subspace of Ran(A) (ii) dim(AV ) â‰¤rank(A) (iii) rank(AB) â‰¤r… • Hello can someone help me with this maths question please. Linear Algebra Problem Requirements A square matrix A is said to be idempotent if A2 = A. Let A be an idempotent matrix. (a) Show that I – A … • Determine the domain and range of the following relations: (Answer in words, set and interval notation) Domain (Words): Range (Words): set Notation D: Set Notation R: Interval Notation D: Ilnterval No… • use the following graph to answer the following a)Identity the dependent and independent variables Dependent Independent b) What does the y-intercept represent in this example?l c) What would the x-in… • . 1030 (15pts) Suppose rref[s1 \| $2\|$3 \| $4] = 0140 . Let A = [$1\|$2\|$3\|$4] and 0 0 01 suppose$1 = (1, 1, 1) whereas $2 = (0, 3, 3) and$4 = (2, 4, 4). What is the third column of A ? Hint: use th… • (Tucker 1.2.8) A company has a budget of $280,000 for computing equipment. Tablets cost$500 each, laptops cost $2000 each, and servers cost$5000 each. The company needs five times as many tablets… • Describe each section of a car’s journey. Include specific quantities of any change. (1 mark) Describing Linear Line Segments 2. Describe each section of a car’s journey. Include specific quantities o… • ) Two ferries travel across a lake 50 miles wide. One ferry goes 5 miles per hour slower than the other. If the slower ferry leaves 1 hour earlier than the faster, but both arrive at the same time, wh… • . (5) Theory questions: Please specify whether the following statements are true or false, and justify your answers. In doing so, you may cite results from the lecture notes or textbook without need… • Question 3 1 pts 3. Suppose u and e are two vectors in R3 and u # v. Which of the following options are possible for Span(u, v)? Select yes or no for each option. (a) A point (b) A line (c) A plane (d… • [25 pts] Remember that for the trace operation tr (v . v ) = Z; v. = v v where v is a vector, and if a square matrix P is orthonormal then P P = / (the identity). Given that v = [1 0 2] , and R is an … • (Tucker 1.2 Example 2) Consider a linear economic supply-demand model described by: Industrial Demand Consumer Total Demand Ener. Const. T rans. Steel Demand Energy : d1 = 0.4×1 + 0.2×2 + 0.2×3 + 0… • (Tucker 1.3.6) Imagine that historical data shows that if the stock market closes up (i.e. increases in value) one day, it has a 60% chance of closing up the next day, a 20% chance of staying flat … • (Tucker 1.3.18) Let Ri and Fi be the population of rabbits and foxes, respectively, in some area at a given point in time. A model for the populations one month in the future, Ri+1 and Fi+1 is give… • Classify each of the following systems of equations as linear or non-linear. If non-linear, explain why. (a) x = 10 + 2t y = âˆ’4t z = 50 âˆ’ 3t 2 (b) x1 + x2 = a x1 âˆ’ x2 = b (c) 2x + 3xy + 4y = 7 x… • . Q 2. Verify that the following matrix is orthogonal. cos 0 0 sine Q ( 0 ) = 0 0 – sin 0 0 cos A What does this matrix do? • . 7. 3 points Determine all values of the parameter h such that h Span = R3. • . 6. [2 points] Consider the traffic flow described by the following diagram. The letters A through D label intersections. The arrows indicate the direction of flow (all roads are one-way) and their… • If you could Â solve this, that’d be great!. 4. [4 points] Set 2 aj = and b = OH Do there exist scalars C1, Ca, C3 ( R such that b = ca, + cza, + caas? You should justify your answer. (If you conclude… • If you could solve this that’d be great.. 3. [4 points] Write down the general solution of the linear system in vector parametric form: T. 2 -1 7 1 -3 2 – 2 -1 -1 • If you could solve this that’d be great :). 2. Compute the following: (a) [1 point] 7 2 (b) [1 point] 2a – =b where a = 1 and b = 2 • If you could solve this, that’d be great :). 1. [3 points] Determine all values of the parameter h such that the linear system m1 – 212 + 2hi3 = = -201 + 4×2 + 413 =-3 is inconsistent. You should just… • . ACTIVITY 1. LINEAR PROGRAMMING A. FOR EACH OF THE FOLLOWING, CREATE A LINEAR PROGRAMMING MODEL. UPLOAD YOUR WORKSHEETS IN WORD DOCUMENT OR PDF. (10 points each) 1. Suppose you own a candy manufact… • If the matrix Î› is symmetric positive definite, then so is B(T)Î›B. Do you agree? Explain. • Solve both parts clearly with original working. Chegg is simply not helpful • . I. Problem Solving. Show your CLEAR AND COMPLETE solution. (10 POINTS EACH). 2): + 3 y = T 1.) Solve the following system of linear equations for x and y graphically: 2x + y = i 2.) Solve the foll… • . 1. [4 marks] Given V is a subspace of R’ consisting of vectors of the form (a + b – c, 2a + c, b + 2c, a + 2b) where a, b, ce R. (i) Express V in terms of a linear span of three vectors. (i.e. Wri… • Please write on paper and solve the Question. 8 Find the inverse of the following matrix using Cayley-Hamilton Theorem: 1 1 1] A = 2 3 4. • Please write on paper and solve the Question. 7 Find the matrix representation of the linear Transformation 7: R3 – R2 defined by T(x, y,z) = (2x – 4y + 9z, 3x + 3y – 2z) Where basis of IR3is B1 = {(1… • please write on paper and solve the Question. 4 Test whether or not the following transformation defined as follows is invertible or not. If invertible then findT-1. Where T = (xty – z, x – 2y + z, x … • Hi, tutor. May I ask for explation for those questions? I’m confused. Thanks. The reference is: https://www.coursehero.com/u/file/118588358/Homework-3-Copypdf/#question (Homework 3 – Copy.pdf). (a) Us… • Solve the matrix equation for a, b, c, and d. 2 1 3 d 9 -2 a IL C • Write the system of linear equations in the form Ax = b and solve this matrix equation for x. X1 – 5X2 + 2X3 = 5 -3X1 + X2 – X3 = -2 -2×2 + 5X3 = -11 X1 5 X2 -2 X3 -11 X1 X2 X3 • let Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â . Let C be a k X k symmetric matrix. Then, which of the following is not true? a) C is diagonalizable. b) Some eigenvalues of C are not complex c) … • Please don’t use the concept of “Rank” I need an explanation. Thank you. Â . Week 4: Practice with ï¬‚amfoï¬nations. For each matrix A below, (a) state the domain and eodomain of TA, (b) ï¬nd TA(81)… • . 4. This problem is about graphs. In each of the following cases, either draw a graph with the given property, or prove that no such graph exists. (For the purpose of this problem, a “graph” me… • Generate the requested sets using onlyr set builder notation and the provided sets. Some problems mayr require you to build one or more sets to be used in the generation of other sets. 3, the set o… • Express the column matrix b as a linear combination of the columns of A. (Use A1, A2, and Ag respectively for the columns of A.) 3 -1 2 A = b -13 = 3 1 1 7 b = • . In writing v as a linear combination of u , , u, &u, ( if possible) given: v = (3,0, -6) , u, = (1, – 1,2) , u, = (2,4, -2) , u,= (1,2, -4) & c, , C,, C, as arbitrary constants find c2 . • as shown.. (Lange Exercise 7.8) Suppose the matrix A = {ajj } is banded in the sense that aij = 0 when i – j > d. Prove that the Cholesky decomposition B = {by } also satisfies the band condition b… • Hi! I was working in my homework, but would like to double check my answers. Could you please solve it? Â Â . Q14 2 Points Consider the following linear transformation T: R2 – R2 T 2 1 – 2 -4×1 + 32 … • Hi! I was working in my homework, but would like to double check my answers. Could you please solve it? Â Â . Q13 4 Points 0 2 0 Consider the 3 X 3 matrix B = 1 2h â€”1 1 2 1 013.1 2 Points Find all … • Hi! I was working in my homework, but would like to double check my answers. Could you please solve it? Â Â . Q12 4 Points LetT: R3 â€”> R3 be a linear transformation, and suppose that 1 0 0 T 1 ?… • Hi! I was working in my homework, but would like to double check my answers. Could you please solve it? Â . Q11 6Points 1 â€”1 3 2 1 The matrixA= 2 â€”2 5 2 4 has reduced rowechelon form â€”1 1 â€”2 0… • Hi! I was working in my homework, but would like to double check my answers. Could you please solve it? Â Â . Q5 2 Points Let T: R4 -> R* be the following linear transformation: 2 1 [1 + 52 20 2 0… • Hi can someone solve and explain on how to get the answer on this? Â Â . 2x+ 3y =7 1.) Solve the following system of linear equations for x and y graphically: 2x+ y =1 2.) Solve the following system … • Consider the following. 3 -1 0 0 1 -3 A = 0 0 , P = 0 4 0 -4 2 -3 1 2 2 (a) Verify that A is diagonalizable by computing P AP. P AP = X (b) Use the result of part (a) and the theorem below to find the… • . 1. [4 marks] Given V is a subspace of R4 consisting of vectors of the form (a+bâ€”c,2a+c,b+20,a+2b) where a,b,c E R. (i) Express V in terms of a linear span of three vectors. (i.e. Write V = spanS… • can you help me with Introduction to Linear Algebra? • You currently work in the company “Neutro” which is dedicated to the production of shower soaps. The company produces four types of soaps: Neutral, Dermatological, Moisturizer and Aloe Vera, which req… • Linear algebra- we would like to construct a simplex tableau where X sub 1 and S sub 1 are the basic variables. We can do this by isolating X sub 1 in row 1 and S sub 1 in row 2.. what is the resultin… • . A particle moving in a planar force field has a position vector x that satisfies x’ = Ax. The 2 x2 matrix A has eigenvalues 7 and 4, with corresponding eigenvectors – 22 V Find the position of the… • Prove that a = C(R”), that is, that set of invertible linear transformations from R” to R” is dense in the set of all linear transformations. This amounts to showing that if A E [.(Rn) and 5 &gt… • Convert the 1st 3 inequalities to equalities using non negative slack variables. What is an upper bound on the number of basic solutions for this system. Explain. Find all basic solutions for this LP…. • Need help with Linear program question. Convert the LP to standard form. Identify the vectors/matrices A, b and c. Is the standard form also in canonical form? Explain.. 1. Consider the following line… • . Let A = xy, where neither r nor y is 0. (a) How many linearly-independent columns does A have? Explain why! (b) What is the rank of A? Explain why! (c) Generate A in MATLAB or PYTHON using random … • What is the relationship between slope and density. Always or just in this case? 6. How precise are your results? What evidence/data supports your answer? • Find a matrix A that induces the transformation T :R 2 â†’R 3 given below. • Find all subspaces of RÂ² which are stable under the linear transformation T : RÂ²Â â†’ RÂ², T(x,y) = (x+y, -x+2y) . Since T(x,y)=(x+y,âˆ’x+2y) is a linear transformation, there must be a non-zero sc… • Find all subspaces of RÂ² which are stable under the linear transformation \ T : \textbf{R}^2 \rightarrow \textbf{R}^2 , T(x,y) = (x + y, -x + 2y) \ Â : Â Since T(x,y) = (x+y, âˆ’x+2y) is a linear tr… • LY BK tead 2bo 1. a) Find the domain and range of the function. b) Determine all the values of x for which f(x) = -5. • AY VX 4. a) Find the domain and range of the function. b) Determine all the values of x for which f(x) = 5. • X 3. a) Find the domain and range of the function. b) Determine all the values of x for which f(x) = 2. • VX 2. a) Find the domain and range of the function. b) Determine all the values of x for which f(x) = 0. • VX 1. a) Find the domain and range of the function. b) Determine all the values of x for which f(x) = -5. • How do I do this?Â 1.Â 2.Â Â . (2′) Determine which of the following are aubspaom: (a) The 301. of all vectors in R5 of the form :n I: 1’3 $4 1’5 where at. least M of the 11,12,13, 3:4,I5 are l]. (… • How do I do this?Â 1.Â 2.Â 3.Â Â . (2) Consider the system = 12 – 13 = T3 + 4 HNNON 15 – 16 CA – 15 +$6 Determine whether it has a unique solution, no solutions, or infinitely many solutions. Jus… • What would be a good first step to graph the solution to the following system • . Find all subspaces of R2 which are stable under the linear transformation T:R2 â€”>R2, T(x,y) = (x +y,â€”x+2y). • . Let a be a real number. Determine the inverse of the matrix 1 0 0 …00 a 1 0 …00 a2 a 1 …00 A: . . . . .. EMn(R)- (Inâ€”2 (Inâ€”3 (Inâ€”4 . .. 1 0 451"" (Inâ€”2 (Inâ€”3 …a 1 • TH 110 : 2021-22 TEST #3a PROBLEMS .1. Let W := RK be the R-vector space of all functions from R to R. A function g E W is odd if the equation g(-x) = -g(x)holds for all x E R (i) Give an example of a… • A triple-threaded worm drives a worm gear. If 100 revolutions of the worm will make the gear revolve 10 revolutions, how many teeth must the gear have? A. 20 B. 40 C. 30 Â 15. Find the diameter o… • question 2: Calculate the Lot and orthogonal projection of the vector v := (3,1,4)* â‚¬ R3 onto the plane formed by the vectors (1,3, -1)’ and (1,1, -1)’ â‚¬ R3 is stretched.. Aufgabe 2. (8 Punkte) Be… • . Exercise 3.3 By summing the Peano-Baker series, compute D(t, 0) for A(t) = A = 0 Exercise 3.4 Compute d(t, 0) for A (t) = 0 • Hi Tutors, Please help me with the following question. No explanations. Â Just want to check my work. Thanks! Â Â . Let S be an infinite subset of a finite dimensional vector space V. Then to compute… • DO NOT HAVE GIZMo and DOES NOT NEED TO BE ANWER BY GIZMO. . Vary c and h to check how c affects logarithmic functions with other bases. You should see that c scales the graph horizontally. What … • Tutors, please help me with the following T/F question. No explanations, or very short one. Â Just want to check my work. Thanks! • Please don’t use the concept of ‘Rank’.Â I need to see the whole work. Thank you Â . 1Week 4: Practice with nunsfomations. For each matrix A below, (a) state the domain and eodomain of TA, (6) ï¬nd … • ALGEBRA I / / UNIT 4 SOLVING EQUATIONS AND INEQUALITIES – 4.1 4.1 READY, SET, GO! Name Period Date READY Topic: Solutions to an equation. Graph the following equations on the coordinate grid. Determin… • Calculate the perpendicular and orthogonal projection of the vector v := (3,1,4)* â‚¬ R3 onto the plane formed by the vectors (1,3, -1)’ and (1,1, -1)’ â‚¬ R3 is stretched. • PROBLEMS T3.1. Let U := RR be the R-vector space of all functions from R to R. A function h E U is even if the equation h(-x) = h(x) holds for all x E R. (i) Give an example of a nonzero even function… • State whether the following e-j are true or false.Â Â . e.) The set of linear combinations of the vectors u = U = W = is all of R3. f.) If A and B are both n x n invertible matrices, then AB is also … • State if the following a-d are true or false:Â Â . a.) For any vectors v, w E R", we have that -1 < v . w < 1. b.) If v and w are perpendicular, then \| \|v – w\|\| = v2. c.) If u I v and u I … • image of MAT 1341. Problem A. Suppose e and f are some parameters. Consider the system of linear equations in unknowns r. y and z: 3x + y + 22 2x – 2y + 29ez = 29f (1) Write down the matrix A of coeff… • . This question will diagonalize a matrix and apply this to the transformation of a curve. EXM.3.AHL. 120.5a [8 mai. Let the matrix M = Find the eigenvalues for M. For each eigenvalue find the set o… • How does the first equation changed into second equation ? Any resources available online to help me better understand the process ? Thank you. . The weights of A, B, A+B are 2,3,4 (Ibs), resp. when m… • Let C2Ã—2 be the C-vector space of all complex (2 Ã— 2)-matrices. Consider the linear map S: C2Ã—2 â†’ C2Ã—2 defined, for all a,b,c,d âˆˆ C, by S ï£±ï£³ ô° a c ô°‚ ï£¼ï£¾ : = ô° c b + d ô°‚ . b … • 4. Let R t) <2 be the R-vector space of polynomials having degree at most 2. Consider the map P: R<2 – R’ defined, for all p(t) E R[t]<2, by [p"(3) Yp(t)] := p(3) p(3) where p'(t) and… • 3. Let Q<3 be the Q-vector space of polynomials having degree at most 3 and consider the ordered basis B := (t + 2,12, 313 -12 + 2t,13 -t + 1) for Q[t] <3. Find the coordinate vector of the p… • A large truck uses 16.5 litres of diesel per 100 kilometres. Calculate how much diesel the truck will need to travel 1 284 km. • please help prove this. thanks. Remark 9.1.1. The hermition transpose satisfies the following properties which are similar to those satisfied by the transpose. If A, BE Max (C) and a EC then (i) (A + … • help prove this. thanks. 1. Let (V, <., . >) be a complex inner product space. (a) Prove the triangle inequality – i.e. if v, u e V we have \| \|7 + will s lull + 1will. (b) Prove the Pythagoras T… • . 3. Use FindRoot to approximate all real solutions to the equation sin (2a) – sin(x) + 0.5x = -0.25. Explain how you know you have obtained approximations to all the real solutions. (Hint: use the … • Solve the followingÂ . Show that the elementary row operations E; + E; Ei, (Ei) (E;) do not change the solution set of a linear system. • could you please provide me solution in paper. Condition for concurrency of three straight lines. please provide me in detail • . 5. Show that the elementary row operations (Ei t XE;) = Ei, (Ei) > (E;) do not change the solution set of a linear system. • Given the following matrices A and B , find a matrix C in M 3,3 so that { A ,Â B ,Â C } is linearly independent and a non-zero matrix D in M 3,3 so that { A ,Â B ,Â D } is linearly dependent: • Suppose that A E RPXP is a real symmetric and positive semidefinite matrix. Find a matrix Z E RPXP such that Z2 = A. (Hint: Use the eigendecomposition of A.) • 1)Â 2) 3). Given A RST ~A LMN, mZR = 65, and m_M = 70, what is the measure of ZI? Enter your answer in the box. mLT. Which pairs of triangles can be shown to be congruent using rigid motions? B R Sel… • Provide an example of a linear transformation 71 : Pi (R) – Pi (R) such that N(T1) = R(T1). (b) Show that there does not exist a linear transformation T2 : P2(R) – P2(R) such that N(T2) = R(T2). (… • What are the missing parts that correctly complete the proof? Given: Point Q is on the perpendicular isector of MN. Prove: Point Q is equidistant from the endpoints of MN. M X Drag the answers into th… • Let V, W, and X be vector spaces. Let Ti : V – W and T2 : W – X be linear transformations. Consider the composite linear transformation T2 0 1 : V – X, defined by (T2 0 T1) ( v) = T2(Ti (v)), VvEV. Pa… • Let T : V – W be a linear transformation where V and W are two finite-dimensional vector spaces. (a) Show that if dim(V) > dim(W), then T cannot be one-to-one. (b) Show that if dim(V) < dim(W), … • . Find an explicit description of Nul A by listing vectors that span the null space. 130 2 0 X A = 001 -3 0 Try again. 000 0 -1 The null space of an m xn matrix A, written as Nul A, is the set of al… • Write the system first as a vector equation and then as a matrix equation. 6X1 + X2 – 3X3 = 9 2×2 + 4X3 = 0 Write the system as a vector equation where the first equation of the system corresponds to … • For each number 1 _ q < co, the so-called l, norm of a vector u E Rk is defined as lull. 1/q . Prove that lull2 < lull, < VEllull2. • . 1. (2 marks) Find two different 2 X 2 matrices A such that A2 = 0. 2. (3 marks) Find three different 2 x 2 matrices A such that A2 = 1. • . Determine if the 7 is a linear transformation. T(X1, X21 Xg) = (-5X2, 8X3) O The function is a linear transformation. O The function is not a linear transformation. If so, identify the matrix A su… • . Find an explicit description of Nul A by listing vectors that span the null space. 130 2 oâ€” A=001â€”30 0000â€”1 A spanning set for Nul A is D. (Use a comma to separate answers as needed.) • . 8. Obtain a factorization of the form P A = LU with Gaussian Elimina tion with partial pivoting for the following matrix: N -1 = 1 2 3 2 1 4 • . 4. Find all values of a that make the following matrix singular. 1 2 -1 A = 1 a 1 2 -1 • . 3. Given the linear system 2×1 – 60×2 = N/ CO CO 3ax1 – 22 = (a) Find value(s) of a for which the system has no solutions. (b) Find value(s) of a for which the system has an infinite number of sol… • . 2. Find the row interchanges that are required to solve the following linear systems using Gaussian Elimination with partial pivoting: $1â€”$2+$3 = 5 7$1+5$2â€”$3 = 8 2$1+$2+$3 = 7 • . 1. Use the Gaussian Elimination Algorithm to solve the following linear system, if possible, and determine whether row interchanges are neces- sary: 21 – X2 + 3X3 = NO 3×1 – 3×2+23 = -1 x1+ 2 = 3 • . Use the given inverse of the coefficient matrix to solve the following system. 5X1 + 2X2 = 6 1 A = -6X1 – 2X2 = 3 3 Select the correct choice below and, if necessary, fill in the answer boxes to c… • WHAT IS THE OVERALL PURPOSE OF SQUARE ROOT METHOD? WHAT IS THE PURPOSE SQUARE ROOT IN SOLVING A QUADRATIC EQUATION…. • . Find an explicit description of Nul A by listing vectors that span the null space. _137 0 014-8 A: A spanning set for Nul A is D. (Use a comma to separate vectors as needed.) • WHAT IS THE OVERALL PURPOSE OF SQUARE ROOT? WHAT IS THE PURPOSE SQUARE ROOT IN SOLVING A QUADRATIC EQUATION…. • . Problem 3: Match each matrix with its determinant! Hints: Subtract R1 from R2 & R4 for D. Use Cofactor expansion for E. Do not use MATLAB except to check your hand calculations. 4 1 1 A = [3 3… • . Problem 2: Row, Column and Null Space The coefficient matrix A and its reduced form B are shown below. Below, R combines all the row- reducing operations into a single, square invertible matrix. X… • . Problem 1: Row, Column and Null Space The coefficient matrix A and its reduced form B are shown below. X1 X2 X4 X1 X2 X3 0 1 1 1 1 0 01 A = 1 1 0 0 0 1 1 = B LO 7 7 0 0 0 X1 X2 a. The transformati… • WHAT IS THE OVERALL PURPOSE OF THE QUADRATIC EQUATION? WHAT IS THE PURPOSE SQUARE ROOT IN SOLVING A QUADRATIC EQUATION…. • ABC manufacturing company produces two types of products: the super and the egular. Resource requirements for production are given below in the table. There are 1,600 hours of assembly worker hours av… • Linear algebra questions 2 and 3. 2 LET AEll, BE A SQUARE MATRIX IF WE DENOTE A+ 2is 1, 5- 2 THEN DEFINE THE TRANSPOSE OF A JA AS THE nXD MATRIX DEFINED as-dji le. The j COLUMN OF A IS THE JI ROW OF A… • Please help to answer the questions. Thank you! Â Â . 1. [-/1 Points] DETAILS HOLTLINALG2 2.1.001-006B.WA.TUT. Suppose u o and v = are vectors in R3. Determine Zu, -v, and u + 2v. 20 = V = u + 2v = A… • Please help to answer the questions. Thank you! Â Â . 1. [-/1 Points] DETAILS HOLTLINALG2 1.3.001. The volume of traffic for a collection of intersections is shown in the figure below. 40 A X3 X2 20 … • Please help to answer the questions. Thank you! Â Â . 11. [0.5/1 Points] DETAILS PREVIOUS ANSWERS HOLTLINALG2 1.2.023. Convert the given system to an augmented matrix. 2×1 + 2X2 – X3 7 -X1 – X2 -3 3x… • (1 point) Find check digit d such that the parity check code with check vector [1, 1, 1, 1] in ZZ? is correct: [1, 1, 1, d] d: • Picture description. Let f : R2 – R be defined as x3 – 13 f (x, y ) = x2+ 1/2 ‘ if (x, y) # (0, 0) 0, if (x, y) = (0, 0) Prove that that fx exists for all (x, y) E R- but fx is not continuous at (0, 0… • Explain in detail how to solve each question. Â . 1 T 3. Â«our marks) The roots, to the nearest hundredth, of y = E x2 + xâ€” E are 4. (two marks) The table shows the curve of a parabolic arch at the … • . Eve and Nina were given a nonhomogeous linear system to solve, and they both solved it correctly. In parametric vector form, Eve’s solution was 1 4 7 2 + s 5 +13 8 . Nina’s solution was ’00 +t… • . For the following linear system, determine k for which the system has exactly one solution, no solution, and infinitely many solutions? Be sure to show work that justifies each case. x + 2y – 3z =… • Write inequalities that describe the following statements. (But don’t solve them!) Â A farmer grows tomatoes and potatoes. At most,$9,000 can be spent on seeding costs and it costs $100/acre to plan… • . Problem 3 (5 pts, 2 pts) When D is an m x n rectangular diagonal matrix, its pseudo-inverse D+ is an n x m rectangular diagonal matrix whose nonzero entries are the reciprocals 1/d of the diagonal… • My answer is incorrect what is the correct way to solve this?. Find the LU factorization of 10 That is, write A = LU where L is a lower triangular matrix with ones on the diagonal, and U is an upper t… • Problem 2: The Gang Formulates Nonlinearly Paddy’s Pub needs to hire employees to serve drinks over the next seven days. The performance of the employees working each day depends on the total number o… • .. Eve and Nina were given a nonhomogeous linear system to solve, and they both solved it correctly. In parametric vector form, Eve’s solution was 1 4 7 2 + s 5 +t 8 . Nina’s solution was 1:0 +t’v… • Find an equation of the line with slope — and y – intercept -3. 3. Find an equation of the line passing through the point (-2,4) with slope 2-. 4. Find an equation of the line passing through the … • This is the full problem, it isn’t missing any information. Thank you.. E Day19-Activity.pdf 1 1 – 125% + Given V = [2, -1) B2 a. Compute Tv to get coordinates in the standard basis B1 b. Compute R(Tv… • . Express every solution of the following system as a sum of a specific solution plus a solution to the associated homogeneous system. 2×1 + X2 – X3 – X4 = -1 3×1+ X2+ X3 – 2×4 -X1 – X2 + 2×3 + X4 W… • . Let A be an m X n matrix. Show the following: If AX = 0 for every n x 1 matrix X, then A = D. • . Background: In this question, we revisit a theme from Assignment #1: using linear algebra to solve non- linear problems. On the previous Assignment, we found a parabola through three points. In th… • (Total 30 marks estion 2 Copy and complete the table below for g (x) =x +2x-15 (5 marks) x 4 -3 -2 -1 0 2 3 4 g (x) -7 -15 -15 0 Using 2 cm to represent 1 unit on the x-axis and I cm to represent 1 un… • . Consider a system of linear equations with augmented matrix A and coefficient matrix C. In each case, either prove the statement or give an example showing that it is false. 1. If there is more th… • Please include all workings. Thanks Â . Find a 2 x 2 matrix such that – -3 1 = 0 2 • The cost function for a particular game in The Toy Store is C(x) =12.60x+750 Determine the unit cost and fixed cost, the cost of purchasing 35 of these games, (c) the number of games purchased if t… • Find an equation of the line with slope – – and y – intercept -3. 3. Find an equation of the line passing through the point (-2, 4) with slope 2- 2 4. Find an equation of the line passing through t… • kindly solve this correctly. 04 Solution Sets and More Applications: Problem 4 (1 point) Write the solution set of the homogeneous system in parametric vector form. You may use r, s, and/or t as param… • . (c) Give a geometrical specification of T.. (b) What property of [T] implies the transformed square in (a) has the same area as the original square, and also involves a reflection? Clearly state t… • . (b) S : P2 + P3 defined by S(p(x)) = xp(x) + Ap(x).. (c) In the case of S, calculate the standard matrix representation. • . (1)) NOW regard M E M435(]F2), Where F2 indicates the use of binary arithmetic. Calculate a basis for the row space of 1M, clearly stating the theory you are using. • S Calculate matrix T = E R eigenvalues when a + q = 4 = det T] • q6 kindly correctly. 05 Linear Independence: Problem 6 (1 point) Consider the vectors For what values of h is us in Span { v1, (2)? For what values of h is {01, us, us} linearly dependent? • q6) kindly solve this correctly. 04 Solution Sets and More Applications: Problem 6 (1 point) Describe all solutions of A = 0 in parametric vector form, where A is row equivalent to the matrix below. Y… • q5) kindly solve this correctly. 04 Solution Sets and More Applications: Problem 5 (1 point) Describe all solutions of AT = 0 in parametric vector form, where A is row equivalent to the matrix below. … • q4) kindly solve this correctly. 4 Solution Sets and More Applications: Problem 4 point) Write the solution set of the homogeneous system in parametric vector form. You may use r, s, and/or as paramet… • The figure shows vectors u, v, and w, along with the images T(u) and T(v) under the action of a linear transformation T: R2-R2. Copy this figure carefully, and draw the image T(w) as accurately as pos… • Carefully graph the two problems below. Use a different color for each section of the graph. Identify whether or not the graph is a function. Then, evaluate the graph at any specified domain values. G… • 1. Let n be a nonnegative integer. For any nonnegative integer k such that 0 < k < n, the Bernstein polynomial is defined to be bkin (1 ) :- ( * ) 14 ( 1 – 1 ) 1- k. (i) Show that the polyno… • find the rank of the following matrix using echelon form • . You are presented with a following LP: Maximize SCI 3421 + 23 ){2 + 29 K3 Subject to: EX1+5X2+3K3Â£52 4K1+2K2+5H3514 Khilihiigiï¬‚ a} J’llpplrllr the simpler: method to solve the problem and calcu… •$ 2y = 4 * + 210) = 4 (4. 0) and the y intercept 4 27 – 4 1+ 24 – 4 2y = 4 y = 4/2 y = 2 , (0. 2) Boundary line = solid (2) Plot the points on the plane and conneed wit a bonded/ its Use Test Point … • Detailed step on How to solve polynomials using synthetic division, give an example • How to prove this. (b) Prove that a symmetric matrix H is a projection matrix if and only if its eigenvalues are 0 or 1. • Find u âˆ’ v , 2( u + 3 v ), and 2 v âˆ’ u . u = (5, 0, âˆ’6, 8),Â Â Â v = (0, 7, 8, 5) (a)Â Â Â u âˆ’ v = Â (b)Â Â Â 2( u + 3 v ) = Â (c)Â Â Â 2 v âˆ’ u = Â Â Â pls help • detail proof. 7. Let A be a square matrix so that A B = BA for every square matrix B of the same order. (6] (i) Prove that A is a diagonal matrix. [Hint: Let B be the matrix whose (i, i)-entry is I an… • detail proof. 7. Let A be a square matrix so that AB = If A for every square matrix B of the same order. (6] () Prove that A is a diagonal matrix. [ Hint: Let # be the matrix whose (i, i)-entry is I a… • MAE 376 Wk1 HW 1 1. (10%) Linearity: X1] -21 a. Let x = X2 , and y = 2 3 x . Show that y is linear in x. X3. 2 1 0 b. Let g(x) = ax + bx+ ci + dx , show that g(x) is linear. • Consider the following system of charges on the y-axis: A negative charge q, = -6 JC at the origin. . A positive charge q2 = +5 pCaty = 7 m. Step 1: Draw a diagram illustrating this charge configurati… • Question 3 Quiz navigation MATH 232A – 2221 -3 2 Incomplete Let A = and B = – 2 2 3 4 5 6 7 answer 3 0 Participants Marked out of 10 11 10.00 Match the given expressions below with the numbers 1, 2, 3… • kindly solve these correctly. 5. ($1.7, 10pts) Consider the following vectors 2 N V1= -5 V2 = and CO a. For what value(s) of h is v3 in Span{v1, V2}? Justify your answer. b. For what value(s) of h is … • kindly solve these correctly. 3. ($1.5, 10pts) Consider the following vectors V and w = a. Find the parametric equation of the line L which is parallel to v and passes through w. b. Find the parametri… • kindly correctly. 2. ($1.5, 10pts) For each of the following matrices, solve Ax = 0 and write the solutions in parametric vector form. a. A = b. A= 3 2 • kindly correctly. 1. ($1.5, 10pts) For each of the following linear system, determine if it has a nontrivial solution. Justify your answers. Hint: write the augmented matrix for each, get it in echelo… • . In Kowalyshyn O. CC Foundations o x 3 Kowalyshyn O. CC Foundations o X Homework Help – Q&A from Onl x \| Mail – Em-Je Vega – Outlook x What’s the difference between a x \| + X C resources.con… • Parts A-D please.. The matrix Re cos (0) – sin(0) = sin(0) cos(0) is called a rotation matrix because given any vector p = E R2, the product Rep is the rotation of p by 0 radians (counter clockwise). … • x + 3y <= 18 If x < 3 then y = 0 else y =5Â Graph the two functions. The first is a line, the second is an if function. Describe how the ending inventory constraint works (the one that includes… • Question 3 only Â . 3. Let T : R2 – R2 be a linear transformation such that T(x) = Ax where A = – 3 Find a basis for R2, B, such that the matrix for the transformation relative to B is diagonal and th… • . -1.64 -2.4 7. Let A = Find a change of coordinate matrix so that A is similar to a matrix of the 1.92 2.2 form C = a -b b a Describe the rotation and scaling relative to a coordinate system define… • . 5. Let A, B, and C be n x n matrices. Show that if B is similar to A and C is similar to A, then B is similar to C. 6. Find the eigenvalues and eigenvectors of the matrix. (a) A= -2 -5 (b) A = 8 • . 2. Let T : R’ – R’ be a linear transformation such that T() = A where A = Let B = {b1, b2} be a basis for R2 where bi = 3 2 and b2 = 2 Find the matrix for the transformation relative to B. • . Allstar oil produces regular and premium blends of gasoline. To obtain these two end products they blend 3 components. The end products have different octane ratings and are sold at different pric… • . 1. Let E = {ei, e2, e3} be the standard basis for R3 and let B = {b1, b2, b3} be a basis for a vector space V. Let T : R3 – V be a linear transformation such that for a = DC 2 ERS, X3 T(x) = (2×3 … • There are many jobs where concepts about graphs and functions can be of a great use if not a requirement. Describe two occupations where graphing lines and/or the concept of functions would be used. P… • . Exercise 10.6. Consider the region B = {(r,y) E lit2 : $2 + y? 1′: Ilia + y 3 D} (the half of the disk of radius 4 centered at the origin that lies on or above the line I + y = [l through the cent… • Hi could I get some help on how to solve this one.. Consider the matrix 1 0 0 1 1 1 1 1 1 0 2 1 3 2 A â€” 2 1 1 2 1 3 and the vector b â€” 4 1 0 0 1 1 1 1 Let v1, . . .V6 be the vectors given by colum… • . Exercise 11.9. Find all points P on the curve C = {(x, y) ( R2 : 3yx2 + 3xy? + y3 + 2×3 = 27} for which the tangent of C at P is parallel to the y-axis. • Find a basis for sI(n, C). What is the dimension of sl(n, C)? P16.3. Let T : V – W be a linear map. (i) Let v1, V2,…, Un be linearly independent vectors in the K-vector space V. When T is inj… • 2. The set of all traceless (n x n)-matrices, sl(n, C) := {A E Cox \| tr(A) = 0}, is a linear subspace of Coxn. Find a basis for sI(n, C). What is the dimension of sl(n, C)? • Let F(x, y) = (x, -y). (a) Sketch the vector field (draw at least 15 vectors). (b) Suppose that at time t = 0, a particle is placed at the point (1, 2) on some fluid with velocity field F(x, y). Fi… • Is it true that every linear programming problem has an optimal solution? Justify your answer: either prove it or give a counterexample. (b) Is it true that if the feasible region of a linear prog… • 3 marks The following dictionary is obtained while solving a standard form LP problem 5. by using the simplex method. What does this dictionary imply to the LP problem? Does it have an optimal solutio… • 8 marks Solve the following LP problems using the simplex method. At each step please state the entering and leaving variables and the current basic feasible solution. Clearly state the optimal sol… • Could you please help me to solve this problem? Thank you so much!. 1. 7 marks (a) \|4 marks Recall the definition of a convex set: We say a set C of points in R" is convex if for every pair x, y … • 3 marks The following dictionary is obtained from solving a standard form LP problem with the simplex method. Find the next dictionary in the simplex method. Z = 7 -21 +2×4 2 = 12 +x1 3 5 +21 -04 = 4 … • Could you please help me to solve this problem? Thank you so much!. 2. 5 marks (a) 2 marks\| Is it true that every linear programming problem has an opti- mal solution? Justify your answer: either prov… • . In this problem, you are given two matrices A, B E R2X2 and a vector x E R2. 1 NO NO A 1 = B = HN 4. (e) What is the projection of a onto the subspace spanned by the columns of A? (f) Let f : R2 -… • Show that if Î¸ is the angle between vectors u and v, then \|\|u âˆ’ v\|\| 2 = \|\|u\|\| 2 + \|\|v\|\| 2 âˆ’ 2 \|\|u\|\| \|\|v\|\| cos Î¸ • PERIOD NAME DATE X 3. The function fis given by f(x) = 160. and X 160 the function g is given by g(x) = 160- () . 120 A =f The graph of fis labeled A and the graph of g is labeled C. 80 B = H If B is … • Show that if Î¸ is the angle between vectors u and v, then ku âˆ’ vk 2 = kuk 2 + kvk 2 âˆ’ 2kukkvk cos Î¸. • Applied Stats Question based on Linear Algebra: Â . 2. Consider a general form of linear models with a data vector y of n observations as y = XB+E, where X is a model (or design) matrix and & is t… • Linear regression model: P: z = Ax+By +C that passes through the three points P(1, 1, 2), Q(1,â€‚2, 6) and R(2, 2, 1) i) Write Â explicit form with the squared error loss function considering P,Q and … • A consumption matrix for forestry, agriculture, and livestock is given by F A L F 0.1 0.5 0.3 A 0.2 0 0.3 L 0.1 0.1 0.2 If a total output was 15 units of forestry, 15 units of agriculture, and 5 un… • A consumption matrix for coal, steel, and chemicals is given by C S CH C 0.4 0.1 0.1 S 0.2 0.3 0.2 CH 0 0 0.6 If the open sector demands 100 units of coal, 108 units of steel, and 128 units of chem… • A consumption matrix for ï¬shery, electricity, crops, and livestock is given by (row / columns are exactly in the order written) F E C L F 0.15 0.1 0.05 0.05 E 0 0.2 0.2 0.1 C 0.4 0.01 0.02 0.1 L … • Help me solve this question pleaseÂ Â . Instructions: (1) Write up detailed solutions (on separate paper) for all problems below – carefully ex- plain/show all your steps. (2) Upload your solution… • . Let A be a 3×3 matrix of coefficients of a linear system of equations A.x = b where b is some vector in R3. Assume that we know that x0 = (1, 2, 3 ) is a solution of that system, and we also kn… • . Exercise 11.10. Figure 11.5.2 shows a yellow ellipsoid 2×2 + 3y2 + 22 = 9 and blue "2-sheeted" hyper- boloid -3×2 + 6y2 + 22 = 7 meeting along a curve C consisting of 2 parts (one per &q… • . Exercise 10.2. Let f (3:, y) : 3:3 â€” 3:1:2 â€” 62:19r + 93: + 3y2. (a) Show that this function f has exactly 2 critical points: (1, 1) and (3, 3). (b) By examining the behavior of f on the lines… • 3. Let T: V – W be a linear map. (i) Let v1, U2, . .., Un be linearly independent vectors in the K-vector space V. When T is injective prove that the vectors T[vi], T[v2], …, T[vn] are linearly … • The set of all traceless (n x n)-matrices, sI(n, C) := {A E Coxn \| tr(A) = 0}, is a linear subspace of Coxn . Find a basis for sI(n, C). What is the dimension of sl(n, C)? • 1. Let n be a nonnegative integer. For any nonnegative integer k such that 0 < k < n, the Bernstein polynomial is defined to be bun ( ! ) : = (" ) 14 ( 1 – 1 7-4 . (i) Show that the pol… • .. Eve and Nina were given a nonhomogeous linear system to solve, and they both solved it correctly. In parametric vector form, Eve’s solution was 1 4 7 2 + s 5 +t 8 . Nina’s solution was ’00 +t?… • For each of the following lists of vectors, decide whether or not they are linearly independent. If a list of vectors is linearly dependent, then list nontrivial relations among the vectors. (a) (3… • For each of the following matrices, write the kernel of the linear transformation T(x) = Ax as the span of a list of vectors. (Make sure that your list of vectors contains no redundant vectors.) NO… • Let Wi and W2 be subspaces of R". Define W1 + W2 to be the subset of all vectors w1 + W2, where wi is in Wi and w2 is in W2. (a) (3 points) Is W1 + W2 always a subspace of R"? If yes, the… • Consider the plane x – y + z = 0 in R3. (a) (2 points) Write down the matrix of a linear transformation T with ker(T) equal to this plane. (b) (2 points) Write down the matrix of a linear transform… • For each of the following matrices, write the image of the linear transformation T(x) = Ax as the span of a list of vectors. (Make sure that your list of vectors contains no redundant vectors.) NO … • . Microsoft Word – A… ld A ATHEMATICAL ASSOCIATION OF AMERICA webwork / math252aculibrk / hw3 / 1 HW3: Problem 1 Previous Problem Problem List Next Problem (1 point) Write X as a product X – E1 E2… • determine a cubic spline approximation that fits through data points:Â Â Â . P1(1; 1); P2(2; 2); P3(3; 1:5); P4(4; 3); P5(5; 2) Spline built with two functions: 3 yl 0,1 ‘I’ (12$ ‘I’ 0,3132 + … • Write DNE if doesn’t exist.. Week 03: Problem 6 Previous Problem Problem List Next Problem (1 point) If possible, write -12 – 11x – 8x as a linear combination of -2 – x – x2, -2 -2x – x2 and 3 + 3x + … • Thank you!. Week 03: Problem 4 Previous Problem Problem List Next Problem (1 point) Let u [ 2 1. and v =[:3] Select all of the vectors that are in the linear combinations of {u, v} . (Check every stat… • Solve properly and originally thanks. 5. The setting for this problem is R3. Suppose you have a plane P and two vectors a and b in P. The task is about the general question, "If you add two vecto… • Solve clearly and originally thanks. 5. The setting for this problem is R3. Suppose you have a plane P and two vectors a and b in P. The task is about the general question, "If you add two vector… • Let U be a subset in R3 with 1, 2 or 3 vectors. For each of the following, explain your reasoning. You may pick a different U for each part.. (c) Find a specific example of U, such that span(U) is a p… • Solve both parts clearly thanks. 4. Let U be a subset in RS with 1, 2 or 3 vectors. For each of the following, explain your reasoning. You may pick a different U for each part. (a) Find a specific exa… • Solve all parts originally and clearly. 4. Let U be a subset in RS with 1, 2 or 3 vectors. For each of the following, explain your reasoning. You may pick a different U for each part. (a) Find a speci… • Solve originally with work please. 3. In this task, you are asked to work with the formal definition of linear independence (algebraic). Let U = {vi, v2, v3, v4} in R", for n 2 4. (a) Write a for… • Solve both parts clearly and well. 3. In this task, you are asked to work with the formal definition of linear independence (algebraic). Let U = {vi, v2, v3, v4 } in R", for n 2 4. (a) Write a fo… • you are asked to create your own examples. assume you are working in the situation with three variables and three equations in R3. Find your own examples of augmented matrices in reduced row echelon f… • create your own examples. assume you are working in the situation with three variables and three equations in R3. Find your own examples of augmented matrices in reduced row echelon form that meet the… • Solve both parts clearly with unique answers. 1. This task is related to Appendix 2, and you are asked to create your own examples. First, assume you are working in the situation with three variables … • Solve both parts clearly thanks. Give an original/unique answer. 1. This task is related to Appendix 2, and you are asked to create your own examples. First, assume you are working in the situation wi… • 3 Linear Algebra Questions Â . 4. Suppose we have 3 X 3 matrices A and B with det(A) = 5 and det(B) = -3. Determine the determinant of the following matrices. a. ABA-1 b. AB-2 A c. 3B 5. Suppose we ha… • Water is flowing through a network of pipes (in thousands of cubic meters per hour), as shown in the figure. (Assume F1 = 600 and F2 = 500. If the system has an infinite number of solutions, express X… • The figure shows the flow of traffic (in vehicles per hour) through a network of streets. Rectangular Sr X1 400 600 X2 X4 x3 300 100 X’s (a) Solve this system for X; / = 1, 2, …, 5. (If the system h… • The graph of a cubic polynomial function has horizontal tangent lines at (1, -2) and (-1, 2). Find an equation for the function. P(X) = Sketch its graph. 10 10- X T X -3 -2 1 2 3 -3 -2 3 -5/ -5 O – 10… Price (USD) \$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8638255596160889, "perplexity": 1145.9882527296181}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500628.77/warc/CC-MAIN-20230207170138-20230207200138-00675.warc.gz"}
https://pdglive.lbl.gov/DataBlock.action?node=S023A&home=sumtabB	#### ${{\mathit \Xi}^{0}}$ DECAY PARAMETERS See the Note on Baryon Decay Parameters'' in the neutron Listings. #### $\alpha$ FOR ${{\mathit \Xi}^{0}}$ $\rightarrow$ ${{\mathit \Lambda}}{{\mathit \pi}^{0}}$ The above average, $\alpha\mathrm {({{\mathit \Xi}^{0}} )}\alpha _{−}({{\mathit \Lambda}}$ ) = $-0.261$ $\pm0.006$ , divided by our current average $\alpha _{−}({{\mathit \Lambda}}$ ) = $0.732$ $\pm0.014$ , gives the following value for $\alpha\mathrm {({{\mathit \Xi}^{0}} )}$: VALUE $\bf{ -0.356 \pm0.011}$ OUR EVALUATION	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9979077577590942, "perplexity": 1050.0760543764995}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943471.24/warc/CC-MAIN-20230320083513-20230320113513-00306.warc.gz"}
https://www.maplesoft.com/support/help/maple/view.aspx?path=RegularChains/ChainTools&L=E	Overview - Maple Help Overview of the RegularChains[ChainTools] Subpackage Calling Sequence RegularChains[ChainTools][command](arguments) command(arguments) Description • The RegularChains[ChainTools] subpackage is a collection of commands for manipulating regular chains. Some commands create regular chains from polynomials and possibly other regular chains. Other commands are used to study the properties of regular chains. • The commands Empty, ListConstruct, Construct, and Chain create regular chains from a list of polynomials representing a regular chain. The commands ListConstruct and Construct try to simplify this regular chain, which may result in splitting it into several ones. On the contrary, Chain returns only one regular chain. • The commands Cut, Polynomial, Under, and Upper, allow the user to extract part of the polynomials defining a regular chain. • The commands IsEmptyChain, IsZeroDimensional, IsAlgebraic, IsStronglyNormalized, Dimension, NumberOfSolutions, and IsPrimitive allow the user to inspect properties of a regular chain. This is often necessary since certain commands apply only to regular chains satisfying a particular property. For instance, the command NormalForm applies only to strongly normalized regular chains whereas SparsePseudoRemainder can be used with any regular chain. See the mathematical definitions below, for a description of some of those properties. • Like the commands RegularGcd and ExtendedRegularGcd, the command ExtendedNormalizedGcd computes a regular GCD of two input polynomials modulo the saturated ideal of a regular chain; in addition, ExtendedNormalizedGcd ensures that the output GCDs and regular chains are normalized. • The command IteratedResultant computes the iterated resultant of a polynomial w.r.t. a regular chain. This type of calculation is motivated by the following fundamental result: a polynomial p is regular w.r.t. the saturated ideal of a regular chain rc if and only if the iterated resultant of p w.r.t. rc is not zero. • The command Regularize provides a regularity test for a polynomial modulo the saturated ideal. In fact, it decomposes the input regular chain such that the input polynomial is either zero or regular modulo the saturated ideal of each output regular chains. • The command NormalizeRegularChain computes a triangular decomposition of a regular chain into other regular chains such that the intersection of the saturated ideals of the output regular chains has the same radical as the saturated ideal of input regular chain. Moreover, each output regular chain is normalized, and optionally, strongly normalized. • The command Squarefree computes a triangular decomposition of a regular chain into other regular chains such that the intersection of the saturated ideals of the output regular chains has the same radical as the saturated ideal of input regular chain. Moreover, each output regular chain is square free. • The command SquareFreeFactorization computes square-free decompositions of polynomials modulo the saturated ideal of a regular chain. • The command Extend computes a triangular decomposition into regular chains of the quasi-component of a triangular set. • The commands ChangeOfOrder, ChangeOfCoordinates, DahanSchostTransform, and Lift perform transformations on a regular chain. Refer to the help pages of those commands for more details. • The commands EquiprojectableDecomposition, SeparateSolutions, and RemoveRedundantComponents perform transformation on a list of regular chains. The first one takes as input a zero-dimensional variety V given as a list of strongly normalized regular chains; it then returns the equiprojectable decomposition of V in the form of another list of regular chains. The second one takes as input two zero-dimensional varieties V1 and V2 given by lists of regular chains; it then returns a list of regular chains pairwise disjoint and encoding the reunion of V1 and V2. RemoveRedundantComponents takes as input a list lrc1 of regular chains and returns another list lrc2 of regular chains such that they both form a triangular decomposition of the same variety (in the sense of Lazard) and the quasi-components of the regular chains in lrc2 are pairwise noninclusive. • The commands IsInSaturate and IsInRadical compare one polynomial p and one regular chain T. The first one decides whether p belongs to the saturated ideal I of T. This is done without computing a system of generators of I but just by pseudo-division. In fact, the RegularChains package never computes explicitly a system of generators for a saturated ideal. The second command decides whether p belongs to the radical of I; this is achieved simply by GCD computations. • The commands EqualSaturatedIdeals and IsIncluded compare two regular chains T1 and T2. More precisely, the first one decides whether the saturated ideals of T1 and T2 are equal or not. If the second one returns true, then the saturated ideal of T1 is contained in that of T2. However, if it returns false, nothing can be said. • The command SubresultantChain returns a data structure which encodes the subresultant chain of two univariate polynomials. The commands LastSubresultant and SubresultantOfIndex inspect such a data structure. Mathematical Definitions • A polynomial p is said to be normalized with respect to a regular chain T if p is constant or if the main variable of p is not algebraic with respect to T and the initial of p is normalized with respect to T. The regular chain T is said to be normalized if every polynomial p of T is normalized with respect to the polynomials of T with main variable less than v, where v is the main variable of p. A regular chain T is said to be strongly normalized if for every polynomial p in T, each variable of in the initial of p is free (that is, not algebraic) with respect to T. Strongly normalized regular chains have the following fundamental property: Let U and X be the set of the free variables and algebraic variables of T. Let K be the field of rational functions with variables in U. and let T be a strongly normalized regular chain. Then T is a lexicographical Groebner basis of the ideal it generates in K[X]. List of RegularChains[ChainTools] Subpackage Commands The following is a list of available commands. References Aubry, P.; Lazard, D. and Moreno Maza, M. "On the theories of triangular sets" J. Symb. Comp., Vol. 28 (1999): 105-124. Boulier, F. and Lemaire, F. "Computing canonical representatives of regular differential ideals" In Proc. ISSAC 2000, St Andrews, Scotland, 2000. Chen, C.; Lemaire, F.; Moreno Maza, M.; Pan, W. and Xie, Y. "Efficient Computations of Irredundant Triangular Decompositions with the RegularChains Library" Proceedings of CASA 2007, Lecture Notes in Computer Science, Vol. 4488, Springer, 2007. Dahan, X. and Schost, E. "Sharp Estimates for Triangular Sets" In Proc. ISSAC 2004, Santander, Spain, ACM Press, 2004. Dahan, X.; Moreno Maza, M.; Schost, E.; Wu, W. and Xie, Y. "Lifting techniques for triangular decompositions" In Proc. ISSAC 2005, Beijing, China, ACM Press, 2005. Dahan, X.; Jin, X.; Moreno Maza, M.; and Schost, E. "Change of ordering for regular chains in positive dimension" J. of Theoretical Computer Science, Vol. 392 (2008): 3765. Della Dora, J.; Discrescenzo, C.; and Duval, D. "About a new method for computing in algebraic number fields" In Proc. EUROCAL 85 Vol. 2, 1985. Kalkbrener, M. "Three contributions to elimination theory" PhD Thesis, University of Linz, Austria, 1991. Kalkbrener, M. "A generalized Euclidean algorithm for computing triangular representations of algebraic varieties" J. Symb. Comp., Vol. 15 (1993): 143-167. Lazard, D. "Solving zero-dimensional algebraic systems" J. Symb. Comp., Vol. 13 (1992) : 117-133. Lemaire, F.; Moreno Maza, M.; Pan, W.; and Xie, Y. "When does T equal Sat(T)?" In Proc. ISSAC 2008, Linz, Austria, ACM Press, 2008. Li, X.; Moreno Maza, M.; and Pan, W. "Computations modulo regular chains." In Proc. ISSAC 2009, Linz, Austria, ACM Press, 2009. Moreno Maza, M. "On Triangular Decompositions of Algebraic Varieties" MEGA-2000 conference Bath, UK, England. Moreno Maza, M. and Rioboo, R. "Polynomial GCD computations over towers of algebraic extensions" In proc. of AAECC-11, Lecture Notes in Comput. Sci, Vol. 948, Springer, 1995. Schost, E. "Complexity results for triangular sets" J. Symb. Comp., Vol. 36 No. 3-4 (2003): 555-594. Wang, D. M. "Decomposing Polynomial Systems into Simple Systems" J. Symb. Comp., Vol. 25 No. 3 (1998): 295-314.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9416409730911255, "perplexity": 2732.282897539496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030334855.91/warc/CC-MAIN-20220926082131-20220926112131-00609.warc.gz"}
http://math.stackexchange.com/questions/266580/show-this-summation-hold-in-term-of-integral	# show this summation hold in term of integral i can show $\sum \|x\|^2=\int_a^b\|f(x)\|^2dx$ in term of integral, or this one $\|\sum x\overline y\|^2=\|\int_a^bf(x)\overline {g(x)}dx\|^2$ but i don't know how to show this one $\sum_{i}^{n-1}\sum_{j=i+1}^{n}\|(x_i\overline y_j-x_j\overline y_i)\|^2$ in term of integral $\sum_{i}^{n-1}\sum_{j=i+1}^{n}\|(x_i\overline y_j-x_j\overline y_i)\|^2=?$ - What are the functions $f,g$, and the numbers $a,b$? What are $x,y$ in your sums? What are you summing over in $\sum \|x\|^2$? – Eckhard Dec 28 '12 at 17:38 How was this question upvoted? As it's written, it makes no sense to me. I downvoted. – mixedmath Dec 29 '12 at 8:36	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.900797426700592, "perplexity": 511.02927039339926}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246656168.61/warc/CC-MAIN-20150417045736-00186-ip-10-235-10-82.ec2.internal.warc.gz"}
https://www.hmc.edu/mathematics/student-mathematics-resources/tex-and-latex-resources/	# TeX and LaTeX Resources The mathematics department has a special interest in promoting the use of TeX and LaTeX for writing pretty much anything —papers, theses, homework, even posters! We have a Tips & Tricks document that gives you some hints about the trickier aspects of using TeX and LaTeX. ## HMC Classes The department has a number of locally developed LaTeX classes for use in formatting documents. These classes include: ## LaTeX Packages The department has developed the following package for internal use. • cmtty.sty • The cmtty package specifies Computer Modern Typewriter as the “typewriter style” font. It is useful when using font packages that set the default monospace font to use Courier. ### Using the Package Simply add the line `\usepackage{cmtty}` to the preamble of your document. ## Online Resources • TUG‘s Getting Started with TeX, LaTeX, and Friends. • Links to lots of resources for getting started learning and using Tex, LaTeX, and, well, Friends! • TUG’s TeX and LaTeX Documentation. • Formatting Information: A Beginner’s Guide to LaTeX, by Peter Flynn, Silmaril Associates • A good, detailed, introductory guide covering the use of TeX and LaTeX on Macs, Windows, and Unix/Linux systems. Available in HTML and PDF. • CTAN—the Comprehensive TeX Archive Network • CTAN maintains a comprehensive archive of TeX and LaTeX packages, formats, fonts, document classes, documentation, and other materials, including complete TeX systems. • The TEX FAQ • Answers many frequently asked questions about TeX and LaTeX. A good first place to check if you’re having problems. • Detexify LaTeX Handwritten Symbol Recognition • Cool site that lets you sketch a symbol using your mouse, then suggests LaTeX symbols that are close to what you’ve drawn. Often a good way to figure out what command to use for a particular symbol. • Comprehensive LaTeX Symbol List • A document showing a huge array of symbols, the commands that produce them, and information about the packages that provide those commands. (Available on many TeX systems as `symbols-a4.pdf` or `symbols-letter.pdf`. Try running `texdoc symbols` for a viewable version or `texdoc symbols-letter` for a printable version.) • LaTeX Search • Allows you to search for equations in Springer journals and gives you the LaTeX code to create those equations. Should be useful both for referencing work in Springer journals and for getting sample code for equations that are similar to what you’re trying to write. • `comp.text.tex` • `comp.text.tex` is the official Usenet newsgroup for discussing TeX and LaTeX questions. Chances are that any problems you’re having, someone else has already had, asked about on `ctt`, and gotten an answer. You can search the newsgroup with Google. • Google is probably your best bet for finding information outside`comp.text.tex`, as well.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9994862675666809, "perplexity": 3641.230200000317}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103642979.38/warc/CC-MAIN-20220629180939-20220629210939-00477.warc.gz"}
http://mathhelpforum.com/math-topics/52234-multistep-equation-problem.html	1. ## Multistep Equation Problem s+300=3s how to solve this equation? 2. Originally Posted by gedprep s+300=3s how to solve this equation? $s+300=3s$ First, subtract s from both sides of the equation. $s+300{\color{red}-s}=3s{\color{red}-s}$ $300=2s$ Finally, divide both sides of the equation by 2. $\frac{300}{{\color{red}2}}=\frac{2s}{{\color{red}2 }}$ $150=s$ $\boxed{s=150}$ 3. ## Thank you very much. Thank You so much.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9557519555091858, "perplexity": 2011.9653695461054}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-51/segments/1512948592972.60/warc/CC-MAIN-20171217035328-20171217061328-00353.warc.gz"}
https://www.physicsforums.com/threads/can-someone-help-me-out.130386/	# Can someone help me out? 1. Aug 31, 2006 ### almost__overnow Mary runs 10 kilometers in 40 minutes. What is her average speed in meters per second? i dont understand and its only like the 4th day of school. 2. Aug 31, 2006 ### lightgrav do you know what speed is? do you know how a kilometer is related to a meter? do you know how a second is related to a minute? Similar Discussions: Can someone help me out?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8985188603401184, "perplexity": 1947.705418427173}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084890771.63/warc/CC-MAIN-20180121135825-20180121155825-00635.warc.gz"}
https://en.wikipedia.org/wiki/Binary_star	# Binary star Not to be confused with Double star. For the hip hop group, see Binary Star (band). Artist's impression of the evolution of a hot high-mass binary star. Hubble image of the Sirius binary system, in which Sirius B can be clearly distinguished (lower left) A binary star is a star system consisting of two stars orbiting around their common barycenter. Systems of two or more stars are called multiple star systems. These systems, especially when more distant, often appear to the unaided eye as a single point of light, and are then revealed as multiple by other means. Research over the last two centuries suggests that half or more of visible stars are part of[1] multiple star systems.[2] The term double star is often used synonymously with binary star; however, double star can also mean optical double star. Optical doubles are so called because the two stars appear close together in the sky as seen from the Earth; they are almost on the same line of sight. Nevertheless, their "doubleness" depends only on this optical effect; the stars themselves are distant from one another and share no physical connection. A double star can be revealed as optical by means of differences in their parallax measurements, proper motions, or radial velocities. Most known double stars have not been studied sufficiently closely to determine whether they are optical doubles or they are doubles physically bound through gravitation into a multiple star system. Binary star systems are very important in astrophysics because calculations of their orbits allow the masses of their component stars to be directly determined, which in turn allows other stellar parameters, such as radius and density, to be indirectly estimated. This also determines an empirical mass-luminosity relationship (MLR) from which the masses of single stars can be estimated. Binary stars are often detected optically, in which case they are called visual binaries. Many visual binaries have long orbital periods of several centuries or millennia and therefore have orbits which are uncertain or poorly known. They may also be detected by indirect techniques, such as spectroscopy (spectroscopic binaries) or astrometry (astrometric binaries). If a binary star happens to orbit in a plane along our line of sight, its components will eclipse and transit each other; these pairs are called eclipsing binaries, or, as they are detected by their changes in brightness during eclipses and transits, photometric binaries. If components in binary star systems are close enough they can gravitationally distort their mutual outer stellar atmospheres. In some cases, these close binary systems can exchange mass, which may bring their evolution to stages that single stars cannot attain. Examples of binaries are Sirius, and Cygnus X-1 (Cygnus X-1 being a well-known black hole). Binary stars are also common as the nuclei of many planetary nebulae, and are the progenitors of both novae and type Ia supernovae. ## Discovery The term binary was first used in this context by Sir William Herschel in 1802,[3] when he wrote:[4] "If, on the contrary, two stars should really be situated very near each other, and at the same time so far insulated as not to be materially affected by the attractions of neighbouring stars, they will then compose a separate system, and remain united by the bond of their own mutual gravitation towards each other. This should be called a real double star; and any two stars that are thus mutually connected, form the binary sidereal system which we are now to consider." By the modern definition, the term binary star is generally restricted to pairs of stars which revolve around a common center of mass. Binary stars which can be resolved with a telescope or interferometric methods are known as visual binaries.[5][6] For most of the known visual binary stars one whole revolution has not been observed yet, they are observed to have travelled along a curved path or a partial arc.[7] This figure shows a system with two stars The more general term double star is used for pairs of stars which are seen to be close together in the sky.[3] This distinction is rarely made in languages other than English.[5] Double stars may be binary systems or may be merely two stars that appear to be close together in the sky but have vastly different true distances from the Sun. The latter are termed optical doubles or optical pairs.[8] Since the invention of the telescope, many pairs of double stars have been found. Early examples include Mizar and Acrux. Mizar, in the Big Dipper (Ursa Major), was observed to be double by Giovanni Battista Riccioli in 1650[9][10] (and probably earlier by Benedetto Castelli and Galileo).[11] The bright southern star Acrux, in the Southern Cross, was discovered to be double by Father Fontenay in 1685.[9] John Tobin and his team found that there are two types of binary star system categories: there either being separated between a distance between less than 300 astronomical units or greater than 1000 astronomical units(Hall, Shannon).[12] John Michell was the first to suggest that double stars might be physically attached to each other when he argued in 1767 that the probability that a double star was due to a chance alignment was small.[13][14] William Herschel began observing double stars in 1779 and soon thereafter published catalogs of about 700 double stars.[15] By 1803, he had observed changes in the relative positions in a number of double stars over the course of 25 years, and concluded that they must be binary systems;[16] the first orbit of a binary star, however, was not computed until 1827, when Félix Savary computed the orbit of Xi Ursae Majoris.[17] Since this time, many more double stars have been catalogued and measured. The Washington Double Star Catalog, a database of visual double stars compiled by the United States Naval Observatory, contains over 100,000 pairs of double stars,[18] including optical doubles as well as binary stars. Orbits are known for only a few thousand of these double stars,[19] and most have not been ascertained to be either true binaries or optical double stars.[20] This can be determined by observing the relative motion of the pairs. If the motion is part of an orbit, or if the stars have similar radial velocities and the difference in their proper motions is small compared to their common proper motion, the pair is probably physical.[21] One of the tasks that remains for visual observers of double stars is to obtain sufficient observations to prove or disprove gravitational connection. ## Classifications Edge-on disc of gas and dust present around the binary star system HD 106906.[22] ### Methods of observation Binary stars are classified into four types according to the way in which they are observed: visually, by observation; spectroscopically, by periodic changes in spectral lines; photometrically, by changes in brightness caused by an eclipse; or astrometrically, by measuring a deviation in a star's position caused by an unseen companion.[5][23] Any binary star can belong to several of these classes; for example, several spectroscopic binaries are also eclipsing binaries. #### Visual binaries Main article: Visual binary A visual binary star is a binary star for which the angular separation between the two components is great enough to permit them to be observed as a double star in a telescope, or even high-powered binoculars. The angular resolution of the telescope is an important factor in the detection of visual binaries, and as better angular resolutions are applied to binary star observations increasing number of visual binaries will be detected. The relative brightness of the two stars is also an important factor, as glare from a bright star may make it difficult to detect the presence of a fainter component. The brighter star of a visual binary is the primary star, and the dimmer is considered the secondary. In some publications (especially older ones), a faint secondary is called the comes (plural comites; companion). If the stars are the same brightness, the discoverer designation for the primary is customarily accepted.[24] The position angle of the secondary with respect to the primary is measured, together with the angular distance between the two stars. The time of observation is also recorded. After a sufficient number of observations are recorded over a period of time, they are plotted in polar coordinates with the primary star at the origin, and the most probable ellipse is drawn through these points such that the Keplerian law of areas is satisfied. This ellipse is known as the apparent ellipse, and is the projection of the actual elliptical orbit of the secondary with respect to the primary on the plane of the sky. From this projected ellipse the complete elements of the orbit may be computed, where the semi-major axis can only be expressed in angular units unless the stellar parallax, and hence the distance, of the system is known.[6] #### Spectroscopic binaries Sometimes, the only evidence of a binary star comes from the Doppler effect on its emitted light. In these cases, the binary consists of a pair of stars where the spectral lines in the light emitted from each star shifts first towards the blue, then towards the red, as each moves first towards us, and then away from us, during its motion about their common center of mass, with the period of their common orbit. In these systems, the separation between the stars is usually very small, and the orbital velocity very high. Unless the plane of the orbit happens to be perpendicular to the line of sight, the orbital velocities will have components in the line of sight and the observed radial velocity of the system will vary periodically. Since radial velocity can be measured with a spectrometer by observing the Doppler shift of the stars' spectral lines, the binaries detected in this manner are known as spectroscopic binaries. Most of these cannot be resolved as a visual binary, even with telescopes of the highest existing resolving power. In some spectroscopic binaries, spectral lines from both stars are visible and the lines are alternately double and single. Such a system is known as a double-lined spectroscopic binary (often denoted "SB2"). In other systems, the spectrum of only one of the stars is seen and the lines in the spectrum shift periodically towards the blue, then towards red and back again. Such stars are known as single-lined spectroscopic binaries ("SB1"). The orbit of a spectroscopic binary is determined by making a long series of observations of the radial velocity of one or both components of the system. The observations are plotted against time, and from the resulting curve a period is determined. If the orbit is circular then the curve will be a sine curve. If the orbit is elliptical, the shape of the curve will depend on the eccentricity of the ellipse and the orientation of the major axis with reference to the line of sight. It is impossible to determine individually the semi-major axis a and the inclination of the orbit plane i. However, the product of the semi-major axis and the sine of the inclination (i.e. a sin i) may be determined directly in linear units (e.g. kilometres). If either a or i can be determined by other means, as in the case of eclipsing binaries, a complete solution for the orbit can be found.[25] Binary stars that are both visual and spectroscopic binaries are rare, and are a precious source of valuable information when found. Visual binary stars often have large true separations, with periods measured in decades to centuries; consequently, they usually have orbital speeds too small to be measured spectroscopically. Conversely, spectroscopic binary stars move fast in their orbits because they are close together, usually too close to be detected as visual binaries. Binaries that are both visual and spectroscopic thus must be relatively close to Earth. #### Eclipsing binaries Algol B orbits Algol A. This animation was assembled from 55 images of the CHARA interferometer in the near-infrared H-band, sorted according to orbital phase. An eclipsing binary star is a binary star in which the orbit plane of the two stars lies so nearly in the line of sight of the observer that the components undergo mutual eclipses. In the case where the binary is also a spectroscopic binary and the parallax of the system is known, the binary is quite valuable for stellar analysis.[26] Algol is the best-known example of an eclipsing binary.[27] In the last decade, measurement of extragalactic eclipsing binaries' fundamental parameters has become possible with 8 meter class telescopes. This makes it feasible to use them to directly measure the distances to external galaxies, a process that is more accurate than using standard candles.[28] Recently, they have been used to give direct distance estimates to the LMC, SMC, Andromeda Galaxy and Triangulum Galaxy. Eclipsing binaries offer a direct method to gauge the distance to galaxies to a new improved 5% level of accuracy.[29] This video shows an artist's impression of an eclipsing binary star system. As the two stars orbit each other they pass in front of one another and their combined brightness, seen from a distance, decreases. Eclipsing binaries are variable stars, not because the light of the individual components vary but because of the eclipses. The light curve of an eclipsing binary is characterized by periods of practically constant light, with periodic drops in intensity. If one of the stars is larger than the other, one will be obscured by a total eclipse while the other will be obscured by an annular eclipse. The period of the orbit of an eclipsing binary may be determined from a study of the light curve, and the relative sizes of the individual stars can be determined in terms of the radius of the orbit by observing how quickly the brightness changes as the disc of the near star slides over the disc of the distant star. If it is also a spectroscopic binary the orbital elements can also be determined, and the mass of the stars can be determined relatively easily, which means that the relative densities of the stars can be determined in this case.[30] #### Non-eclipsing binaries that can be detected through photometry Close non-eclipsing binaries can also be photometrically detected by observing how the stars affect each other in three ways. First is by observing extra light which the stars reflect from their companion. Second is by observing ellipsoidal light variations which are caused by deformation of the star's shape by their companions. The third method is by looking at how relativistic beaming affects the apparent magnitude of the stars. Detecting binaries with these methods requires accurate photometry.[31] #### Astrometric binaries Astronomers have discovered some stars that seemingly orbit around an empty space. Astrometric binaries are relatively nearby stars which can be seen to wobble around a point in space, with no visible companion. The same mathematics used for ordinary binaries can be applied to infer the mass of the missing companion. The companion could be very dim, so that it is currently undetectable or masked by the glare of its primary, or it could be an object that emits little or no electromagnetic radiation, for example a neutron star.[32] The visible star's position is carefully measured and detected to vary, due to the gravitational influence from its counterpart. The position of the star is repeatedly measured relative to more distant stars, and then checked for periodic shifts in position. Typically this type of measurement can only be performed on nearby stars, such as those within 10 parsecs. Nearby stars often have a relatively high proper motion, so astrometric binaries will appear to follow a wobbly path across the sky. If the companion is sufficiently massive to cause an observable shift in position of the star, then its presence can be deduced. From precise astrometric measurements of the movement of the visible star over a sufficiently long period of time, information about the mass of the companion and its orbital period can be determined.[33] Even though the companion is not visible, the characteristics of the system can be determined from the observations using Kepler's laws.[34] This method of detecting binaries is also used to locate extrasolar planets orbiting a star. However, the requirements to perform this measurement are very exacting, due to the great difference in the mass ratio, and the typically long period of the planet's orbit. Detection of position shifts of a star is a very exacting science, and it is difficult to achieve the necessary precision. Space telescopes can avoid the blurring effect of Earth's atmosphere, resulting in more precise resolution. ### Configuration of the system Detached Semidetached Contact Configurations of a binary star system with a mass ratio of 3. The black lines represent the inner critical Roche equipotentials, the Roche lobes. Another classification is based on the distance of the stars, relative to their sizes:[35] Detached binaries are binary stars where each component is within its Roche lobe, i.e. the area where the gravitational pull of the star itself is larger than that of the other component. The stars have no major effect on each other, and essentially evolve separately. Most binaries belong to this class. Semidetached binary stars are binary stars where one of the components fills the binary star's Roche lobe and the other does not. Gas from the surface of the Roche-lobe-filling component (donor) is transferred to the other, accreting star. The mass transfer dominates the evolution of the system. In many cases, the inflowing gas forms an accretion disc around the accretor. A contact binary is a type of binary star in which both components of the binary fill their Roche lobes. The uppermost part of the stellar atmospheres forms a common envelope that surrounds both stars. As the friction of the envelope brakes the orbital motion, the stars may eventually merge.[36] Discovered in 1903 by Gustav Muller and Paul Kempf that W Ursae Majoris contained an intensity that varied by about 0.7 magnitiude.[37] ### Cataclysmic variables and X-ray binaries Artist's conception of a cataclysmic variable system When a binary system contains a compact object such as a white dwarf, neutron star or black hole, gas from the other (donor) star can accrete onto the compact object. This releases gravitational potential energy, causing the gas to become hotter and emit radiation. Cataclysmic variable stars, where the compact object is a white dwarf, are examples of such systems.[38] In X-ray binaries, the compact object can be either a neutron star or a black hole. These binaries are classified as low-mass or high-mass according to the mass of the donor star. High-mass X-ray binaries contain a young, early-type, high-mass donor star which transfers mass by its stellar wind, while low-mass X-ray binaries are semidetached binaries in which gas from a late-type donor star or a white dwarf overflows the Roche lobe and falls towards the neutron star or black hole.[39] Probably the best known example of an X-ray binary is the high-mass X-ray binary Cygnus X-1. In Cygnus X-1, the mass of the unseen companion is estimated to be about nine times that of the Sun,[40] far exceeding the Tolman–Oppenheimer–Volkoff limit for the maximum theoretical mass of a neutron star. It is therefore believed to be a black hole; it was the first object for which this was widely believed.[41] ## Orbital period Orbital periods can be less than an hour (for AM CVn stars), or a few days (components of Beta Lyrae), but also hundreds of thousands of years (Proxima Centauri around Alpha Centauri AB). ### Variations in period Main article: Applegate mechanism The Applegate mechanism explains long term orbital period variations seen in certain eclipsing binaries. As a main-sequence star goes through an activity cycle, the outer layers of the star are subject to a magnetic torque changing the distribution of angular momentum, resulting in a change in the star's oblateness. The orbit of the stars in the binary pair is gravitationally coupled to their shape changes, so that the period shows modulations (typically on the order of ∆P/P ∼ 10−5) on the same time scale as the activity cycles (typically on the order of decades).[42] Another phenomenon observed in some Algol binaries has been monotonic period increases. This is quite distinct from the far more common observations of alternating period increases and decreases explained by the Applegate mechanism. Monotonic period increases have been attributed to mass transfer, usually (but not always) from the less massive to the more massive star[43] ## Designations ### A and B Artist’s impression of the binary star system AR Scorpii.[44] The components of binary stars are denoted by the suffixes A and B appended to the system's designation, A denoting the primary and B the secondary. The suffix AB may be used to denote the pair (for example, the binary star α Centauri AB consists of the stars α Centauri A and α Centauri B.) Additional letters, such as C, D, etc., may be used for systems with more than two stars.[45] In cases where the binary star has a Bayer designation and is widely separated, it is possible that the members of the pair will be designated with superscripts; an example is Zeta Reticuli, whose components are ζ1 Reticuli and ζ2 Reticuli.[46] ### Discoverer designations Double stars are also designated by an abbreviation giving the discoverer together with an index number.[47] α Centauri, for example, was found to be double by Father Richaud in 1689, and so is designated RHD 1.[9][48] These discoverer codes can be found in the Washington Double Star Catalog.[49] ### Hot and cold The components of a binary star system may be designated by their relative temperatures as the hot companion and cool companion. Examples: • Antares (Alpha Scorpii) is a red supergiant star in a binary system with a hotter blue main-sequence star Antares B. Antares B can therefore be termed a hot companion of the cool supergiant.[50] • Symbiotic stars are binary star systems composed of a late-type giant star and a hotter companion object. Since the nature of the companion is not well-established in all cases, it may be termed a "hot companion".[51] • The luminous blue variable Eta Carinae has recently been determined to be a binary star system. The secondary appears to have a higher temperature than the primary and has therefore been described as being the "hot companion" star. It may be a Wolf–Rayet star.[52] • R Aquarii shows a spectrum which simultaneously displays both a cool and hot signature. This combination is the result of a cool red supergiant accompanied by a smaller, hotter companion. Matter flows from the supergiant to the smaller, denser companion.[53] • NASA's Kepler mission has discovered examples of eclipsing binary stars where the secondary is the hotter component. KOI-74b is a 12,000 K white dwarf companion of KOI-74 (KIC 6889235), a 9,400 K early A-type main-sequence star.[54][55][56] KOI-81b is a 13,000 K white dwarf companion of KOI-81 (KIC 8823868), a 10,000 K late B-type main-sequence star.[54][55][56] ## Evolution ### Formation While it is not impossible that some binaries might be created through gravitational capture between two single stars, given the very low likelihood of such an event (three objects are actually required, as conservation of energy rules out a single gravitating body capturing another) and the high number of binaries, this cannot be the primary formation process. Also, the observation of binaries consisting of pre main-sequence stars, supports the theory that binaries are already formed during star formation. Fragmentation of the molecular cloud during the formation of protostars is an acceptable explanation for the formation of a binary or multiple star system.[57][58] The outcome of the three-body problem, where the three stars are of comparable mass, is that eventually one of the three stars will be ejected from the system and, assuming no significant further perturbations, the remaining two will form a stable binary system. ### Mass transfer and accretion As a main-sequence star increases in size during its evolution, it may at some point exceed its Roche lobe, meaning that some of its matter ventures into a region where the gravitational pull of its companion star is larger than its own.[59] The result is that matter will transfer from one star to another through a process known as Roche lobe overflow (RLOF), either being absorbed by direct impact or through an accretion disc. The mathematical point through which this transfer happens is called the first Lagrangian point.[60] It is not uncommon that the accretion disc is the brightest (and thus sometimes the only visible) element of a binary star. If a star grows outside of its Roche lobe too fast for all abundant matter to be transferred to the other component, it is also possible that matter will leave the system through other Lagrange points or as stellar wind, thus being effectively lost to both components.[61] Since the evolution of a star is determined by its mass, the process influences the evolution of both companions, and creates stages that cannot be attained by single stars.[62][63] Studies of the eclipsing ternary Algol led to the Algol paradox in the theory of stellar evolution: although components of a binary star form at the same time, and massive stars evolve much faster than the less massive ones, it was observed that the more massive component Algol A is still in the main sequence, while the less massive Algol B is a subgiant at a later evolutionary stage. The paradox can be solved by mass transfer: when the more massive star became a subgiant, it filled its Roche lobe, and most of the mass was transferred to the other star, which is still in the main sequence. In some binaries similar to Algol, a gas flow can actually be seen.[64] ### Runaways and novae Artist's Illustration of Scenario for Plasma Ejections from V Hydrae[65] It is also possible for widely separated binaries to lose gravitational contact with each other during their lifetime, as a result of external perturbations. The components will then move on to evolve as single stars. A close encounter between two binary systems can also result in the gravitational disruption of both systems, with some of the stars being ejected at high velocities, leading to runaway stars.[66] If a white dwarf has a close companion star that overflows its Roche lobe, the white dwarf will steadily accrete gases from the star's outer atmosphere. These are compacted on the white dwarf's surface by its intense gravity, compressed and heated to very high temperatures as additional material is drawn in. The white dwarf consists of degenerate matter, and so is largely unresponsive to heat, while the accreted hydrogen is not. Hydrogen fusion can occur in a stable manner on the surface through the CNO cycle, causing the enormous amount of energy liberated by this process to blow the remaining gases away from the white dwarf's surface. The result is an extremely bright outburst of light, known as a nova.[67] In extreme cases this event can cause the white dwarf to exceed the Chandrasekhar limit and trigger a supernova that destroys the entire star, and is another possible cause for runaways.[68][69] An example of such an event is the supernova SN 1572, which was observed by Tycho Brahe. The Hubble Space Telescope recently took a picture of the remnants of this event. ## Astrophysics A simulated example of a binary star, where two bodies with similar mass orbit around a common barycenter in elliptic orbits Binaries provide the best method for astronomers to determine the mass of a distant star. The gravitational pull between them causes them to orbit around their common center of mass. From the orbital pattern of a visual binary, or the time variation of the spectrum of a spectroscopic binary, the mass of its stars can be determined, for example with the binary mass function. In this way, the relation between a star's appearance (temperature and radius) and its mass can be found, which allows for the determination of the mass of non-binaries. Because a large proportion of stars exist in binary systems, binaries are particularly important to our understanding of the processes by which stars form. In particular, the period and masses of the binary tell us about the amount of angular momentum in the system. Because this is a conserved quantity in physics, binaries give us important clues about the conditions under which the stars were formed. ### Calculating the center of mass in binary stars In a simple binary case, r1, the distance from the center of the first star to the center of mass or barycenter, is given by: ${\displaystyle r_{1}=a\cdot {m_{2} \over m_{1}+m_{2}}={a \over 1+m_{1}/m_{2}}}$ where: a is the distance between the two stellar centers and m1 and m2 are the masses of the two stars. If a is taken to be the semi-major axis of the orbit of one body around the other, then r1 will be the semimajor axis of the first body's orbit around the center of mass or barycenter, and r2 = ar1 will be the semimajor axis of the second body's orbit. When the center of mass is located within the more massive body, that body will appear to wobble rather than following a discernible orbit. ### Center of mass animations Main article: Barycenter Images are representative, not simulated. The position of the red cross indicates the center of mass of the system. (a.) Two bodies of similar mass orbiting around a common center of mass, or barycenter. (b.) Two bodies with a difference in mass orbiting around a common barycenter, like the Charon-Pluto system (c.) Two bodies with a major difference in mass orbiting around a common barycenter (similar to the Earth–Moon system) (d.) Two bodies with an extreme difference in mass orbiting around a common barycenter (similar to the Sun–Earth system) (e.) Two bodies with similar mass orbiting in an ellipse around a common barycenter. ### Research findings It is estimated that approximately 1/3 of the star systems in the Milky Way are binary or multiple, with the remaining 2/3 consisting of single stars.[70] There is a direct correlation between the period of revolution of a binary star and the eccentricity of its orbit, with systems of short period having smaller eccentricity. Binary stars may be found with any conceivable separation, from pairs orbiting so closely that they are practically in contact with each other, to pairs so distantly separated that their connection is indicated only by their common proper motion through space. Among gravitationally bound binary star systems, there exists a so-called log normal distribution of periods, with the majority of these systems orbiting with a period of about 100 years. This is supporting evidence for the theory that binary systems are formed during star formation.[71] In pairs where the two stars are of equal brightness, they are also of the same spectral type. In systems where the brightnesses are different, the fainter star is bluer if the brighter star is a giant star, and redder if the brighter star belongs to the main sequence.[72] Artist's impression of the sight from a (hypothetical) moon of planet HD 188753 Ab (upper left), which orbits a triple star system. The brightest companion is just below the horizon. The mass of a star can be directly determined only from its gravitational attraction. Apart from the Sun and stars which act as gravitational lenses, this can be done only in binary and multiple star systems, making the binary stars an important class of stars. In the case of a visual binary star, after the orbit and the stellar parallax of the system has been determined, the combined mass of the two stars may be obtained by a direct application of the Keplerian harmonic law.[73] Unfortunately, it is impossible to obtain the complete orbit of a spectroscopic binary unless it is also a visual or an eclipsing binary, so from these objects only a determination of the joint product of mass and the sine of the angle of inclination relative to the line of sight is possible. In the case of eclipsing binaries which are also spectroscopic binaries, it is possible to find a complete solution for the specifications (mass, density, size, luminosity, and approximate shape) of both members of the system. #### Planets Science fiction has often featured planets of binary or ternary stars as a setting, for example George Lucas' Tatooine from Star Wars, and one notable story, "Nightfall", even takes this to a six-star system. In reality, some orbital ranges are impossible for dynamical reasons (the planet would be expelled from its orbit relatively quickly, being either ejected from the system altogether or transferred to a more inner or outer orbital range), whilst other orbits present serious challenges for eventual biospheres because of likely extreme variations in surface temperature during different parts of the orbit. Planets that orbit just one star in a binary system are said to have "S-type" orbits, whereas those that orbit around both stars have "P-type" or "circumbinary" orbits. It is estimated that 50–60% of binary systems are capable of supporting habitable terrestrial planets within stable orbital ranges.[74] Simulations have shown that the presence of a binary companion can actually improve the rate of planet formation within stable orbital zones by "stirring up" the protoplanetary disk, increasing the accretion rate of the protoplanets within.[74] Detecting planets in multiple star systems introduces additional technical difficulties, which may be why they are only rarely found.[75] Examples include the white dwarf-pulsar binary PSR B1620-26, the subgiant-red dwarf binary Gamma Cephei, and the white dwarf-red dwarf binary NN Serpentis. More planets around binaries are listed in: [Muterspaugh; Lane; Kulkarni; Maciej Konacki; Burke; Colavita; Shao; Hartkopf; Boss (2010). "The PHASES Differential Astrometry Data Archive. V. Candidate Substellar Companions to Binary Systems". The Astronomical Journal. 140 (6): 1657. arXiv:1010.4048. Bibcode:2010AJ....140.1657M. doi:10.1088/0004-6256/140/6/1657.]. A study of fourteen previously known planetary systems found three of these systems to be binary systems. All planets were found to be in S-type orbits around the primary star. In these three cases the secondary star was much dimmer than the primary and so was not previously detected. This discovery resulted in a recalculation of parameters for both the planet and the primary star.[76] ## Examples The two visibly distinguishable components of Albireo. The large distance between the components, as well as their difference in color, make Albireo one of the easiest observable visual binaries. The brightest member, which is the third-brightest star in the constellation Cygnus, is actually a close binary itself. Also in the Cygnus constellation is Cygnus X-1, an X-ray source considered to be a black hole. It is a high-mass X-ray binary, with the optical counterpart being a variable star.[77] Sirius is another binary and the brightest star in the night time sky, with a visual apparent magnitude of −1.46. It is located in the constellation Canis Major. In 1844 Friedrich Bessel deduced that Sirius was a binary. In 1862 Alvan Graham Clark discovered the companion (Sirius B; the visible star is Sirius A). In 1915 astronomers at the Mount Wilson Observatory determined that Sirius B was a white dwarf, the first to be discovered. In 2005, using the Hubble Space Telescope, astronomers determined Sirius B to be 12,000 km (7,456 mi) in diameter, with a mass that is 98% of the Sun.[78] Luhman 16, the third closest star system, contains two brown dwarfs. An example of an eclipsing binary is Epsilon Aurigae in the constellation Auriga. The visible component belongs to the spectral class F0, the other (eclipsing) component is not visible. The last such eclipse occurred from 2009–2011, and it is hoped that the extensive observations that will likely be carried out may yield further insights into the nature of this system. Another eclipsing binary is Beta Lyrae, which is a semi-detached binary star system in the constellation of Lyra. Other interesting binaries include 61 Cygni (a binary in the constellation Cygnus, composed of two K class (orange) main-sequence stars, 61 Cygni A and 61 Cygni B, which is known for its large proper motion), Procyon (the brightest star in the constellation Canis Minor and the eighth-brightest star in the night time sky, which is a binary consisting of the main star with a faint white dwarf companion), SS Lacertae (an eclipsing binary which stopped eclipsing), V907 Sco (an eclipsing binary which stopped, restarted, then stopped again) and BG Geminorum (an eclipsing binary which is thought to contain a black hole with a K0 star in orbit around it). ## Multiple star examples Systems with more than two stars are termed multiple stars. Algol is the most noted ternary (long thought to be a binary), located in the constellation Perseus. Two components of the system eclipse each other, the variation in the intensity of Algol first being recorded in 1670 by Geminiano Montanari. The name Algol means "demon star" (from Arabic: الغول‎‎ al-ghūl), which was probably given due to its peculiar behavior. Another visible ternary is Alpha Centauri, in the southern constellation of Centaurus, which contains the fourth-brightest star in the night sky, with an apparent visual magnitude of −0.01. This system also underscores the fact that binaries need not be discounted in the search for habitable planets. Alpha Centauri A and B have an 11 AU distance at closest approach, and both should have stable habitable zones.[79] There are also examples of systems beyond ternaries: Castor is a sextuple star system, which is the second-brightest star in the constellation Gemini and one of the brightest stars in the nighttime sky. Astronomically, Castor was discovered to be a visual binary in 1719. Each of the components of Castor is itself a spectroscopic binary. Castor also has a faint and widely separated companion, which is also a spectroscopic binary. The Alcor–Mizar visual binary in Ursa Majoris also consists of six stars, four comprising Mizar and two comprising Alcor. ## Notes and references 1. ^ "Gale - Enter Product Login". go.galegroup.com. Retrieved 2016-10-03. 2. ^ Filippenko, Alex, Understanding the Universe (of The Great Courses on DVD), Lecture 46, time 1:17, The Teaching Company, Chantilly, VA, USA, 2007 3. ^ a b The Binary Stars, Robert Grant Aitken, New York: Dover, 1964, p. ix. 4. ^ Herschel, William (1802). "Catalogue of 500 New Nebulae, Nebulous Stars, Planetary Nebulae, and Clusters of Stars; With Remarks on the Construction of the Heavens". Philosophical Transactions of the Royal Society of London. 92: 477–528 [481]. Bibcode:1802RSPT...92..477H. doi:10.1098/rstl.1802.0021. JSTOR 107131. 5. ^ a b c Heintz, W. D. (1978). Double Stars. Dordrecht: D. Reidel Publishing Company. pp. 1–2. ISBN 90-277-0885-1. 6. ^ a b "Visual Binaries". University of Tennessee. 7. ^ Heintz, W. D. (1978). Double Stars. Dordrecht: D. Reidel Publishing Company. p. 5. ISBN 90-277-0885-1. 8. ^ Heintz, W. D. (1978). Double Stars. D. Reidel Publishing Company, Dordrecht. p. 17. ISBN 90-277-0885-1. 9. ^ a b c The Binary Stars, Robert Grant Aitken, New York: Dover, 1964, p. 1. 10. ^ Vol. 1, part 1, p. 422, Almagestum Novum, Giovanni Battista Riccioli, Bononiae: Ex typographia haeredis Victorij Benatij, 1651. 11. ^ A New View of Mizar, Leos Ondra, accessed on line May 26, 2007. 12. ^ "Gale - Enter Product Login". go.galegroup.com. Retrieved 2016-10-03. 13. ^ pp. 10–11, Observing and Measuring Double Stars, Bob Argyle, ed., London: Springer, 2004, ISBN 1-85233-558-0. 14. ^ pp. 249–250, An Inquiry into the Probable Parallax, and Magnitude of the Fixed Stars, from the Quantity of Light Which They Afford us, and the Particular Circumstances of Their Situation, John Michell,Philosophical Transactions (1683–1775) 57 (1767), pp. 234–264. 15. ^ Heintz, W. D. (1978). Double Stars. Dordrecht: D. Reidel Publishing Company. p. 4. ISBN 90-277-0885-1. 16. ^ Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing, William Herschel, Philosophical Transactions of the Royal Society of London 93 (1803), pp. 339–382. 17. ^ p. 291, French astronomers, visual double stars and the double stars working group of the Société Astronomique de France, E. Soulié, The Third Pacific Rim Conference on Recent Development of Binary Star Research, proceedings of a conference sponsored by Chiang Mai University, Thai Astronomical Society and the University of Nebraska-Lincoln held in Chiang Mai, Thailand, 26 October-1 November 1995, ASP Conference Series 130 (1997), ed. Kam-Ching Leung, pp. 291–294, Bibcode1997ASPC..130..291S. 18. ^ "Introduction and Growth of the WDS", The Washington Double Star Catalog, Brian D. Mason, Gary L. Wycoff, and William I. Hartkopf, Astrometry Department, United States Naval Observatory, accessed on line August 20, 2008. 19. ^ Sixth Catalog of Orbits of Visual Binary Stars, William I. Hartkopf and Brian D. Mason, United States Naval Observatory, accessed on line August 20, 2008. 20. ^ The Washington Double Star Catalog, Brian D. Mason, Gary L. Wycoff, and William I. Hartkopf, United States Naval Observatory. Accessed on line December 20, 2008. 21. ^ Heintz, W. D. (1978). Double Stars. Dordrecht: D. Reidel Publishing Company. pp. 17–18. ISBN 90-277-0885-1. 22. ^ "Planet-hunting SPHERE Images First Circumbinary Planet System with Disc". Retrieved 26 October 2015. 23. ^ "Binary Stars". Cornell Astronomy. 24. ^ The Binary Stars, Robert Grant Aitken, New York: Dover, 1964, p. 41. 25. ^ Herter, T. "Stellar Masses". Cornell University. Archived from the original on June 17, 2012. 26. ^ Bruton, D. "Eclipsing Binary Stars". Stephen F. Austin State University. 27. ^ Bruton, D. "Eclipsing Binary Stars". Stephen F. Austin State University. 28. ^ Wilson, R. E. (1 January 2008). "Eclipsing Binary Solutions in Physical Units and Direct Distance Estimation". The Astrophysical Journal. 672 (1). Bibcode:2008ApJ...672..575W. doi:10.1086/523634. 29. ^ Bonanos, Alceste Z. (2006). "Eclipsing Binaries: Tools for Calibrating the Extragalactic Distance Scale". Proceedings of the International Astronomical Union. 2. arXiv:astro-ph/0610923. doi:10.1017/S1743921307003845. 30. ^ Worth, M. "Binary Stars" . Stephen F. Austin State University. 31. ^ Lev Tal-Or; Simchon Faigler; Tsevi Mazeh (2014). "Seventy-two new non-eclipsing BEER binaries discovered in CoRoT lightcurves and confirmed by RVs from AAOmega". arXiv:1410.3074. 32. ^ Bock, D. "Binary Neutron Star Collision". NCSA. 33. ^ Asada, H.; T. Akasaka; M. Kasai (27 September 2004). "Inversion formula for determining parameters of an astrometric binary". Publ.Astron.Soc.Jap. 56: L35–L38. arXiv:astro-ph/0409613. Bibcode:2004PASJ...56L..35A. doi:10.1093/pasj/56.6.L35. 34. ^ "Astrometric Binaries". University of Tennessee. 35. ^ Nguyen, Q. "Roche model". San Diego State University. 36. ^ Voss, R.; T.M. Tauris (2003). "Galactic distribution of merging neutron stars and black holes". Monthly Notices of the Royal Astronomical Society. 342 (4): 1169–1184. arXiv:0705.3444. Bibcode:2003MNRAS.342.1169V. doi:10.1046/j.1365-8711.2003.06616.x. 37. ^ Encyclopedia of the History of Astronomy and Astrophysics. Cambridge University Press. 2013. 38. ^ Robert Connon Smith (November 2006). "Cataclysmic Variables". Contemporary Physics. 47 (6): 363–386. arXiv:astro-ph/0701654. Bibcode:2007astro.ph..1654C. doi:10.1080/00107510601181175. 39. ^ Neutron Star X-ray binaries, A Systematic Search of New X-ray Pulsators in ROSAT Fields, Gian Luca Israel, Ph. D. thesis, Trieste, October 1996. 40. ^ Iorio, Lorenzo (July 24, 2007). "On the orbital and physical parameters of the HDE 226868/Cygnus X-1 binary system". Astrophysics and Space Science. 315 (1–4): 335. arXiv:0707.3525. Bibcode:2008Ap&SS.315..335I. doi:10.1007/s10509-008-9839-y. 41. ^ Black Holes, Imagine the Universe!, NASA. Accessed on line August 22, 2008. 42. ^ Applegate, James H. (1992). "A mechanism for orbital period modulation in close binaries". Astrophysical Journal, Part 1. 385: 621–629. Bibcode:1992ApJ...385..621A. doi:10.1086/170967. 43. ^ Hall, Douglas S. (1989). "The relation between RS CVn and Algol". Space Science Reviews. 50: 219–233. Bibcode:1989SSRv...50..219H. doi:10.1007/BF00215932. 44. ^ "White Dwarf Lashes Red Dwarf with Mystery Ray". www.eso.org. Retrieved 28 July 2016. 45. ^ Heintz, W. D. (1978). Double Stars. Dordrecht: D. Reidel Publishing Company. p. 19. ISBN 90-277-0885-1. 46. ^ "Binary and Multiple Star Systems". Lawrence Hall of Science at the University of California. 47. ^ pp. 307–308, Observing and Measuring Double Stars, Bob Argyle, ed., London: Springer, 2004, ISBN 1-85233-558-0. 48. ^ Entry 14396-6050, discoverer code RHD 1AB,The Washington Double Star Catalog, United States Naval Observatory. Accessed on line August 20, 2008. 49. ^ References and discoverer codes, The Washington Double Star Catalog, United States Naval Observatory. Accessed on line August 20, 2008. 50. ^ [1] – see essential notes: "Hot companion to Antares at 2.9arcsec; estimated period: 678yr." 51. ^ Kenyon, S. J.; Webbink, R. F. (1984). "The nature of symbiotic stars". Astrophysical Journal. 279: 252–283. Bibcode:1984ApJ...279..252K. doi:10.1086/161888. 52. ^ Iping, Rosina C.; Sonneborn, George; Gull, Theodore R.; Massa, Derck L.; Hillier, D. John (2005). "Detection of a Hot Binary Companion of η Carinae". The Astrophysical Journal. 633 (1): L37–L40. arXiv:astro-ph/0510581. Bibcode:2005ApJ...633L..37I. doi:10.1086/498268. 53. ^ Nigel Henbest; Heather Couper (1994). The guide to the galaxy. ISBN 978-0-521-45882-5. 54. ^ a b Rowe, Jason F.; Borucki, William J.; Koch, David; Howell, Steve B.; Basri, Gibor; Batalha, Natalie; Brown, Timothy M.; Caldwell, Douglas; Cochran, William D.; Dunham, Edward; Dupree, Andrea K.; Fortney, Jonathan J.; Gautier, Thomas N.; Gilliland, Ronald L.; Jenkins, Jon; Latham, David W.; Lissauer, Jack J.; Marcy, Geoff; Monet, David G.; Sasselov, Dimitar; Welsh, William F. (2010). "Kepler Observations of Transiting Hot Compact Objects". The Astrophysical Journal Letters. 713 (2): L150–L154. arXiv:1001.3420. Bibcode:2010ApJ...713L.150R. doi:10.1088/2041-8205/713/2/L150. 55. ^ a b van Kerkwijk, Marten H.; Rappaport, Saul A.; Breton, René P.; Justham, Stephen; Podsiadlowski, Philipp; Han, Zhanwen (2010). "Observations of Doppler Boosting in Kepler Light Curves". The Astrophysical Journal. 715 (1): 51–58. arXiv:1001.4539. Bibcode:2010ApJ...715...51V. doi:10.1088/0004-637X/715/1/51. 56. ^ a b Borenstein, Seth (4 January 2010). "Planet-hunting telescope unearths hot mysteries" (6:29 pm EST). 57. ^ Boss, A. P. (1992). "Formation of Binary Stars". In J. Sahade; G. E. McCluskey; Yoji Kondo. The Realm of Interacting Binary Stars. Dordrecht: Kluwer Academic. p. 355. ISBN 0-7923-1675-4. 58. ^ Tohline, J. E.; J. E. Cazes; H. S. Cohl. "The Formation of Common-Envelope, Pre-Main-Sequence Binary Stars". Louisiana State University. 59. ^ Kopal, Z. (1989). The Roche Problem. Kluwer Academic. ISBN 0-7923-0129-3. 60. ^ "Contact Binary Star Envelopes" by Jeff Bryant, Wolfram Demonstrations Project. 61. ^ "Mass Transfer in Binary Star Systems" by Jeff Bryant with Waylena McCully, Wolfram Demonstrations Project. 62. ^ Boyle, C.B. (1984). "Mass transfer and accretion in close binaries – A review". Vistas in Astronomy. 27: 149–169. Bibcode:1984VA.....27..149B. doi:10.1016/0083-6656(84)90007-2. 63. ^ Vanbeveren, D.; W. van Rensbergen; C. de Loore (2001). The Brightest Binaries. Springer. ISBN 0-7923-5155-X. 64. ^ Blondin, J. M.; M. T. Richards; M. L. Malinowski. "Mass Transfer in the Binary Star Algol". American Museum of Natural History. 65. ^ "Artist's Illustration of Scenario for Plasma Ejections from V Hydrae". www.spacetelescope.org. Retrieved 12 October 2016. 66. ^ Hoogerwerf, R.; de Bruijne, J.H.J.; de Zeeuw, P.T. (December 2000). "The Origin of Runaway Stars". Astrophysical Journal. 544 (2): L133. arXiv:astro-ph/0007436. Bibcode:2000ApJ...544L.133H. doi:10.1086/317315. 67. ^ Prialnik, D. (2001). "Novae". Encyclopaedia of Astronomy and Astrophysics. pp. 1846–1856. 68. ^ Icko, I. (1986). "Binary Star Evolution and Type I Supernovae". Cosmogonical Processes. p. 155. 69. ^ Fender, R. (2002). "Relativistic outflows from X-ray binaries (a.k.a. 'Microquasars')". Lect.Notes Phys. Lecture Notes in Physics. 589 (101): 101. arXiv:astro-ph/0109502. Bibcode:2002LNP...589..101F. doi:10.1007/3-540-46025-X_6. ISBN 978-3-540-43518-1. 70. ^ Most Milky Way Stars Are Single, Harvard-Smithsonian Center for Astrophysics 71. ^ Hubber, D. A.; A.P. Whitworth (2005). "Binary Star Formation from Ring Fragmentation". Astronomy & Astrophysics. 437: 113–125. arXiv:astro-ph/0503412. Bibcode:2005A&A...437..113H. doi:10.1051/0004-6361:20042428. 72. ^ Schombert, J. "Birth and Death of Stars". University of Oregon. 73. ^ "Binary Star Motions". Cornell Astronomy. 74. ^ a b Elisa V. Quintana; Jack J. Lissauer (2007). "Terrestrial Planet Formation in Binary Star Systems". arXiv:0705.3444 [astro-ph]. 75. ^ Schirber, M (17 May 2005). "Planets with Two Suns Likely Common". Space.com. 76. ^ Daemgen, S.; Hormuth, F.; Brandner, W.; Bergfors, C.; Janson, M.; Hippler, S.; Henning, T. (2009). "Binarity of transit host stars – Implications for planetary parameters" (PDF). Astronomy and Astrophysics. 498 (2): 567–574. arXiv:0902.2179. Bibcode:2009A&A...498..567D. doi:10.1051/0004-6361/200810988. 77. ^ See sources at Cygnus X-1 78. ^ McGourty, C. (2005-12-14). "Hubble finds mass of white dwarf". BBC News. Retrieved 2010-01-01. 79. ^ Elisa V. Quintana; Fred C. Adams; Jack J. Lissauer & John E. Chambers (2007). "Terrestrial Planet Formation around Individual Stars within Binary Star Systems". Astrophysical Journal. 660: 807. arXiv:astro-ph/0701266. Bibcode:2007ApJ...660..807Q. doi:10.1086/512542.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8132157325744629, "perplexity": 2184.997817547407}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719027.25/warc/CC-MAIN-20161020183839-00119-ip-10-171-6-4.ec2.internal.warc.gz"}
https://dsp.stackexchange.com/questions/14299/correlation-and-convolution?noredirect=1	# Correlation and Convolution [duplicate] Both Correlation and Convolution are displacement function, i.e they are used to slide the filter mask across the image. Convolution is same as correlation except that the filter mask is rotated 180 degree before computing the sum of products. We mostly use convolution. why it is widely used? why convolution is preferred over correlation? We have, Convolution in time domain = Multiplication in Frequency domain Do we have any similar relation for correlation ? Correlation in time domain = _______________ in Frequency domain ? ## marked as duplicate by Dilip Sarwate, Paul R, penelope, jonsca, Peter K.♦Feb 24 '14 at 16:45 Please note that for complex-valued signals, complex conjugates are needed for correlation, but neither signal is conjugated in the convolution. So it is only partially true for real valued signals that Convolution is same as correlation except that the filter mask is rotated 180 degree before computing the sum of products.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8204131126403809, "perplexity": 1486.3545525930253}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570987836368.96/warc/CC-MAIN-20191023225038-20191024012538-00281.warc.gz"}
https://proceedings.mlr.press/v97/katiyar19a.html	# Robust Estimation of Tree Structured Gaussian Graphical Models Ashish Katiyar, Jessica Hoffmann, Constantine Caramanis Proceedings of the 36th International Conference on Machine Learning, PMLR 97:3292-3300, 2019. #### Abstract Consider jointly Gaussian random variables whose conditional independence structure is specified by a graphical model. If we observe realizations of the variables, we can compute the covariance matrix, and it is well known that the support of the inverse covariance matrix corresponds to the edges of the graphical model. Instead, suppose we only have noisy observations. If the noise at each node is independent, we can compute the sum of the covariance matrix and an unknown diagonal. The inverse of this sum is (in general) dense. We ask: can the original independence structure be recovered? We address this question for tree structured graphical models. We prove that this problem is unidentifiable, but show that this unidentifiability is limited to a small class of candidate trees. We further present additional constraints under which the problem is identifiable. Finally, we provide an O(n^3) algorithm to find this equivalence class of trees.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9412253499031067, "perplexity": 265.9630773768457}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323588246.79/warc/CC-MAIN-20211028003812-20211028033812-00462.warc.gz"}
http://math.stackexchange.com/users/10463/palio?tab=summary	palio Reputation 2,531 Next privilege 3,000 Rep. 1 6 25 Impact ~65k people reached ### Questions (230) 9 Why does a limit at infinity not exist? 9 reducible polynomial modulo every prime 9 rational points of an algebraic variety 6 The special orthogonal group is a manifold 5 subgroup generated by two subgroups ### Reputation (2,531) +5 Is every group homomorphism of the rationals an isomorphism +5 Needing a double inclusion to determine the kernel of a matrix +5 Extension of a linear map to a commutative graded algebra +5 A wrong proof that the kernel and image are always complementary 4 If $gf$ is 1-1, is $g$ 1-1? 4 The fundamental group of the product of a 3-sphere and circle 3 A question on arcwised connected spaces 3 The field of fractions of a field $F$ is isomorphic to $F$ 2 To show a set is open ### Tags (80) 6 algebraic-topology × 55 1 group-theory × 48 5 general-topology × 70 1 ring-theory × 11 4 abstract-algebra × 30 1 elementary-number-theory × 8 4 field-theory × 7 1 real-analysis × 3 4 proof-verification × 3 0 linear-algebra × 29 ### Accounts (2) Mathematics 2,531 rep 1625 Cross Validated 101 rep	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8346301317214966, "perplexity": 1365.9569485584072}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246648209.18/warc/CC-MAIN-20150417045728-00280-ip-10-235-10-82.ec2.internal.warc.gz"}
http://sewintriguing.blogspot.com/2006/07/its-three-star-day.html	## Tuesday, July 04, 2006 ### It's A Three Star Day! Yes, I did get my third star on PatternReview today, thanks to all of you who helped. If anyone wants a simpler explanation of the "star-figuring" method they use, let me know. For now, I'll just say that the strangest thing about it is that giving someone anything other than a Very Helpful rating on a review will actually hurt them. I never dreamed that rating someone Helpful was not a nice thing to do. Oh well, live and learn. Now that I have 3 stars, what will I do with them? I have no idea, but it's nice, just the same. Again, thank you all, I couldn't have done it without you, and I really didn't mean to whine and beg (well, not much anyhow!)	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.880622386932373, "perplexity": 826.9561916976758}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1398223207985.17/warc/CC-MAIN-20140423032007-00195-ip-10-147-4-33.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/work-and-power-homework-problem.264034/	# Work and Power Homework problem 1. Oct 13, 2008 ### Santorican This question, I have absolutely no idea what to do. Question: Tarzan, in one tree, sights Jane in another tree. He grabs the end of a vine with length 20 m that makes an angle of 45 degrees with the vertical, steps off his tree limb, and swings down and then up to Jane's open arms. When he arrives, his vine makes an angle of 30 degrees with the vertical. Part A) Calculate Tarzan's speed just before he reaches Jane. You can ignore air resistance and the mass of the vine. I figured that I am going to have to use the work energy theorm, W=change in K and then break it down into smaller bits, but there is no mass and I have no idea what work is equal to so I am totally stumped. 2. Oct 13, 2008 ### Redbelly98 Staff Emeritus Try using conservation of energy. The vine does not actually do any work on Tarzan. 3. Oct 13, 2008 ### Santorican So by using Ue=Uk and breaking it down I can figure out the answer!? That is so beautifully simply! Thank you 4. Oct 14, 2008 ### Santorican Okay so I tried using the conservation of energy Ue=Ke and then broke it down to mgh=(mv^2)/2 and then simplified to find velocity but this is where I am confused. I tried using V=(2gh)^1/2 and it was wrong. Then I tried to find the component of gravity at a 30 degree angle with respect to y and used V=(2(sin30g)h)^1/2 and it was wrong. Then again with respect to x, don't have a clue at this point, V=(2g(cos30)h)^1/2 and it too was wrong. 5. Oct 14, 2008 ### Redbelly98 Staff Emeritus Well, for conservation of energy you need 4 quantities to set up the equation: initial kinetic energy and initial potential energy final kinetic energy and final potential energy	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9349810481071472, "perplexity": 898.8305678468913}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988717783.68/warc/CC-MAIN-20161020183837-00006-ip-10-171-6-4.ec2.internal.warc.gz"}
http://www.demon-software.com/public_html/support/htmlug/ug-node115.html	### Keyword GEOSURFACE This keyword controls the calculation and plotting of geometrical surfaces such as spheres or ellipsoids. See the keyword BOX (4.10.7) for the definition of the geosurface boundary. Options: SPHERE / ELLIPSOID SPHERE A sphere of radius around the origin is constructed. ELLIPSOID An ellipsoid with semi-axes , , and around the origin is constructed. BINARY / ASCII / TABLE BINARY A binary output of the geometrical surface is written in the file LAT.bin using the VU file format. The VU control file deMon.pie is also written. This is the default. ASCII An ascii output of the geometrical surface is written in the file deMon.lat. TABLE A function table of the geometrical surface coordinates is written in the output file deMon.out. TOL=Real Tolerance for data reduction. Description: The options BINARY, ASCII, TABLE, and TOL are identical in definition to the corresponding options of the ISOSURFACE keyword. Despite the syntax, specification of SPHERE or ELLIPSOID is mandatory for the GEOSURFACE keyword. The origin and radius or semi-axes of the sphere or ellipsoid must be defined in the keyword body of GEOSURFACE. For a sphere with the origin at and a radius of the input has the form: GEOSURFACE SPHERE 1.0 3.0 4.0 7.0 The units are defined by the unit definition in the GEOMETRY keyword (see Section 4.1.1). The following input is used to generate the sphere, shown just below on the right, on which the molecular electrostatic potential is plotted: GEOSURFACE SPHERE 0.0 0.0 0.0 3.0 # BOX READ 100 100 100 -6.0 -6.0 -6.0 6.0 -6.0 -6.0 -6.0 6.0 -6.0 -6.0 -6.0 6.0 # Geometry Z-Matrix C H 1 RCH H 1 RCH 2 AHCH H 1 RCH 2 AHCH 3 AHCH RAA H 1 RCH 2 AHCH 3 -AHCH RAA VARIABLES RCH 1.094 CONSTANTS AHCH 109.47122063449 Note the explicit box definition that must enclose the sphere. For an ellipsoid with semi-axes 7.0, 1.0, and 0.5 in the current unit system and centered at , the following input form must be used: GEOSURFACE ELLIPSOID 1.0 3.0 4.0 7.0 1.0 0.5 For the option ELLIPSOID the definition of all three semi-axes is mandatory.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8538892865180969, "perplexity": 2514.219290462973}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103645173.39/warc/CC-MAIN-20220629211420-20220630001420-00040.warc.gz"}
https://www.wikihow.com/Add-Font-in-Microsoft-Word	# How to Add Font in Microsoft Word This wikiHow teaches you how to install a font on your Windows or Mac computer in order to use the font in Microsoft Word. ### Part 1 of 3: On Windows 1. 1 Download the font from a trusted website. Fonts are a common way to transmit viruses, so take care to only download fonts from trusted sources, and avoid any fonts that come in EXE format. Fonts typically come packed into ZIP files, or in TTF or OTF format. A few of the more popular font sites include the following: • dafont.com • fontspace.com • fontsquirrel.com • 1001freefonts.com 2. 2 Extract the font file if necessary. If your font downloads in a ZIP folder, double-click the folder, then click Extract at the top of the window, click Extract all, and click Extract at the bottom of the window. • Skip this step if the font downloads in TTF or OTF format, not in a ZIP folder. 3. 3 Double-click the font file. Doing so will open the font in a preview window. 4. 4 Click Install. It's at the top of the preview window. 5. 5 Click Yes if prompted. Since installing a font requires administrator permissions, you may be prompted to confirm this step. • If you aren't on an administrator account, you can't install a font. 6. 6 Wait for the font to install. This will usually only take a few seconds. Once the font is installed on your computer, it will be accessible by any programs which use system fonts, including Microsoft Word. ### Part 2 of 3: On Mac 1. 1 • dafont.com • fontspace.com • fontsquirrel.com • 1001freefonts.com 2. 2 Extract the font file if necessary. Since most font files download in a ZIP folder, you'll need to unzip the folder by double-clicking it and waiting for the extracted folder to open. • Skip this step if your font downloads as a TTF or OTF file, not a ZIP folder. 3. 3 Double-click the font file. Doing so will bring up a preview window. 4. 4 Click Install Font. It's at the top of the preview window. This will install your font for all text-based programs on your Mac, thus making it usable with Microsoft Word.[1] ### Part 3 of 3: Accessing the Font in Word 1. 1 Note the installed font's name. Since fonts in Word are listed in alphabetical order, you'll need to know the first few letters of your font in order to find it. 2. 2 Open Microsoft Word. Its app icon resembles a white "W" on a dark-blue background. • If Microsoft Word was already open, close it and then re-open it. Failing to do this may prevent your font from showing up until you restart Word. 3. 3 Click Blank document. You'll find this option in the upper-left side of the launch page. Doing so opens a new Word document. 4. 4 Click the Home tab. It's at the top of the Word window. 5. 5 Open the "Font" drop-down menu. Click to the right of the current font's name in the toolbar. You should see a drop-down menu appear. 6. 6 7. 7 Test the font. Click the font's name, then try typing using the font. You may need to adjust the size of the font in order for it to appear normal. ## Community Q&A Search • Question What can I do if my fonts don't appear in the dropdown? Closing Microsoft Word and then opening it again will usually refresh the font list. If this doesn't work, try re-installing the font. • Question It says it does not appear to be a valid font and cannot install. What should I do? Instead of copying onto the control panel, double click on the extracted font file. You will then see either an open type or a true type file, double click on this and a preview of the font will show up. On the top left area of the window, there will be a button saying install. Click that. • Question How do I rename the font? Never rename a font while it's in the installed folder; it could get corrupted. Delete the font, and reinstall it. Before putting it into the fonts folder, right-click it, and select the option "Rename." After naming it whatever you want, put it into the fonts folder normally. • Question I cannot find my font in Word at all. What's up? Make sure all Microsoft Office/other Office-like applications are closed. Install the font via the instructions listed, then open Word and search for your font. • Question I cannot find my font when I reach to the Control Panel Go into "Control Panel," then press the "View By] button. The font file should be there. Open it. Then drag your selected font from your desktop into the font file. • Question When I drag the font file into the fonts page, the font that I want does not install. It says that the font is not valid. What do I do? If there is more than one file inside of the zip folder, you may be installing the text document. That is a description of the font itself. You need to install the TrueType font file. • Question I have added my font to Word but when I try to use it it still just types in a normal font. What should I do? Even if you have downloaded and selected the font, Word might go back to the default font (it depends on the theme of the document). Make sure that when you are typing the font is still selected. If this still doesn't work, try right-clicking (PC) or two-finger-clicking (other computers) the text you typed. The font settings will appear and you can change the font there. You can also just type in a default font and then select the text you want to change the font of. Once the text is selected, you can change the font in the font box located in the toolbar. 200 characters left ## Tips • After a font is installed, it will be available in all Microsoft Office programs. Thanks! • If you plan on sending your Word document to someone else, save it as a PDF to ensure that your custom font is retained. You can save as a PDF by clicking the "Save as type" (Windows) or "Format" (Mac) drop-down box on the "Save" window and then selecting PDF. Thanks! ## Warnings • Some symbols aren't available in all fonts. Thanks! JL Written by: Tech Specialist This article was written by Jack Lloyd. Jack Lloyd is a Technology Writer and Editor for wikiHow. He has over two years of experience writing and editing technology-related articles. He is technology enthusiast and an English teacher. This article has been viewed 1,691,781 times. Co-authors: 24 Updated: February 6, 2020 Views: 1,691,781 Article SummaryX 2. Unzip the font if necessary. 3. Double-click the font. 4. Click Install or Install Font. Thanks to all authors for creating a page that has been read 1,691,781 times.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9282085299491882, "perplexity": 3846.8009490175473}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-31/segments/1627046154486.47/warc/CC-MAIN-20210803222541-20210804012541-00178.warc.gz"}
https://papers.nips.cc/paper/2012/hash/00411460f7c92d2124a67ea0f4cb5f85-Abstract.html	#### Authors Marcelo Fiori, Pablo Musé, Guillermo Sapiro #### Abstract Graphical models are a very useful tool to describe and understand natural phenomena, from gene expression to climate change and social interactions. The topological structure of these graphs/networks is a fundamental part of the analysis, and in many cases the main goal of the study. However, little work has been done on incorporating prior topological knowledge onto the estimation of the underlying graphical models from sample data. In this work we propose extensions to the basic joint regression model for network estimation, which explicitly incorporate graph-topological constraints into the corresponding optimization approach. The first proposed extension includes an eigenvector centrality constraint, thereby promoting this important prior topological property. The second developed extension promotes the formation of certain motifs, triangle-shaped ones in particular, which are known to exist for example in genetic regulatory networks. The presentation of the underlying formulations, which serve as examples of the introduction of topological constraints in network estimation, is complemented with examples in diverse datasets demonstrating the importance of incorporating such critical prior knowledge.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8264899253845215, "perplexity": 782.1350944735223}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585504.90/warc/CC-MAIN-20211022084005-20211022114005-00101.warc.gz"}
http://mathhelpforum.com/trigonometry/136291-rotation-triangle.html	# Math Help - Rotation of a triangle 1. ## Rotation of a triangle (a) A triangle has vertices at the points A(3,−1), B(−2,1) and C(2,3). Suppose that the triangle is to be moved so that B is at the origin and BA lies along the positive x-axis. One isometry that achieves this transformation is the composite of a translation followed by a rotation. (You may find it helpful to sketch the triangle.) (i) Determine the translation that moves B to the origin, giving your answer in the form $t_{a,b}$. Write down a formal definition of this translation in two-line notation. (ii) Find the images A' of A and C' of C under the translation in part (i) (iii) Let rθ be the rotation that completes the required transformation, where θ lies in the interval (−π,π]. Find the exact values of tan θ,cos θand sin θ, and hence write down a formal definition of rθ using two-line notation. (There is no need to work out the value of the angle θ.) (iv) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values. I have done (i) and (ii) see below, but am unsure about how to work out the rotation without knowing what angle it rotates by. Thanks for your pointers. (i) $t_{a,b}:R^2 \rightarrow R^2$ $(x,y) \mapsto (x+2,y-1)$ (ii) A' (5,-2) and C' (4,2) (iii) ? 2. You need to find the rotation matrix in 2 dimensions. See: Rotation matrix - Wikipedia, the free encyclopedia for details. All you need is the equation for x' and y' in the section "Dimension 2". To find the values of cos(theta) and sin(theta), just substitute x for cos(theta) and sqrt(1-x^2) for sin(theta). Then use algebra to force the y component of point A to be zero. 3. Originally Posted by cozza (iii) Let rθ be the rotation that completes the required transformation, where θ lies in the interval (−π,π]. Find the exact values of tan θ,cos θand sin θ, and hence write down a formal definition of rθ using two-line notation. (There is no need to work out the value of the angle θ.) (iv) Find the coordinates of the images of A' and C' under the rotation rθ. Give your answers as exact values. I have done (i) and (ii) see below, but am unsure about how to work out the rotation without knowing what angle it rotates by. Thanks for your pointers. (i) $t_{a,b}:R^2 \rightarrow R^2$ $(x,y) \mapsto (x+2,y-1)$ (ii) A' (5,-2) and C' (4,2) (iii) ? This question is still driving me crazy! I thought I had it, but obviously not. My tutor added the following comments: From (i) and (ii) you know the coordinates of A'B'C', so you can plot these. You then need the rotation that makes BA lie along the positive x-axis. If you draw the traingle A'B'C' you will have formed a right-angled traingle and the required theta is one of the angles of the traingle. You can then work out sin, cos and tan of this angle from the triangle. These values can be used in the formal two line definition of a rotation to answer (iii). Then use this rule to find out the images of A' and C'. We have a software program that by trial and error I worked out the rotation θ = π/8, but I am not sure how to show this. Please help, I am really struggling 4. So you started off with: A = (3,-1) B = (-2,1) C = ( 2,3) You then translated by -B to get A' = (5, -2) B' = (0, 0) C' = (4,2) Now you need to rotate by a 2D rotation matrix $\begin{bmatrix}cos(\theta)&- sin(\theta)\\sin(\theta)&cos(\theta)\end{bmatrix} \begin{bmatrix}5\\-2\end{bmatrix}=\begin{bmatrix}\sqrt(29)\\0\end{bma trix}$ to get the y-coordinate of A to be zero. Note the sqrt(29) comes from the distance of point A from the origin. Rotations won't change radial distances. If you solve this for cos(theta), you should get cos(theta) = 5/sqrt(29), if I haven't made any calculation errors. From here you can get sin and tan, etc. You also use the same rotation matrix on C' to find the image of C' after rotation.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8973667025566101, "perplexity": 608.856316207778}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-41/segments/1410657129409.8/warc/CC-MAIN-20140914011209-00090-ip-10-196-40-205.us-west-1.compute.internal.warc.gz"}
https://alstatr.blogspot.com/2013/06/measures-of-skewness-and-kurtosis.html	### R: Measures of Skewness and Kurtosis Skewness and kurtosis in R are available in the moments package (to install a package, click here), and these are: • Skewness - skewness; and, • Kurtosis - kurtosis. Example 1. Mirra is interested on the elapse time (in minutes) she spends on riding a tricycle from home, at Simandagit, to school, MSU-TCTO, Sanga-Sanga for three weeks (excluding weekends). She obtain the following data: 19.09, 19.55, 17.89, 17.73, 25.15, 27.27, 25.24, 21.05, 21.65, 20.92, 22.61, 15.71, 22.04, 22.60, and 24.25. Compute and interpret the skewness and kurtosis. Interpretation: The skewness here is -0.01565162. This value implies that the distribution of the data is slightly skewed to the left or negatively skewed. It is skewed to the left because the computed value is negative, and is slightly, because the value is close to zero. For the kurtosis, we have 2.301051 implying that the distribution of the data is platykurtic, since the computed value is less than 3. Graphical illustration of the data is in Figure 1. Figure 1. Histogram of the Time Elapsed Figure 1 confirms the numerical findings above, it is clear that the histogram is slightly skewed to the left, and is platykurtic. Below is the codes of the said figure, Example 2. Simulate 10000 samples from a normal distribution with mean 55, and standard deviation 4.5, then compute and interpret the skewness and kurtosis, and plot the histogram. Interpretation: The skewness of the simulated data is -0.008525844. This concludes that the data is close to bell shape but slightly skewed to the left. The computed kurtosis is 2.96577, which means the data is mesokurtic. Figure 2 is the histogram of the simulated data with empirical PDF. Figure 2. Histogram of the Simulated Data	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8136171102523804, "perplexity": 789.2790061791487}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267160145.76/warc/CC-MAIN-20180924050917-20180924071317-00530.warc.gz"}
https://www.physicsforums.com/threads/geometric-series.69985/	# Geometric Series 1. Apr 4, 2005 ### Briggs I am having a little trouble with some questions on geometric series' For example, Find the values of x for which the following geometric series converge I have done the first one easy enough $$2+4x+8x^2+16x^3.....$$ $$r=2x$$ $$\|r\|<1$$ $$\|x\|<\frac{1}_{2}$$ $$\frac{-1}{2} < x < \frac{1}_{2}$$ But then it gets more difficult and adds more to the question for example $$5+25(3x+4)+125(3x+4)^2$$ I get $$r=5(3x+4)$$ so $$\|5(3x+4)\|<1$$ But then I am stuck as to what to do next any hints on how to go about these questions would be appreciated. 2. Apr 4, 2005 ### dextercioby It's the same kind of story to explicitate the modulus... $$\left\|a\right\|<b\Leftrightarrow -b<a<b$$ Thankfully,the ratio is linear in "x"... Daniel. EDITalls,this is the kind of terminology i had been used to in high school.Always the teacher said about "modulus explicitation" (sic!)... Last edited: Apr 4, 2005 3. Apr 4, 2005 ### HallsofIvy Staff Emeritus "explicate the modulus"?? Wow! Them big words is too much for me! Briggs, what dextercioby means is: \|5(3x+4)\|< 1 is exactly the same as -1< 5(3x+4)< 1. Now solve for x. 4. Apr 4, 2005 ### Data It's explicitate! :tongue: 5. Apr 4, 2005 ### Briggs Thanks for the help, it seems quite simple now that you have shown me the next step, I get $$-1<15x+20<1$$ $$\frac{-1-20}{15}<x<\frac{1-20}{15}$$ $$\frac{-7}{5}<x<\frac{-19}{15}$$ Which is the correct answer so thanks a lot for the help 6. Apr 4, 2005 ### Gokul43201 Staff Emeritus To help vizualize this geometrically, imagine a point P on the number line. Then, all points X that are strictly inside a distance d from the fixed point P, are given by the equation : \|X-P\| < d In other words, this means that X is the set of points in (P-d, P+d). This follows directly from the definition of the modulus : Definition : \|x\| = x if x > 0 and \|x\| = -x if x <= 0 Derivation of Geometric Statement : \|X-P\| = X-P if X > P and \|X-P\| = P - X, otherwise (from def'n.) In the first case (when X > P), \|X-P\| < d means that X-P < d or X < P+d. So : P < X < P+d In the second case (when X <= P) \|X-P\| < d means that P-X < d or X > P-d. So : P-d < X <= P Combining the above two cases, you see that \|X-P\| < d gives P-d < X < P+d. Example : \|ax + b\| < d is the same as writing \|ax - (-b)\| < d, which translates as "the distance of points ax from the point b is less than d. So, b-d < ax < b+d, or (b-d)/a < x < (b+d)/a Similar Discussions: Geometric Series	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8122845888137817, "perplexity": 1664.2838447644974}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-47/segments/1510934805417.47/warc/CC-MAIN-20171119061756-20171119081756-00727.warc.gz"}
https://www.newtonproject.ox.ac.uk/view/texts/diplomatic/NATP00345	<1r> ## Probleme An equation being given expressing the Relation of two or more lines x, y & z described in the same time by two or more moving bodies A, B, C, & to find the relation of their velocities p, q, r, &c ## Solution Set all ye terms on one side of the Equation that they become equal to nothing. Multiply each term by so many times $\frac{p}{x}$ as x hath dimensions in that term. Secondly multiply each term by so many times $\frac{q}{y}$ as y hath dimensions in it. Thirdly multiply each term by so many times $\frac{r}{z}$ as z hath dimensions in it &c. The summ of all these products shall be equal to nothing. Which Equation gives the relation of p, q, r, &c. ## Or more generally thus. Order the Equation according to the dimensions of x, & putting a & b for any two numbers whether rational or not, multiply the thers thereof by any part of this Progression $\frac{ap-3bp}{x}.\frac{ap-2bp}{x}.\frac{ap-bp}{x}.\frac{ap}{x}.\frac{ap+bp}{x}.\frac{ap+2bp}{x}.\frac{ap+3bp}{x}.&c.$ Also order the Equation according to the dimensions of y & multiply the terms of it by this Progression $\frac{aq-3bq}{y}.\frac{aq-2bq}{y}.\frac{aq-bq}{y}.\frac{aq}{y}.\frac{aq+bq}{y}.\frac{aq+2bq}{y}.\frac{aq+3bq}{y}.&c$ Also order it according to the dimensions of z & multiply its terms by this Progression $\frac{ar-3br}{z}.\frac{ar-2br}{z}.\frac{ar-br}{z}.\frac{ar}{z}.\frac{ar+br}{z}.\frac{ar+2br}{z}.\frac{ar+3br}{z}$. &c. The summ of these products shall bee equal to nothing. Which equation gives the relation of p, q, r, &c. ## Examples. Example 1. If the propounded Equation be ${x}^{3}-2xxy+4xx-7xyy-{y}^{3}-103=0$. By the precedent Rule the first operation will produce $3xxp-4xyp+8xp+7yyp$. The second produceth $-2xxq+14xyq-3yyq$. And these two added together make $3xxp-4xyp+8xp+7yyp-2xxq+14xyq-3yyq=0$. Example 2. If the Equation be ${x}^{3}-2aay+zzx-yyx+zyy-{z}^{3}=0$ the first operation will produce $3pxx+pzz-pyy$, the second $-2aaq-2yxq+2zyq$, the third $+2zxr-yyr-3zzr$. The summ of wch is $3pxx+pzz-pyy-2aaq-2yxq+2zyq+2zxr-yyr-3zzr=0$. ## Demonstration If two bodies A & B move uniformly, the one successively \succesively/ from a to c, e, g &c the other in the same times from b to d, f, h &c then are the lines ac & bd, ce & df, eg & fh &c in the same proportion to one another as are the velocities p & q wherewith the bodies describe them <2r> And tho they move not uniformly yet are the infinitely little lines wch each moment they describe, as the velocities are which they have while they describe them. As if the body A with the velocity p describe the infinitely little line po in one moment, the body B with the velocity q will describe the line {illeg}qo in the same moment For p.q:::po. {illeg}qo. So that if the described lines be x & y in one moment they will be x+po & y+{illeg}qo in the next moment. Now if the equation expressing the relation between the lines x & y be for instance $rx+xx-yy=0$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 16, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.965681791305542, "perplexity": 868.8327564039193}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300343.4/warc/CC-MAIN-20220117061125-20220117091125-00514.warc.gz"}
http://tex.stackexchange.com/questions/103536/label-for-package-fancyvrb	# label for package: fancyvrb I have a latex code for a computer program code like as specified in here. I want to write a label to this code. I searched for it but was unable to find how to add label for the source code. Any help will be great! To write a code in latex, you can use \begin{verbatim} \end{verbatim} environment. A similar approach is by doing this \usepackage{fancyvrb} \DefineVerbatimEnvironment{code}{Verbatim}{fontsize=\small} \DefineVerbatimEnvironment{example}{Verbatim}{fontsize=\small} and then you can type your code in the body \begin{code} c+=1; \end{code} - I think I didn't understand properly. But listings may a worth try. – Harish Kumar Mar 20 '13 at 23:03 Do you mean a TeX label, which you can reference elsewhere in the document with \ref? One way is to use the listings package instead of fancyvrb. Then, you can include your code in the document like this: \begin{lstlisting}[caption={Cool code},label=cool] c+=1; \end{lstlisting} As you would expect, the caption option provides a caption under the code listing (à la "Listing 1: Cool code"), and the label option provides a label which can be referenced like \ref{cool}. A caption must be provided for the label to work, at least in my installation. A MWE: \documentclass{article} \usepackage{listings} \begin{document} \begin{lstlisting}[caption={Cool code},label=cool] c+=1; \end{lstlisting} \begin{lstlisting}[caption={Wow},label=wow] c-=1; \end{lstlisting} As you can see in Listing~\ref{cool}, we can increment variables. Having thought better of it, Listing~\ref{wow} allows us to retreat. \end{document} If, instead, you wish to reference a particular line number of a code listing, you can stick with the fancyvrb package and do something like this example from the fancyvrb documentation: \begin{Verbatim}[commandchars=\\\{\},numbers=left,numbersep=2pt] First verbatim line. Second line.\label{vrb:Important} Third verbatim line. \end{Verbatim} As I previously showed in line~\ref{vrb:Important}, it is... You could make this simple to use by including the options in your \DeclareVerbatimEnvironment command, like so: \documentclass{article} \usepackage{fancyvrb} \DefineVerbatimEnvironment{code}{Verbatim}{fontsize=\small,commandchars=\\\{\},numbers=left,numbersep=2pt} \begin{document} \begin{code} c+=1;\label{cool} c-=1;\label{wow} \end{code} As you can see on Line~\ref{cool}, we can increment variables. Having thought better of it, Line~\ref{wow} allows us to retreat. \end{document} - Since it is tagged with cross-refencing I think you are right. – Speravir Mar 20 '13 at 23:11 Hey, with your edit you stole my answer !!! ;-) The label could, of course, also set in the first place of fancyvrb, and so the functionality should be very similar to listings then. – Speravir Mar 20 '13 at 23:20	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8454474806785583, "perplexity": 2175.666030071847}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-15/segments/1397609530895.48/warc/CC-MAIN-20140416005210-00601-ip-10-147-4-33.ec2.internal.warc.gz"}
http://mathhelpforum.com/statistics/171952-variation-location.html	# Math Help - Variation+Location 1. ## Variation+Location Q: Height of women have a bell shaped distribution with a mean of 157 cm and standard deviation 7 cm. Using Chebyshev's theorem, what do we know about the percentage of women with heights that are within 2 standards deviations of the mean? What are the minimum and maximum heights that are within 2 standard deviations of the mean? A: Atleast __% of women have heights within 2 standard deviations of 157 cm. (Round to the nearest percentage as needed) The minimum height that is within 2 standard deviation of the mean is __ cm? The maximum height that is within 2 standard deviation of the mean is __ cm? 2. Originally Posted by catchdillon Q: Height of women have a bell shaped distribution with a mean of 157 cm and standard deviation 7 cm. Using Chebyshev's theorem, what do we know about the percentage of women with heights that are within 2 standards deviations of the mean? What are the minimum and maximum heights that are within 2 standard deviations of the mean? A: Atleast __% of women have heights within 2 standard deviations of 157 cm. (Round to the nearest percentage as needed) The minimum height that is within 2 standard deviation of the mean is __ cm? The maximum height that is within 2 standard deviation of the mean is __ cm? What have you tried? Where are you stuck?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8815869092941284, "perplexity": 361.7086521739716}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701162035.80/warc/CC-MAIN-20160205193922-00216-ip-10-236-182-209.ec2.internal.warc.gz"}
http://demonstrations.wolfram.com/TrisectingAnAngleUsingTheCycloidOfCeva/	# Trisecting an Angle Using the Cycloid of Ceva Requires a Wolfram Notebook System Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products. Requires a Wolfram Notebook System Edit on desktop, mobile and cloud with any Wolfram Language product. The cycloid of Ceva has the polar equation . To trisect the angle , construct a line parallel to the polar axis (the positive axis). Let be the point of intersection of the cycloid and the line. Then the angle is one-third of the angle . Proof: let angle be and let the point on the axis be such that . Let be the orthogonal projection of on the line . The angle , so . Since , , . So angle equals , but . Contributed by: Izidor Hafner (October 2013) Open content licensed under CC BY-NC-SA ## Permanent Citation Izidor Hafner Feedback (field required) Email (field required) Name Occupation Organization Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback. Send	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8918366432189941, "perplexity": 2420.828403605961}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-47/segments/1542039744348.50/warc/CC-MAIN-20181118093845-20181118115845-00150.warc.gz"}
https://science.sciencemag.org/content/337/6101/1511	Report Kepler-47: A Transiting Circumbinary Multiplanet System See allHide authors and affiliations Science 21 Sep 2012: Vol. 337, Issue 6101, pp. 1511-1514 DOI: 10.1126/science.1228380 A Pair of Planets Around a Pair of Stars Most of the planets we know about orbit a single star; however, most of the stars in our galaxy are not single. Based on data from the Kepler space telescope, Orosz et al. (p. 1511, published online 28 August) report the detection of a pair of planets orbiting a pair of stars. These two planets are the smallest of the known transiting circumbinary planets and have the shortest and longest orbital periods. The outer planet resides in the habitable zone—the “goldilocks” region where the temperatures could allow liquid water to exist. This discovery establishes that, despite the chaotic environment around a close binary star, a system of planets can form and persist. Abstract We report the detection of Kepler-47, a system consisting of two planets orbiting around an eclipsing pair of stars. The inner and outer planets have radii 3.0 and 4.6 times that of Earth, respectively. The binary star consists of a Sun-like star and a companion roughly one-third its size, orbiting each other every 7.45 days. With an orbital period of 49.5 days, 18 transits of the inner planet have been observed, allowing a detailed characterization of its orbit and those of the stars. The outer planet’s orbital period is 303.2 days, and although the planet is not Earth-like, it resides within the classical “habitable zone,” where liquid water could exist on an Earth-like planet. With its two known planets, Kepler-47 establishes that close binary stars can host complete planetary systems. View Full Text	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8916020393371582, "perplexity": 1479.0501851862373}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107900860.51/warc/CC-MAIN-20201028191655-20201028221655-00377.warc.gz"}
https://baas.aas.org/pub/2022n3i110p90/release/1	# Radiative Relativistic MHD Simulations of Neutron Star Column Accretion in Cartesian Geometry Presentation #110.90 in the session “Stellar/Compact (Poster)”. Published onApr 01, 2022 Radiative Relativistic MHD Simulations of Neutron Star Column Accretion in Cartesian Geometry Strongly magnetized neutron stars in X-ray binaries accreting at supercritical luminosities form radiation pressure dominated accretion columns near their magnetic poles. Such columns are known to be intrinsically unstable and therefore cannot form stationary structures. We perform radiative relativistic MHD simulations to investigate the dynamics of these columns. We find that the overall vertical structure is maintained through nonlinear oscillations, regardless of the existence of the photon bubble instability. The oscillations are mainly driven by an imbalance between gravity and radiation support in the sinking region, where the accretion power would otherwise fail to replenish the radiative losses. The existence of this oscillatory behavior therefore facilitates the redistribution of the accretion power to maintain the column structure. When the photon bubble instability is resolved in the simulation, an incoherent spatial structure forms below the shock front. Such a structure creates cavities at the shock front and allows the radiation to penetrate the shock front, which leads to the formation of a secondary shock structure above. However, this complicated dynamical effect seems not to alter the oscillation period. In the future, we will post-process these simulations to predict spectral, polarization, and timing observables. We are also pursuing more global simulations in order to examine if the above dynamics still holds.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9651795625686646, "perplexity": 1438.8565129280332}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-27/segments/1656103034877.9/warc/CC-MAIN-20220625065404-20220625095404-00274.warc.gz"}
http://www.msri.org/workshops/202/schedules/1160	Mathematical Sciences Research Institute Home » Workshop » Schedules » Efficient quantum algorithms for estimating Gauss sums (with Gadiel Seroussi) Efficient quantum algorithms for estimating Gauss sums (with Gadiel Seroussi) Quantum Algorithms and Complexity September 23, 2002 - September 27, 2002 September 25, 2002 (11:00 AM PDT - 11:30 AM PDT) Speaker(s): Willem van Dam Location: MSRI: Simons Auditorium Video No Consent	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8117492198944092, "perplexity": 4194.523598133483}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-22/segments/1432207930866.20/warc/CC-MAIN-20150521113210-00074-ip-10-180-206-219.ec2.internal.warc.gz"}
https://www.atmos-chem-phys.net/19/9595/2019/	Journal topic Atmos. Chem. Phys., 19, 9595–9611, 2019 https://doi.org/10.5194/acp-19-9595-2019 Atmos. Chem. Phys., 19, 9595–9611, 2019 https://doi.org/10.5194/acp-19-9595-2019 Research article 31 Jul 2019 Research article \| 31 Jul 2019 # Observation of absorbing aerosols above clouds over the south-east Atlantic Ocean from the geostationary satellite SEVIRI – Part 1: Method description and sensitivity Observation of absorbing aerosols above clouds over the south-east Atlantic Ocean from the geostationary satellite SEVIRI – Part 1: Method description and sensitivity Fanny Peers1, Peter Francis2, Cathryn Fox2, Steven J. Abel2, Kate Szpek2, Michael I. Cotterell1,2, Nicholas W. Davies1,2, Justin M. Langridge2, Kerry G. Meyer3, Steven E. Platnick3, and Jim M. Haywood1,2 Fanny Peers et al. • 1College of Engineering, Mathematics, and Physical Sciences, University of Exeter, Exeter, UK • 2Met Office, Fitzroy Road, Exeter, UK • 3NASA Goddard Space Flight Center, Greenbelt, Maryland, USA Correspondence: Fanny Peers ([email protected]) Abstract High-temporal-resolution observations from satellites have a great potential for studying the impact of biomass burning aerosols and clouds over the south-east Atlantic Ocean (SEAO). This paper presents a method developed to simultaneously retrieve aerosol and cloud properties in aerosol above-cloud conditions from the geostationary instrument Meteosat Second Generation/Spinning Enhanced Visible and Infrared Imager (MSG/SEVIRI). The above-cloud aerosol optical thickness (AOT), the cloud optical thickness (COT) and the cloud droplet effective radius (CER) are derived from the spectral contrast and the magnitude of the signal measured in three channels in the visible to shortwave infrared region. The impact of the absorption from atmospheric gases on the satellite signal is corrected by applying transmittances calculated using the water vapour profiles from a Met Office forecast model. The sensitivity analysis shows that a 10 % error on the humidity profile leads to an 18.5 % bias on the above-cloud AOT, which highlights the importance of an accurate atmospheric correction scheme. In situ measurements from the CLARIFY-2017 airborne field campaign are used to constrain the aerosol size distribution and refractive index that is assumed for the aforementioned retrieval algorithm. The sensitivities in the retrieved AOT, COT and CER to the aerosol model assumptions are assessed. Between 09:00 and 15:00 UTC, an uncertainty of 40 % is estimated on the above-cloud AOT, which is dominated by the sensitivity of the retrieval to the single-scattering albedo. The absorption AOT is less sensitive to the aerosol assumptions with an uncertainty generally lower than 17 % between 09:00 and 15:00 UTC. Outside of that time range, as the scattering angle decreases, the sensitivity of the AOT and the absorption AOT to the aerosol model increases. The retrieved cloud properties are only weakly sensitive to the aerosol model assumptions throughout the day, with biases lower than 6 % on the COT and 3 % on the CER. The stability of the retrieval over time is analysed. For observations outside of the backscattering glory region, the time series of the aerosol and cloud properties are physically consistent, which confirms the ability of the retrieval to monitor the temporal evolution of aerosol above-cloud events over the SEAO. 1 Introduction Until recently, there has been a relative dearth of observations of biomass burning above clouds as passive sensor retrievals of aerosol and cloud are generally mutually exclusive. In past studies, biases in cloud properties derived from passive shortwave measurements were expected because the impact of aerosol absorption above clouds was not taken into account in the retrievals (Haywood et al., 2004). Over the last decade, techniques have been developed for the observation of aerosols above clouds. POLDER (Polarization measurements from POLarization and Directionality of the Earth's Reflectances) has been used to detect aerosols above clouds and to characterize the aerosol and the cloud layers by exploiting the sensitivity in polarized measurements (Waquet et al., 2013a, b; Peers et al., 2015). In the case of fine-mode absorbing aerosols overlying clouds, the absorption Ångström exponent leads to a greater impact on radiances reflected by the clouds at shorter wavelengths than longer ones (De Graaf et al., 2012; Torres et al., 2012). The “colour-ratio” approach has been applied to OMI (Ozone Monitoring Instrument – Torres et al., 2012) and MODIS (Moderate Resolution Imaging Spectroradiometer – Jethva et al., 2013) to simultaneously retrieve the aerosol and the cloud optical thicknesses over the SEAO. Using a similar technique, the MODIS retrieval developed by Meyer et al. (2015) takes advantage of the six channels of the instrument from the UV to the shortwave infrared (SWIR) range to characterize not only the aerosol and cloud optical thicknesses, but also the cloud droplet effective radius. For the first time, these studies have provided large-scale observations of aerosols above clouds in the SEAO. However, these approaches have been applied to satellite instruments on polar-orbiting platforms that provide only two observations per day for MODIS (on the Aqua and Terra platforms) and one for OMI and POLDER. The cloud cover over the SEAO has an important diurnal cycle which modulates the DRE of aerosols during the day (Min and Zhang, 2014). Therefore, the study of the SEAO cloud and above-cloud aerosol optical properties would benefit from the high-temporal-resolution observations provided by geostationary satellite platforms. Chang and Christopher (2016) have highlighted the ability of SEVIRI (Spinning Enhanced Visible and Infrared Imager) to identify absorbing aerosols above clouds at high temporal resolution. The instrument is on board the geostationary satellite MSG (Meteosat Second Generation) and provides a full-disc observation every 15 min, offering a unique opportunity to monitor the evolution of the cloud cover and to track aerosol plumes over the SEAO. The objective of this two-part paper is to demonstrate the potential of this instrument to simultaneously retrieve aerosol and cloud properties in the case of absorbing aerosols above clouds. In this first contribution, we describe the approach used to derive the above-cloud aerosol optical thickness (AOT), the cloud optical thickness (COT) and the cloud droplet effective radius (CER) and discuss the accuracy of the retrievals. The algorithm, as well as the atmospheric correction scheme and the assumed aerosol model, are presented in Sect. 2. The sensitivities in the retrieved quantities to the water vapour profile and the aerosol property assumptions are assessed in Sect. 3. The evaluation of the stability of the retrieval is shown in Sect. 4 and conclusions are drawn in Sect. 5. In a second companion paper, we will compare our SEVIRI-based retrievals of cloud and aerosol properties with those from MODIS products (Meyer et al., 2015) more comprehensively and also compare against in situ aircraft observations from the CLARIFY-2017 field campaign. Figure 1Radiance ratio R0.64R0.81 as a function of the radiance at 0.81 µm for absorbing aerosols above clouds simulated with the adding–doubling method (De Haan et al., 1987). COTs and AOTs are indicated at 0.55 µm. 2 Retrieval method ## 2.1 Principle The approach used to retrieve aerosol and cloud properties from satellite spectral radiance measurements relies on the colour-ratio effect (Jethva et al., 2013). The signal backscattered by a liquid cloud is almost spectrally neutral from the UV to the near-infrared (NIR) ranges. Conversely, the absorption from biomass burning aerosols is typically larger at shorter wavelengths. Therefore, the presence of absorbing aerosols above clouds modifies the apparent colour of clouds. This enhancement of the spectral contrast can be detected by any passive remote-sensing instrument with two channels with enough separation in the UV–NIR region. The SEVIRI instrument, aboard the MSG satellite (Aminou et al., 1997), has channels centred at 0.64, in the visible, and at 0.81 µm, in the NIR ranges. Figure 1 plots the 0.81 µm radiance (R0.81) against the ratio of the 0.64 to 0.81 µm radiances (R0.64R0.81), for absorbing aerosols above clouds over an ocean surface for several aerosol and cloud optical thicknesses. Throughout this paper, the radiances R refer to the normalized quantity as defined by Herman et al. (2005) and the optical thicknesses (i.e. AOT, COT) are given at 0.55 µm. The simulations have been performed with the adding–doubling method (De Haan et al., 1987), considering a viewing geometry of 20 for the solar zenith angle, 50 for the viewing zenith angle and 140 for the relative azimuth. The cloud is located between 0 and 1 km and the aerosol layer is between 2 and 3 km. Aerosols have a refractive index of 1.54–0.025i and the size distribution follows a lognormal with a geometric mean radius of 0.1 µm. The cloud droplets have an effective radius of 10 µm. Rayleigh scattering has been accounted for but the simulations do not include the absorption from atmospheric gases. A Lambertian surface with an albedo of 0.05 is assumed. For AOT = 0, the radiance ratio is around 1 and weakly depends on the COT. As the AOT increases, the radiance at 0.81 µm as well as the radiance ratio decreases, indicating that the attenuation from the aerosol layer is larger at 0.64 µm. This attenuation is mainly due to the absorption from the aerosol layer, which means that it is primarily correlated to the absorption AOT (AAOT). Figure 2Simulated radiances at 1.64 and 0.81 µm for clouds with varying COTs and CERs (µm), without (blue) and with (orange and red) overlying absorbing aerosols above. The viewing geometry, the aerosol and the cloud properties are the same as in Fig. 1. As in the Nakajima and King technique (1990), the sensitivity of the retrieval to the CER comes from the measurements of the shortwave infrared (SWIR) channel of SEVIRI centred at 1.64 µm. Figure 2 shows the radiances at 0.81 and 1.64 µm for several COTs and CERs as well as the impact of overlying absorbing aerosols. The simulations without aerosol are plotted in blue and represent the signal typically used by cloud property retrievals that do not include light absorption from overlying aerosols. The orange and red grids are associated with an AOT of 0.5 and 1.5 at 0.55 µm. Compared to the no-aerosol case, these grids are shifted towards the upper left, which means that the presence of aerosols decreases the NIR radiance and increases the SWIR signal. As highlighted by Haywood et al. (2004), not taking into account the aerosol absorption above clouds leads to low biases in both the COT and the CER. These biases depend on the aerosol loading as well as on the brightness of the underlying cloud. Although the aerosol microphysical properties have some influence on the signal measured by satellites, this kind of approach requires us to assume an aerosol model. Fundamentally, the algorithm developed here aims to retrieve the above-cloud AOT, the COT and the CER from the magnitude and the gradient of the radiances measured by SEVIRI at 0.64, 0.81 and 1.64 µm using a basic look-up table (LUT) approach and appropriate assumptions about the aerosol model for the region (Haywood et al., 2003) that have been refined based on measurements from the CLARIFY-2017 observational campaign (Zuidema et al., 2016). Figure 3Spectral response function of the SEVIRI bands at 0.64 (a), 0.81 (b) and 1.64 µm (c) with the corresponding MODIS ones (dashed lines) as well as the atmospheric transmittance within the spectral range (in colour). The transmittances have been calculated with the SOCRATES radiative transfer scheme (Manners et al., 2015; Edwards and Slingo, 1996) assuming a humidity profile measured during SAFARI (Keil and Haywood, 2003). In the legend of each plot, the transmittance weighted by the spectral response function is given for the main absorbing gases. ## 2.2 Atmospheric correction The SEVIRI channels chosen for the retrieval are fairly standard in atmospheric science and have been widely used for aerosol and cloud analysis (e.g. Brindley and Ignatov, 2006; Thieuleux et al., 2005; Watts et al., 1998). However, the SEVIRI bandwidths are much larger than other state-of-the-art instruments such as MODIS. Hence, SEVIRI radiances are significantly more impacted by the absorption from various atmospheric gases. The spectral response functions for the 0.64, 0.81 and 1.64 µm SEVIRI channels are plotted in Fig. 3 together with the equivalent MODIS bands. The main absorbing gases in these spectral bands are ozone, water vapour, methane and carbon dioxide, gases which are typically produced and transported within biomass burning plumes (Browell et al., 1996; Koppmann et al., 2005). The contributions of each gas to the atmospheric absorption are also shown in Fig. 3 and the two-way transmittances (i.e. from the top of the atmosphere to the cloud top and from the cloud top to the top of the atmosphere) weighted by the spectral response function have been calculated. For the sake of simplicity, the two-way transmittances will be referred to as transmittances. Although the MODIS bandwidths are narrower than the SEVIRI ones, the weighted transmittances are similar for the 0.64 and 1.64 µm channels. In the NIR, the MODIS central wavelength (0.86 µm) is slightly larger than for SEVIRI (0.81 µm) and the spectral band is only weakly impacted by the humidity, with a weighted transmittance of 0.989. Within the SEVIRI band, water vapour absorption is much higher, with a transmittance of 0.931. As a result, humidity has an impact on the spectral contrast between the VIS and the NIR, and therefore on the above-cloud AOT retrieval. The atmospheric correction, especially for the water vapour, is essential to accurately retrieve the aerosol and cloud properties from SEVIRI. In order to correct the SEVIRI measurements for atmospheric absorption, the transmittances Tatm,λ are calculated for each spectral band λ from the cloud top height to the top of the atmosphere using the fast radiative transfer model RTTOV (Matricardi et al., 2004; Hocking et al., 2014). The cloud top height is derived from the Met Office cloud property algorithm, which uses the 10.8, 12.0 and 13.4 µm channels of SEVIRI (Francis et al., 2008; Hamann et al., 2014). Water vapour profiles come from the operational forecast configuration of the global Met Office Unified Model (Brown et al., 2012). This forecast is assimilated according to the scheme described by Clayton et al. (2013) that uses humidity data from various sources, including radiosondes and remote-sensing sounding data from many meteorological satellites. The forecast is run every 6 h and the humidity profile used for the atmospheric correction comes from the latest time-appropriate forecast field available. The profiles of the remaining gases – including ozone, carbon dioxide and methane – are those implicitly assumed by the RTTOV calculations (Matricardi, 2008). The radiance measured by SEVIRI Ratm,λ is finally corrected using $\begin{array}{}\text{(1)}& {R}_{\mathrm{atm},\mathit{\lambda }}={T}_{\mathrm{atm},\mathit{\lambda }}{R}_{\mathit{\lambda }},\end{array}$ where Rλ is the radiance corrected from the gaseous absorption. ## 2.3 Aerosol model The choice of the aerosol microphysical properties to use for the retrieval is similar to that of Haywood et al. (2003), but based on more comprehensive in situ measurements acquired during the CLARIFY-2017 field campaign. The Facility for Airborne Atmospheric Measurements (FAAM) BAe 146 aircraft was deployed in August–September 2017 operating from Ascension Island, with a main objective of studying biomass burning aerosol interactions with both radiation and clouds over the SEAO. This analysis focuses on flight C050, performed on 4 September, 2017. A profile descent from 7.3 to 1.9 km altitude was performed in order to sample the aerosol layer above clouds. The aerosol dry extinction and absorption were measured with the EXSCALABAR instrument (EXtinction, SCattering and Absorption of Light for AirBorne Aerosol Research), which consists of a series of cavity ring-down and photoacoustic absorption cells operating at different wavelengths (Davies et al., 2018). From these in situ measurements, the single-scattering albedo (SSA) has been calculated at the instrument wavelengths of 405 and 658 nm. The uncertainty in SSA calculations is related to the corresponding uncertainties in the extinction and absorption coefficients measured by EXSCALABAR. This error analysis has been performed previously and the reader is directed to Davies et al. (2019). Briefly, the measured extinction has an accuracy of ∼2 %, and we use a 2 % extinction uncertainty in the analysis here. The errors in absorption measurements using photoacoustic spectroscopy depend on uncertainties in the ozone calibration, microphone pressure dependence and the background response from laser scattering/absorption on the windows of the photoacoustic cell. We have shown in recent publications that our calibration uncertainties are ∼5 % (Cotterell et al., 2019; Davies et al., 2018), and the uncertainty in the pressure-dependent microphone response is 1.2 % (Davies et al., 2019). The background response from laser-window interactions is from 0.27 to 0.54 Mm−1. Thus, the total absorption uncertainty, propagating all the above uncertainties, is absorption-dependent and ranges from 29.0 % to 55.0 % (dependent on PAS measurement wavelength) at 1 Mm−1 and 8.1 % at 100 Mm−1 (independent of PAS measurement wavelength). We propagated these total measurement uncertainties for both extinction and absorption measurements to derive the standard deviation σ in our calculated SSA values. We find that the mean SSA uncertainties are 0.013 and 0.018 at the measurement wavelengths of 405 and 658 nm respectively. The aerosol size distribution was characterized between 0.05 and 1.50 µm radius using a wing-mounted passive cavity aerosol spectrometer probe (PCASP). Before and after the campaign, the bin sizes of the PCASP were calibrated using aerosolized diethyhexyl sebacate and polystyrene latex of known size and refractive index (Rosenberg et al., 2012). Further calculations based on Mie-scattering theory are performed in order to determine the bin sizes at the refractive index of the biomass burning aerosol sample. Partial evaporation of water is expected in the PCASP due to the heating of the probe, which may decrease the aerosol size. However, the sonde dropped during the flight indicates an average relative humidity above clouds of 29.2 % with a maximum of 38.6 %. According to Magi and Hobbs (2003), the light scattering coefficient of an aged African biomass burning plume only increases by a factor of 1.01 for a relative humidity of 40 %. For this reason, the impact of humidity on the PCASP and EXSCALABAR measurements is neglected. Three sources of errors have been taken into account on the PCASP measurements: the error on the bin concentration is calculated according to Poisson counting statistics, the sample flow rate error is assumed to be 10 % and a bin edge calibration error of half a bin has been considered. Figure 4Normalized size distribution (a) and SSA (b) measured above clouds during flight C050 of the CLARIFY-2017 campaign (black). The grey shaded area represents the PCASP measurement and calibration uncertainties. Blue lines represent the fitted aerosol model, the orange lines correspond to the aged aerosol size distribution from SAFARI (Haywood et al., 2003), and the dashed lines show the contribution of each mode. CLARIFY-2017 aerosol model: [Rfine, σfine, Nfine; Rcoarse, σcoarse, Ncoarse] = [0.12 µm, 1.42, 0.9996; 0.62 µm, 2.23, 0.0004], refractive index = 1.51–0.029i. SAFARI aged aerosol model: [R1, σ1, N1; R2, σ2, N2; R3, σ3, N3] = [0.12 µm, 1.30, 0.996; 0.26 µm, 1.50, 0.0033; 0.80 µm, 1.90, 0.0007]. The aerosol properties needed for the SEVIRI retrieval include the size distribution and the complex refractive index. The normalized number size distribution (dN∕dln r) is commonly represented by a combination of lognormal modes: $\begin{array}{}\text{(2)}& \frac{\mathrm{d}\phantom{\rule{0.125em}{0ex}}N}{\mathrm{d}\phantom{\rule{0.125em}{0ex}}\mathrm{ln}r}=\phantom{\rule{0.125em}{0ex}}\sum _{i}\frac{{N}_{i}}{\sqrt{\mathrm{2}\mathit{\pi }}}\frac{\mathrm{1}}{\mathrm{ln}{\mathit{\sigma }}_{i}}\phantom{\rule{0.125em}{0ex}}\mathrm{exp}\left[\frac{-\left(\mathrm{ln}{r}_{i}-\mathrm{ln}r{\right)}^{\mathrm{2}}}{\mathrm{2}\left(\mathrm{ln}{\mathit{\sigma }}_{i}{\right)}^{\mathrm{2}}}\right],\end{array}$ where Ni, ri and σi are the number fraction, the geometric mean radii and the standard deviation of the mode i respectively. As in most remote-sensing applications, it has been chosen to represent the particle size distribution for the aerosol during CLARIFY-2017 with fine- and coarse-mode contributions. The aerosol optical properties are calculated using the Mie theory, as the spherical approximation is expected to be valid for biomass burning particles from 1 h after being released in the atmosphere (Martins et al., 1998). The aerosol model is selected by iteratively adjusting the refractive index and fitting the PCASP measurements (Fig. 4a) until the aerosol model matches the SSA from EXSCALABAR (Fig. 4b). In order to obtain the most suitable aerosol optical parameters for the retrieval, it is important to accurately fit the PCASP measurements where the aerosols contribute the most to the SEVIRI signal. Each bin of the PCASP has been assigned a weight for the fit of the bimodal distribution. The weights have been calculated in a similar way to Haywood et al. (2003), which means that they are proportional to the contribution of each bin to the total aerosol extinction in the 0.6 µm band. The bins corresponding to the 0.15 to 0.25 µm radius range contribute about 77 % of the extinction. Consequently, these bins have been assigned appropriate larger weights during the fitting process of the size distribution. Due to the small fraction of coarse-mode aerosols, the standard deviation of this mode σcoarse could not be reliably fitted and has been set to a value of 2.23, which is within the same order of magnitude as the one assumed for absorbing aerosol (∼2.12) in the MODIS Dark Target operational algorithm (Levy et al., 2009). The aerosol model that best represents the PCASP and EXSCALABAR measurements is shown in blue in Fig. 4a, b. A refractive index of 1.51–0.029i has been obtained, associated with an SSA of 0.85 at 0.55 µm, which is within the range of SSA measured over the SEAO during the SAFARI and the DABEX campaigns (Johnson et al., 2008) and on the upper end of the values from Ascension Island reported by Zuidema et al. (2018). Regarding the refractive index, it should be noted that the SSA is not very sensitive to the real part, suggesting that the value of 1.51 is not particularly well constrained. However, a real part of 1.51 is consistent with the AERONET retrievals for African biomass burning particles (Sayer et al., 2014) and is adopted here. The best-fit size distribution is characterized by [Rfine, σfine, Nfine; Rcoarse, σcoarse, Ncoarse] = [0.12 µm, 1.42, 0.9996; 0.62 µm, 2.23, 0.0004]. By way of comparison, the three-mode lognormal distribution obtained for aged biomass burning aerosols during the SAFARI 2000 campaign (Haywood et al., 2003), defined by [R1, σ1, N1; R2, σ2, N2; R3, σ3, N3] = [0.12 µm, 1.30, 0.996; 0.26 µm, 1.50, 0.0033; 0.80 µm, 1.90, 0.0007], is plotted in orange in Fig. 4a. The radius associated with the first mode is consistent with the CLARIFY-2017 model. The absence of the second fine mode in this study is compensated for by a larger standard deviation for the fine mode. Finally, the radius of the CLARIFY-2017 coarse mode is slightly smaller than the SAFARI-2000 one but the coarse-mode fractions of the two models are close to each other. The uncertainties on the aerosol properties have been estimated using the errors on the PCASP and EXSCALABAR measurements. The uncertainty on the imaginary part of the refractive index is 0.02 for the real part and 0.004 for the imaginary part. For the size distribution, the uncertainty is 0.016 µm, 0.09 and 0.00045 for the radius, standard deviation and number fraction of the fine mode respectively. Table 1Aerosol and cloud properties used to compute the radiances LUT of the SEVIRI retrieval. Note that 0.55 µm does not correspond to a SEVIRI channel. ## 2.4 Algorithm The algorithm relies on the comparison of the corrected SEVIRI signal at 0.64, 0.81 and 1.64 µm with precomputed radiances. The simulations have been performed using an adding–doubling radiative transfer code (De Haan et al., 1987). The surface is assumed to be Lambertian with an albedo of 0.05 at all wavelengths, which is typical of the sea surface albedo under diffuse radiation conditions. The aerosol and cloud properties assumed for the LUT are summarized in Table 1. The truncation of the cloud droplet phase function has been carried out using the delta-M method (Wiscombe, 1977) and the TMS correction (Nakajima and Tanaka, 1988) has been applied. The cloud layer is assumed to be located between 0 and 1 km and the aerosol layer between 2 and 3 km. The sensitivity of the algorithm to the altitudes of the aerosol and cloud layers is expected to be negligible due to the small contribution of the Rayleigh scattering to the signal at the SEVIRI wavelengths. We have evaluated the error due to the fixed aerosol and cloud altitudes to be lower than 2.5 % on the AOT and 0.3 % on the cloud properties. The cloud droplets are assumed to follow a gamma law distribution characterized by an effective variance of 0.06. When the cloud is optically thin and/or the cloud droplets are too small, it is not possible to separate the contribution to the optical signal arising from aerosols from that of clouds. Therefore, the minimum values for the CER and the COT in the LUT are 4 and 3 µm respectively. This also justifies the assumption of a relatively simple sea surface reflectance parameterization as, at COTs exceeding 3, the sea surface has little impact on the upwelling radiances above clouds. Clouds associated with lower COT and/or CER are rejected. The aerosol model corresponds to the CLARIFY-2017 model mentioned above, assuming the same refractive index at the three SEVIRI wavelengths. Figure 5RGB composite (a), above-cloud AOT at 0.55 µm (b) and cloud properties (c and d) retrieved from SEVIRI measurements on 28 August 2017 at 10:12 UTC over the SEAO. The retrieval of the above-cloud AOT, COT and CER is performed simultaneously. The result corresponds to the parameters that minimize the difference ε between the simulated radiances Rsim and the corrected satellite signal Rλ: $\begin{array}{}\text{(3)}& \mathit{\epsilon }=\phantom{\rule{0.125em}{0ex}}\sum _{\mathit{\lambda }}{\left(\frac{{R}_{\mathit{\lambda }}-{R}_{\mathrm{sim},\mathit{\lambda }}}{{R}_{\mathit{\lambda }}}\right)}^{\mathrm{2}}.\end{array}$ When the simulated signal is not close enough to the satellite measurements (i.e. ε>0.0006), the result is rejected. The retrieval of the above-cloud AOT is highly uncertain at the cloud edges and for inhomogeneous clouds. In order to remove these results, the products are aggregated onto a $\mathrm{0.1}{}^{\circ }×\mathrm{0.1}{}^{\circ }$ grid and the standard deviation of the AOT and the CER are calculated. Note that each grid cell represents approximately 12 SEVIRI pixels. The inhomogeneity parameter ρ is defined by the ratio of the standard deviation of a parameter to the average value of this parameter. The results corresponding to a standard deviation of the AOT larger than 0.7 and/or ρCER>0.2 as well as grid cells associated with fewer than nine successful retrievals are rejected. It is important to realize that the uncertainties that we quantify here are structural and parametric uncertainties related to assumptions made in the retrieval algorithm. When using a fixed aerosol model, no account is made for natural variability in the aerosol optical parameters and the associated uncertainty; this is dealt with in the uncertainty analysis that follows. Figure 6Above-cloud AOT at 0.55 µm (a) and cloud properties (b and c) retrieved from MODIS Terra with the MOD06ACAERO algorithm (Meyer et al., 2015) on 28 August 2017. 3 Results and uncertainty analysis ## 3.1 Case study The algorithm has been applied to an event of biomass burning aerosols above clouds captured by SEVIRI on 28 August 2017 at 10:12 UTC. The RGB composite and the retrieved above-cloud AOT, COT and CER over the SEAO region are shown in Fig. 5. The largest AOTs are observed off the coast of Angola, with a local average value of 1.0 and a maximum of 1.6 at 0.55 µm. The AERONET site of Lubango (14.96 S–13.45 E) measured an average AOT of 0.75 that day with an Ångström exponent of 1.83, indicating the expected domination of fine-mode biomass burning aerosols. A gradient of AOT is observed towards the south-west, as we move away from the source as might be expected from a pre-campaign analysis of satellite retrievals (Zuidema et al., 2016). Absorbing aerosols above clouds are also detected in the north-west part of the region. Around Ascension Island (7.98 S–14.42 W), the above-cloud AOT from SEVIRI is around 0.37 while the AERONET site indicates a value of 0.48 associated with an Ångström exponent of 1.271. This suggests that coarse-mode aerosols, such as sea salt within the boundary layer but generally below cloud, are contributing to the total column aerosol load. The cloud properties retrieved are within the range of values typically observed for marine stratocumulus (Szczodrak et al., 2001) with more than 90 % of the COT lower than 25 and 99 % of the CER between 4 and 20 µm. As a comparison, Fig. 6 shows the equivalent aerosol and cloud properties retrieved from MODIS Terra with the MOD06ACAERO algorithm (Meyer et al., 2015) for the 10:00 and 11:30 UTC overpasses. The MODIS above-cloud AOT pixels associated with an uncertainty larger than 100 % have been removed. A good spatial agreement is observed between the two satellite products. The above-cloud AOT from MODIS is also 1.0 on average close to the coast. On average over the area, the MODIS above-cloud AOT is larger by 0.05 compared to SEVIRI. Considering that MODIS is less sensitive to the atmospheric absorption and that the two algorithms are based on the same principle, the small differences observed between the two above-cloud AOT tend to validate the atmospheric correction applied to the SEVIRI measurements for that case. There is a good consistency between the MODIS and the SEVIRI COT. Finally, the CER retrieved with the MOD06ACAERO algorithm is larger by 2.2 µm compared to the SEVIRI CER. This almost systematic difference is mainly due to differences in the satellite instruments, and especially the difference in the channels used for the retrieval (Platnick, 2000). A fully statistical analysis against the MODIS algorithm, and against airborne remote-sensing and in situ measurements will be presented in a companion paper. ## 3.2 Atmospheric correction The atmospheric transmittances above clouds used to correct the SEVIRI measurements from the gas absorption are calculated based on forecast water vapour profiles. In order to assess the sensitivity of the retrieval to the atmospheric correction, new transmittances have been calculated for the event studied here, modifying the specific humidity by ±10 %. The aerosol and cloud properties retrieved with the modified atmospheric corrections are aggregated on a $\mathrm{0.1}{}^{\circ }×\mathrm{0.1}{}^{\circ }$ grid. Figure 7 compares the retrieved aerosol and cloud properties from SEVIRI-measured radiances using the original specific humidity forecast with the perturbed specific humidity (+10 % in orange and −10 % in blue). The uncertainty on the water vapour content impacts mainly the retrieval of the above-cloud AOT, and then the COT, because of its effect on the radiance ratio. A +10 %/−10 % bias on the humidity leads to an overestimation/underestimation of the AOT and COT respectively. On average, errors of 18.5 %, 5.5 % and 2.3 % have been calculated for the AOT, COT and CER respectively, based on biases of 10 % in the specific humidity forecast. These errors are likely upper estimates because forecast errors in specific humidity are unlikely to reach these values owing to the extensive assimilation of satellite data and sonde profiles by the data assimilation process used in the Met Office forecast model as previously mentioned. However, the differences between forecast model specific humidities and those of simple standard atmosphere climatological values (e.g. those of McClatchey et al., 1972) frequently exceed 10 %, indicating that accurate retrievals of aerosol and cloud need synergistic retrievals or data-assimilated forecasts of specific humidity. Figure 7Uncertainty in the retrieved above-cloud AOT (a), COT (b) and CER (c) due to an error of +10 % in orange and −10 % in blue on the specific humidity profile compared to the original forecast for 28 August 2017 at 10:12 UTC. ## 3.3 Aerosol model The LUT used for the SEVIRI retrieval uses an assumed aerosol model based on in situ measurements from CLARIFY-2017. However, the absorption property and the size of biomass burning particles are expected to vary during the fire season and across the SEAO (e.g. Eck et al., 2003). Here, we analyse the impact of the aerosol assumptions on the retrieved aerosol and cloud properties. Figure 8Histograms of the SSA (a) and asymmetry factor g (b) at 0.55 µm simulated from a range of size distributions and refractive indices (orange) and retrieved by AERONET (blue) over southern Africa. Dashed lines represent the mean ± the standard deviation. In order to create a range of aerosol optical properties, a thousand aerosol models have been processed using the Mie theory. The radius and the standard deviation of the fine mode and the real and imaginary part of the refractive index of the models are random values following a normal distribution. Their mean corresponds to the CLARIFY model values provided in Table 1, with standard deviations of 0.01 µm and 0.1 for the radius and the standard deviation of the fine mode, 0.02 for the real part of the refractive index, and 0.008 for the imaginary part. Figure 8a and b show the histograms of the simulated SSA and asymmetry factor g at 0.55 µm in orange. As a comparison, histograms of the AERONET SSA and g are plotted in blue. The data correspond to the AERONET level 2.0 retrievals for August–September, from 1997 to 2018 and for inland sites of southern Africa (10–35 S, 10–40 E). Only data associated with an Ångström exponent larger than 1.0 have been used in order to remove measurements dominated by coarse-mode particles (such as dust and sea salt) that are less likely to be observed above clouds in the SEAO. The mean SSA (0.862) and the mean g (0.620) from AERONET are respectively slightly larger and smaller than the CLARIFY model. Small differences between above-cloud and full column aerosol properties could be explained by the contribution of aerosol within the boundary layer, such as pollution, desert dust and sea salt. The dashed lines in Fig. 8a and b represent the mean ± the standard deviation of SSA and g. The AERONET standard deviation is 0.023 for the SSA and 0.024 for g while the simulation produces a standard deviation of 0.036 for the SSA and 0.041 for g. The simulated range of both optical properties is larger than the range observed by AERONET. Therefore, the variation in the aerosol microphysical properties used for the simulations is wide enough to cover the range of observed aerosol optical properties. Figure 9Impact of the assumption on the SSA and the asymmetry factor g on the retrieved aerosol and cloud properties. AOT, AAOT, COT and CER obtained for 28 August 2017 at 10:12 UTC with the CLARIFY-2017 model are plotted against the properties retrieved with the modified aerosol models. Table 2Aerosol properties used to test the sensitivity of the SEVIRI retrieval to the aerosol model assumption. SSA and g are given at 0.55 µm. From the simulated standard deviation σ of g and SSA, eight aerosol models have been defined and their properties are summarized in Table 2. The first four are used to test the sensitivity of the retrieval to g and SSA independently ([SSACLARIFY±σSSA, gCLARIFY] and [SSACLARIFY, gCLARIFY±σg]) and the sensitivity to both parameters will be assessed with the last four ([SSACLARIFY±σSSA, gCLARIFY±σg]). New LUTs have been processed with these modified aerosol models and used to re-process the case study from Sect. 3a. After aggregating the data on a $\mathrm{0.1}{}^{\circ }×\mathrm{0.1}{}^{\circ }$ grid, the AOT as well as the absorption AOT (AAOT), the COT and the CER are compared against those obtained with the standard CLARIFY-2017 aerosol model. Results are shown in Figs. 9 and 10. For each aerosol and cloud property, a linear relationship is observed between the retrieval using the standard CLARIFY-2017 aerosol model and the modified one. The retrieval of cloud properties (Figs. 9c, d and 10c, d) appears to be weakly sensitive to the assumed aerosol model, with g having a slightly larger impact. On average, differences lower than 4.1 % are observed on the COT and lower than 2.4 % on the CER. As expected, the choice of the aerosol model has much more influence on the AOT retrieval. The uncertainty on the AOT is dominated by the SSA assumption. When aerosols are more absorbing than the CLARIFY model, the algorithm overestimates the AOT by 25.7 %. Conversely, the retrieved AOT is underestimated by 32.6 % when aerosols are less absorbing than the CLARIFY model. The impact of g alone on the retrieved AOT is far less significant and lower than 4.3 %. Figure 9a, which shows the impact of a perturbation on both the SSA and g, confirms that the SSA is the parameter with the strongest influence on the AOT retrieval. The largest overestimation (27.5 %) is observed when both the SSA and g are overestimated (Fig. 10a), while the largest underestimation (−33.3 %) is obtained when the SSA is underestimated and g is overestimated. The retrieval of the above-cloud AOT depends mostly on the aerosol absorption of the light reflected by the cloud. Therefore, it is expected that the retrieved AAOT is less sensitive to the absorbing property of the aerosol than the AOT. The sensitivity of the AAOT to the assumed aerosol properties is shown in Figs. 9b and 10b. The uncertainty in the AAOT due to an error in g is similar to the uncertainty in the AOT (<5 %). However, the influence of the SSA assumption alone on the AAOT is smaller than the influence on the AOT, with differences of 1.9 % and −8.7 %. This means that a perturbation of the SSA primarily impacts the scattering AOT. The largest overestimation of the AAOT (2.7 %) is obtained when the assumed aerosol model overestimates g. An underestimation of the SSA and an overestimation of g lead to the largest underestimation of the AAOT (−5.1 %). Figure 10Similar to Fig. 9 but for the combined impact and the SSA. The variation in the solar zenith angle, and therefore in the satellite observation geometry during the day, can impact the sensitivity of the retrieval to the aerosol assumptions. Therefore, the 15 min SEVIRI observations for 28 August have been processed using the eight aerosol models described above and compared to the aerosol and cloud properties retrieved with the CLARIFY aerosol model. The difference Δxi of a product x is defined as $\mathrm{\Delta }{x}_{i}=\phantom{\rule{0.125em}{0ex}}\left({x}_{\mathrm{CLARIFY}}-{x}_{i}\right)/{x}_{i}×\mathrm{100}\phantom{\rule{0.125em}{0ex}}\mathit{%},$ where xCLARIFY and xi is the mean product x retrieved over the SEVIRI slot with the aerosol CLARIFY model and the modified model i respectively. Figure 11 shows the time series of ΔAOT (a), ΔAAOT (b), ΔCOT (c) and ΔCER (d) obtained with the modified aerosol models. The sensitivity of the retrieved cloud properties to the aerosol model assumptions remains small (lower than 5.6 % for the COT and 2.6 % for the CER) and dominated by the sensitivity to g. Apart from a small decrease in ΔCOT at midday when g is overestimated (solid blue line) and an increase in ΔCOT in late afternoon when the SSA is underestimated (solid red line), no significant trend is observed in the cloud property sensitivities. As observed previously, the uncertainty on the AOT is led by the SSA assumption, with the AOT being overestimated (respectively underestimated) when the assumed SSA is overestimated (respectively underestimated). Until 15:00, ΔAOT stays within ±40 %, with the sensitivity to the SSA being slightly larger at midday. Then it increases up to 60 % when the SSA is overestimated and g is underestimated (dashed blue line). Similar trends are observed in ΔAAOT, with generally lower values than ΔAOT. An increase in the uncertainty is observed on the AAOT after 15:00, which reaches up to 27 % at 16:30. Before 15:00, there is a larger AAOT sensitivity to the SSA around midday (+8.9 % −15.2 %), but there is no evident evolution of the sensitivity to g with time. The case that leads to the largest biases on the AAOT is when the SSA is underestimated and g overestimated (dashed green lines), with an underestimation of up to 23 %. However, it should be noted that 0 % of the AERONET observations used in Fig. 8 are associated with an SSA lower than SSACLARIFYσSSA and a g larger than gCLARIFYσg. Otherwise, the sensitivity of the AAOT to the aerosol property assumptions stays between −16.6 and +9 % before 15:00. Figure 11Time series (UTC) of the difference Δ (%) of the above-cloud AOT (a), AAOT (b), COT (c) and CER (d) retrieved with the CLARIFY model and the modified aerosol models for 28 August 2017. In conclusion, the retrieved AOT is less sensitive to the aerosol property assumption before 15:00, with an uncertainty of 40 %. This uncertainty is dominated by the sensitivity of the retrieval to the SSA. An overestimation (respectively underestimation) of the AOT is expected when the observed aerosols are more (respectively less) absorbing than the aerosol model assumed for the retrieval. A better accuracy is obtained on the retrieved AAOT, with an uncertainty generally lower than 17 % before 15:00. The sensitivity of the cloud properties to the aerosol model assumption remains small all day long, with an uncertainty of 5.6 % on the COT and 2.6 % on the CER. Figure 12Above-cloud AOT retrieved on 5 September 2017 at 11:42, 12:12 and 12:42 UTC. The red square represents the area over which the SEVIRI products have been averaged. 4 Assessing the stability of the retrieval One of the major benefits from using SEVIRI is the ability to track both aerosol and cloud events at high temporal resolution. Therefore, it is important to evaluate how consistent the retrieval is over time. For that purpose, 2 d of continuous observations (i.e. 5 and 6 September 2017) have been analysed and the retrieved properties have been averaged over 20 and 10 S and 5 and 15 E, which correspond to the red square on the maps in Fig. 12. The above-cloud AOT, COT and CER time series are presented in Fig. 13a, b, c. The studied area is located next to the coast, where the AOT is typically the highest. The above-cloud AOT is around 0.66 and 0.72 for 5 and 6 September respectively. As expected, the transport of the aerosol plume from east to west is slow, resulting in a small evolution of the above-cloud AOT. On both days, a peak is observed at 12:12 with an anomaly larger than the AOT variability. This localized discontinuity in the above-cloud AOT is shown in the 11:42, 12:12 and 12:42 UTC maps for 5 September 2017 in Fig. 12. The evolution of the cloud properties is slightly more complex. A small decrease is observed in both the COT and CER until 14:00. After 15:00, both properties sharply increase. The clouds are strongly affected by the diurnal cycle and a shoaling of the cloud cover is expected from early morning to late afternoon. As the thinnest clouds vanish, the cloud fraction decreases together with the number of retrievals in the area. This results in a larger contribution of the thickest clouds to the mean value in the late afternoon. As for the above-cloud AOT, large variations in the CER are observed around noon. At that time, the sun and the satellite are almost aligned and the scattering angle (Fig. 13d) reaches values larger than 175, which corresponds to the region where the glory phenomenon is typically observed. Several reasons can explain why the retrieval does not perform well in backscattering direction. The first one is the uncertainty in the LUT due to the truncation of the cloud phase function. Although the TMS correction gives good results, biases still remain in the glory aureole (Iwabushi and Suzuki, 2009). Also, the radiances in the glory are more sensitive to the cloud droplet microphysics (Mayer et al., 2004). The assumption on the variance of the droplet size distribution may induce biases in the retrieval. Therefore, the accuracy of the retrieval cannot be guaranteed within the glory aureole and these observations should be discarded. In Fig. 13, the time spans corresponding to the MODIS Aqua and Terra overpasses in the region are highlighted in orange. This shows that MODIS measurements are typically performed before and after SEVIRI observes the glory backscattering over the SEAO, usually allowing comparisons between these instruments. Figure 13Time series (UTC) of the above-cloud AOT (a), COT (b), CER (c) and scattering angle (d) averaged between 20 and 10 S and 5 and 15 E for 5 and 6 September 2017. The grey area represents scattering angles larger than 175 and the orange areas show the typical overpass times of MODIS Aqua and Terra over the region. The performance of the algorithm is further assessed by evaluating the stability of the retrieved above-cloud AOT at pixel level. As noted by Chang and Christopher (2016), in this region over these scales, aerosols are expected to have a limited temporal variability and the variation in the above-cloud AOT is expected to be small between t=0 and t±15 min. The differences between the AOT retrieved at t=0 and the running mean estimated between t−15 and t+15 min have been calculated at pixel level for observations between 09:00 and 15:00 UTC, removing measurements within the glory backscattering region. Figure 14 shows the histogram of the AOT differences calculated over a 12 d period (1 to 12 September 2017). The differences follow a normal distribution centred around 0.0 with a standard deviation of 0.1. This short-term variability can be attributed to several sources of uncertainties, such as the total amount of water vapour, its vertical distribution, the retrieved cloud top height and the numerical fitting procedure. This analysis indicates that the retrieval of the above-cloud AOT remains relatively stable, with an observed variability of ±0.1 between consecutive observations. Except for the glory backscattering, the stability observed on the retrieved aerosol and cloud properties reinforces the reliability of the algorithm. Figure 14Histogram of the difference between AOT retrieved at t=0 and the running mean calculated between t−15 and t+15 min from 1 to 12 September 2017. Observations within the glory region have been removed. Dashed lines represent the mean ± the standard deviation. 5 Conclusions Recently, progress has been made in the remote-sensing field in order to fill the lack of aerosol above-cloud observations. Techniques have been developed to retrieve aerosol and cloud properties over the SEAO from passive remote-sensing instruments. These algorithms take advantage of the colour-ratio effect (Jethva et al., 2013), which is the spectral contrast produced by the aerosol absorption above clouds. Although OMI (Torres et al., 2012), MODIS (Jethva et al., 2013; Meyer et al., 2015) and POLDER (Peers et al., 2015) already provide useful information about aerosols above clouds, these instruments are on polar-orbiting satellites and their low temporal resolutions prevent monitoring the diurnal variation in the cloud cover and in the DRE of aerosols over the SEAO. For the first time, we have applied a similar algorithm to geostationary measurements from the SEVIRI instrument, which has a repeat cycle of 15 min. The method consists of a LUT approach, using the channels at 0.64, 0.81 and 1.64 µm in order to simultaneously retrieve the above-cloud AOT, COT and CER. Compared to other satellite instruments, the SEVIRI measurements are more sensitive to the absorption from atmospheric gases because of their wider spectral bands. Therefore, an efficient atmospheric correction scheme is essential in order to separate the absorption from the aerosols and from the atmospheric gases. Atmospheric transmittances are calculated with the fast radiative transfer model RTTOV based on the cloud top height observed by SEVIRI and the forecasted water vapour profiles from the Met Office Unified Model. The water vapour correction has the largest impact on the above-cloud aerosol retrieval. The impact of errors in the atmospheric correction has been evaluated by modulating the humidity profile for a case study. A positive bias of both the AOT and the COT is observed when the water vapour is overestimated, and vice versa. On average, an 18.5 % bias on the AOT and a 5.5 % bias on the COT are expected for a 10 % error on the water vapour profile. Although a good accuracy is expected from the forecast model, this limitation should be kept in mind when utilizing or further developing SEVIRI products. In the companion paper, the humidity from the forecast will be compared against the dropsonde measurements from the CLARIFY-2017 campaign. The choice of the aerosol model used to produce the LUT is also a key feature of the method. In situ measurements of aerosols above clouds have been performed off the coast of Ascension Island during the CLARIFY-2017 field campaign. An aerosol model optimized for the SEVIRI spectral bands has been obtained by analysing the vertical profiles of extinction and absorption from EXSCALABAR together with the size distribution from a PCASP. A bimodal lognormal distribution has shown to adequately reproduce the observations. A fine-mode radius of 0.12 µm has been obtained, which is in good agreement with the biomass burning measured over the SEAO during SAFARI 2000 (Haywood et al., 2003). The refractive index has been evaluated at 1.51–0.029i. The corresponding SSA of 0.85 at 0.55 µm is consistent with both in situ and remote-sensing observations of African biomass burning aerosols (Johnson et al., 2008; Sayer et al., 2014). In addition to the uncertainty associated with the estimation of the aerosol model, a seasonal dependence is expected in the biomass burning properties as well as modifications due to ageing processes during their transport over the SEAO. We have evaluated the impact of applying a single model assumption on both aerosol and cloud properties. Retrievals have been performed considering aerosol models with modified SSA and asymmetry factor g. It has been shown that the sensitivity of the retrieved cloud properties to the aerosol model assumption is small, with errors lower than 5.6 % on the COT and 2.6 % on the CER. As expected the impact of the assumed aerosol properties is much larger on the above-cloud AOT, with an uncertainty estimated at 40 % before 15:00 UTC. This uncertainty is led by the sensitivity of the retrieval to the SSA. Because the method relies on the impact of the aerosol absorption on the light reflected by the clouds, the perturbation of the SSA has primarily an impact on the scattering contribution of the AOT. Therefore, a better accuracy is obtained on the retrieved AAOT, with biases generally lower than 17 % before 15:00 UTC. After that time, an increase in the uncertainty on both the AOT and the AAOT has been observed, and users are advised to be careful when using the late afternoon aerosol product. For any satellite retrievals based on the colour-ratio technique, aerosol properties, including the SSA, have to be assumed and the same order of magnitude can be expected on the sensitivity of their AOT. This analysis highlights the importance of a suitable constraint on the SSA. Despite the wider channels and the narrower spectral range of SEVIRI, it has been demonstrated that the geostationary instrument has the potential to detect and quantify the absorbing aerosol plumes transported above the clouds of the SEAO. Except from observations within the glory backscattering for which the retrieval has shown to be unstable, a good consistency has been observed on the aerosol and cloud properties. The stability of the results during the day is promising for future uses of the SEVIRI algorithm. In the companion paper, the reliability of the retrieved aerosol and cloud properties will be further assessed by analysing the consistency with the MODIS retrievals and comparing with direct measurements from the CLARIFY-2017 field campaign. The potential of such a retrieval is obvious. The 15 min resolution will aid in tracking the fate of above-cloud biomass burning aerosol and will prove invaluable for assessing models of the emission, transport and deposition of biomass burning aerosol, with implications for accurate determination of the direct radiative effects of biomass burning aerosol at high temporal resolution. Data availability Data availability. The data used for this study are available from the corresponding author, FP, upon reasonable request. Author contributions Author contributions. FP, PF and JMH developed the concept and the ideas for this paper. PF implemented the atmospheric correction scheme and FP the retrieval algorithm. CF, SJA, KS, MIC, NWD and JMH operated, calibrated and prepared the in situ measurements from EXSCALABAR and the PCASP. The reliability of the retrieved products was analysed throughout the development of the algorithm with the help of KGM and SEP. FP carried out the analysis and prepared the paper with contributions from all co-authors. Competing interests Competing interests. The authors declare that they have no conflict of interest. Special issue statement Special issue statement. This article is part of the special issue “New observations and related modelling studies of the aerosol–cloud–climate system in the Southeast Atlantic and southern Africa regions (ACP/AMT inter-journal SI)”. It is not associated with a conference. Acknowledgements Acknowledgements. We thank the Natural Environment Research Council (NERC) and the Norwegian Research Council for their financial support. Airborne data were obtained using the BAe-146 Atmospheric Research Aircraft operated by Directflight Ltd. and managed by Facility for Airborne Atmospheric Measurements (FAAM), which is jointly supported by NERC and the Met Office. The authors acknowledge the dedicated work of FAAM and Directflight during the aircraft campaign. We thank the AERONET PIs, Paola Formenti, Derek Griffith, Brent Holben, Nichola Knox, Gillian Maggs-Kölling, Stuart Piketh, Carlos Ribeiro, Venkataram Sivakumar and Rick Wagener for their efforts in establishing and maintaining the Ascension Island, Bethlehem, Bonanza, DRAGON Henties, Durban UKZN, Elandsfontein, Etosha Pan, Gobabeb, Gorongosa, Henties Bay, HESS, Huambo, Inhaca, Joberg, Kaoma, Loskop Dam, Lubango, Maun Tower, Mongu, Mwinilunga, Namibe, Ndola, Paardefontein, Pietersburg, Possession Island, Potchefstroom, Pretoria CSIR-DPSS, Senanga, Sesheke, Skukuza, Solwezi, Swakopmund, Tsumkwe, Upington, Walvis Bay airport, Windhoek-NUST, Windpoort and Wits University sites. Financial support Financial support. This research has been supported by the Natural Environment Research Council (NERC) via the CLARIFY project (grant no. NE/L013479/1) and the Research Council of Norway via the projects AC/BC (grant no. 240372) and NetBC (grant no. 244141). Review statement Review statement. This paper was edited by Nikolaos Mihalopoulos and reviewed by Ian Chang and one anonymous referee. References Aminou, D. M. A., Jacquet, B., and Pasternak, F.: Characteristics of the Meteosat Second Generation (MSG) radiometer/imager: SEVIRI, Proc. SPIE, 3221, 19–31, https://doi.org/10.1117/12.298084, 1997. Brindley, H. and Ignatov, A.: Retrieval of mineral aerosol optical depth and size information from Meteosat Second Generation SEVIRI solar reflectance bands, Remote Sens. Environ., 102, 344–363, https://doi.org/10.1016/j.rse.2006.02.024, 2006. Browell, E. V., Fenn, M. A., Butler, C. F., Grant, W. B., Clayton, M. B., Fishman, J., Bachmeier, A. S., Anderson, B. E., Gregory, G. L., Fuelberg, H. E., Bradshaw, J. D., Sandholm, S. T., Blake, D. R., Heikes, B. G., Sachse, G. W., Singh, H. B., and Talbot, R. W.: Ozone and aerosol distributions and air mass characteristics over the South Atlantic Basin during the burning season, J. Geophys. Res.-Atmos., 101, 24043–24068, https://doi.org/10.1029/95JD02536, 1996. Brown, A., Milton, S., Cullen, M., Golding, B., Mitchell, J., and Shelly, A.: Unified Modeling and Prediction of Weather and Climate: A 25-Year Journey, B. Am. Meteorol. Soc., 93, 1865–1877, https://doi.org/10.1175/BAMS-D-12-00018.1, 2012. Chang, I. and Christopher, S. A.: Identifying Absorbing Aerosols Above Clouds From the Spinning Enhanced Visible and Infrared Imager Coupled With NASA A-Train Multiple Sensors, IEEE T. Geosci. Remote, 54, 3163–3173, https://doi.org/10.1109/TGRS.2015.2513015, 2016. Clayton, A. M., Lorenc, A. C., and Barker, D. M.: Operational implementation of a hybrid ensemble/4D-Var global data assimilation system at the Met Office, Q. J. Roy. Meteor. Soc., 139, 1445–1461, https://doi.org/10.1002/qj.2054, 2013. Cotterell, M. I., Orr-Ewing, A. J., Szpek, K., Haywood, J. M., and Langridge, J. M.: The impact of bath gas composition on the calibration of photoacoustic spectrometers with ozone at discrete visible wavelengths spanning the Chappuis band, Atmos. Meas. Tech., 12, 2371–2385, https://doi.org/10.5194/amt-12-2371-2019, 2019. Davies, N. W., Cotterell, M. I., Fox, C., Szpek, K., Haywood, J. M., and Langridge, J. M.: On the accuracy of aerosol photoacoustic spectrometer calibrations using absorption by ozone, Atmos. Meas. Tech., 11, 2313–2324, https://doi.org/10.5194/amt-11-2313-2018, 2018. Davies, N. W., Fox, C., Szpek, K., Cotterell, M. I., Taylor, J. W., Allan, J. D., Williams, P. I., Trembath, J., Haywood, J. M., and Langridge, J. M.: Evaluating biases in filter-based aerosol absorption measurements using photoacoustic spectroscopy, Atmos. Meas. Tech., 12, 3417–3434, https://doi.org/10.5194/amt-12-3417-2019, 2019. de Graaf, M., Tilstra, L. G., Wang, P., and Stammes, P.: Retrieval of the aerosol direct radiative effect over clouds from spaceborne spectrometry, J. Geophys. Res.-Atmos., 117, D07207, https://doi.org/10.1029/2011JD017160, 2012. de Haan, J. F., Bosma, P., and Hovenier, J.: The adding method for multiple scattering calculations of polarized light, Astron. Astrophys., 183, 371–391, 1987. Eck, T. F., Holben, B. N., Ward, D. E., Mukelabai, M. M., Dubovik, O., Smirnov, A., Schafer, J. S., Hsu, N. C., Piketh, S. J., Queface, A., Roux, J. L., Swap, R. J., and Slutsker, I.: Variability of biomass burning aerosol optical characteristics in southern Africa during the SAFARI 2000 dry season campaign and a comparison of single scattering albedo estimates from radiometric measurements, J. Geophys. Res.-Atmos., 108, 8477, https://doi.org/10.1029/2002JD002321, 2003. Edwards, J. M. and Slingo, A.: Studies with a flexible new radiation code. I: Choosing a configuration for a large-scale model, Q. J. Roy. Meteor. Soc., 122, 689–719, https://doi.org/10.1002/qj.49712253107, 1996. Francis, P. N., Hocking, J. A., and Saunders, R. W.: Improved diagnosis of low-level cloud from MSG SEVIRI data for assimilation into Met Office limited area models, in: Proceedings of the 2008 EUMETSAT Meteorological Satellite Conference, Darmstadt, 2008. Gordon, H., Field, P. R., Abel, S. J., Dalvi, M., Grosvenor, D. P., Hill, A. A., Johnson, B. T., Miltenberger, A. K., Yoshioka, M., and Carslaw, K. S.: Large simulated radiative effects of smoke in the south-east Atlantic, Atmos. Chem. Phys., 18, 15261–15289, https://doi.org/10.5194/acp-18-15261-2018, 2018. Hamann, U., Walther, A., Baum, B., Bennartz, R., Bugliaro, L., Derrien, M., Francis, P. N., Heidinger, A., Joro, S., Kniffka, A., Le Gléau, H., Lockhoff, M., Lutz, H.-J., Meirink, J. F., Minnis, P., Palikonda, R., Roebeling, R., Thoss, A., Platnick, S., Watts, P., and Wind, G.: Remote sensing of cloud top pressure/height from SEVIRI: analysis of ten current retrieval algorithms, Atmos. Meas. Tech., 7, 2839–2867, https://doi.org/10.5194/amt-7-2839-2014, 2014. Haywood, J. M., Osborne, S. R., Francis, P. N., Keil, A., Formenti, P., Andreae, M. O., and Kaye, P. H.: The mean physical and optical properties of regional haze dominated by biomass burning aerosol measured from the C-130 aircraft during SAFARI 2000, J. Geophys. Res.-Atmos., 108, 8473, https://doi.org/10.1029/2002JD002226, 2003. Haywood, J. M., Osborne, S. R., and Abel, S. J.: The effect of overlying absorbing aerosol layers on remote sensing retrievals of cloud effective radius and cloud optical depth, Q. J. Roy. Meteor. Soc., 130, 779–800, https://doi.org/10.1256/qj.03.100, 2004. Herman, M., Deuzé, J.-L., Marchand, A., Roger, B., and Lallart, P.: Aerosol remote sensing from POLDER/ADEOS over the ocean: Improved retrieval using a nonspherical particle model, J. Geophys. Res.-Atmos., 110, https://doi.org/10.1029/2004JD004798, 2005. Hocking, J., Rayer, P., Rundle, D., Saunders, R., Matricardi, M., Geer, A., Brunel, P., and Vidot, J.: RTTOV v11 Users Guide, NWP-SAF Report, Met Office, Exeter, UK, 2014. Iwabuchi, H. and Suzuki, T.: Fast and accurate radiance calculations using truncation approximation for anisotropic scattering phase functions, J. Quant. Spectrosc. Ra., 110, 1926–1939, https://doi.org/10.1016/j.jqsrt.2009.04.006, 2009. Jethva, H., Torres, O., Remer, L. A., and Bhartia, P. K.: A Color Ratio Method for Simultaneous Retrieval of Aerosol and Cloud Optical Thickness of Above-Cloud Absorbing Aerosols From Passive Sensors: Application to MODIS Measurements, IEEE T. Geosci. Remote, 51, 3862–3870, https://doi.org/10.1109/TGRS.2012.2230008, 2013. Johnson, B. T., Osborne, S. R., Haywood, J. M., and Harrison, M. A. J.: Aircraft measurements of biomass burning aerosol over West Africa during DABEX, J. Geophys. Res.-Atmos., 113, D00C06, https://doi.org/10.1029/2007JD009451, 2008. Keil, A. and Haywood, J. M.: Solar radiative forcing by biomass burning aerosol particles during SAFARI 2000: A case study based on measured aerosol and cloud properties, J. Geophys. Res.-Atmos., 108, 8467, https://doi.org/10.1029/2002JD002315, 2003. Koppmann, R., von Czapiewski, K., and Reid, J. S.: A review of biomass burning emissions, Part I: gaseous emissions of carbon monoxide, methane, volatile organic compounds, and nitrogen containing compounds, Atmos. Chem. Phys. Discuss., 5, 10455–10516, https://doi.org/10.5194/acpd-5-10455-2005, 2005. Koren, I., Kaufman, Y. J., Remer, L. A., and Martins, J. V.: Measurement of the Effect of Amazon Smoke on Inhibition of Cloud Formation, Science, 303, 1342–1345, https://doi.org/10.1126/science.1089424, 2004. Levy, R. C., Remer, L. A., Tanre, D., Mattoo, S., and Kaufman, Y. J.: Algorithm for remote sensing of tropospheric aerosol over dark targets from MODIS: Collections 005 and 051: Revision 2, Feb 2009, MODIS algorithm theoretical basis document, 2009. Lu, Z., Liu, X., Zhang, Z., Zhao, C., Meyer, K., Rajapakshe, C., Wu, C., Yang, Z., and Penner, J. E.: Biomass smoke from southern Africa can significantly enhance the brightness of stratocumulus over the southeastern Atlantic Ocean, P. Natl. Acad. Sci. USA, 115, 2924–2929, https://doi.org/10.1073/pnas.1713703115, 2018. Magi, B. I. and Hobbs, P. V.: Effects of humidity on aerosols in southern Africa during the biomass burning season, J. Geophys. Res.-Atmos., 108, 8495, https://doi.org/10.1029/2002JD002144, 2003. Manners, J.: Socrates technical guide suite of community radiative transfer codes based on edwards and slingo, in: Tech. Rep., Met Office, FitzRoy Rd, Exeter EX1 3PB, 2015. Martins, J. V., Hobbs, P. V., Weiss, R. E., and Artaxo, P., Sphericity and morphology of smoke particles from biomass burning in Brazil, J. Geophys. Res., 103, 32051– 32057, https://doi.org/10.1029/98JD01153, 1998. Matricardi, M.: The generation of RTTOV regression coefficients for IASI and AIRS using a new profile training set and a new line-by-line database, European Centre for Medium-Range Weather Forecasts, 47 pp., 2008. Matricardi, M., Chevallier, F., Kelly, G., and Thépaut, J.-N.: An improved general fast radiative transfer model for the assimilation of radiance observations, Q. J. Roy. Meteor. Soc., 130, 153–173, https://doi.org/10.1256/qj.02.181, 2004. Mayer, B., Schröder, M., Preusker, R., and Schüller, L.: Remote sensing of water cloud droplet size distributions using the backscatter glory: a case study, Atmos. Chem. Phys., 4, 1255–1263, https://doi.org/10.5194/acp-4-1255-2004, 2004. McClatchey, R. A., Fenn, R. W., Selby, J. A., Volz, F., and Garing, J.: Optical properties of the atmosphere, Tech. Rep., Air Force Cambridge Research Labs Hanscom AFB MA, 113 pp., 1972. Meyer, K., Platnick, S., and Zhang, Z.: Simultaneously inferring above-cloud absorbing aerosol optical thickness and underlying liquid phase cloud optical and microphysical properties using MODIS, J. Geophys. Res.-Atmos., 120, 5524–5547, https://doi.org/10.1002/2015JD023128, 2015. Min, M. and Zhang, Z.: On the influence of cloud fraction diurnal cycle and sub-grid cloud optical thickness variability on all-sky direct aerosol radiative forcing, J. Quant. Spectrosc. Ra., 142, 25–36, https://doi.org/10.1016/j.jqsrt.2014.03.014, 2014. Nakajima, T. and King, M. D.: Determination of the Optical Thickness and Effective Particle Radius of Clouds from Reflected Solar Radiation Measurements. Part I: Theory, J. Atmos. Sci., 47, 1878–1893, https://doi.org/10.1175/1520-0469(1990)047<1878:DOTOTA>2.0.CO;2, 1990. Nakajima, T. and Tanaka, M.: Algorithms for radiative intensity calculations in moderately thick atmospheres using a truncation approximation, J. Quant. Spectrosc. Ra., 40, 51–69, https://doi.org/10.1016/0022-4073(88)90031-3, 1988. Peers, F., Waquet, F., Cornet, C., Dubuisson, P., Ducos, F., Goloub, P., Szczap, F., Tanré, D., and Thieuleux, F.: Absorption of aerosols above clouds from POLDER/PARASOL measurements and estimation of their direct radiative effect, Atmos. Chem. Phys., 15, 4179–4196, https://doi.org/10.5194/acp-15-4179-2015, 2015. Peers, F., Bellouin, N., Waquet, F., Ducos, F., Goloub, P., Mollard, J., Myhre, G., Skeie, R. B., Takemura, T., Tanré, D., Thieuleux, F., and Zhang, K.: Comparison of aerosol optical properties above clouds between POLDER and AeroCom models over the South East Atlantic Ocean during the fire season, Geophys. Res. Lett., 43, 3991–4000, https://doi.org/10.1002/2016GL068222, 2016. Platnick, S.: Vertical photon transport in cloud remote sensing problems, J. Geophys. Res., 105, 22919–22935, 2000. Rosenberg, P. D., Dean, A. R., Williams, P. I., Dorsey, J. R., Minikin, A., Pickering, M. A., and Petzold, A.: Particle sizing calibration with refractive index correction for light scattering optical particle counters and impacts upon PCASP and CDP data collected during the Fennec campaign, Atmos. Meas. Tech., 5, 1147–1163, https://doi.org/10.5194/amt-5-1147-2012, 2012. Rosenfeld, D.: Suppression of Rain and Snow by Urban and Industrial Air Pollution, Science, 287, 1793–1796, https://doi.org/10.1126/science.287.5459.1793, 2000. Sayer, A. M., Hsu, N. C., Eck, T. F., Smirnov, A., and Holben, B. N.: AERONET-based models of smoke-dominated aerosol near source regions and transported over oceans, and implications for satellite retrievals of aerosol optical depth, Atmos. Chem. Phys., 14, 11493–11523, https://doi.org/10.5194/acp-14-11493-2014, 2014. Szczodrak, M., Austin, P. H. and Krummel, P. B.: Variability of Optical Depth and Effective Radius in Marine Stratocumulus Clouds, J. Atmos. Sci., 58, 2912–2926, https://doi.org/10.1175/1520-0469(2001)058<2912:VOODAE>2.0.CO;2, 2001. Thieuleux, F., Moulin, C., Bréon, F. M., Maignan, F., Poitou, J., and Tanré, D.: Remote sensing of aerosols over the oceans using MSG/SEVIRI imagery, Ann. Geophys., 23, 3561–3568, https://doi.org/10.5194/angeo-23-3561-2005, 2005. Torres, O., Jethva, H., and Bhartia, P. K.: Retrieval of Aerosol Optical Depth above Clouds from OMI Observations: Sensitivity Analysis and Case Studies, J. Atmos. Sci., 69, 1037–1053, https://doi.org/10.1175/JAS-D-11-0130.1, 2012. Twomey, S.: Pollution and the planetary albedo, Atmos. Environ., 8, 1251–1256, https://doi.org/10.1016/0004-6981(74)90004-3, 1974. Waquet, F., Cornet, C., Deuzé, J.-L., Dubovik, O., Ducos, F., Goloub, P., Herman, M., Lapyonok, T., Labonnote, L. C., Riedi, J., Tanré, D., Thieuleux, F., and Vanbauce, C.: Retrieval of aerosol microphysical and optical properties above liquid clouds from POLDER/PARASOL polarization measurements, Atmos. Meas. Tech., 6, 991–1016, https://doi.org/10.5194/amt-6-991-2013, 2013a. Waquet, F., Peers, F., Ducos, F., Goloub, P., Platnick, S., Riedi, J., Tanré, D., and Thieuleux, F.: Global analysis of aerosol properties above clouds, Geophys. Res. Lett., 40, 5809–5814, https://doi.org/10.1002/2013GL057482, 2013b. Watts, P., Mutlow, C., Baran, A., and Zavody, A.: Study on cloud properties derived from Meteosat Second Generation observations, EUMETSAT ITT, no 97/181, 344 pp., 1998. Wilcox, E. M.: Stratocumulus cloud thickening beneath layers of absorbing smoke aerosol, Atmos. Chem. Phys., 10, 11769–11777, https://doi.org/10.5194/acp-10-11769-2010, 2010. Wilcox, E. M.: Direct and semi-direct radiative forcing of smoke aerosols over clouds, Atmos. Chem. Phys., 12, 139–149, https://doi.org/10.5194/acp-12-139-2012, 2012. Wiscombe, W.: The Delta–M method: Rapid yet accurate radiative flux calculations for strongly asymmetric phase functions, J. Atmos. Sci., 34, 1408–1422, 1977. Zuidema, P., Redemann, J., Haywood, J., Wood, R., Piketh, S., Hipondoka, M., and Formenti, P.: Smoke and Clouds above the Southeast Atlantic: Upcoming Field Campaigns Probe Absorbing Aerosol's Impact on Climate, B. Am. Meteorol. Soc., 97, 1131–1135, https://doi.org/10.1175/BAMS-D-15-00082.1, 2016. Zuidema, P., Sedlacek III, A. J., Flynn, C., Springston, S., Delgadillo, R., Zhang, J., Aiken, A. C., Koontz, A., and Muradyan, P.: The Ascension Island Boundary Layer in the Remote Southeast Atlantic is Often Smoky, Geophys. Res. Lett., 45, 4456–4465, https://doi.org/10.1002/2017GL076926, 2018.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 7, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8831558227539062, "perplexity": 4079.437830517369}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-29/segments/1593655911092.63/warc/CC-MAIN-20200710144305-20200710174305-00037.warc.gz"}
https://www.physicsforums.com/threads/normal-force-of-a-hoop-rolling-down-an-inclined-plane.686622/	Homework Help: Normal Force of a hoop rolling down an inclined plane. 1. Apr 19, 2013 mercrave 1. The problem statement, all variables and given/known data I have a question from my homework. My homework is completed, but I've been running some thought experiments lately and I wish to conceptually discuss this. The problem has to do with a hoop rolling down an inclined plane. I had to find the Normal Force using Lagrange's Equations. Hoop has mass M, radius R, the hypotenuse of the incline is L and the angle from the bottom of the incline is α. 2. Relevant equations I'm not sure what to place here, but I have set my origin to be at the tip of the inclined plane, and my x axis lies parallel to the bottom of the incline whereas the y axis is perpendicular to my x-axis. I have set θ to be the angle through which the hoop rotates. 3. The attempt at a solution When all is said and done I get the following (the ' refers to a time derivative); T = .5 m x'2 + .5 m y'2 + .5 m R2 θ'2 U = mgy g1[x,y,θ] = y - x Tan[α] (= 0)<- hoop has to stay on the plane g2[x,y,θ] = L - x Sec[α] - Rθ (= 0) <- hoop is rolling without slipping (L - x Sec[α] therefore refers to the portion of the plane that is traversed by the hoop) I plug into Lagrange's equations and obtain the following; λ1 = .5 mg(1+Cos2[α]) (normal force) λ2 = -.5 mg Sin[α] (acceleration down the incline) I did some checks in my head for the normal force; at α = .5$\pi$, we have -.5 mg for acceleration down the incline and normal force is at a minimum (.5 mg) whereas at α = 0, no acceleration and the normal force is effectively the mass of the hoop. I am basically wondering if this is correct. My thought process right now is that hypothetically, it can roll down a .5$\pi$ incline, and to do that it requires a frictional force which then requires a normal force. It also means that it effectively doesn't roll when there is no incline (flat surface) so normal force is at a maximum and it is just supporting the hoop. Am I correct in thinking so? Thank you very much, and I hope to hear from you. I am fairly confident in my answer, but my friends get conflicting results (notably, they get mgCos[α] for the normal force, which makes sense to me to some extent until we factor in the rolling) and we were supposedly supposed to recognize this other force. It could be possible that I need to find the x and y components of this force using the constraints on Lagrange's Equation's and probably find the magnitude to round it off, but I'm just curious to hear what everyone here thinks. Great forum by the way, it has helped me through numerous homeworks in the past! 2. Apr 19, 2013 haruspex Why would the rolling make it otherwise? That adds a frictional force up the slope, so has no effect on the normal force. Maybe you could post your working. 3. Apr 19, 2013 mercrave I basically took double-time derivatives of each of the first two constraints, then used lagranges equations alongside them. My friends and I all double checked lagranges equations, so at this point I am wondering if the constraints are the issue or, better yet, the lagrangian? It also helped that all algebra we did was performed on Mathematica. Also, I thought rolling friction was normal force dependent even still. 4. Apr 19, 2013 TSny I get the same expressions for the lagrange multipliers λ1 and λ2. But λ1 does not represent the normal force. Note that in setting up your constraint g2[x,y,θ] for rolling without slipping, you invoked the constraint g1[x,y,θ] to get the x Sec[α] part. So, both g1 and g2 include the constraint to stay on the plane. You can find the normal component of each of the constraint forces. For example, the normal component of the constraint force associated with g1 is $\lambda_1\frac{\partial{g_1}}{\partial n}=\lambda_1 (\frac{\partial{g_1}}{\partial x}\frac{\partial{x}}{\partial n}+\frac{\partial{g_1}}{\partial y}\frac{\partial{y}}{\partial n} ) = \lambda_1 (-\frac{\partial{g_1}}{\partial x}sin\alpha+\frac{\partial{g_1}}{\partial y}cos\alpha )$ where $n$ indicates normal direction. Similarly you can find the normal component of the force due to the second constraint. If you add the two normal components from the two constraints, you should get the total normal force.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 2, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8524641394615173, "perplexity": 532.7825787106145}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589557.39/warc/CC-MAIN-20180717031623-20180717051623-00248.warc.gz"}
https://stacks.math.columbia.edu/tag/04B1	Lemma 10.137.20. Let $R$ be a ring and let $I \subset R$ be an ideal. Let $R/I \to \overline{S}$ be a smooth ring map. Then there exists elements $\overline{g}_ i \in \overline{S}$ which generate the unit ideal of $\overline{S}$ such that each $\overline{S}_{g_ i} \cong S_ i/IS_ i$ for some (standard) smooth ring $S_ i$ over $R$. Proof. By Lemma 10.137.10 we find a collection of elements $\overline{g}_ i \in \overline{S}$ which generate the unit ideal of $\overline{S}$ such that each $\overline{S}_{g_ i}$ is standard smooth over $R/I$. Hence we may assume that $\overline{S}$ is standard smooth over $R/I$. Write $\overline{S} = (R/I)[x_1, \ldots , x_ n]/(\overline{f}_1, \ldots , \overline{f}_ c)$ as in Definition 10.137.6. Choose $f_1, \ldots , f_ c \in R[x_1, \ldots , x_ n]$ lifting $\overline{f}_1, \ldots , \overline{f}_ c$. Set $S = R[x_1, \ldots , x_ n, x_{n + 1}]/(f_1, \ldots , f_ c, x_{n + 1}\Delta - 1)$ where $\Delta = \det (\frac{\partial f_ j}{\partial x_ i})_{i, j = 1, \ldots , c}$ as in Example 10.137.8. This proves the lemma. $\square$ Comment #773 by Keenan Kidwell on This doesn't actually make any difference, but to be consistent with 00T8 and the definition of standard smooth in terms of Jacobian determinants, the matrix of which $\Delta$ is the determinant should be $(\partial f_j/\partial x_i)_{i,j=1,\ldots,c}$. In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).	{"extraction_info": {"found_math": true, "script_math_tex": 2, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 2, "x-ck12": 0, "texerror": 0, "math_score": 0.9908446073532104, "perplexity": 169.63046238420065}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-21/segments/1652662526009.35/warc/CC-MAIN-20220519074217-20220519104217-00090.warc.gz"}
https://www.groundai.com/project/lasing-of-donor-bound-excitons-in-znse-microdisks/	Lasing of donor-bound excitons in ZnSe microdisks # Lasing of donor-bound excitons in ZnSe microdisks A. Pawlis M. Panfilova D. J. As K. Lischka Dept. of Physics, University of Paderborn, Warburger Str. 100, 33098 Paderborn, Germany K. Sanaka T. D. Ladd Y. Yamamoto Edward L. Ginzton Laboratory, Stanford University, Stanford, California 94305-4088, USA National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan ###### Abstract Excitons bound to flourine atoms in ZnSe have the potential for several quantum optical applications. Examples include optically accessible quantum memories for quantum information processing and lasing without inversion. These applications require the bound-exciton transitions to be coupled to cavities with high cooperativity factors, which results in the experimental observation of low-threshold lasing. We report such lasing from fluorine-doped ZnSe quantum wells in 3 and 6 micron microdisk cavities. Photoluminescence and selective photoluminescence spectroscopy confirm that the lasing is due to bound-exciton transitions. ###### pacs: 78.55.Et, 78.67.-n, 42.55.Sa, 42.50.Pq, 03.67.-a Quantum interference of the two optical pathways of an optical -system, in which two long-lived ground states are optically coupled to a single excited state, provides a powerful mechanism for a number of useful applications. These applications include lasing without inversion Scully and Fleischhauer (1994), electromagnetically induced transparency Harris et al. (1990), and optically addressable quantum memory for quantum information processing Cirac et al. (1997); Yao et al. (2005); Childress et al. (2005); Waks and Vuckovic (2006); Clark et al. (2007); T. D. Ladd et al. (2006). The development of scalable technologies based on these effects would be facilitated by solid-state implementations, which are generally more difficult than atomic demonstrations due to optical inhomogeneity and decoherence. In semiconductors, donor impurities and charged quantum dots both provide promising realizations of such -systems. The long-lived ground states are provided by the bound electron spin in high magnetic field, while the optically excited state is formed by the lowest-energy donor-bound exciton or trion state. For applications involving quantum memory, however, quantum dots have the disadvantage of severe inhomogeneity in optical transition frequencies and electron magnetic moments. Such inhomogeneity may inhibit the ability to incorporate many optically interacting quantum-dot-based -systems into a scalable quantum-information-processing device. In contrast, the potentials provided by donor impurities in a perfect crystal are very homogeneous, as demonstrated by high-quality samples of GaAs V. A. Karasyuk et al. (1994) and Si A. Yang et al. (2006). Homogeneous ensembles of donor-bound excitons in bulk GaAs have been demonstrated to exhibit coherent-population trapping K. M. C. Fu et al. (2005), which is one effect exhibiting quantum coherence between the two emission pathways of the optical -system. Donor-bound excitons in ZnSe are of particular interest because ZnSe may be isotopically purified to feature only spin-0 substrate nuclei, avoiding the disadvantage of decoherence limited by the dynamics of nuclear spins in iii-v semiconductor systems R. de Sousa and S. Das Sarma (2003). Silicon-based systems also have the potential to overcome nuclear decoherence Tyryshkin et al. (2003), but silicon is optically dark due to its indirect bandgap. In a ii-vi system such as ZnSe, the nuclear spins may be entirely removed from the substrate as in the case in silicon, and the donor-bound exciton emission is optically bright as in the case of GaAs. The fluorine donor is of particular interest because it provides a 100% abundant spin-1/2 nucleus, which may be employed for long-lived storage of quantum information. ZnSe is a wide-bandgap semiconductor which emits light in the blue region ( nm), where lasing is traditionally difficult to obtain. Early studies of lasing in ZnSe for the purpose of blue lasers were limited because the material is too soft to be used in high-current or high-excitation regimes, so that heating effects degenerate the active medium L.-L. Chao et al. (1997); H. Okuyama (2000). However, low-threshold lasing is achievable when a high-, low-volume optical microcavity introduces a high ratio of stimulated emission to spontaneous emission, known as the -factor Yamamoto et al. (1991). Such low-threshold lasing is seen in recent developments in microcavity quantum dot lasers S. Strauf et al. (2006). Further, if the lasing medium is provided by discrete donor-bound exciton transitions rather than the continuum of free-excitons, the coherence provided by the spin-ground states of the donor-spins could allow the possibility of lasing without inversion Scully and Fleischhauer (1994), further decreasing the input power requirements. Lasing in ZnSe donor-bound excitons may be particularly useful as a component in quantum information processing devices. The design of quantum computers Clark et al. (2007) or quantum repeaters T. D. Ladd et al. (2006) based on donor-bound excitons in optical microcavities requires a low-noise source laser nearly resonant the bound-exciton transitions used for qubit initialization, control, and read-out. A laser based on the same medium as the qubit provides promising pathways for device integration. Bound-exciton emission in ZnSe was observed many years ago Merz et al. (1972), and the optical transitions of single, isolated ZnSe acceptors were recently measured S. Strauf et al. (2002). However, coherent applications involving -systems are more likely to be achieved from donors, due to the longer coherence time of electrons over holes. The F-donor has been studied earlier in bulk samples A. Pawlis et al. (2006). In this Letter, we report evidence of F-donors optically coupled to microcavities exhibiting low-threshold lasing of the donor-bound exciton transitions. Previous work on microcavity lasers in the wide-bandgap ii-vi semiconductors have focused on lasers based on CdSe quantum dots J. Renner et al. (2006a, b) or ZnCdSe quantum wells (QW’s) M. Hovinen et al. (1993) confined by quaternary alloys of ZnMgSSe and ZnCdSSe, which are lattice-matched to a GaAs substrate. To reduce alloy fluctuations, we confined a ZnSe QW in ternary ZnMgSe barriers with low Mg-concentration. Lasing due to confined free-excitons seeded by excitonic molecules has been previously observed in similar systems Kreller et al. (1995); V. Kozlov et al. (1996). Here we show lasing directly on discrete, bound-exciton transitions in samples with a fluorine -doped (F:ZnSe) layer at the center of the QW. The ZnMgSe/ZnSe/F:ZnSe/ZnSe/ZnMgSe samples were grown by molecular beam epitaxy on GaAs-(001) substrates. For optimal interface properties, a 15 nm buffer of undoped ZnSe was first deposited on the substrate, followed by 28 nm of ZnMgSe with Mg content of about 8%. The ZnSe QW was 10 nm thick. For doped samples, the QW was -doped in the center using a ZnF evaporation cell with a net molecular flux of approximately cms, equivalent to a sheet donor concentration of cm. Photoluminescence (PL) spectra of unstructured F-doped and undoped samples were measured at 5 K, with excitation provided by a blue light-emitting diode of wavelength 365 nm [Fig. 1(a)]. The PL spectra of these strained samples show two peaks. The emission at 2.810 eV corresponds to recombination of heavy-hole free excitions (FX-hh) from the compressively strained ZnSe QW (lattice mismatch of f(ZnSe)=); this energy is consistent with the finite barrier model in Refs. 27 and 28. Light-hole free-exciton (FX-lh) emission at 2.829 eV is not observed at low temperature. The lower energy peak seen at 2.804 eV in Fig. 1 is due to the heavy-hole donor-bound exciton (DX-hh) recombination, as confirmed by the expected 6 meV separation and the 3-fold increase in this line provided by the fluorine -doped (red curve) versus the undoped sample (black curve). The widths of these peaks indicate inhomogeneous broadening, likely due to strain effects. However, the broadening is still less than typically seen in ensembles of quantum dots, and may be useful for the isolation of individual impurities. Micro-disks with diameters of 3 m and 6 m were defined by photolithography, reactive ion etching, and wet etching of the underlying GaAs substrate. The inset in Fig. 1(b) shows a scanning electron micrograph picture of such a microdisk. The absence of cracks and lateral deformation indicates a homogeneous release of strain along the radial direction of the disks; however, considering the volume ratio of ZnMgSe vs. ZnSe in the disks, the ZnSe QW is likely to be tensile strained on ZnMgSe in the periphery of the disks. This results in a substantial band gap narrowing effect along the radial direction of the disk from the center to the edge which enhances the charge carrier transfer to the periphery, where the optical whispering gallery modes (WGMs) are localized. As a result of the tensile strain, the relative positions of the FX-lh and FX-hh energies are reversed and both states contribute to the PL. Bandgap narrowing and strain relaxation are confirmed by the corresponding PL-spectrum in Fig. 1(b) taken from a microdisk with 6 m diameter. The modified transition energies are found by fitting the above-band PL to 5 Gaussian peaks. These energies are again consistent with the finite barrier model reported in Refs. K. L. Teo et al., 1994 and T.-Y. Chung et al., 1997 with a remaining induced tensile lattice mismatch of the quantum well of f(ZnSe)= in the etched structure. The FX-hh and DX-hh transitions at 2.81 eV and 2.804 eV are only slightly shifted and broadened; these peaks come from PL around the center of the disk, where the material remains fully strained. No cavity modes are ever observed in this region of the spectrum. The DX-hh luminescence is now stronger than the FX-hh luminescence due to surface recombination of the FX, and satellite transitions are now visible at a position of 21 meV beneath the DX-hh luminescence. This energy is in agreement with the calculated position of the two electron satellite (TES) of the DP. J. Dean et al. (1981). In the relaxed periphery, the FX-lh emission moves to the lower energy position of about 2.799 eV; the DX transitions again appear 6 meV lower than their corresponding FX transitions, but are substantially broadened due to the strain gradient across the disk. The association of the spectral peaks in structured samples is also confirmed by selective PL (SPL), in which the FX-hh transition and the DX-hh transition are resonantly excited by a scanning narrow-band Ti:sapphire ring laser doubled by an LBO crystal in a high-finesse cavity. Fig. 1(b) shows the SPL spectra of a 6 m micro-disk. By resonant excitation of the FX-hh transition at 2.809 eV (green curve), the TES-hh lines are enhanced relative to above-band excitation. Resonant excitation of the DX-hh peak (red curve) results in the appearance of a new satellite peak at 2.774 eV which we identify as TES-lh transitions. These observations of TES lines in SPL strongly confirm the expected contribution of F-donor-bound excitons to the band-edge luminescence, as opposed to emission from quantum dots due to surface fluctuations of the QW. We also repeatedly observe a sharp peak at 2.79 eV when resonantly exciting the DX-hh line. This peak corresponds to a WGM of the microdisk. The WGM modes are enhanced with high-power pumping. Figure 2 shows a series of PL spectra taken from a micro-disk with 3 m diameter and measured with different average excitation intensities varying between 9 and 32 W cm. For small power densities exceeding 5 W cm, superlinear increase of a sharp peak at 2.790 eV (mode M1 in Fig. 2) is observed. This peak corresponds to a WGM of the micro-disk excited by DX-lh emission. The linewidth indicates a cavity- of about 2500. Additional modes emerge at energies of 2.783 eV and 2.785 eV (mode M2) and at 2.776 eV (mode M3) for pump power densities larger than 12 W cm. These modes are too closely spaced to correspond to the free spectral range of the lowest order WGMs; the multiple modes seen are likely due to higher order radial modes of the disk. The strain gradient and slight warping of these disks prevent a simple quantitative evaluation of their spacing. Mode M2 consists of two competing modes close to each other which might stem from anisotropy of the disk shape. At higher excitation density the overall intensity of mode M1 and mode M2 is transferred to mode M3, which is resonant with the TES transition energies of the bound exciton states. The WGMs are only observed in the lower energy part of the spectra corresponding to emission from donor bound excitons and their related transitions, resulting from the bandgap narrowing at the periphery of the disk. The peaks corresponding to WGMs are not visible in low-power PL using either a continuous wave 405 nm diode laser or the doubled 408 nm output of a mode-locked Ti:sapphire laser. We believe this is because absorption due to lower-energy transitions in the inhomogeneously broadened distribution damps the cavity-. However, at higher excitation powers, these lower-energy transitions become saturated and the cavity modes appear. Figure 3(a) shows the power dependence of mode M3 as a function of the pump power. A clear lasing threshold is seen at about 15 W cm. This threshold is substantially smaller than observed in previous ii-vi-semiconductor microdisk lasers based on quantum dots J. Renner et al. (2006b). The ratio of the slope of output power vs. input power indicates the fraction of spontaneous emission coupled into the lasing mode to be . Lasing was also observed in 6 m micro-disks, revealing threshold power densities larger than 100 W cm for the TES transition with a corresponding -factor of about 0.03. Lasing was confirmed by the power dependence of the second order photon correlation function , measured at the TES lasing mode M3 with a standard Hanbury-Brown Twiss (HBT) experiment and shown in Fig. 3(b). Below threshold the spontaneous emission into mode M3 follows a nearly Poisson distribution with . Near threshold we observe significant photon bunching, manifested as . Such bunching occurs as the coherence length of the photons increases near threshold but before the fully coherent output of above-threshold lasing. Far above threshold (powers exceeding 30 W cm), the HBT signal converges back to , indicating the generation of a coherent state. The observed factors allow us to estimate the cooperativity factors of the microdisk cavities. In microdisks such as these that are very wide and very thin relative to the emission wavelength, the spontaneous emission rate into all cavity modes including unguided modes sum to very nearly the spontaneous emission rate in the bulk. Therefore, the factor for one of the WGMs is approximately , where is the ratio of the emission rate into a vanishing- WGM normalized by the bulk emission rate and the cooperativity factor manifests as the Purcell effect. We estimate the mode volume using the approximation presented in Ref. Chin et al., 1994 in the large-radius limit. We find that for disks as thin as these, the WGM’s are quite wide, allowing about 500 impurities to contribute to the lasing in a 3 m disk. This model yields , for an expected cooperativity factor of about , which is consistent with estimates based on a of 2500 and the calculated mode volume. Both and scale inversely with the disk radius, consistent with the smaller observed factor of the 6 m disk. The appearance of low-threshold lasing in -doped F:ZnSe QW micro disks indicates that small ensembles of F-Donors have been successfully coupled to a cavity with cooperativity factors high enough to allow some schemes for quantum communication and quantum computation T. D. Ladd et al. (2006). The reduction of inhomogeneous broadening by preserving the uniform strain acquired during MBE growth may allow such applications as lasing without inversion. Further work will involve the isolation of individual impurities in smaller microcavities, the measurement of the donor-spin coherence times, and the measurement of single F nuclear spin states. This work was supported by nict. We thank Kristiaan De Greve and Shinichi Koseki for valuable discussions and experimental assistance. ## References • Scully and Fleischhauer (1994) M. O. Scully and M. Fleischhauer, Science 263, 337 (1994). • Harris et al. (1990) S. E. Harris, J. E. Field, and A. Imamoglu, Phys. Rev. Lett. 64, 1107 (1990). • Cirac et al. (1997) J. I. Cirac, P. Zoller, H. J. Kimble, and H. Mabuchi, Phys. Rev. Lett. 78, 3221 (1997). • Yao et al. (2005) W. Yao, R.-B. Liu, and L. J. Sham, Phys. Rev. Lett. 95, 30504 (2005). • Childress et al. (2005) L. Childress, J. M. Taylor, A. S. Sørensen, and M. D. Lukin, Phys. Rev. A 72, 52330 (2005). • Waks and Vuckovic (2006) E. Waks and J. Vuckovic, Phys. Rev. Lett. 96, 153601 (2006). • Clark et al. (2007) S. M. Clark, K. M. C. Fu, T. D. Ladd, and Y. Yamamoto, Phys. Rev. Lett. 99, 40501 (2007). • T. D. Ladd et al. (2006) T. D. Ladd et al., New J. Phys. 8, 184 (2006). • V. A. Karasyuk et al. (1994) V. A. Karasyuk et al., Phys. Rev. B 49, 16381 (1994). • A. Yang et al. (2006) A. Yang et al., Phys. Rev. Lett. 97, 227401 (2006). • K. M. C. Fu et al. (2005) K. M. C. Fu et al., Phys. Rev. Lett. 95, 187405 (2005). • R. de Sousa and S. Das Sarma (2003) R. de Sousa and S. Das Sarma, Phys. Rev. B 68, 115322 (2003). • Tyryshkin et al. (2003) A. M. Tyryshkin, S. A. Lyon, A. V. Astashkin, and A. M. Raitsimring, Phys. Rev. B 68, 193207 (2003). • L.-L. Chao et al. (1997) L.-L. Chao et al., Appl. Phys. Lett. 70, 535 (1997). • H. Okuyama (2000) H. Okuyama, IECE Trans. Electron. E83-C, 536 (2000). • Yamamoto et al. (1991) Y. Yamamoto, S. Machida, and G. Björk, Phys. Rev. A 44, 657 (1991). • S. Strauf et al. (2006) S. Strauf et al., Phys. Rev. Lett. 96, 127404 (2006). • Merz et al. (1972) J. L. Merz, H. Kukimoto, K. Nassau, and J. W. Shiever, Phys. Rev. B 6, 545 (1972). • S. Strauf et al. (2002) S. Strauf et al., Phys. Rev. Lett. 89, 177403 (2002). • A. Pawlis et al. (2006) A. Pawlis et al., Semicond. Sci. Technol. 21, 1412 (2006). • J. Renner et al. (2006a) J. Renner et al., Appl. Phys. Lett. 89, 091105 (2006a). • J. Renner et al. (2006b) J. Renner et al., Appl. Phys. Lett. 89, 231104 (2006b). • M. Hovinen et al. (1993) M. Hovinen et al., Appl. Phys. Lett. 63, 3128 (1993). • Kreller et al. (1995) F. Kreller, M. Lowisch, J. Puls, and F. Henneberger, Phys. Rev. Lett. 75, 2420 (1995). • V. Kozlov et al. (1996) V. Kozlov et al., Phys. Rev. B 53, 10837 (1996). • K. L. Teo et al. (1994) K. L. Teo et al., Semicond. Sci. Technol. 9, 349 (1994). • T.-Y. Chung et al. (1997) T.-Y. Chung et al., Semicond. Sci. Technol. 12, 701 (1997). • P. J. Dean et al. (1981) P. J. Dean et al., Phys. Rev. B 23, 4888 (1981). • Chin et al. (1994) M. K. Chin, D. Y. Chu, and S.-T. Ho, J. Appl. Phys. 75, 3302 (1994). You are adding the first comment! How to quickly get a good reply: • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made. • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements. • Your comment should inspire ideas to flow and help the author improves the paper. The better we are at sharing our knowledge with each other, the faster we move forward. The feedback must be of minimum 40 characters and the title a minimum of 5 characters	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8644379377365112, "perplexity": 3717.918191658926}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-10/segments/1581875145654.0/warc/CC-MAIN-20200222054424-20200222084424-00521.warc.gz"}
https://it.mathworks.com/help/stats/prob.multinomialdistribution.html	# MultinomialDistribution Multinomial probability distribution object ## Description A `MultinomialDistribution` object consists of parameters and a model description for a multinomial probability distribution. The multinomial distribution is a generalization of the binomial distribution. While the binomial distribution gives the probability of the number of “successes” in n independent trials of a two-outcome process, the multinomial distribution gives the probability of each combination of outcomes in n independent trials of a k-outcome process. The probability of each outcome in any one trial is given by the fixed probabilities p1, ..., pk. The multinomial distribution uses the following parameter. ParameterDescriptionSupport `Probabilities`Outcome probabilities$0\le \text{Probabilities}\left(i\right)\le 1\text{\hspace{0.17em}};\text{\hspace{0.17em}}\sum _{\text{all}\left(i\right)}\text{Probabilities}\left(i\right)=1$ ## Creation Create a `MultinomialDistribution` probability distribution object with specified parameter values by using `makedist`. ## Properties expand all ### Distribution Parameter Outcome probabilities for the multinomial distribution, stored as a vector of scalar values in the range `[0,1]`. The values in `Probabilities` must sum to 1. Data Types: `single` \| `double` ### Distribution Characteristics Logical flag for truncated distribution, specified as a logical value. If `IsTruncated` equals `0`, the distribution is not truncated. If `IsTruncated` equals `1`, the distribution is truncated. Data Types: `logical` Number of parameters for the probability distribution, specified as a positive integer value. Data Types: `double` Distribution parameter values, specified as a vector of scalar values. Data Types: `single` \| `double` Truncation interval for the probability distribution, specified as a vector of scalar values containing the lower and upper truncation boundaries. Data Types: `single` \| `double` ### Other Object Properties Probability distribution name, specified as a character vector. Data Types: `char` Distribution parameter descriptions, specified as a cell array of character vectors. Each cell contains a short description of one distribution parameter. Data Types: `char` Distribution parameter names, specified as a cell array of character vectors. Data Types: `char` ## Object Functions `cdf` Cumulative distribution function `icdf` Inverse cumulative distribution function `iqr` Interquartile range of probability distribution `mean` Mean of probability distribution `median` Median of probability distribution `pdf` Probability density function `plot` Plot probability distribution object `random` Random numbers `std` Standard deviation of probability distribution `truncate` Truncate probability distribution object `var` Variance of probability distribution ## Examples collapse all Create a multinomial distribution object using the default parameter values. `pd = makedist('Multinomial')` ```pd = MultinomialDistribution Probabilities: 0.5000 0.5000 ``` Create a multinomial distribution object for a distribution with three possible outcomes. Outcome 1 has a probability of 1/2, outcome 2 has a probability of 1/3, and outcome 3 has a probability of 1/6. `pd = makedist('Multinomial','Probabilities',[1/2 1/3 1/6])` ```pd = MultinomialDistribution Probabilities: 0.5000 0.3333 0.1667 ``` Generate a random outcome from the distribution. ```rng('default'); % for reproducibility r = random(pd)``` ```r = 2 ``` The result of this trial is outcome 2. By default, the number of trials in each experiment, $n$, equals 1. Generate random outcomes from the distribution when the number of trials in each experiment, $n$, equals 1, and the experiment is repeated ten times. ```rng('default'); % for reproducibility r = random(pd,10,1)``` ```r = 10×1 2 3 1 3 2 1 1 2 3 3 ``` Each element in the array is the outcome of an individual experiment that contains one trial. Generate random outcomes from the distribution when the number of trials in each experiment, $n$, equals 5, and the experiment is repeated ten times. ```rng('default'); % for reproducibility r = random(pd,10,5)``` ```r = 10×5 2 1 2 2 1 3 3 1 1 1 1 3 3 1 2 3 1 3 1 2 2 2 2 1 1 1 1 2 2 1 1 1 2 2 1 2 3 1 1 2 3 2 2 3 2 3 3 1 1 2 ``` Each element in the resulting matrix is the outcome of one trial. The columns correspond to the five trials in each experiment, and the rows correspond to the ten experiments. For example, in the first experiment (corresponding to the first row), 2 of the 5 trials resulted in outcome 1, and 3 of the 5 trials resulted in outcome 2. ## Version History Introduced in R2013a	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 4, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9825195074081421, "perplexity": 1154.4346317187271}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-14/segments/1679296943698.79/warc/CC-MAIN-20230321131205-20230321161205-00420.warc.gz"}
https://eng.libretexts.org/Courses/Canada_College/Circuits_and_Devices/04%3A_Analysis_Theorems_and_Techniques/4.06%3A_Maximum_Power_Transfer_Theorem	# 4.6: Maximum Power Transfer Theorem Given a simple voltage source with internal resistance, a useful question to ask is “What value of load resistance will yield the maximum amount of power in the load?” While it is not true that maximizing load power is a goal of all circuit designs, it is a goal of a portion of them and thus worth a closer look. Consider the basic circuit depicted in Figure 6.6.1 with source $$E$$, source internal resistance $$R_i$$ and load resistance $$R$$. Figure 6.6.1 : Defining maximum power transfer. We would like to describe the load power in terms of the load resistance. To make the job easier, we may normalize the voltage source $$E$$ to 1 volt and the source resistance $$R_i$$ to 1 Ohm. By doing this, $$R$$ also becomes a normalized value, that is, it no longer represents a simple resistance value but rather represents a ratio in comparison to $$R_i$$. In this way the analysis will work for any set of source values. Note that the value of $$E$$ will equally scale the power in both $$R_i$$ and $$R$$, so a precise value is not needed, and thus, we may as well chose 1 volt for convenience. The power in the load can be determined by using $$I^2 R$$ where $$I = E / (R_i+R)$$. Using our normalized values $$I = 1 / (1+R)$$ and thus the load power is: $P = \left( \frac{1}{1+R} \right)^2 R \text{ or after expanding, } P = \frac{R}{R^2+2 R+1} \nonumber$ We now have an equation that describes the load power in terms of the load resistance. Before we go any further, take a look at what this equation tells you, in general. It is obvious that maximum power will not occur at the extremes. If $$R = 0$$ or $$R = \infty$$ (i.e., shorted or opened load) the load power is zero. While shorting the load will yield the maximum load current, it will also yield zero load voltage and thus no power. Similarly, opening the load will yield maximum load voltage but it will also yield zero load current, and again, no load power. To find the precise value that produces the maximum load power, the proof can be divided into two major portions. The first involves graphing the function and the second requires differential calculus to solve for a precise value. We shall proceed with the graphing portion which will lead us to the answer. The more rigorous proof of the second method is detailed in Appendix C. ## Graphing the Power Function $$P = R / (R^2+2R+1)$$ This may be done in parts, looking at the contribution of each term, and then combining to form the final result. First, consider the denominator $$R^2 + 2R + 1$$. It consists of three segments. The most simple is the horizontal line at +1. The $$2R$$ term creates a straight line with slope 2 starting at the origin. The $$R^2$$ term creates a simple curve with increasing slope that starts at the origin, crosses the horizontal +1 line at $$R = 1$$ and also crosses the $$2R$$ line at $$R = 2$$. These items are drawn individually and then added together as illustrated in Figure 6.6.2 . Figure 6.6.2 : The three terms of the power equation plotted individually and then summed. Of course, we must remember that we really want $$1 / ( R^2 + 2R + 1)$$ and so we plot the reciprocal of the combo as shown in Figure 6.6.3 (red curve). We also include the numerator term, $$R$$. This is shown as a straight line (blue) with a slope of 1. Finally, these two curves are multiplied together to produce the load power equation (black). Figure 6.6.3 : The power equation and components plotted. A close examination of the power curve will show that the peak occurs at $$R = 1$$. This is easier to see if we plot the completed power curve using a logarithmic horizontal axis and also scale the vertical axis to 100%, as shown in Figure 6.6.4 . The peak is more apparent and the curve is symmetrical in shape rather than lopsided. This reinforces the idea that the ratio of the resistances is what matters. Figure 6.6.4 : The load power curve with logarithmic axis showing symmetry. Finally, we can state: $\text{Maximum load power will be achieved when the load resistance is equal to the internal resistance of the driving source.} \nonumber$ No other value of load resistance will produce a higher load power. For the circuit of Figure 6.6.1 , this means that $$R$$ must equal $$R_i$$. As an exercise, we can try substituting a few values around the peak to verify this. For example, given $$E$$ = 1 V and $$R_i$$ = 1 $$\Omega$$, we calculate $$P$$ for $$R$$ = 0.5 $$\Omega$$, 1 $$\Omega$$, and 2 $$\Omega$$: $P_{0.5} = 0.5/(0.5^2+2 \cdot 0.5+1) = 2/9 \nonumber$ $P_1 = 1/(1^2+2 \cdot 1+1) = 2/8 = 1/4 \nonumber$ $P_2 = 2/(2^2+2 \cdot 2+1) = 2/9 \nonumber$ You can also try this with extremely small variations such as $$R$$ = 0.9999 $$\Omega$$ along with $$R$$ = 1.0001 $$\Omega$$ and you will not achieve a value equal to or greater than 1/4 Watt. While matching the resistance produces the maximum load power, it does not produce maximum load current or maximum load voltage. In fact, this condition produces a load voltage and a load current that are half of their maximums. Their product, however, is at the maximum. Further, efficiency at maximum load power is only 50% (i.e., only half of all generated power goes to the load with the other half being wasted internally). Values of $$R$$ greater than $$R_i$$ will achieve higher efficiency but at reduced load power. Sometimes we favor efficiency over maximal load power. As any linear two point network can be reduced to something like Figure 6.6.1 by using Thévenin's theorem, combining the two theorems allows us to determine maximum power conditions for any resistor in a complex circuit.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9116451740264893, "perplexity": 314.9368346219663}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058450.44/warc/CC-MAIN-20210927120736-20210927150736-00516.warc.gz"}
https://scholars.ncu.edu.tw/en/publications/an-adaptive-homomorphic-aperture-photometry-algorithm-for-merging	# An adaptive homomorphic aperture photometry algorithm for merging galaxies J. C. Huang, C. Y. Hwang Research output: Contribution to journalArticlepeer-review 1 Scopus citations ## Abstract We present a novel automatic adaptive aperture photometry algorithm for measuring the total magnitudes of merging galaxies with irregular shapes. First, we use a morphological pattern recognition routine for identifying the shape of an irregular source in a background-subtracted image. Then, we extend the shape of the source by using the Dilation image operation to obtain an aperture that is quasi-homomorphic to the shape of the irregular source. The magnitude measured from the homomorphic aperture would thus have minimal contamination from the nearby background. As a test of our algorithm, we applied our technique to the merging galaxies observed by the Sloan Digital Sky Survey and the Canada-France-Hawaii Telescope. Our results suggest that the adaptive homomorphic aperture algorithm can be very useful for investigating extended sources with irregular shapes and sources in crowded regions. Original language English 034001 Publications of the Astronomical Society of the Pacific 129 973 https://doi.org/10.1088/1538-3873/aa5786 Published - 1 Mar 2017 ## Keywords • Techniques: image processing • Techniques: photometric ## Fingerprint Dive into the research topics of 'An adaptive homomorphic aperture photometry algorithm for merging galaxies'. Together they form a unique fingerprint.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9233251810073853, "perplexity": 3179.108627814241}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320301264.36/warc/CC-MAIN-20220119064554-20220119094554-00194.warc.gz"}
http://www.zora.uzh.ch/id/eprint/108305/	# Observation of the resonant character of the $Z(4430)^-$ state LHCb Collaboration; Bernet, R; Müller, K; Steinkamp, O; Straumann, U; Vollhardt, A; et al (2014). Observation of the resonant character of the $Z(4430)^-$ state. Physical Review Letters, 112(222002):online. ## Abstract Resonant structures in $B^0\to\psi'\pi^-K^+$ decays are analyzed by performing a four-dimensional fit of the decay amplitude, using $pp$ collision data corresponding to $\rm 3 fb^{-1}$ collected with the LHCb detector. The data cannot be described with $K^+\pi^-$ resonances alone, which is confirmed with a model-independent approach. A highly significant $Z(4430)^-\to\psi'\pi^-$ component is required, thus confirming the existence of this state. The observed evolution of the $Z(4430)^-$ amplitude with the $\psi'\pi^-$ mass establishes the resonant nature of this particle. The mass and width measurements are substantially improved. The spin-parity is determined unambiguously to be $1^+$. ## Abstract Resonant structures in $B^0\to\psi'\pi^-K^+$ decays are analyzed by performing a four-dimensional fit of the decay amplitude, using $pp$ collision data corresponding to $\rm 3 fb^{-1}$ collected with the LHCb detector. The data cannot be described with $K^+\pi^-$ resonances alone, which is confirmed with a model-independent approach. A highly significant $Z(4430)^-\to\psi'\pi^-$ component is required, thus confirming the existence of this state. The observed evolution of the $Z(4430)^-$ amplitude with the $\psi'\pi^-$ mass establishes the resonant nature of this particle. The mass and width measurements are substantially improved. The spin-parity is determined unambiguously to be $1^+$. ## Statistics ### Citations 122 citations in Web of Science® 120 citations in Scopus® ### Altmetrics Detailed statistics Item Type: Journal Article, refereed, original work 07 Faculty of Science > Physics Institute 530 Physics English 2014 24 Feb 2015 14:06 27 Apr 2017 23:46 American Physical Society 0031-9007 https://doi.org/10.1103/PhysRevLett.112.222002	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9851714372634888, "perplexity": 1969.6730438322209}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-34/segments/1502886105291.88/warc/CC-MAIN-20170819012514-20170819032514-00496.warc.gz"}
https://wordpandit.com/lcm-hcf-formula-application/	Select Page In the previous articles on LCM and HCF, we studied the basic concepts and formulas required to solve questions. Now, we study some important shortcuts, tricks and formulas that are important from exam’s point of view. 1. LCM – LCM of two or more fractions is given by Example: Find the LCM and HCF of 3/14, 4/15 and 5/12 Solution: 1. For two numbers a and b their product is equal to the product of their LCM and HCF hence Example: The LCM of two numbers is 520 and their HCF is 4. If one of the numbers is 52, then the other number is (1) 40 (2) 42 (3) 50 (4) 52 Solution: (1) HCF × LCM = First number × second number 52 * second number = 520 4 Second number = 40 1. HCF of a given set of numbers is always a factor of LCM. As we know HCF is calculated by considering the least power of the numbers, which will automatically be a part of their LCM. Example: LCM of 12 and 18: 12= 223 18=233 LCM will be highest power ; LCM = 2233 =36 HCF of 12 and 18 : 12= 223 18=233 HCF = 2 3 = 6 Which is a factor of 36. 1. If ‘h’ is the HCF of two numbers ‘a’ and ‘b’ then ‘h’ is also HCF of ‘a’ and ‘a + b’. If ‘h’ is the HCF of two numbers ‘a’ and ‘b’ then ‘h’ is also HCF of ‘a’ and ‘ab’. If ‘h’ is the HCF of two numbers ‘a’ and ‘b’ then ‘h’ is also HCF of ‘a’ and ‘ab’ and ‘a + b’. Example: HCF of 12 and 18 is 6 . so HCF of 12 and 12+18=30 will also be 6 . 12= 223 30=235 HCF – 3 2= 6 1. If for two numbers HCF = LCM then two numbers must be equal to each other. 6. For a Number n = apbqcr (here a, b & c are prime factors and p, q and r are integers then number of pairs of two number such that their LCM is N is given by: 1. For three numbers a, b and c if LCM and HCF of (a, b), (b, c) and (c, a) is Lab, Lbc, Lbc, Lca, Hab, Hbc and Hca respectively then Example: prove the above property for 12,18 and 20 Solution: 12 = 223 18= 2 33 20=225 LCM= 22335=180 HCF = 2 1. For two numbers A and B if HCF is h then we can assume A = hx and B = hy here x and y are co-prime to each other, and also their LCM is given by “hxy”. Example: How many pairs of number exist such that their H.C.F is 4 & L.C.M is 48? Solution: Since H.C.F of two number is 4. Hence we can assume two numbers as 4x & 4y. (Here x and y are co-prime to each other) We know for two numbers: H.C.F × L.C.M = Product of two numbers Hence 4 × 48 = 4x × 4y Or xy = 12 = 3 × 4 x = 3 and y=4 OR x=4 ,y= 3 so two pairs will exist Example: How many pairs of two digit numbers exist such that their H.C.F is 8 & L.C.M is 120? Solution: Let numbers are 8x & 8then 8x × 8y = 8 × 120 Or xy= 15 So pair of x & y is 3 & 5 that will give numbers as 24 & 40. If we take a & b as 1 & 15 then we will get the numbers as 8 & 120 but here one of them 8 is not a two digit number. Here only one such pair exists. 1. HCF of two consecutive natural numbers is always 1. HCF of two consecutive even natural number is always 2 Let’s apply these concepts to some more questions. EXERCISE: Question 1. The H.C.F. of two numbers is 96 and their L.C.M. is 1296. If one of the numbers is 864, the other is (1) 132 (2) 135 (3) 140 (4) 144 Solution: (4) HCF × LCM = First number × second number ⇒ 864 × second number= 96 × 1296 ⇒ Second number Question 2. The smallest square number divisible by 10, 16 and 24 is (1) 900 (2) 1600 (3) 2500 (4) 3600 Solution: (4) Question 3. The sum of two numbers is 84 and their HCF is 12. Total number of such pairs of numbers is (1) 2 (2) 3 (3) 4 (4) 5 Solution: (2) Question 4. HCF and LCM of two numbers are 7 and 140 respectively. If the numbers are between 20 and 45, the sum of the numbers, is : (1) 70 (2) 77 (3) 63 (4) 56 Let the numbers be 7a and 7b where a and b are co-prime. Now, LCM of 7a and 7b = 7ab Question 5. The LCM of .two numbers is 4 times their HCF. The sum of LCM and HCF is 125. If one of the numbers is 100, then the other number is (1) 5 (2) 25 (3) 100 (4) 125	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8390204310417175, "perplexity": 1526.1555375725645}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-49/segments/1669446710462.59/warc/CC-MAIN-20221128002256-20221128032256-00719.warc.gz"}
https://theinformaticists.com/2019/09/16/lecture-9-multiple-hypothesis-testing-more-examples/	# Lecture 9: Multiple Hypothesis Testing: More Examples (Warning: These materials may be subject to lots of typos and errors. We are grateful if you could spot errors and leave suggestions in the comments, or contact the author at [email protected].) In the last lecture we have seen the general tools and concrete examples of reducing the statistical estimation problem to multiple hypothesis testing. In these examples, the loss function is typically straightforward and the construction of the hypotheses is natural. However, other than statistical estimation the loss function may become more complicated (e.g., the excess risk in learning theory), and the hypothesis constructions may be implicit. To better illustrate the power of the multiple hypothesis testing, this lecture will be exclusively devoted to more examples in potentially non-statistical problems. 1. Example I: Density Estimation This section considers the most fundamental problem in nonparametric statistics, i.e., the density estimation. It is well-known in the nonparametric statistics literature that, if a density on the real line has Hölder smoothness parameter $s>0$, it then can be estimated within $L_2$ accuracy $\Theta(n^{-s/(2s+1)})$ based on $n$ samples. However, in this section we will show that this is no longer the correct minimax rate when we generalize to other notions of smoothness and other natural loss functions. Fix an integer $k\ge 1$ and some norm parameter $p\in [1,\infty]$, the Sobolev space $\mathcal{W}^{k,p}(L)$ on $[0,1]$ is defined as $$\mathcal{W}^{k,p}(L) := \{f\in C[0,1]: \\|f\\|_p + \\|f^{(k)}\\|_p \le L \},$$ where the derivative $f^{(k)}$ is defined in terms of distributions ($f$ not necessarily belongs to $C^k[0,1]$). Our target is to determine the minimax rate $R_{n,k,p,q}^\star$ of estimating the density $f\in \mathcal{W}^{k,p}(L)$ based on $n$ i.i.d. observations from $f$, under the general $L_q$ loss with $q\in [1,\infty)$. For simplicity we suppress the dependence on $L$ and assume that $L$ is a large positive constant. The main result of this section is as follows: $$R_{n,k,p,q}^\star = \begin{cases} \Omega(n^{-\frac{k}{2k+1}}) & \text{if } q<(1+2k)p, \\ \Omega((\frac{\log n}{n})^{\frac{k-1/p + 1/q}{2(k-1/p) + 1/q}} ) & \text{if } q \ge (1+2k)p. \end{cases}$$ This rate is also tight. We see that as $k,p$ are fixed and $q$ increases from $1$ to $\infty$, the minimax rate for density estimation increases from $n^{-k/(2k+1)}$ to $(\log n/n)^{(k-1/p)/(2(k-1/p)+1) }$. We will show that these rates correspond to the “dense” and the “sparse” cases, respectively. 1.1. Dense case: $q<(1+2k)p$ We first look at the case with $q<(1+2k)p$, which always holds for Hölder smoothness where the norm parameter is $p=\infty$. The reason why it is called the “dense” case is that in the hypothesis construction, the density is supported everywhere on $[0,1]$, and the difficulty is to classify the directions of the bumps around some common density. Specifically, let $h\in (0,1)$ be a bandwidth parameter to be specified later, we consider $$f_v(x) := 1 + \sum_{i=1}^{h^{-1}} v_i\cdot h^sg\left(\frac{x - (i-1)h}{h}\right)$$ for $v\in \{\pm 1\}^{1/h}$, where $g$ is a fixed smooth function supported on $[0,1]$ with $\int_0^1 g(x)dx=0$, and $s>0$ is some tuning parameter to ensure that $f_v\in \mathcal{W}^{k,p}(L)$. Since $$f_v^{(k)}(x) = \sum_{i=1}^{h^{-1}} v_i\cdot h^{s-k}g^{(k)}\left(\frac{x - (i-1)h}{h}\right),$$ we conclude that the choice $s=k$ is sufficient. To invoke the Assoaud’s lemma, first note that for $q=1$ the separation condition is fulfilled with $\Delta = \Theta(h^{k+1})$. Moreover, for any $v,v'\in \{\pm 1\}^{1/h}$ with $d_{\text{H}}(v,v') = 1$, we have $$\chi^2(f_v^{\otimes n}, f_{v'}^{\otimes n}) + 1 \le \left( 1 + \frac{ \\|f_v - f_{v'}\\|_2^2 }{1 - h^k\\|g\\|_\infty} \right)^n = \left(1 + \frac{ 4h^{2k+1}\\|g\\|_2^2 }{1 - h^k\\|g\\|_\infty} \right)^n.$$ Hence, the largest $h>0$ to ensure that $\chi^2(f_v^{\otimes n}, f_{v'}^{\otimes n}) = O(1)$ is $h = \Theta(n^{-1/(2k+1)})$, which for $q=1$ gives the minimax lower bound $$R_{n,k,p,q}^\star = \Omega(n^{-\frac{k}{2k+1}} ).$$ This lower bound is also valid for the general case $q\ge 1$ due to the monotonicity of $L_q$ norms. 1.2. Sparse case: $q\ge (1+2k)p$ Next we turn to the sparse case $q\ge (1+2k)p$, where the name of “sparse” comes from the fact that for the hypotheses we will construct densities with only one bump on a small interval, and the main difficulty is to locate that interval. Specifically, let $h\in (0,1), s>0$ be bandwidth and smoothness parameters to be specified later, and we consider $$f_i(x) := h^sg\left(\frac{x-(i-1)h}{h}\right)\cdot 1\left((i-1)h where $i\in [h^{-1}]$, and $g$ is a fixed smooth density on $[0,1]$. Clearly $f_i(x)$ is a density on $[0,1]$ for all $i\in [M]$, and $$f_i^{(k)}(x) = h^{s-k}g^{(k)} \left(\frac{x-(i-1)h}{h}\right)\cdot 1\left((i-1)h Consequently, $\\|f_i^{(k)}\\|_p = h^{s-k+1/p}\\|g^{(k)}\\|_p$, and the Sobolev ball requirement leads to the choice $s = k-1/p$. Next we check the conditions of Fano’s inequality. Since $$\\|f_i - f_j\\|_q = h^{k-1/p+1/q} \cdot 2^{1/q}\\|g\\|_q, \qquad \forall i\neq j\in [h^{-1}],$$ the separation condition is fulfilled with $\Delta = \Theta(h^{k-1/p+1/q})$. Moreover, since $$D_{\text{KL}}(f_i \\| f_j) \le \frac{\\|f_i - f_j\\|_2^2}{1-h^s} = \frac{2\\|g\\|_2^2}{1-h^s}\cdot h^{2(k-1/p)+1},$$ we conclude that $I(V;X)= O(nh^{2(k-1/p)+1})$. Consequently, Fano’s inequality gives $$R_{n,k,p,q}^\star = \Omega\left(h^{k-1/p+1/q}\left(1 - \frac{O(nh^{2(k-1/p)+1})+\log 2}{\log(1/h)} \right) \right),$$ and choosing $h = \Theta((\log n/n)^{1/(2(k-1/p)+1)} )$ gives the desired result. 2. Example II: Aggregation In estimation or learning problems, sometimes the learner is given a set of candidate estimators or predictors, and she aims to aggregate them into a new estimate based on the observed data. In scenarios where the candidates are not explicit, aggregation procedures can still be employed based on sample splitting, where the learner splits the data into independent parts, uses the first part to construct the candidates and the second part to aggregate them. In this section we restrict ourselves to the regression setting where there are $n$ i.i.d. observations $x_1,\cdots,x_n\sim P_X$, and $$y_i = f(x_i )+ \xi_i, \qquad i\in [n],$$ where $\xi_i\sim \mathcal{N}(0,1)$ is the independent noise, and $f: \mathcal{X}\rightarrow {\mathbb R}$ is an unknown regression function with $\\|f\\|_\infty \le 1$. There is also a set of candidates $\mathcal{F}=\{f_1,\cdots,f_M\}$, and the target of aggregation is to find some $\hat{f}_n$ (not necessarily in $\mathcal{F}$) to minimize $$\mathop{\mathbb E}_f \left(\\|\hat{f}_n - f\\|_{L_2(P_X)}^2 - \inf_{\lambda\in \Theta} \\|f_\lambda - f\\|_{L_2(P_X)}^2\right),$$ where the expectation is taken over the random observations $(x_1,y_1),\cdots,(x_n,y_n)$, $f_\lambda(x) := \sum_{i=1}^M \lambda_i f_i(x)$ for any $\lambda \in {\mathbb R}^M$, and $\Theta\subseteq {\mathbb R}^p$ is a suitable subset corresponding to different types of aggregation. For a fixed data distribution $P_X$, the minimax rate of aggregation is defined as the minimum worst-case excess risk over all bounded functions $f$ and candidate functions $\{f_1,\cdots,f_M\}$: $$R_{n,M}^{\Theta} := \sup_{f_1,\cdots,f_M} \inf_{\hat{f}_n} \sup_{\\|f\\|_\infty\le 1} \mathop{\mathbb E}_f \left(\\|\hat{f}_n - f\\|_{L_2(P_X)}^2 - \inf_{\lambda\in \Theta} \\|f_\lambda - f\\|_{L_2(P_X)}^2\right).$$ Some special cases are in order: 1. When $\Theta = {\mathbb R}^M$, the estimate $\hat{f}_n$ is compared with the best linear aggregates and this is called linear aggregation, with optimal rate denoted as $R^{\text{L}}$ 2. When $\Theta = \Lambda^M = \{\lambda\in {\mathbb R}_+^M: \sum_{i=1}^M \lambda_i\le 1 \}$, the estimate $\hat{f}_n$ is compared with the best convex combination of candidates (and zero) and this is called convex aggregation, with optimal rate denoted as $R^{\text{C}}$ 3. When $\Theta = \{e_1,\cdots,e_M\}$, the set of canonical vectors, the estimate $\hat{f}_n$ is compared with the best candidate in $\mathcal{F}$ and this is called model selection aggregation, with optimal rate denoted as $R^{\text{MS}}$ The main result in this section is summarized in the following theorem. Theorem 1 If there is a cube $\mathcal{X}_0\subseteq \mathcal{X}$ such that $P_X$ admits a density lower bounded from below w.r.t. the Lebesgue measure on $\mathcal{X}_0$, then $$R_{n,M}^\Theta = \begin{cases} \Omega(1\wedge \frac{M}{n}) &\text{if } \Theta = {\mathbb R}^M, \\ \Omega(1\wedge (\frac{M}{n} + \sqrt{\frac{1}{n}\log(\frac{M}{\sqrt{n}}+1)}) ) & \text{if } \Theta = \Lambda^M, \\ \Omega(1\wedge \frac{\log M}{n}) & \text{if } \Theta = \{e_1,\cdots,e_M\}. \end{cases}$$ We remark that the rates in Theorem 1 are all tight. In the upcoming subsections we will show that although the loss function of aggregation becomes more complicated, the idea of multiple hypothesis testing can still lead to tight lower bounds. 2.1. Linear aggregation Since the oracle term $\inf_{\lambda \in {\mathbb R}^d} \\|f_\lambda - f\\|_{L^2(P_X)}^2$ is hard to deal with, a natural idea would be to consider a well-specified model such that this term is zero. Since $P_X$ admits a density, we may find a partition $\mathcal{X} = \cup_{i=1}^M \mathcal{X}_i$ such that $P_X(\mathcal{X}_i) = M^{-1}$ for all $i$. Consider the candidate functions $f_i(x) = 1(x\in \mathcal{X}_i)$, and for $v\in \{0, 1\}^M$, let $$f_v(x) := \gamma\sum_{i=1}^M v_if_i(x),$$ where $\gamma\in (0,1)$ is to be specified. To apply the Assoaud’s lemma, note that for the loss function $$L(f,\hat{f}) := \\|\hat{f} - f\\|_{L_2(P_X)}^2 - \inf_{\lambda\in {\mathbb R}^p} \\|f_\lambda - f\\|_{L_2(P_X)}^2,$$ the orthogonality of the candidates $(f_i)_{i\in [M]}$ implies that the separability condition holds for $\Delta = \gamma^2/2M$. Moreover, $$\max_{d_{\text{H}}(v,v')=1 } D_{\text{KL}}(P_{f_v}^{\otimes n} \\| P_{f_{v'}}^{\otimes n}) = \max_{d_{\text{H}}(v,v')=1 } \frac{n\\|f_v - f_{v'}\\|_{L_2(P_X)}^2}{2} = O\left(\frac{n\gamma^2}{M}\right),$$ Therefore, the Assoud’s lemma (with Pinsker’s inequality) gives $$R_{n,M}^{\text{L}} \ge \frac{\gamma^2}{4}\left(1 - O\left( \sqrt{\frac{n\gamma^2}{M}}\right)\right).$$ Choosing $\gamma = 1\wedge \sqrt{M/n}$ completes the proof. 2.2. Convex aggregation The convex aggregation differs only with the linear aggregation in the sense that the linear coefficients must be non-negative and the sum is at most one. Note that the only requirement in the previous arguments is the orthogonality of $(f_i)_{i\in [M]}$ under $L_2(P_X)$, we may choose any orthonormal functions $(f_i)_{i\in [M]}$ under $L_2(dx)$ with $\max_i\\|f_i\\|_\infty = O(1)$ (existence follows from the cube assumption) and use the density lower bound assumption to conclude that $\\|f_i\\|_{L_2(P_X)} = \Omega(\\|f_i\\|_2)$ (to ensure the desired separation). Then the choice of $\gamma$ above becomes $O(n^{-1/2})$, and we see that the previous arguments still hold for convex aggregation if $M = O(\sqrt{n})$. Hence, it remains to prove that when $M = \Omega(\sqrt{n})$ $$R_{n,M}^{\text{C}} = \Theta\left(1\wedge \sqrt{\frac{1}{n}\log\left(\frac{M}{\sqrt{n}}+1\right) }\right).$$ Again we consider the well-specified case where $$f_v(x) := \gamma\sum_{i=1}^M v_if_i(x),$$ with the above orthonormal functions $(f_i)$, a constant scaling factor $\gamma = O(1)$, and a uniform size-$m$ subset of $(v_i)$ being $1/m$ and zero otherwise ($m\in \mathbb{N}$ is to be specified). Since the vector $v$ is no longer the vertex of a hypercube, we apply the generalized Fano’s inequality instead of Assoaud. First, applying Lemma 4 in Lecture 8 gives $$I(V; X) \le \mathop{\mathbb E}_V D_{\text{KL}}(P_{f_V}^{\otimes n} \\| P_{f_0}^{\otimes n}) = O\left(\frac{n}{m}\right).$$ Second, as long as $m\le M/3$, using similar arguments as in the sparse linear regression example in Lecture 8, for $\Delta = c/m$ with a small constant $c>0$ we have $$p_\Delta := \sup_{f} \mathop{\mathbb P}\left(\\|f_V - f\\|_{L^2(P_X)}^2 \le \Delta \right) \le \left(\frac{M}{m}+1\right)^{c'm},$$ with $c'>0$. Therefore, the generalized Fano’s inequality gives $$R_{n,M}^{\text{C}} \ge \frac{c}{m}\left(1 - \frac{O(n/m) + \log 2}{c'm\log\left(\frac{M}{m}+1\right)} \right),$$ and choosing $m \asymp \sqrt{n/\log(M/\sqrt{n}+1)} = O(M)$ completes the proof. 2.3. Model selection aggregation As before, we consider the well-specified case where we select orthonormal functions $(\phi_i)_{i\in [M]}$ on $L_2(dx)$, and let $f_i = \gamma \phi_i$, $f\in \{f_1,\cdots,f_M\}$. The orthonormality of $(\phi_i)$ gives $\Delta = \Theta(\gamma^2)$ for the separation condition of the original Fano’s inequality, and $$\max_{i\neq j} D_{\text{KL}}(P_{f_i}^{\otimes n}\\| P_{f_j}^{\otimes n}) = \max_{i\neq j} \frac{\\|f_i - f_j\\|_{L_2(P_X)}^2}{2} = O\left(n\gamma^2\right).$$ Hence, Fano’s inequality gives $$R_{n,M}^{\text{MS}} = \Omega\left(\gamma^2 \left(1 -\frac{n\gamma^2 + \log 2}{\log M}\right)\right),$$ and choosing $\gamma^2 \asymp 1\wedge n^{-1}\log M$ completes the proof. We may show a further result that any model selector cannot attain the above optimal rate of model selection aggregation. Hence, even to compare with the best candidate function in $\mathcal{F}$, the optimal aggregate $\hat{f}$ should not be restricted to $\mathcal{F}$. Specifically, we have the following result, whose proof is left as an exercise. Exercise 1 Under the same assumptions of Theorem 1, show that $$\sup_{f_1,\cdots,f_M} \inf_{\hat{f}_n\in\{f_1,\cdots,f_M\} } \sup_{\\|f\\|_\infty\le 1} \mathop{\mathbb E}_f \left(\\|\hat{f}_n - f\\|_{L_2(P_X)}^2 - \min_{i\in [M]} \\|f_i - f\\|_{L_2(P_X)}^2\right) = \Omega\left(1\wedge \sqrt{\frac{\log M}{n}}\right).$$ 3. Example III: Learning Theory Consider the classification problem where there are $n$ training data $(x_1,y_1), \cdots, (x_n, y_n)$ i.i.d. drawn from an unknown distribution $P_{XY}$, with $y_i \in \{\pm 1\}$. There is a given collection of classifiers $\mathcal{F}$ consisting of functions $f: \mathcal{X}\rightarrow \{\pm 1\}$, and given the training data, the target is to find some classifier $\hat{f}\in \mathcal{F}$ with a small excess risk $$R_{\mathcal{F}}(P_{XY}, \hat{f}) = \mathop{\mathbb E}\left[P_{XY}(Y \neq \hat{f}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))\right],$$ which is the difference in the performance of the chosen classifer and the best classifier in the function class $\mathcal{F}$. In the definition of the excess risk, the expectation is taken with respect to the randomness in the training data. The main focus of this section is to characterize the minimax excess risk of a given function class $\mathcal{F}$, i.e., $$R_{\text{pes}}^\star(\mathcal{F}) := \inf_{\hat{f}}\sup_{P_{XY}} R_{\mathcal{F}}(P_{XY}, \hat{f}).$$ The subscript “pes” here stands for “pessimistic”, where $P_{XY}$ can be any distribution over $\mathcal{X} \times \{\pm 1\}$ and there may not be a good classifier in $\mathcal{F}$, i.e., $\inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))$ may be large. We also consider the optimistic scenario where there exists a perfect (error-free) classifer in $\mathcal{F}$. Mathematically, denoting by $\mathcal{P}_{\text{opt}}(\mathcal{F})$ the collection of all probability distributions $P_{XY}$ on $\mathcal{X}\times \{\pm 1\}$ such that $\inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))=0$, the minimax excess risk of a given function class $\mathcal{F}$ in the optimistic case is defined as $$R_{\text{opt}}^\star(\mathcal{F}) := \inf_{\hat{f}}\sup_{P_{XY} \in \mathcal{P}_{\text{opt}}(\mathcal{F})} R_{\mathcal{F}}(P_{XY}, \hat{f}).$$ The central claim of this section is the following: Theorem 2 Let the VC dimension of $\mathcal{F}$ be $d$. Then $$R_{\text{\rm opt}}^\star(\mathcal{F}) = \Omega\left(1\wedge \frac{d}{n}\right), \qquad R_{\text{\rm pes}}^\star(\mathcal{F}) = \Omega\left(1\wedge \sqrt{\frac{d}{n}}\right).$$ Recall that the definition of VC dimension is as follows: Definition 3 For a given function class $\mathcal{F}$ consisting of mappings from $\mathcal{X}$ to $\{\pm 1\}$, the VC dimension of $\mathcal{F}$ is the largest integer $d$ such that there exist $d$ points from $\mathcal{X}$ which can be shattered by $\mathcal{F}$. Mathematically, it is the largest $d>0$ such that there exist $x_1,\cdots,x_d\in \mathcal{X}$, and for all $v\in \{\pm 1\}^d$, there exists a function $f_v\in \mathcal{F}$ such that $f_v(x_i) = v_i$ for all $i\in [d]$ VC dimension plays a significant role in statistical learning theory. For example, it is well-known that for the empirical risk minimization (ERM) classifier $$\hat{f}^{\text{ERM}} = \arg\min_{f\in \mathcal{F}} \frac{1}{n}\sum_{i=1}^n 1(y_i \neq f(x_i)),$$ we have $$\mathop{\mathbb E}\left[P_{XY}(Y \neq \hat{f}^{\text{ERM}}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X))\right] \le 1\wedge \begin{cases} O(d/n) & \text{in optimistic case}, \\ O(\sqrt{d/n}) & \text{in pessimistic case}. \end{cases}$$ Hence, Theorem 2 shows that the ERM classifier attains the minimax excess risk for all function classes, and the VC dimension exactly characterizes the difficulty of the learning problem. 3.1. Optimistic case We first apply the Assoaud’s lemma to the optimistic scenario. By the definition of VC dimension, there exist points $x_1,\cdots,x_d\in \mathcal{X}$ and functions $f_v\in \mathcal{F}$ such that for all $v\in \{\pm 1\}^d$ and $i\in [d]$, we have $f_v(x_i) = v_i$. Consider $2^{d-1}$ hypotheses indexed by $u\in \{\pm 1\}^{d-1}$: the distribution $P_X$ is always $$P_X(\{x_i\}) = p > 0, \quad \forall i\in [d-1], \qquad P_X(\{x_d\}) = 1-p(d-1),$$ where $p\in (0,(d-1)^{-1})$ is to be specified later. For the conditional distribution $P_{Y\|X}$, let $Y = f_{(u,1)}(X)$ hold almost surely under the joint distribution $P_u$. Clearly this is the optimistic case, for there always exists a perfect classifier in $\mathcal{F}$. We first examine the separation condition in Assoaud’s lemma, where the loss function here is $$L(P_{XY}, \hat{f}) := P_{XY}(Y \neq \hat{f}(X)) - \inf_{f^\star \in \mathcal{F}} P_{XY}(Y \neq f^\star(X)).$$ For any $u,u'\in \{\pm 1\}^{d-1}$ and any $\hat{f}\in \mathcal{F}$, we have $$\begin{array}{rcl} L(P_u, \hat{f}) + L(P_{u'}, \hat{f}) &=& p\cdot \sum_{i=1}^{d-1} 1(\hat{f}(x_i) \neq f_u(x_i)) + 1(\hat{f}(x_i) \neq f_{u'}(x_i)) \\ &=& p\cdot \sum_{i=1}^{d-1} 1(\hat{f}(x_i) \neq u_i) + 1(\hat{f}(x_i) \neq u_i') \\ &\ge & p\cdot d_{\text{H}}(u,u'), \end{array}$$ and therefore the separation condition holds for $\Delta = p$. Moreover, if $u$ and $u'$ only differ in the $i$-th component, then $P_u$ and $P_{u'}$ are completely indistinguishable if $x_i$ does not appear in the training data. Hence, by coupling, $$\max_{d_{\text{H}}(u,u')=1 } \\|P_u^{\otimes n} - P_{u'}^{\otimes n}\\|_{\text{\rm TV}} \le \left(1 - p\right)^n.$$ Therefore, Assoaud’s lemma gives $$R_{\text{opt}}^\star(\mathcal{F}) \ge \frac{(d-1)p}{2}\left(1 - (1-p)^n\right),$$ and choosing $p = \min\{n^{-1},(d-1)^{-1}\}$ yields to the desired lower bound. 3.2. Pessimistic case The analysis of the pessimistic case only differs in the construction of the hypotheses. As before, fix $x_1,\cdots,x_d\in \mathcal{X}$ and $f_v\in \mathcal{F}$ such that $f_v(x_i) = v_i$ for all $v\in \{\pm 1\}^d$ and $i\in [d]$. Now let $P_X$ be the uniform distribution on $\{x_1,\cdots,x_d\}$, and under $P_v$, the conditional probability $P_{Y\|X}$ is $$P_v(Y= v_i \| X=x_i) = \frac{1}{2} + \delta, \qquad \forall i\in [d],$$ and $\delta\in (0,1/2)$ is to be specified later. In other words, the classifier $f_v$ only outperforms the random guess by a margin of $\delta$ under $P_v$. Again we apply the Assoaud’s lemma for the lower bound of $R_{\text{pes}}^\star(\mathcal{F})$. First note that for all $v\in \{\pm 1\}^d$ $$\min_{f^\star \in \mathcal{F}} P_v(Y \neq f^\star(X)) = \frac{1}{2} - \delta.$$ Hence, for any $v\in \{\pm 1\}^d$ and any $f\in \mathcal{F}$, $$\begin{array}{rcl} L(P_v, f) &=& \frac{1}{d}\sum_{i=1}^d \left[\frac{1}{2} + \left(2\cdot 1(f(x_i) = v_i) - 1\right)\delta \right] - \left(\frac{1}{2}-\delta\right) \\ &=& \frac{2\delta}{d}\sum_{i=1}^d 1(f(x_i) = v_i). \end{array}$$ By triangle inequality, the separation condition is fulfilled with $\Delta = 2\delta/d$. Moreover, direct computation yields $$\max_{d_{\text{H}}(v,v')=1 } H^2(P_v, P_{v'}) = \frac{1}{d}\left(\sqrt{\frac{1}{2}+\delta} - \sqrt{\frac{1}{2}-\delta}\right)^2 = O\left(\frac{\delta^2}{d}\right),$$ and therefore tensorization gives $$\max_{d_{\text{H}}(v,v')=1 } H^2(P_v^{\otimes n}, P_{v'}^{\otimes n}) = 2 - 2\left(1 - O\left(\frac{\delta^2}{d}\right) \right)^n.$$ Finally, choosing $\delta = (1\wedge \sqrt{d/n})/2$ gives the desired result. 3.3. General case We may interpolate between the pessimistic and optimistic cases as follows: for any given $\varepsilon\in (0,1/2]$, we restrict to the set $\mathcal{P}(\mathcal{F},\varepsilon)$ of joint distributions with $$\sup_{P_{XY}\in \mathcal{P}(\mathcal{F},\varepsilon) }\inf_{f^\star\in \mathcal{F}} P_{XY}(Y\neq f^\star(X))\le \varepsilon.$$ Then we may define the minimax excess risk over $\mathcal{P}(\mathcal{F},\varepsilon)$ as $$R^\star(\mathcal{F},\varepsilon) := \inf_{\hat{f}}\sup_{P_{XY} \in \mathcal{P}(\mathcal{F},\varepsilon)} R_{\mathcal{F}}(P_{XY}, \hat{f}).$$ Clearly the optimistic case corresponds to $\varepsilon =0$, and the pessimistic case corresponds to $\varepsilon = 1/2$. Similar to the above arguments, we have the following lower bound on $R^\star(\mathcal{F},\varepsilon)$, whose proof is left as an exercise. Exercise 2 Show that when the VC dimension of $\mathcal{F}$ is $d$, then $$R^\star(\mathcal{F},\varepsilon) = \Omega\left(1\wedge \left(\frac{d}{n} + \sqrt{\frac{d}{n}\cdot \varepsilon}\right) \right).$$ 4. Example IV: Stochastic Optimization Consider the following oracle formulation of the convex optimizaton: let $f: \Theta\rightarrow {\mathbb R}$ be the convex objective function, and we aim to find the minimum value $\min_{\theta\in\Theta} f(\theta)$. To do so, the learner can query some first-order oracle adaptively, where given the query $\theta_t\in \Theta$ the oracle outputs a pair $(f(\theta_t), g(\theta_t))$ consisting of the objective value at $\theta_t$ (zeroth-order information) and a sub-gradient of $f$ at $\theta_t$ (first-order information). The queries can be made in an adaptive manner where $\theta_t$ can depend on all previous outputs of the oracle. The target of convex optimization is to determine the minimax optimization gap after $T$ queries defined as $$R_T^\star(\mathcal{F}, \Theta) = \inf_{\theta_T}\sup_{f\in \mathcal{F}} \left( f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta) \right),$$ where $\mathcal{F}$ is a given class of convex functions, and the final query $\theta_{T+1}$ can only depend on the past outputs of the functions. Since there is no randomness involved in the above problem, the idea of multiple hypothesis testing cannot be directly applied here. Therefore, in this section we consider a simpler problem of stochastic optimization, and we postpone the general case to later lectures. Specifically, suppose that the above first-order oracle $\mathcal{O}$ is stochastic, i.e., it only outputs $(\widehat{f}(\theta_t), \widehat{g}(\theta_t))$ such that $\mathop{\mathbb E}[\widehat{f}(\theta_t)] = f(\theta_t)$, and $\mathop{\mathbb E}[\widehat{g}(\theta_t)] \in \partial f(\theta_t)$. Let $\mathcal{F}$ be the set of all convex $L$-Lipschitz functions (in $L_2$ norm), $\Theta = \{\theta\in {\mathbb R}^d: \\|\theta\\|_2\le R \}$, and assume that the subgradient estimate returned by the oracle $\mathcal{O}$ always satisfies $\\|\widehat{g}(\theta_t)\\|_2\le L$ as well. Now the target is to determine the quantity $$R_{T,d,L,R}^\star = \inf_{\theta_T} \sup_{f \in \mathcal{F}} \sup_{\mathcal{O}} \left(\mathop{\mathbb E} f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta)\right),$$ where the expectation is taken over the randomness in the oracle output. The main result in this section is summarized in the following theorem: Theorem 4 $$R_{T,d,L,R}^\star = \Omega\left(\frac{LR}{\sqrt{T}}\right).$$ Since it is well-known that the stochastic gradient descent attains the optimization gap $O(LR/\sqrt{T})$ for convex Lipschitz functions, the lower bound in Theorem 4 is tight. Now we apply the multiple hypothesis testing idea to prove Theorem 4. Since the randomness only comes from the oracle, we should design the stochastic oracle carefully to reduce the optimization problem to testing. A natural way is to choose some function $F(\theta;X)$ such that $F(\cdot;X)$ is convex $L$-Lipschitz for all $X$, and $f(\theta) = \mathop{\mathbb E}_P[F(\theta;X)]$. In this way, the oracle can simply generate i.i.d. $X_1,\cdots,X_T\sim P$, and reveals the random vector $(F(\theta_t;X_t), \nabla F(\theta_t;X_t))$ to the learner at the $t$-th query. Hence, the proof boils down to find a collection of probability distributions $(P_v)_{v\in \mathcal{V}}$ such that they are separated apart while hard to distinguish based on $T$ observations. We first look at the separation condition. Since the loss function of this problem is $L(f,\theta_{T+1}) = f(\theta_{T+1}) - \min_{\theta\in\Theta} f(\theta)$, we see that $$\inf_{\theta_{T+1}\in \Theta} L(f,\theta_{T+1}) + L(g,\theta_{T+1}) = \min_{\theta\in\Theta} (f(\theta) + g(\theta)) - \min_{\theta\in\Theta} f(\theta) - \min_{\theta\in\Theta} g(\theta),$$ which is known as the optimization distance $d_{\text{opt}}(f,g)$. Now consider $$F(\theta;X) = L\left\|\theta_i - \frac{Rx_i}{\sqrt{d}}\right\| \qquad \text{if }x = \pm e_i,$$ and for $v\in \{\pm 1\}^d$, let $f_v(\theta) = \mathop{\mathbb E}_{P_v}[F(\theta;X)]$ with $P_v$ defined as $$P_v(X = v_ie_i) = \frac{1+\delta}{2d}, \quad P_v(X= -v_ie_i) = \frac{1-\delta}{2d}, \qquad \forall i\in [d].$$ Clearly $F(\cdot;X)$ is convex and $L$-Lipschitz, and for $\\|\theta\\|_\infty \le R/\sqrt{d}$ we have $$f_v(\theta) = \frac{LR}{\sqrt{d}} - \frac{\delta L}{d}\sum_{i=1}^d v_i\theta_i.$$ Hence, it is straightforward to verify that $\min_{\\|\theta\\|_2\le R} f_v(\theta) = LR(1-\delta)/\sqrt{d}$, and $$d_{\text{opt}}(f_v,f_{v'}) = \frac{2\delta LR}{d}\cdot d_{\text{H}}(v,v').$$ In other words, the separation condition in the Assoaud’s lemma is fulfilled with $\Delta = 2\delta LR/d$. Moreover, simple algebra gives $$\max_{d_{\text{H}}(v,v') =1} D_{\text{KL}}(P_v^{\otimes T}\\| P_{v'}^{\otimes T}) = O(T\delta^2),$$ and therefore Assoaud’s lemma gives $$R_{T,d,L,R}^\star \ge d\cdot \frac{\delta LR}{d}\left(1 - O(\delta\sqrt{T})\right),$$ and choosing $\delta \asymp T^{-1/2}$ completes the proof of Theorem 4. In fact, the above arguments also show that if the $L_2$ norm is replaced by the $L_\infty$ norm, then $$R_{T,d,L,R}^\star = \Theta\left(LR\sqrt{\frac{d}{T}}\right).$$ 5. Bibliographic Notes The example of nonparametric density estimation over Sobolev balls is taken from Nemirovski (2000), which also contains the tight minimax linear risk among all linear estimators. For the upper bound, linear procedures fail to achieve the minimax rate whenever $q>p$, and some non-linearities are necessary for the minimax estimators either in the function domain (Lepski, Mammen and Spokoiny (1997)) or the wavelet domain (Donoho et al. (1996)). The linear and convex aggregations are proposed in Nemirovski (2000) for the adaptive nonparametric thresholding estimators, and the concept of model selection aggregation is due to Yang (2000). For the optimal rates of different aggregations (together with upper bounds), we refer to Tsybakov (2003) and Leung and Barron (2006). The examples from statistical learning theory and stochastic optimization are similar in nature. The results of Theorem 2 and the corresponding upper bounds are taken from Vapnik (1998), albeit with a different proof language. For general results of the oracle complexity of convex optimization, we refer to the wonderful book Nemirovksi and Yudin (1983) and the lecture note Nemirovski (1995). The current proof of Theorem 4 is due to Agarwal et al. (2009). 1. Arkadi Nemirovski, Topics in non-parametric statistics. Ecole d’Eté de Probabilités de Saint-Flour 28 (2000): 85. 2. Oleg V. Lepski, Enno Mammen, and Vladimir G. Spokoiny, Optimal spatial adaptation to inhomogeneous smoothness: an approach based on kernel estimates with variable bandwidth selectors. The Annals of Statistics 25.3 (1997): 929–947. 3. David L. Donoho, Iain M. Johnstone, Gérard Kerkyacharian, and Dominique Picard, Density estimation by wavelet thresholding. The Annals of Statistics (1996): 508–539. 4. Yuhong Yang, Combining different procedures for adaptive regression. Journal of multivariate analysis 74.1 (2000): 135–161. 5. Alexandre B. Tsybakov, Optimal rates of aggregation. Learning theory and kernel machines. Springer, Berlin, Heidelberg, 2003. 303–313. 6. Gilbert Leung, and Andrew R. Barron. Information theory and mixing least-squares regressions. IEEE Transactions on Information Theory 52.8 (2006): 3396–3410. 7. Vlamimir Vapnik, Statistical learning theory. Wiley, New York (1998): 156–160. 8. Arkadi Nemirovski, and David Borisovich Yudin. Problem complexity and method efficiency in optimization. 1983. 9. Arkadi Nemirovski, Information-based complexity of convex programming. Lecture Notes, 1995. 10. Alekh Agarwal, Martin J. Wainwright, Peter L. Bartlett, and Pradeep K. Ravikumar, Information-theoretic lower bounds on the oracle complexity of convex optimization. Advances in Neural Information Processing Systems, 2009. ## 2 thoughts on “Lecture 9: Multiple Hypothesis Testing: More Examples” 1. Jonathan Scarlett These notes are excellent! I think it’d be worth considering combining them into a survey monograph 1. Yanjun Han Really glad you like them Jonathan! We’ll definitely combine these lecture notes at the end (also plan to teach a course based on these materials soon). Also this is not the end of the lecture series – more (tools and examples) to follow! This site uses Akismet to reduce spam. Learn how your comment data is processed.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 320, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9757575392723083, "perplexity": 265.72936223149713}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-40/segments/1600400206329.28/warc/CC-MAIN-20200922161302-20200922191302-00563.warc.gz"}
https://tex.stackexchange.com/questions/245979/numberwithinequationsubsection-fails-for-subsections-0?noredirect=1	# \numberwithin{equation}{subsection} fails for subsections "0" \numberwithin{equation}{subsection} is in the preamble of an article. Equations in sections before subsection 1 of that section begin numbering as x.0.y+1, where "y" is the number of the last equation of the last subsection of the previous section. (Numbers are OK in subsections, as 2.1.1, etc.) I can't see how such behavior would ever be desirable. Example: \documentclass[12pt]{article} \usepackage{amsmath} \numberwithin{equation}{subsection} \begin{document} \section{1} $$1$$ \subsection{1} $$1$$ \subsection{1} $$1$$ $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \end{document} produces equations numbered (1.0.1), (1.1.1), (1.2.1), (1.2.2), (2.0.3), (2.1.1), (3.0.2), (3.1.1). • The counter is now reset each time a new subsection is used, but not if a new section is used ;-) If there is no subsection its value is 0 of course. – user31729 May 20, 2015 at 12:07 Update Since the release of LaTeX from 2018/04/01 the mentioned package chngcntr is now part of the LaTeX kernel -- it's not necessary to load it separately. Using \numberwithin{equation}{subsection} changes the counter output format and shifts the resetting of the equation counter from section to subsection level (i.e. each time \stepcounter{subsection} is used. However, this means that an orphane equation after a \section, but before \subsection will just use the old counter value, from a previous equation. This resets the counter of equation number each time a new \section is used, but will not change the \theequation command (being the counter formatting macro) \makeatletter \makeatother As an alternative to this low-level variant, it's possible to use \usepackage{chngcntr} \counterwithin{equation}{section} in the preamble. However, this will not solve the issue with trailing 0 if the subsection number is 0. This can be attacked with a conditional within \theequation or -- the better way -- avoid equations outside of \subsection if subsection is the driver counter. \documentclass[12pt]{article} \usepackage{amsmath} \makeatletter \makeatother \numberwithin{equation}{subsection} \begin{document} \section{1} $$1$$ \subsection{1} $$1$$ \subsection{1} $$1$$ $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \end{document} • Using the chngcntr package is easier: \counterwithin{equation}{section} and \counterwithin(equation}{subsection} is all you need. The first is starred since we don't want to perform the redefinition of \theequation that would be overridden with the next command. May 20, 2015 at 16:41 • @egreg: I mentioned the chngcntr package (however, it's only a small comment. Perhaps I'll change my answer later on, but thanks for notifying me – user31729 May 20, 2015 at 17:49 I do not see the benefit of using such complicated equation numbers. Numbering equations within sections is more than enough IMHO. Nevertheless, the additional level can be suppressed, if the subsection counter is zero. Also \numberwithin does not define a transitive closure. Thus the equation number needs to be reset for each \section, too. \documentclass[12pt]{article} \usepackage{amsmath} \numberwithin{equation}{section}% reset equation counter for sections \numberwithin{equation}{subsection} % Omit .0 in equation numbers for non-existent subsections. \renewcommand{\theequation}{% \ifnum\value{subsection}=0 % \thesection \else \thesubsection \fi .\arabic{equation}% } \begin{document} \section{1} $$1$$ \subsection{1} $$1$$ \subsection{1} $$1$$ $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \section{1} $$1$$ \subsection{1} $$1$$ \end{document} This behaviour was actually changed somewhere back in 2014 (shortly before this question was asked, interestingly enough). When a counter is incremented, subordinate (“within”) counters are now reset recursively. (This is done by first setting the subordinate counters to −1 and then incrementing them.) With an up to date version of the LaTeX kernel, the MWE in the question now produces equations tagged “(1.0.1)”, “(1.1.1)”, “(1.2.1)”, “(1.2.2)”, “(2.0.1)”, “(2.1.1)”, “(3.0.1)” and “(3.1.1)”, as one would likely expect. The following was taking from the change log for LaTeX2ε: # 24 Changes between LATEX releases 2014/05/01 and 2015/01/01 […] ## 24.12 Within counters only reset next level down (pr4393) This is actually implicitly documented behavior in the LATEX Manual that states that \stepcounter resets all counters marked “within”. However it means that if, for example, theorems are numbered within sections and you start a new chapter in a book, the section counter is reset to zero but the theorem counter is not until the first section appears. Thus a theorem directly within the chapter body (without a new section) would show an incremented number relative to the last theorem of the previous chapter. For this reason we are now resetting all levels of within in one go even if that means that some of these resets may happen several times unnecessarily.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 24, "x-ck12": 0, "texerror": 0, "math_score": 0.9181889891624451, "perplexity": 2111.046904232872}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-40/segments/1664030336674.94/warc/CC-MAIN-20221001132802-20221001162802-00198.warc.gz"}
https://www.mysciencework.com/publication/show/measurement-qcd-analysis-proton-structure-function-f_2-x-q-2-hera-9a25e9d8	A Measurement and QCD Analysis of the Proton Structure Function $F_2(x,Q^2)$ at HERA Authors Type Published Article Publication Date Submission Date Identifiers DOI: 10.1016/0550-3213(96)00211-8 arXiv ID: hep-ex/9603004 Source arXiv A new measurement of the proton structure function $F_2(x,Q^2)$ is reported for momentum transfers squared $Q^2$ between 1.5 GeV$^2$ and 5000 GeV$^2$ and for Bjorken $x$ between $3\cdot 10^{-5}$ and 0.32 using data collected by the HERA experiment H1 in 1994. The data represent an increase in statistics by a factor of ten with respect to the analysis of the 1993 data. Substantial extension of the kinematic range towards low $Q^2$ and $x$ has been achieved using dedicated data samples and events with initial state photon radiation. The structure function is found to increase significantly with decreasing $x$, even in the lowest accessible $Q^2$ region. The data are well described by a Next to Leading Order QCD fit and the gluon density is extracted.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9234704971313477, "perplexity": 993.9294894132031}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891812556.20/warc/CC-MAIN-20180219072328-20180219092328-00624.warc.gz"}
https://www.physicsforums.com/threads/infinities-and-the-many-worlds-theory.572291/	# Infinities and the Many Worlds theory 1. Jan 30, 2012 ### Gordon Imroth Can anyone help me understand a feature of the Many Worlds interpretation of quantum theory, please? Many people (on this forum and elsewhere) say that the Many World's theory implies infinity parallel universes. I can understand why people think so. For example: Suppose a photon is sent through a slit to hit a screen at any point. There are infinity points (of the power of the continuum) where the photon can hit the screen, so the Many Worlds theory implies the continuum number of universes, in each of which the photon hits at only one place. Alternatively, we might say that the number of choices for a photon at any quantum event is always finite, however large. There may be infinity mathematical points on a screen where the photon can hit, but there are not infinity real physical points in space. This implies that space and time are atomic, however, and the problem with that idea is that, in the Many World's theory, classical physics is true in each parallel universe, and classical physics allows the infinite divisibility of space and time. So my question is: does the Many Worlds theory imply infinity parallel universes, or is it an entirely finite theory? And if Many Worlds is entirely finite, then what happens to classical physics in each universe? Gordon. P.s. Suppose we could design an experiment so the trajectory of the photon can follow every possible curve from source to target. As the infinity of curves is greater than the infinity of points on a surface, the Many Worlds theory implies an infinity of greater power than the continuum number. And as for curves, so for surfaces (perhaps employing multiple photons); and the infinity of surfaces is even bigger than the infinity of curves. If the Many Worlds theory implies a physical infinity, then what size infinity does it imply, or all of them? 2. Jan 30, 2012 ### Ken G I would say the MWI approach is an interpretation of some other theory, so if the other theory is discrete, then MWI is discrete, and if the other theory is continuous, then so is its MWI. None of that means that there really are an infinity of worlds in MWI-- it seems unlikely that any continuous theory is an exact representation of actual reality. (I feel the same could be said of MWI, but that's a matter of opinion.) I don't think there is any meaningful physical way to think about an "infinity of points" on a surface. We can treat the surface mathematically that way, but any actual experiment we use to test our theory is not going to be able to distinguish an infinity of different locations on the screen. "Infinity" doesn't mean anything in physics beyond "a number so large that making it any larger wouldn't have any important consequences." The same can be said about the continuity of classical physics. 3. Jan 30, 2012 ### Gordon Imroth Ken G, Thanks very much for your reply, and so quickly. I need to go away and think about what you said. For now, I can say I understand your point that MWI is an interpretation and whether it is discrete or continuous depends on the underlying theory; in which case, what basis do people have for saying MWI implies infinity parallel universes, or (I presume) others say it does not? What I am looking for is an objection to MWI, which I thought might be that MWI implies infinity parallel universes of many different powers all at the same time. Thanks again, Gordon. Similar Discussions: Infinities and the Many Worlds theory	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.958686351776123, "perplexity": 457.7477742348941}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-05/segments/1516084889542.47/warc/CC-MAIN-20180120083038-20180120103038-00691.warc.gz"}
http://aas.org/node/4816	## Search form Printing is different from typewriting, and TeX is different from other word processing tools. This file consists of reminders about things that require special attention so that TeX can format the input properly. # 1. Running text • In TeX, the ends of words and sentences are marked by spaces, and it does not matter how many spaces are typed; one is as good as 100. TeX treats the end of a line in the input file as a space. Paragraphs are separated by blank lines. • TeX will automatically take care of hyphenating and breaking words at the end of lines, so you do not need to break words with hyphens. Do, however, hyphenate modifiers within a line of text, e.g., "author-prepared copy." • Quotation marks should be typed as pairs of opening and closing single quotes, e.g., ‘‘quoted text’’; do not use double quotes ("bad form"). • Do not underline. In printing, text is emphasized by changing the type style, usually to slanted or italic type. • A number of common characters are interpreted as commands; therefore, if you want to use them in text, they must be preceded by a backslash (\): $& % # { and } must be typed \$ \& \% \# \{ and \}. • Refrain from adding vertical or horizontal space. Concentrate on the content of the document and identifying its components with the structural markup commands rather than worrying about producing perfectly formatted pages. # 2. Math Mathematical expressions that are part of the running text are delimited by a single dollar sign ($), e.g.,$\pi r^2$yields πr2. To get the appropriately sized superscript or subscript in the roman font, use the \rm command, e.g.,$J_{\rm HF}(t)$produces JHF(t). Displayed equations can be delimited in several ways. See the User Guide for details on display equation markup. While it is possible for authors to assign their own equation numbers, it is easier to let LaTeX number them automatically. By default, LaTeX will number equations sequentially from the beginning of the paper to the end. # 3. Cross-referencing Cross-referencing equations, tables, and figures in text depends upon the use of "keys," which are defined by the user. The \label command is used to define cross-reference keys for LaTeX; \ref is used to refer to them. Keys are simply text strings that serve to label equations, tables, and figures, so that they may be referred to symbolically in the text. For sections and the like, place \label commands immediately after the markup command that starts the structure being referenced. For figure and table captions, place the \label command \caption, \tablecaption, or \figcaption , e.g., \tablecaption{This is a caption.\label{tab1}} Do not put references to page numbers in your paper. LaTeX keeps track of autonumbered counters and cross-reference information by maintaining an auxiliary file in the same working directory as the source file. The auxiliary file will have an extension of .aux. This file should not be deleted, since subsequent LaTeX processing uses the auxiliary data to resolve references, etc. The auxiliary file mechanism makes it necessary to run LaTeX on a given source file more than once to ensure that the cross-reference information has been properly resolved. When changes are made that affect the number or the placement of equations, tables, and the like. LaTeX will issue a warning message that advises the user to "rerun to get cross-references right," in which case, LaTeX should be run again. # 4. Useful External Packages • ## emulateapj Alexey Vikhlinin has created a LateX class file that mimics the tight, type set look of a published AASTeX paper. The class file, emulateapj.cls, and documentation can be obtained here • ## lineno To add line numbering to your paper obtain the lineno package. After obtaining the lineno.sty file, add these commands before the \begin{document} call near the top of your LaTeX manuscript: \usepackage{lineno} \linenumbers This places continuously running numbers in the left margin. See the user manual to determine how to use lineno's many other options. Note that while lineno works with AASTeX (v5.2) in both single and double column preprint modes, it does not mark both columns when used with the emulateapj classfile. Only one of the two columns are marked. • ## Trackchanges The Trackchanges style file allows multiple authors to edit and annotate a LaTeX document. After obtaining the trackchanges style files, simply add: \usepackage{trackchanges} \addeditor{ABC} % where ABC is an editor identifier to the top of your LaTeX manuscript before the \begin{document} call. The five commands used to most often are: \note[editor]{The note} \annote[editor]{Text to annotate}{The note} \add[editor]{Text to add} \remove[editor]{Text to remove} \change[editor]{Text to remove}{Text to add} where "editor" is the editor identifier defined earlier. The user manual provides numerous examples of the output and descriptions of the different options available. • ## Highlighting new and changed text in revised manuscripts with bold or colored font. Some authors highlight new or modified text in revised manuscripts using bold face to make it easier for the editor and referee to pick out the changes. This is a very useful practice that should be encouraged but its widespread acceptance may be hampered by the time required to remove the bold highlighting once the paper has been accepted. While there is no package currently available to do this here are a set of LaTeX commands to add this functionality to your papers. LaTeX command Function \newcommand{\btxt}[1]{{\bf #1}} Put the argument in bold \newcommand{\btxt}[1]{{\it \textcolor{blue}{ #1}}} Put the argument in blue italics \newcommand{\btxt}[1]{{#1}} Turn off the mark up. Generally commented out until needed. Note that the use of color requires the color style file. This should be a standard package in all modern LaTeX distributions but the style file has to be specifically called before color can be used. Place \usepackage{color} at the beginning of the manuscript before the \begin{document} call to allow color. Additional colors or fonts can be supplied by the user by modifying the commands accordingly. This functionality can be turned off by commenting out the initial command and using the last command. To use this functionality place one of the first two commands in the table above at the beginning of the manuscript before the \begin{document} call. The third command can be included but should be commented out, i.e. preface with a "%". Next, wrap any text that you want to highlight with the "\btxt" command. For example: "We have obtained \btxt{$BVRI\$} photometry of the \btxt{exceptionally} long lived nova V723 Cas." will produce "We have obtained BVRI photometry of the exceptionally long lived nova V723 Cas." To turn off the bold highlighting simply activate the third definition and comment out the first, e.g. %\newcommand{\btxt}[1]{{\bf #1}} \newcommand{\btxt}[1]{{#1}} • ## Watermarking There are various style files available to add watermarks to LaTeX manuscripts. One of the best in terms of number of options available to control the watermarking features is draftwatermark. The user can set the angle, size, lightness and text within the style file. There is also an option to only mark the first page. Note that manuscripts submitted to the AAS Journals should not include any watermarking. • ## Merge multiple postscript files into a single file The assemble perl script by Robert Lupton will merge multiple postscript files into a single postscript file. K.Z. Stanek provides some documentation on its use here. Stanek's main use of assemble.pl is to create meeting posters but the code can also be used to make more compact paper figures by merging multiple component figures into a single one. • ## Merge multiple latex components into a single file For authors who use the \input and \include commands to reference external latex files such as tables and sections, it is sometimes necessary to "flatten" the LaTeX structure by merging all the external files into one. A strong advantabe of the single, flat LaTeX file is that the AAS journal's metric counters (e.g. ApJL manuscript length calculator) only accepted a single, combined LaTeX manuscript. Two source codes are available to merge LaTeX structures. One is a C code called flatten and the other is a Perl script called latexexpand. • ## Excel to LaTeX tables converters Two programs are available to covert Microsoft Excel tables into LaTeX tabular format: excel2latex and xl2latex. • ## Word to latex manuscripts converters Similarly, many programs are available to convert Microsoft Word files into LaTeX format. The programs are WordML2LaTeX, catdoc, word-to-latex, and word2latex. Share:	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8366628289222717, "perplexity": 3196.2300099974996}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-49/segments/1416931005028.11/warc/CC-MAIN-20141125155645-00155-ip-10-235-23-156.ec2.internal.warc.gz"}
http://proteinsandwavefunctions.blogspot.se/2012/09/microstates-macrostates-and-boltzmann.html	## Sunday, September 9, 2012 ### Microstates, macrostates, and the Boltzmann distribution This is Figure 10.1 from Dill and Bromberg's Molecular Driving Forces, which is my favorite book on statistical mechanics. It is a beautiful example from a beautiful book. The Boltzmann distribution says that, at equilibrium, the probability of being in a microstate (also sometimes referred to as an energy state) is $$p_i=\frac{e^{-\varepsilon_i/kT}}{q}\text{ where }q=\sum_i^{\begin{array}{ c }\text{micro} \\ \text{states} \\ \end{array}} e^{-\varepsilon_i/kT}$$ Let's apply this formula to the simple four-bead polymer example shown in the figure above. Each of the five conformations represents a microstate. There is one low-energy conformation with a bead-bead contact and four higher-energy conformations where the bead-bead contact is lost. Because higher energy conformations have a common feature (the lack of bead-bead contact) and the same energies we group them into one common macrostate (a state comprised of more than one microstate). I'll refer to this macrostate at the "unfolded" state and the to ground state the "folded" state. According to the Boltzmann distribution the probability of the folded state is $$p_f=\frac{1}{q}\text{ where }q=1+4e^{-\varepsilon_0/kT}$$ The probability is the unfolded state is the sum of the (equal) probabilities of being in one of the four unfolded microstates $$p_{uf}=p_{uf1}+p_{uf2}+p_{uf3}+p_{uf4}=\frac{4e^{-\varepsilon_0/kT}}{q}$$ This figure shows a plot of the probabilities of the folded and unfolded state, together with the probability of being in one of the four unfolded microstates, as a function of temperature and for $N_A\varepsilon_0=2.5$ kJ/mol. The polymer can be said to unfold at 217 Kelvin where the unfolded state starts to become more probable than the folded state. Why does the polymer unfold beyond 217 K? Answer 1. The plot shows that if there is only one unfolded microstate then the polymer would never unfold (i.e. unfolded state would never be more probable than the folded state). Therefore, the polymer unfolds because there are more unfolded microstates than folded microstates (4 compared to 1). Answer 2. The number of microstates with the same energy is known as the degeneracy, so one can also say that the polymer unfolds because the unfolded state has a higher degeneracy than the folded state (4 compared to 1). Answer 3. The ratio of probabilities can be written as $$\frac{p_{uf}}{p_f}=4e^{-\varepsilon_0/kT}=e^{-( \varepsilon_0-kT\ln(4))/kT}$$The latter form clearly shows that the unfolded state will become more probable for temperatures were $T(k\ln(4))\gt\varepsilon_0$. The similarly of $k\ln(4)$ to the entropy $S=k\ln(W)$ is no accident. The entropy of the unfolded state is given by $$S_{uf}=k\ln\left(\frac{N_{uf}!}{N_{uf1}!N_{uf2}!N_{uf3}!N_{uf4}!}\right)$$where $N_{uf1}=N_{uf2}=N_{uf3}=N_{uf4}=\frac{1}{4}N_{uf}$ is the number of polymers in each of the unfolded microstates. For large $N_{uf}$, where Stirling's approximation [$x!\approx(x/e)^x$] holds, it is easy to show that this expression reduces to $$S_{uf}=N_{uf}k\ln(4)$$So one can also say that the polymer unfolds because the unfolded state has a higher entropy than the folded state. A simulation of the unfolding of a slightly longer polymer can be found here This work (except the first figure which is © by Garland Science) is licensed under a Creative Commons Attribution 3.0 Unported License. Post a Comment	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8946590423583984, "perplexity": 853.2434171101246}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": false}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-43/segments/1508187822851.65/warc/CC-MAIN-20171018085500-20171018105500-00023.warc.gz"}
https://standards.globalspec.com/std/3842795/astm-d1677-02-2011	### This is embarrasing... An error occurred while processing the form. Please try again in a few minutes. ### This is embarrasing... An error occurred while processing the form. Please try again in a few minutes. # ASTM International - ASTM D1677-02(2011) ## Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation inactive Organization: ASTM International Publication Date: 1 April 2011 Status: inactive Page Count: 4 ICS Code (Mica based materials): 29.035.50 ##### significance And Use: Accurate determination of thickness is important for identification purposes. Thickness is related to weight and must be known in order to calculate apparent density and the dielectric... View More ##### scope: 1.1 These methods cover procedures for sampling and testing untreated mica paper to be used as an electrical insulator or as a constituent of a composite material used for electrical insulating purposes. 1.2 The procedures appear in the following order: ### Document History November 1, 2017 Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation 1.1 These methods cover procedures for sampling and testing untreated mica paper to be used as an electrical insulator or as a constituent of a composite material used for electrical insulating... ASTM D1677-02(2011) April 1, 2011 Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation Accurate determination of thickness is important for identification purposes. Thickness is related to weight and must be known in order to calculate apparent density and the dielectric strength.... October 1, 2007 Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation 1.1 These methods cover procedures for sampling and testing untreated mica paper to be used as an electrical insulator or as a constituent of a composite material used for electrical insulating... March 10, 2002 Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation 1.1 These methods cover procedures for sampling and testing untreated mica paper to be used as an electrical insulator or as a constituent of a composite material used for electrical insulating... March 10, 2002 Standard Methods for Sampling and Testing Untreated Mica Paper Used for Electrical Insulation 1.1 These methods cover procedures for sampling and testing untreated mica paper to be used as an electrical insulator or as a constituent of a composite material used for electrical insulating...	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8804669380187988, "perplexity": 3327.108176896719}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2023-06/segments/1674764500273.30/warc/CC-MAIN-20230205161658-20230205191658-00820.warc.gz"}
http://math.stackexchange.com/questions/164174/clarifications-needed-in-divergence-theorem	# Clarifications needed in Divergence Theorem Consider this problem: $F=[4x,x^2y,-x^2z]$ and $S$ is the tetrahedron described by $(0,0,0);(1,0,0);(0,1,0);(0,0,1)$ we have to find the surface integral. Question 1: Applying divergence theorem $\nabla\cdot F =4$ which implies the surface integral turns into $\int \int \int 4 dV$ or otherwise 4 times the volume of tetrahedron. We know volume of tetrahedron is $\sqrt{2}a^3/12$ which means the answer should be $\sqrt{2}/3$. But the answer is not that. Question 2: If we split $dV$ into $dxdydz$ and then integrate, the limit my book says should be $0 to 1-y-z$,$0 to 1$,$0 to 1$ respectively. Can you talk a bit more about it. Intuitively I got an understanding, but how is it different the limits had it been from x+y+z=1? Question 3: Just for the kicks I tried integrating this on the surface of the tetrahedron. I get the normal vector for only one surface, i.e the plane x+y+z =1 the others, turn into an equation where one of the variable (x,y,z) is zero [as the tetrahedron is bounded by that plane] which implies when turning into parametric form of u-v plane, the r(u,v) will be dependent on only 1 variable. For instance, one of the planes will be $x+z\leq 1$ => r(u,v)=[u,0,1-u] which implies $r_v=0$, hence cross product of $r_u$ and $r_v$ will be zero, hence the integral will be zero on that surface. This was my modus operandi. So I surface integrated over only $x+y+z \leq 1$ and I am getting a pretty way off the answer. Can someone lend me a hand here. - The volume of a simplex in $\mathbb{R}^n$ (tetrahedron in $\mathbb{R}^3$), one of whose vertices is the origin, is $\frac{1}{n!}$ times the determinant of the matrix with the other vertices as columns. Thus, the volume of your tetrahedron is $$\frac{1}{3!}\det\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}=\frac16$$ and $4\cdot\frac16=\frac23$, which is hopefully the correct answer. The volume integral should be $$\int_0^1\int_0^{1-z}\int_0^{1-z-y}4\,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z$$ That is, $z$ can take on any value in $[0,1]$. The integral in $y$ is taken over each particular value of $z$, so $y$ can take any value in $[0,1-z]$. The integral in $x$ is taken over each particular value of $z$ and $y$, so $x$ can take any value in $[0,1-z-y]$. On the surface $x+y+z=1$, the surface normal is $\nabla(x+y+z-1)=(1,1,1)$. Normalizing this gives the unit normal: $n=\frac{1}{\sqrt{3}}(1,1,1)$. To integrate over this surface, the integral would be \begin{align} &\int_0^1\int_0^{1-y}F\cdot n\sqrt{\left(\frac{\partial z}{\partial x}\right)^2+\left(\frac{\partial z}{\partial y}\right)^2+1}\;\mathrm{d}x\,\mathrm{d}y\\ &=\int_0^1\int_0^{1-y}F\cdot n\sqrt{3}\;\mathrm{d}x\,\mathrm{d}y\\ &=\int_0^1\int_0^{1-y}(4x,x^2y,-x^2z)\cdot(1,1,1)\frac{1}{\sqrt{3}}\sqrt{3}\;\mathrm{d}x\,\mathrm{d}y\\ &=\int_0^1\int_0^{1-y}(4x+x^2y-x^2z)\;\mathrm{d}x\,\mathrm{d}y\\ &=\int_0^1\int_0^{1-y}(4x+x^2y-x^2(1-x-y))\;\mathrm{d}x\,\mathrm{d}y\\ &=\int_0^1\int_0^{1-y}(x^3-x^2+4x+2x^2y)\;\mathrm{d}x\,\mathrm{d}y \end{align} The other surfaces should be much simpler. - Q1: The volume of a regular tetrahedron of side length $a$ may be $\sqrt{2}a^3/12$; but the tetrahedron $T$ at stake here is not regular. It can be viewed as a pyramid with base area ${1\over2}$ and height $1$, so its volume is ${1\over6}$. This implies that the flux $\Phi$ computed as a volume integral amounts to ${2\over3}$. Q2: The triple integral in question is $$\Phi=\int_0^1\Bigl(\int_0^{1-x}\bigl(\int_0^{1-x-y}4\ dz\bigr)\ dy\Bigr)\ dx\ .$$ To compute it you have to begin with the innermost integral: For given $(x,y)\in\Delta_1$ (the base triangle) the allowable $z$ have to satisfy $z\geq0$ and $x+y+z\leq1$. This leads to the integration interval $[0,1-x-y]$ for $z$, and the value of the innermost integral is therefore $=4(1-x-y)=:g(x,y)$. In the second step we have to integrate the function $g$ over $\Delta_1$. For given $x\in[0,1]$ (the base of $\Delta_1$) the allowable $y$ have to satisfy $y\geq0$ and $x+y\leq1$. This leads to the integration interval $[0,1-x]$ for $y$. The value of the integral $\int_0^{1-x} g(x,y)\ dy$ is a function $x\mapsto h(x)$ which in the third step has to be integrated from $0$ to $1$. Q3: The surface $\partial T$ of our tetrahedron consists of four triangles. In order to compute the total flux $\Phi:=\int_{\partial T}{\bf F}\cdot d{\bf \vec\omega}$ we therefore have to compute four separate integrals over four parameter triangles. For the base triangle $\Delta_1$ we use the trivial parametrization ${\bf f}:(x,y)\mapsto(x,y,0)$ with $d\vec\omega=(0,0,-1)\ {\rm d}(x,y)$ (we need the outward normal). Therefore we get $$\int_{\Delta_1}{\bf F}\cdot d\vec\omega=\int_0^1\int_0^{1-x}\bigl(4x,x^2y,0)\cdot(0,0,-1)\ dy\ dx=0\ .$$ I hope you can do the other three yourself. - Regarding Q#2, I was not actually asking about the way to integrate but rather the limits. How did they come decide upon the limits – Soham Jun 28 '12 at 17:36 @Soham: See my edit. – Christian Blatter Jun 28 '12 at 18:02	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9903349876403809, "perplexity": 132.62380683084126}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701147841.50/warc/CC-MAIN-20160205193907-00060-ip-10-236-182-209.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/14107/how-to-calculate-likelihood-to-succeed-knowing-attempts-and-successful-attempts?answertab=active	# How to calculate likelihood to succeed knowing attempts and successful attempts amounts? Say I have two algorithms that I don't know how they work but I know what they are meant to achieve. I tried algorithm A once and it succeeded. And tried algorithm B 100 times and succeeded 99 times. So A succeeded 100% times and B succeeded 99% times, but my common sense tells me B is still more "reliable" than A. How can I express this mathematically, or, how can I calculate which algorithm should I choose if I have to choose one when all I know about each of them is how many times it has been tested and how many times it succeeded (if it didn't that just means it failed for what I need)? Please be as simple as possible I'm a newbie at math. - Wouldn't (SuccessfulAttempts - 1) / Tries be a good approximation? – jsoldi Dec 13 '10 at 9:15 ${\rm SuccessfulAttempts}/{\rm Tries}$ is a good approximation provided that ${\rm Tries}$ is sufficiently large. – Shai Covo Dec 14 '10 at 17:12 I do the -1 so that cases with small Tries don't get a very high number, so that, for example, 1 / 1 doesn't get 100% – jsoldi Dec 14 '10 at 17:17	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.895480751991272, "perplexity": 321.059536839052}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-23/segments/1406510267075.55/warc/CC-MAIN-20140728011747-00425-ip-10-146-231-18.ec2.internal.warc.gz"}
https://math.stackexchange.com/questions/3341449/relation-between-gaussian-width-and-its-squared-version/3341819	# Relation between Gaussian width and its squared version I'm currently reading through Roman Vershynin's High Dimensional Probability and working through one of the exercises (7.6.1). Consider a set $$T \subseteq \mathbf{R}^n$$ and define its Gaussian width $$w(T)$$, as $$w(T) := \mathbb{E} \sup_{x \in T} \langle g, x\rangle, \quad g \sim \mathcal{N}(0, I_n).$$ A closely related version, $$h(T)$$, is defined similarly: $$h(T) := \sqrt{\mathbb{E}\left[ \sup_{x \in T} \langle g, x \rangle^2 \right]}.$$ Now, Exercise 7.6.1 in the book asks the reader to show that $$h(T - T) \leq w(T - T) + C_1 \mathrm{diam}(T), \quad ()$$ with $$T - T := \left\{u - v : u, v \in T \right\}$$, and the hint is to use Gaussian concentration. I have been unable to use this hint, and only end up with a trivial upper bound where $$C_1 = \sqrt{n}$$, as follows: $$h(T - T)^2 = \mathbb{E} \sup_{x \in T - T} \langle g, x \rangle^2 = \mathbb{E} \left( \sup_{x \in T - T} \left\langle g, \frac{x}{\\| x \\|_2} \right\rangle^2 \\| x \\|_2^2 \right) \\ \leq \sup_{x \in T - T} \\| x \\|_2^2 \mathbb{E} \\| g \\|_2^2 = \mathrm{diam}^2(T) \cdot n,$$ followed by taking square roots. Question: How does one use Gaussian concentration to show the bound $$()$$? I tried showing that $$g \mapsto \sqrt{\sup_{x \in T - T} \langle g, x \rangle^2} - \sup_{y \in T - T} \langle g, y \rangle$$ is Lipschitz, but couldn't get anything useful since there is a square root involved. • @GabrielRomon: thank you so much, sorry for the late reply! The part I was missing was moving the square out of the supremum. – VHarisop Sep 9 '19 at 0:50 For fixed $$g$$ note that $$\sup\limits_{x,y\in T} \langle g,x-y \rangle = \sup\limits_{x,y\in T} \|\langle g,x-y \rangle\|$$, hence $$\left(\sup\limits_{x,y\in T} \langle g,x-y \rangle \right)^2=\left(\sup\limits_{x,y\in T} \|\langle g,x-y \rangle\|\right)^2=\sup\limits_{x,y\in T} \|\langle g,x-y \rangle\|^2=\sup\limits_{x,y\in T} \langle g,x-y \rangle^2$$ Let $$F:g\mapsto \sup\limits_{x,y\in T} \langle g,x-y \rangle$$. The previous equalities show that $$h(T-T)^2=\mathbb E(F(g)^2)$$, and of course $$w(T-T)=\mathbb E(F(g))$$. Let us prove that $$F$$ is $$\mathrm{diam}(T)$$-Lipschitz: for $$g,g'\in \mathbb R^n$$, $$\langle g,x-y \rangle = \langle g-g',x-y \rangle + \langle g',x-y \rangle \leq \\|g-g'\\|\mathrm{diam}(T) + F(g')$$ hence $$F(g) - F(g')\leq \\|g-g'\\|\mathrm{diam}(T)$$ and the claim is obtained by symmetry. Gaussian concentration provides an upper bound on $$\mathbb V(F(g))$$. Indeed $$\mathbb V(F(g)) = \int_0^\infty P(\| F(g)- \mathbb E(F(g))\|\geq \sqrt t)\leq 2\int_0^\infty e^{-t/(2 \mathrm{diam}(T)^2)} = 4\mathrm{diam}(T)^2$$ Thus $$h(T-T)=\sqrt{\mathbb E(F(g)^2)}\leq \sqrt{w(T-T)^2 + 4\mathrm{diam}(T)^2}\leq w(T-T) + 2\mathrm{diam}(T)$$. Using the Gaussian Poincaré inequality one can get the stronger inequality $$h(T-T)\leq w(T-T) + \mathrm{diam}(T)$$ Regarding the other inequalities, $$w(T-T)\leq h(T-T)$$ follows from Jensen's inequality: $$h(T-T)=\sqrt{\mathbb E\left[\left(\sup\limits_{x,y\in T} \|\langle g,x-y \rangle\| \right)^2\right]}\geq \mathbb E (\sup\limits_{x,y\in T} \|\langle g,x-y \rangle\|) = w(T-T)$$ The last inequality $$w(T-T)+2\mathrm{diam}(T) \leq Cw(T-T)$$ follows from Proposition 7.5.2 of the book: $$w(T-T)+2\mathrm{diam}(T)\leq w(T-T)+ 2\sqrt{2\pi}w(T) = \left(1+\sqrt{2\pi} \right)w(T-T)$$ Using the tighter bound on the variance, the last constant can be improved to $$1+\sqrt{\frac \pi 2}$$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 31, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9931372404098511, "perplexity": 212.7985869940877}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-05/segments/1579250593937.27/warc/CC-MAIN-20200118193018-20200118221018-00396.warc.gz"}
http://mathhelpforum.com/calculus/21105-gradient-field.html	# Math Help - gradient field Hi I have a quiestion : What thoes a gradient field means? 2. Originally Posted by MarianaA Hi I have a quiestion : What thoes a gradient field means? for a scalar function $f(x,y)$ of two variables, we define $\nabla f$ as the following: $\nabla f = f_x(x,y) ~\bold{i} + f_y(x,y)~ \bold{j}$ similarly, for a function $f(x,y,z)$ of three variables, $\nabla f = f_x(x,y,z) ~\bold{i} + f_y(x,y,z)~\bold{j} + f_z(x,y,z)~\bold{k}$ we call $\nabla f$ the gradient of $f$ and it is a vector field on $\mathbb{R}^2$ or $\mathbb{R}^3$ for two or three variables respectively. specifically, it is called a gradient vector field, as it assigns each point $(x,y)$ in domain $f(x,y)$ (respectively each point $(x,y,z)$ in the domain of $f(x,y,z)$) with a 2-dimensional vector in $\mathbb{R}^2$ (respectively, a 3-Dimensional vector in $\mathbb{R}^3$) 3. Originally Posted by Jhevon for a scalar function $f(x,y)$ of two variables, we define $\nabla f$ as the following: $\nabla f = f_x(x,y) ~\bold{i} + f_y(x,y)~ \bold{j}$ similarly, for a function $f(x,y,z)$ of three variables, $\nabla f = f_x(x,y,z) ~\bold{i} + f_y(x,y,z)~\bold{j} + f_z(x,y,z)~\bold{k}$ we call $\nabla f$ the gradient of $f$ and it is a vector field on $\mathbb{R}^2$ or $\mathbb{R}^3$ for two or three variables respectively. specifically, it is called a gradient vector field, as it assigns each point $(x,y)$ in domain $f(x,y)$ (respectively each point $(x,y,z)$ in the domain of $f(x,y,z)$) with a 2-dimensional vector in $\mathbb{R}^2$ (respectively, a 3-Dimensional vector in $\mathbb{R}^3$) Ok Thanks	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 30, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9328304529190063, "perplexity": 238.9891681744338}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783398216.41/warc/CC-MAIN-20160624154958-00150-ip-10-164-35-72.ec2.internal.warc.gz"}
http://en.wikipedia.org/wiki/Phase_detector	# Phase detector Four phase detectors. Signal flow is from left to right. In the upper left is a Gilbert cell, which works well for sine waves and square waves, but less well for pulses. In the case of square waves it acts as an XOR gate, which can also be made from NAND gates. On the middle left are two phase detectors: adding feedback and removing one NAND gate produces a time frequency detector. The delay line avoids a dead band. On the right is a charge pump with a filter at its output. A phase detector or phase comparator is a frequency mixer, analog multiplier or logic circuit that generates a voltage signal which represents the difference in phase between two signal inputs. It is an essential element of the phase-locked loop (PLL). Detecting phase difference is very important in many applications, such as motor control, radar and telecommunication systems, servo mechanisms, and demodulators. ## Types Phase detectors for phase-locked loop circuits may be classified in two types.[1] A Type I detector is designed to be driven by analog signals or square-wave digital signals and produces an output pulse at the difference frequency. The Type I detector always produces an output waveform, which must be filtered to control the phase-locked loop voltage-controlled oscillator (VCO). A type II detector is sensitive only to the relative timing of the edges of the input and reference pulses, and produces a constant output proportional to phase difference when both signals are at the same frequency. This output will tend not to produce ripple in the control voltage of the VCO. ### Analog phase detector The phase detector needs to compute the phase difference of its two input signals. Let α be the phase of the first input and β be the phase of the second. The actual input signals to the phase detector, however, are not α and β, but rather sinusoids such as sin(α) and cos(β). In general, computing the phase difference would involve computing the arcsine and arccosine of each normalized input (to get an ever increasing phase) and doing a subtraction. Such an analog calculation is difficult. Fortunately, the calculation can be simplified by using some approximations. Assume that the phase differences will be small (much less than 1 radian, for example). The small-angle approximation for the sine function and the sine angle addition formula yield: $\alpha - \beta \approx \sin(\alpha-\beta) = \sin \alpha \cos\beta - \sin \beta \cos \alpha$ The expression suggests a quadrature phase detector can be made by summing the outputs of two multipliers. The quadrature signals may be formed with phase shift networks. Two common implementations for multipliers are the double balanced diode mixer (diode ring) and the four-quadrant multiplier (Gilbert cell). Instead of using two multipliers, a more common phase detector uses a single multiplier and a different trigonometric identity: $\sin \alpha \cos \beta = {\sin(\alpha - \beta) \over 2} + {\sin(\alpha + \beta) \over 2} \approx {\alpha - \beta \over 2} + {\sin(\alpha + \beta) \over 2}$ The first term provides the desired phase difference. The second term is a sinusoid at twice the reference frequency, so it can be filtered out. In the case of general waveforms the phase detector output is described with phase detector characteristic A mixer-based detector (e.g., a Schottky diode-based double-balanced mixer) provides "the ultimate in phase noise floor performance" and "in system sensitivity." since it does not create finite pulse widths at the phase detector output.[2] Another advantage of a mixer-based PD is its relative simplicity.[2] Both the quadrature and simple multiplier phase detectors have an output that depends on the input amplitudes as well as the phase difference. In practice, the input amplitudes are normalized. ### Digital phase detector An Example CMOS Digital Phase Frequency Detector (Transistor Variety). Inputs are R and V while the outputs UP and DN feed to a charge pump. A phase detector suitable for square wave signals can be made from an exclusive-OR (XOR) logic gate. When the two signals being compared are completely in-phase, the XOR gate's output will have a constant level of zero. When the two signals differ in phase by 1°, the XOR gate's output will be high for 1/180th of each cycle — the fraction of a cycle during which the two signals differ in value. When the signals differ by 180° — that is, one signal is high when the other is low, and vice versa — the XOR gate's output remains high throughout each cycle. The XOR detector compares well to the analog mixer in that it locks near a 90° phase difference and has a square-wave output at twice the reference frequency. The square-wave changes duty-cycle in proportion to the phase difference resulting. Applying the XOR gate's output to a low-pass filter results in an analog voltage that is proportional to the phase difference between the two signals. It requires inputs that are symmetrical square waves, or nearly so. The remainder of its characteristics are very similar to the analog mixer for capture range, lock time, reference spurious and low-pass filter requirements. Digital phase detectors can also be based on a sample and hold circuit, a charge pump, or a logic circuit consisting of flip-flops (see figure (Where?)). When a phase detector that's based on logic gates is used in a PLL, it can quickly force the VCO to synchronize with an input signal, even when the frequency of the input signal differs substantially from the initial frequency of the VCO. Such phase detectors also have other desirable properties, such as better accuracy when there are only small phase differences between the two signals being compared. This is because a digital phase detector has a nearly infinite pull-in range in comparison to an XOR detector. ### Phase-frequency detector A phase-frequency detector is an asynchronous sequential logic circuit originally made of four flip-flops (i.e., the phase-frequency detectors found in both the RCA CD4046 and the motorola MC4344 ICs introduced in the 1970s). The logic determines which of the two signals has a zero-crossing earlier or more often. When used in a PLL application, lock can be achieved even when it is off frequency and is known as a Phase Frequency Detector. Such a detector has the advantage of producing an output even when the two signals being compared differ not only in phase but in frequency. A phase frequency detector prevents a "false lock" condition in PLL applications, in which the PLL synchronizes with the wrong phase of the input signal or with the wrong frequency (e.g., a harmonic of the input signal).[3] A bang-bang charge pump phase detector supplies current pulses with fixed total charge, either positive or negative, to the capacitor acting as an integrator. A phase detector for a bang-bang charge pump must always have a dead band where the phases of inputs are close enough that the detector fires either both or neither of the charge pumps, for no total effect. Bang-bang phase detectors are simple, but are associated with significant minimum peak-to-peak jitter, because of drift within the dead band. In 1976 it was shown that by using a three-state phase detector configuration (using only two flip-flops) instead of the original RCA/Motorola twelve-state configurations, this problem could be elegantly overcome.[citation needed] For other types of phase-frequency detectors other, though possibly less-elegant, solutions exist to the dead zone phenomenon.[3] Other solutions are necessary since the three-state phase-frequency detector does not work for certain applications involving randomized signal degradation, which can be found on the inputs to some signal regeneration systems (e.g., clock recovery designs).[4] A proportional phase detector employs a charge pump that supplies charge amounts in proportion to the phase error detected. Some have dead bands and some do not. Specifically, some designs produce both "up" and "down" control pulses even when the phase difference is zero. These pulses are small, nominally the same duration, and cause the charge pump to produce equal-charge positive and negative current pulses when the phase is perfectly matched. Phase detectors with this kind of control system don't exhibit a dead band and typically have lower minimum peak-to-peak jitter when used in PLLs. In PLL applications it is frequently required to know when the loop is out of lock. The more complex digital phase-frequency detectors usually have an output that allows a reliable indication of an out of lock condition. ## Electronic phase detector Some signal processing techniques such as those used in radar may require both the amplitude and the phase of a signal, to recover all the information encoded in that signal. One technique is to feed an amplitude-limited signal into one port of a product detector and a reference signal into the other port; the output of the detector will represent the phase difference between the signals. ## Optical phase detectors Phase detectors are also known in optics as interferometers. For pulsed (amplitude modulated) light, it is said to measure the phase between the carriers. It is also possible to measure the delay between the envelopes of two short optical pulses by means of cross correlation in a nonlinear crystal. And it is possible to measure the phase between the envelope and the carrier of an optical pulse, by sending a pulse into an nonlinear crystal. There the spectrum gets wider and at the edges the shape depends significantly on the phase.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 2, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8054727911949158, "perplexity": 1091.3947663131542}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-35/segments/1408500825010.41/warc/CC-MAIN-20140820021345-00027-ip-10-180-136-8.ec2.internal.warc.gz"}
http://www.sfb45.de/publications-1/a-short-note-on-vector-bundles-on-curves	##### Personal tools You are here: Home A short note on vector bundles on curves # A short note on vector bundles on curves Martin Kreidl Number 12 Martin Kreidl 2010 Beauville and Laszlo give an interpretation of the affine Grassmannian for Gl_n over a field k as a moduli space of, loosely speaking, vector bundles over a projective curve together with a trivialization over the complement of a fixed closed point. In order to establish this correspondence, they have to show that descent for vector bundles holds in a situation which is not a classical fpqc-descent situation. They prove this as a consequence of an abstract descent lemma. It turns out, however, that one can avoid this descent lemma by using a simple approximation-argument, which leads to a more direct prove of the above mentioned correspondence. « October 2019 » October MoTuWeThFrSaSu 123456 78910111213 14151617181920 21222324252627 28293031	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.980525016784668, "perplexity": 563.6271880890613}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-43/segments/1570986684854.67/warc/CC-MAIN-20191018204336-20191018231836-00232.warc.gz"}
http://mathhelpforum.com/math-topics/109497-choosing-coefficients.html	1. ## choosing coefficients Hello: I did take some university math years ago, but now approaching 50 yrs old, so am pretty rusty on a problem I'm trying to solve. I am trying to choose coefficients for a 2nd equation to make the curve of it's solution as close as possible to a first equation, particulary in the region of interest for 15<x<75. The first equation is y=10(1/((x/15) - 1)) for (15<x<300) The second equation is y=-Tln(1 - d/(x*0.002)), also for (15<x<300). I need to keep d > 0.03, T can be any reasonable positive value. Thanks, PT 2. Unfortunately, you must define what it is you mean by "close". Some possibilities are: 1) Total area between the two curves. 2) Total absolute distance at 37 selected points. 3) Matching Exactly for a few points and maybe matching some derivatives elsewhere? There are endless possibilities. 3. ## idea on closeness defined TKHunny: thanks for the ideas. My definition of closeness would be to define a region of greatest interest within the entire x=15 to 300 range, say x= from 15 to 75, and find the % error at each integer value of x compared to the result from the first equation. Whether the point generated by the second equation was above or below the point generated by the first equation wouldn't matter, only the % error. A curve from the 2nd equation that never touched the curve from the first equation but always stayed within say 7% would be better than a curve that had approx same values in one segment of the curve but was off by 15% at other segments. I've been using Microsoft Excel and varying T and d. I've seen that varying d does little to the shape of the curve, but does change it's offset above the x axis. Changing T through a large range changes both the value it approaches asymptotically, and the shape/slope a lot. If anyone is interested you can email me at [email protected], and I can send you the MS Excel files for some varying T values. 4. Calling both eqyations 'y' is rather inconvenient. I rewrote them as H(x), your target equation, and R(x), your equation with T and d parameters. I defined a minimiztion function: $\displaystyle \sum_{n=16}^{75}\frac{\|H(n)-R(n)\|}{R(n)}$ You should be able to do that in a spreadsheet. Look at it carefully. It it the absolute percent error at each point - avoiding x = 15. I just started playing with it. It becomes clear quickly that you can't move 'd' very far from 0.03, since the problemat x = 15 keeps moving to the right. I began with: d = 0.031 and T = 1 -- Total Error Measure = 55.283 Increasing T by 1 T = 2 -- Total Error Measure = 50.566 etc. T = 11 -- Total Error Measure = 8.315 T = 12 -- Total Error Measure = 7.826 T = 13 -- Total Error Measure = 9.746 Okay, perhaps something optimal is in there, somewhere. The closest I got was T = 11.55, givign Total Error Measure = 7.583 I couldn't find any advantage by changing 'd'. Note: A different error measure will produce a different result. Note: There may also be discontinuities in my sequence. I stopped at T = 13 because the error measure started increasing. Who can say that it won't start decreasing at T = 25?	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9155867099761963, "perplexity": 951.7664536929074}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-17/segments/1524125948029.93/warc/CC-MAIN-20180425232612-20180426012612-00252.warc.gz"}
http://physics.stackexchange.com/questions/11340/small-change-in-theta-polar-coordinates	# Small change in theta - polar coordinates I'm reading (I'm trying to read) Schutz's "A first course in general relativity" (1985). On page 126 he mentions that a small change in angle theta in polar coordinates is given by: I can't see why this is. Can anyone please explain. Thanks - The polar angle is given by $$\theta = \arctan(y/x)$$ the total derivative ("a small change in theta") of theta is given by $$d\theta = \frac{\partial \theta}{\partial x}dx + \frac{\partial \theta}{\partial y}dy$$ this works out to $$\frac{\partial \theta}{\partial x} = \frac{-y}{x^2+y^2} = \frac{-y}{r^2}$$ $$\frac{\partial \theta}{\partial y} = \frac{x}{x^2+y^2} = \frac{x}{r^2}$$ therfore the result is as given in the book. Edit: in case it is not clear: $$\frac{\partial}{\partial x}\arctan x = \frac{1}{x^2+1}$$ - luksen: yes, eventually I understand. Thank you. – Peter4075 Jun 20 '11 at 18:03 Wishing to avoid inv trig like Carl, I've always preferred to think of it as differentiating $\tan\theta = y/x$ instead. Since $x = r\cos\theta$, the LHS is $(\sec^2\theta)d\theta = d\theta(r/x)^2$, and RHS is $(xdy-ydx)/x^2$ by quotient rule. Hence the book's result: $r^2d\theta = xdy-ydx$, rearranged. – Stan Liou Aug 13 '11 at 23:42 I'll give what I think is a more beautiful explanation. First of all, calculate $dr$ in terms of $dx$ and $dy$: $$r^2 = x^2 + y^2$$ $$2r\;dr = 2x\;dx + 2y\;dy$$ $$dr = \frac{x\;dx}{r} + \frac{y\;dy}{r}$$ With regard to the above, note that $(x/r,y/r)$ is a unit vector in the $dr$ direction. Since $d\theta$ is perpendicular to $dr$; we first look for a unit vector that is perpendicular to $(x/r, y/r)$. The answer is clearly $(y/r,-x/r)$ or its negative. To get the right sign, note that when $y=0$, we are on the x-axis and we want $d\theta$ to point in the $+y$ direction so we use $(-y/r,x/r)$. And we need an overall scale. The circumference of the unit circle is $2\pi$ and this is what you get from integrating $d\theta$ from $0$ to $2\pi$. For larger radii, we need to multiply by the radius. So we get: $$r\;d\theta \;\;=\;\; - \frac{y\;dx}{r} + \frac{x\;dy}{r},$$ $$d\theta \;\;=\;\; - \frac{y\;dx}{r^2} + \frac{x\;dy}{r^2}.$$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9713872671127319, "perplexity": 230.50546188221088}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1393999652570/warc/CC-MAIN-20140305060732-00045-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.vosesoftware.com/riskwiki/Ellipticalcopulas-NormalandT.php	# Elliptical copulas - Normal and T Elliptical copulas are simply the copulas of elliptically contoured (or elliptical) distributions. The most commonly used elliptical distributions are the multivariate normal and Student-t distributions. The key advantage of elliptical copula is that one can specify different levels of correlation between the marginals and the key disadvantages are that elliptical copulas do not have closed form expressions and are restricted to have radial symmetry. For elliptical copulas the relationship between the linear correlation coefficient r and Kendall's tau t is given by: The Normal and Student T copulas are described below. ### Normal copula The normal copula is an elliptical copula given by: where F-1 is the inverse of the univariate standard Normal distribution function and r, the linear correlation coefficient, is the copula parameter. The relationship between Kendall's tau t and the Normal copula parameter r is given by: This Copula is implemented in ModelRisk as VoseCopulaBiNormal for the bivariate case, and as VoseCopulaMultiNormal for the multivariate case. ### The T copula The Student-t copula is an elliptical copula defined as: where n (the number of degrees of freedom) and r (linear correlation coefficient) are the parameters of the copula. When the number of degrees of freedom n is large (around 30 or so), the copula converges to the Normal copula just as the Student distribution converges to the Normal. But for a limited number of degrees of freedom the behavior of the copulas is different: the t-copula has more points in the tails than the Gaussian one and a star like shape. A Student-t copula with n = 1 is sometimes called a Cauchy copula. As in the normal case (and also for all other elliptical copulas), the relationship between Kendall's tau t and the T copula parameter r is given by: This Copula is implemented in ModelRisk as VoseCopulaBiT for the bivariate case, and as VoseCopulaMultiT for the multivariate case.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8961142301559448, "perplexity": 914.3683856886619}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-09/segments/1550247481612.36/warc/CC-MAIN-20190217031053-20190217053053-00328.warc.gz"}
http://mathhelpforum.com/calculus/109614-solved-improper-integrals.html	1. ## [SOLVED] Improper integrals Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent." I don't get what to do next, but turns out the integral is arctan(x) I don't know where to continue from there. 2. Because $\frac{1}{1+x^{2}}$ is an 'even function' we can swap x with -x and obtain... $\int_{-\infty}^{1}\frac{dx}{1+x^{2}} = \int_{-1}^{\infty}\frac{dx}{1+x^{2}}= \int_{-1}^{0}\frac{dx}{1+x^{2}} + \int_{0}^{\infty}\frac{dx}{1+x^{2}}$ (1) Because for $n-1, $n\ge 1$ integer is... $\frac{1}{1+x^{2}} < \frac{1}{n^{2}}$ (2) ... is... $\int_{0}^{\infty}\frac{dx}{1+x^{2}} < \sum_{n=1}^{\infty}\frac{1}{n^{2}} = \frac{\pi^{2}}{6}$ (3) ... so that the integral (1) converges. Its value can be computed in 'standard form'... $\int_{-1}^{\infty} \frac{1}{1+x^{2}} = \lim_{x \rightarrow \infty} (tan^{-1} x + \frac{\pi}{4}) = \frac{\pi}{2} + \frac{\pi}{4} = \frac{3\pi}{4}$ (4) Kind regards $\chi$ $\sigma$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 9, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9562233090400696, "perplexity": 2422.8958488695525}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-40/segments/1474738661155.56/warc/CC-MAIN-20160924173741-00273-ip-10-143-35-109.ec2.internal.warc.gz"}
https://cdr-cvuc.cm/gw97qz/d3be88-anova-f-test-regression	In our example, F(2,27) = 6.15. Returns F array, shape = [n_features,] The set of F values. Because their expected values suggest how to test the null hypothesis H0: β1 = 0 against the alternative hypothesis HA: β1 ≠ 0. ). Why use the F-test in regression analysis . ANOVA is short for ANalysis Of Variance. Compute the ANOVA F-value for the provided sample. We’ll study its use in linear regression. Therefore, we reject the null hypothesis. How can I use the margins command to understand multiple interactions in regression and anova? (Stated another way, this says that at least one of the means is different from the others.) That's because the ratio is known to follow an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom. The alternative hypothesis is HA: β1 ≠ 0. The appropriateness of the multiple regression model as a whole can be tested by this test. That's because the ratio is known to follow an F distribution with 1 numerator degree of freedom and n-2 denominator degrees of freedom. ... (F). ANOVA as Regression • It is important to understand that regression and ANOVA are identical approaches except for the nature of the explanatory variables (IVs). eval(ez_write_tag([[336,280],'explorable_com-box-4','ezslot_2',261,'0','0']));However one assumption of the t-test is that the variance of the two populations is equal; in this case the two populations are the populations of heights for male and female students. If the associated p-value is small i.e. are special cases of linear models or a very close approximation. If the variation among groups (the group mean square) is high relative to the variation within groups, the test statistic is large and therefore unlikely to occur by chance. You can use it freely (with some kind of link), and we're also okay with people reprinting in publications like books, blogs, newsletters, course-material, papers, wikipedia and presentations (with clear attribution). The fact that you had 2 levels, or groups (0 and 1) implies that your F-test results and group means will be identical between slope-intercept regression and ANOVA. Find the F statistic: the ratio of Between Group Variation to Within Group Variation. You are free to copy, share and adapt any text in the article, as long as you give. anova— Analysis of variance and covariance 5. regress, baselevels Source SS df MS Number of obs = 10 F( 3, 6) = 21.46 Model 5295.54433 3 1765.18144 Prob > F = 0.0013 The F-test can be used to test the hypothesis that the population variances are equal. We have now completed our investigation of all of the entries of a standard analysis of variance table for simple linear regression. For this reason, it is often referred to as the analysis of variance F-test. Years ago, statisticians discovered that when pairs of samples are taken from a normal population, the ratios of the variances of the samples in each pair will always follow the same distribution. In attempting to reach decisions, we always begin by specifying the null hypothesis against a complementary hypothesis called the alternative hypothesis. For our example, we are testing the following hypothesis. The ANOVA F-test is known to be nearly optimal in the sense of minimizing false negative errors for a fixed rate of false positive errors ... More complex techniques use regression. A t-test compares means, while the ANOVA compares variances between populations. In other words, a lower p-value reflects a value that is more significantly different across populations. F-Test and One-Way ANOVA F-distribution. \| Stata FAQ. It is used when the sample size is small i.e. Privacy and Legal Statements In most cases, when people talk about the F-Test, what they are actually talking about is The F-Test to Compare Two Variances. ANOVA is (in part) a test of statistical significance. We should follow up on the significant test with pairwise comparisons at female equals zero. Two-Way ANOVA: A statistical test used to determine the effect of two nominal predictor variables on a continuous outcome variable. That is it. The ANOVA table provides a formal F test for the factor effect. The following section summarizes the ANOVA F-test. The following section summarizes the ANOVA F-test. If all assumptions are met, F follows the F-distribution shown below. The one-way ANOVA procedure calculates the average of each of the four groups: 11.203, 8.938, 10.683, and 8.838. Definition. However, the ANOVA does not tell you where the difference lies. Related posts: How to do One-Way ANOVA in Excel and How to do Two-Way ANOVA in Excel. All statistics software packages provide these p-values. Similarly, we obtain the "regression mean square (MSR)" by dividing the regression sum of squares by its degrees of freedom 1: $MSR=\frac{\sum(\hat{y}_i-\bar{y})^2}{1}=\frac{SSR}{1}.$. For correlation coefficients use . You could technically perform a series of t-tests on your data. you can see the parameters that were estimated. We don’t even need to crunch the numbers to see why this is the case. Note that, because β1 is squared in E(MSR), we cannot use the ratio MSR/MSE: We can only use MSR/MSE to test H0: β1 = 0 versus HA: β1 ≠ 0. MS is the Mean Square, it is basically SS divided by DF. Remember that in a one-way anova, the test statistic, F s, is the ratio of two mean squares: the mean square among groups divided by the mean square within groups. For the moment, the main point to note is that you can look at the results from aov() in terms of the linear regression that was carried out, i.e. These values are used to answer the question “Do the independent variables reliably predict the dependent variable?”. Recall that there were 49 states in the data set. If we only compare two means, then the t-test (independent samples) will give the same results as the ANOVA. The formula for each entry is summarized for you in the following analysis of variance table: However, we will always let statistical software do the dirty work of calculating the values for us. pval array, shape = [n_features,] The set of p-values. For this reason, it is often referred to as the analysis of variance F-test. However, it does not indicate It is used to compare the means of more than two samples. ANOVA assumes that the residuals are normally distributed, and that the variances of all groups are equal. Irrespective of the type of F-test used, one assumption has to be met: the populations from which the samples are drawn have to be normal. ANOVA is a statistical test for estimating how a quantitative dependent variable changes according to the levels of one or more categorical independent variables. A Student’s t-test will tell you if there is a significant variation between groups. The test of prog at female equal one (females) was not significant. Correlations. In linear regression, the F-test can be used to answer the following questions: Will you be able to improve your linear regression model by making it more complex i.e. We would like to show you a description here but the site won’t allow us. In our example -3 groups of n = 10 each- that'll be F(2,27). In the analysis of variance (ANOVA), alternative tests include Levene's test, Bartlett's test, and the Brown–Forsythe test.However, when any of these tests are conducted to test the underlying assumption of homoscedasticity (i.e. H 0: All individual batch means are equal. Cohen suggests that f values of 0.1, 0.25, and 0.4 represent small, medium, and large effect sizes respectively. A significant F value indicates a linear relationship between Y and at least one of the Xs. n < 30. For example, suppose one is interested to test if there is any significant difference between the mean height of male and female students in a particular college. You don't need our permission to copy the article; just include a link/reference back to this page. The ANOVA F-test for the slope parameter β 1 Most of the common statistical models (t-test, correlation, ANOVA; chi-square, etc.) Some of the more common types are outlined below. The test for equality of several means is carried out by the technique called ANOVA. This project has received funding from the, Select from one of the other courses available, Creative Commons-License Attribution 4.0 International (CC BY 4.0), ANOVA - Statistical Test - The Analysis Of Variance, Statistical Variance - A Measure of Data Distribution, Factor Analysis - Categorizing Common Variables, One-Way ANOVA - Testing Multiple Levels of a Factor, European Union's Horizon 2020 research and innovation programme. For a one-way ANOVA effect size is measured by f where . This means you're free to copy, share and adapt any parts (or all) of the text in the article, as long as you give appropriate credit and provide a link/reference to this page. Contact the Department of Statistics Online Programs, $$SSR=\sum_{i=1}^{n}(\hat{y}_i-\bar{y})^2$$, ‹ 3.4 - Analysis of Variance: The Basic Idea, Lesson 1: Statistical Inference Foundations, Lesson 2: Simple Linear Regression (SLR) Model, 3.1 - Inference for the Population Intercept and Slope, 3.4 - Analysis of Variance: The Basic Idea, 3.5 - The Analysis of Variance (ANOVA) table and the F-test, 3.7 - Decomposing The Error When There Are Replicates, 3.8 - The Lack of Fit F-test When There Are Replicates, Lesson 4: SLR Assumptions, Estimation & Prediction, Lesson 5: Multiple Linear Regression (MLR) Model & Evaluation, Lesson 6: MLR Assumptions, Estimation & Prediction, Lesson 12: Logistic, Poisson & Nonlinear Regression, Website for Applied Regression Modeling, 2nd edition. ANOVA in R: A step-by-step guide. In such a situation, a t-test for difference of means can be used. In that case, we say that the test is significant at 1%. The means of these groups spread out around the global mean (9.915) of all 40 data points. Unless this assumption is true, the t-test for difference of means cannot be carried out. chi2. H a: At least one batch mean is not equal to the others. Revised on January 7, 2021. What is an F Test? There are different types of t-tests for different purposes. For example, suppose that an experimenter wishes to test the efficacy of a drug at three levels: 100 mg, 250 mg and 500 mg. A test is conducted among fifteen human subjects taken at random, with five subjects being administered each level of the drug. F-test for testing significance of regression is used to test the significance of the regression model. At least one of the means is different. ANOVA tests whether there is a difference in means of the groups at each level of the independent variable. Now, why do we care about mean squares? Similarly, it has been shown that the average (that is, the expected value) of all of the MSEs you can obtain equals: These expected values suggest how to test H0: β1 = 0 versus HA: β1 ≠ 0: These two facts suggest that we should use the ratio, MSR/MSE, to determine whether or not β1 = 0. This beautiful simplicity means that there is less to learn.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8424866199493408, "perplexity": 636.8242064473684}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585380.70/warc/CC-MAIN-20211021005314-20211021035314-00110.warc.gz"}
http://mathhelpforum.com/algebra/27799-solving-trinomial.html	1. ## Solving trinomial Help! Solve algebraically: 10x^3+x^2-69x+54 = 0 How do i solve this? How do i find one root so i can find the other 2? Thanks so much for any help, mike 2. Hello, Mike! Solve algebraically: . $10x^3+x^2-69x+54 \:= \:0$ We have the Rational Roots Theorem. If a polynomial $P(x)$ has a rational root, it is of the form $\frac{n}{d}$ . . where $n$ is a factor of the constant term . . and $d$ is a factor of the leading coefficient. The constant term is $54$ with factors: . $\pm1,\:\pm2,\:\pm3,\:\pm6,\:\pm9,\:\pm18,\:\pm27,\ :\pm54$ The leading coefficient is $10$ with factors: . $\pm1,\:\pm2,\:\pm5,\:\pm10$ That's a lot of fractions to test, but we may get lucky . . . Try $P(1) \:=\:10\!\cdot\!1^3 + 1^2 = 60\!\cdot\!1 + 54 \:=\:-4$ . . . no Try $P(2) \;=\;10\!\cdot\!2^3 + 2^2 - 69\!\cdot\!2 + 54 \:=\:0$ . . . YES! Since $x = 2$ is a zero of $P(x)$, then $(x-2)$ is a factor. Using long division, we have: . $10x^3 + x^2 - 69x + 54 \;=\;(x-2)(10x^2 + 21x - 27)$ . . And we find that the quadratic also factors. Therefore: . $(x-2)(x+3)(10x-9) \;=\;0\quad\Rightarrow\quad x \;=\;2,\,-3,\,\frac{9}{10}$ 3. We can also try a little factoring. Rewrite as $10x^{3}-9x^{2}+10x^{2}-9x-60x+54$ Now group: $(10x^{3}+10x^{2}-60x)-(9x^{2}+9x-54)$ Now, factor: $10x(x^{2}+x-6)-9(x^{2}+x-6)$ $(10x-9)(x^{2}+x-6)$ $(10x-9)(x-2)(x+3)$	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 21, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9685693383216858, "perplexity": 602.36936184638}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2013-48/segments/1387345773230/warc/CC-MAIN-20131218054933-00098-ip-10-33-133-15.ec2.internal.warc.gz"}
https://www.physicsforums.com/threads/regarding-double-well-potential.787506/	# Homework Help: Regarding double well potential Tags: 1. Dec 14, 2014 ### peripatein A particle is in the following double-well potential with E<0: V(x)=0 for x<-a, x>a; -V0 for -a<x<-b, b<x<a; 0 for -b<x<b I am asked to show that the eigenvalues conditions may be written in the form: tan(q(a-b))=qα(1+tanh(αb)/(q22tanh(αb))$$and tan(q(a-b))=qα(1+coth(αb)/(q22coth(αb))$$ for the even and odd solutions, where -E=ħ2α2/2m and E+V02q2/2m. I first tried to define the wave function in the various regions, focusing on the positive x axis only and demanding odd solutions: Ψ(x)=Ae-αx for x>a; Bsin(q(x-b)) for b<x<a; Ce-αx + Deαx for 0<x<b Is that correct thus far? 2. Dec 14, 2014 ### Staff: Mentor Looks good so far. 3. Dec 14, 2014 ### peripatein Alright, but as I impose continuity of wave function and its derivatives I obtain the four following equations, from which I am unable to derive the desired expression: (1) Ae-αa = Bsin(q(a-b)) (2) -αAe-αa = Bqcos(q(a-b)) (3) 0 = Ce-αb + Deαb (4) Bq = -αCe-αb + Dαeαb What am I doing wrong? 4. Dec 14, 2014 ### Staff: Mentor You could explain what you did. Where does equation 3 come from? Where is the odd/even condition? 5. Dec 14, 2014 ### peripatein Well, equation 3 stems from the demand for continuity at x=b. I have decided to focus on the odd solutions first, hence the use of sine. Is that wrong? 6. Dec 14, 2014 ### Staff: Mentor Ah, I see the issue. The way you defined the wave function in the well, you get a mixture of odd and even functions (the sine would have some non-zero value at x=0). If you want odd functions, you need something like sin(qx). 7. Dec 14, 2014 ### peripatein So which function ought I to choose for Ψ(x) in the well? Ought it to be a combination of sine and cosine at once (like harmonic oscillator)? 8. Dec 14, 2014 ### peripatein In other words, should I then simply choose sin(qx) for the odd and cos(qx) for the even, within the well? 9. Dec 14, 2014 ### Staff: Mentor You can do that, if you choose them without that offset it is easier to split them into odd and even contributions. That is the easierst approach I think. 10. Dec 15, 2014 ### peripatein Yet please note the expressions I am expected to arrive at. It seems I am indeed expected to use forms as sin(q(x-b)). Wouldn't you agree? 11. Dec 15, 2014 ### Staff: Mentor I don't see how you get that impression. Yes there is an q(a-b), but that can appear in other ways, too.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9590193629264832, "perplexity": 3084.5942800261573}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-30/segments/1531676589179.32/warc/CC-MAIN-20180716041348-20180716061348-00541.warc.gz"}
https://ora.ox.ac.uk/objects/uuid:958397f6-58c5-400e-b730-aad8152a1b56	Journal article Bayesian rose trees Abstract: Hierarchical structure is ubiquitous in data across many domains. There are many hierarchical clustering methods, frequently used by domain experts, which strive to discover this structure. However, most of these methods limit discoverable hierarchies to those with binary branching structure. This limitation, while computationally convenient, is often undesirable. In this paper we explore a Bayesian hierarchical clustering algorithm that can produce trees with arbitrary branching structure at... Publication status: Published Authors Blundell, C More by this author More by this author Institution: University of Oxford Department: Oxford, MPLS, Statistics Heller, KA More by this author Journal: Proceedings of the 26th Conference on Uncertainty in Artificial Intelligence, UAI 2010 Pages: 65-72 Publication date: 2010-12-01 URN: Source identifiers: 353227 Local pid: pubs:353227	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8427125215530396, "perplexity": 4762.2628427735235}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107882103.34/warc/CC-MAIN-20201024080855-20201024110855-00522.warc.gz"}
https://www.earthdoc.org/content/papers/10.3997/2214-4609.20130395	1887 ### Abstract The frequency content of continuous passive seismic recordings of hydraulic fracturing are sometimes used to gain information on the numerous microseismic events occurring during the fluid injection. In addition, resonance frequencies are frequently recorded as well. Resonances can be used in many ways due to their multiple origins possible. In every circumstance, all possible sources have to be reviewed to define their respective influence. We present two different experiments showing resonances with presumably different origins. For the first experiment, the low-frequency resonances (5-50 Hz) are only recorded by downhole geophones and broadband stations on the surface that are close to the injection well. For the second experiment, four resonances at 17, 35, 51 and 60 Hz are detected. The fluid injection being at approximately the same depth as the receivers, the path effect influence will be limited and the resonances coming from receiver effects are anticipated to be outside the frequency range of the observed resonances. A possible source would be the resonance of fluid-filled cracks. The size of a crack corresponding to a resonance of 17 Hz is calculated to be 17 m. These resonances would then correspond to mesoscale deformation of the reservoir. /content/papers/10.3997/2214-4609.20130395 2013-06-10 2022-01-29	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8921844959259033, "perplexity": 1085.3118266232466}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-05/segments/1642320300573.3/warc/CC-MAIN-20220129062503-20220129092503-00578.warc.gz"}
http://www.ck12.org/book/CK-12-Physical-Science-Concepts-For-Middle-School/r7/section/4.24/	<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" /> # 4.24: Calculating Acceleration from Force and Mass Difficulty Level: At Grade Created by: CK-12 Estimated2 minsto complete % Progress Practice Calculating Acceleration from Force and Mass MEMORY METER This indicates how strong in your memory this concept is Progress Estimated2 minsto complete % Estimated2 minsto complete % MEMORY METER This indicates how strong in your memory this concept is Xander goes airborne on his scooter as he exits a half-pipe at Newton’s Skate Park. How did he gain enough speed in the half-pipe to fly into the air when he got to the top? His increase in speed was due partly to the force of gravity. ### Acceleration, Force, and Mass A change in an object’s motion—such as Xander speeding up on his scooter—is called acceleration. Acceleration occurs whenever an object is acted upon by an unbalanced force. The greater the net force acting on the object, the greater its acceleration will be, but the mass of the object also affects its acceleration. The smaller its mass is, the greater its acceleration for a given amount of force. Newton’s second law of motion summarizes these relationships. According to this law, the acceleration of an object equals the net force acting on it divided by its mass. This can be represented by the equation: Acceleration=NetforceMass\begin{align}\mathrm{Acceleration=\frac{Net\;force}{Mass}}\end{align} or a=Fm\begin{align}\mathrm{a=\frac{F}{m}}\end{align} ### Calculating Acceleration This equation for acceleration can be used to calculate the acceleration of an object that is acted on by a net force. For example, Xander and his scooter have a total mass of 50 kilograms. Assume that the net force acting on Xander and the scooter is 25 Newtons. What is his acceleration? Substitute the relevant values into the equation for acceleration: a=Fm=25N50kg=0.5Nkg\begin{align}\mathrm{a=\frac{F}{m}=\frac{25N}{50kg}=\frac{0.5N}{kg}}\end{align} The Newton is the SI unit for force. It is defined as the force needed to cause a 1-kilogram mass to accelerate at 1 m/s2. Therefore, force can also be expressed in the unit kg ∙ m/s2. This way of expressing force can be substituted for Newtons in Xander’s acceleration so the answer is expressed in the SI unit for acceleration, which is m/s2: a=0.5Nkg=0.5kgm/s2kg=0.5m/s2\begin{align}\mathrm{a=\frac{0.5N}{kg}=\frac{0.5 kgm/s^2}{kg}=0.5 m/s^2}\end{align} Q: Why are there no kilograms in the final answer to this problem? A: The kilogram units in the numerator and denominator of the fraction cancel out. As a result, the answer is expressed in the correct SI units for acceleration. ### Calculating Force It’s often easier to measure the mass and acceleration of an object than the net force acting on it. Mass can be measured with a balance, and average acceleration can be calculated from velocity and time. However, net force may be a combination of many unseen forces, such as gravity, friction with surfaces, and air resistance. Therefore, it may be more useful to know how to calculate the net force acting on an object from its mass and acceleration. The equation for acceleration above can be rewritten to solve for net force as: Net Force = Mass x Acceleration, or F = m x a Look at Xander in the picture below. He’s riding his scooter down a ramp. Assume that his acceleration is 0.8 m/s2. How much force does it take for him to accelerate at this rate? Substitute the relevant values into the equation for force to find the answer: F = m x a = 50 kg x 0.8 m/s2 = 40 kg ∙ m/s2, or 40 N Q: If Xander and his scooter actually had a mass of 40 kg instead of 50 kg, how much force would it take for him to accelerate at 0.8 m/s2? A: It would take only 32 N of force (40 kg x 0.8 m/s2). At the URL below, you can review how to calculate net force from mass and acceleration and try some practice problems. If you want to try more challenging problems, click on the relevant links at the bottom of the Web page. ### Summary • According to Newton’s second law of motion, the acceleration of an object equals the net force acting on it divided by its mass, or a=Fm\begin{align}\mathrm{a=\frac{F}{m}}\end{align*}. • This equation for acceleration can be used to calculate the acceleration of an object when its mass and the net force acting on it are known. • The equation for acceleration can be rewritten as F = m x a to calculate the net force acting on an object when its mass and acceleration are known. ### Vocabulary • acceleration: Measure of the change in velocity of a moving object. ### Practice At the following URL, solve the six problems on page 2. ### Review 1. What is the equation for calculating the acceleration of an object when its mass and the net force acting on it are known? 2. Xander’s friend Corey has a skateboard that he rides at Newton’s Skate Park. That’s Corey doing a jump in the picture below. The combined mass of Corey and his skateboard is 60 kg. At the top of his jump, the net force acting on him is 30 Newtons. What is his acceleration at that moment? 3. What net force would have to act on Cory for him to have an acceleration of 1 m/s2? ### Notes/Highlights Having trouble? Report an issue. Color Highlighted Text Notes ### Vocabulary Language: English acceleration Measure of the change in velocity of a moving object. Show Hide Details Description Difficulty Level: Authors: Tags: Subjects:	{"extraction_info": {"found_math": true, "script_math_tex": 5, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9390816688537598, "perplexity": 578.7359606389446}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-50/segments/1480698542765.40/warc/CC-MAIN-20161202170902-00366-ip-10-31-129-80.ec2.internal.warc.gz"}
https://cp4space.hatsya.com/2013/06/13/plateau/	# Plateau Given two identical circles lying in parallel planes with a common axis of symmetry, what is the minimal surface that connects them? Initially, one might suppose that it is a cylinder (blue in the diagram below). However, we can quickly establish that a cylinder is not favourable. For a minimal surface, optimal solutions have zero mean curvature, whereas the cylinder has nonzero mean curvature. The correct solution is a catenoid (green), which is the revolute of the curve $cx = \cosh{cz}$ about the z-axis; this can be demonstrated by the calculus of variations. Another minimal surface is the helicoid, obtained by taking the surface swept out by a horizontal line undergoing a continuous screwing about a vertical axis. Indeed, there is actually a continuous distance-preserving map from the helicoid to the catenoid, describable as a complex rotation: Obviously, a boring example of a minimal surface is the plane. Indeed, when the two circles are sufficiently separated, the catenoid is no longer optimal and a pair of disjoint discs is instead minimal. Suppose that the circles are replaced with impenetrable discs, and we inject a prescribed volume of air into the catenoid. The requirement is no longer that the surface have zero mean curvature everywhere, but rather constant mean curvature. In this generalisation of the problem, the cylinder is optimal for a particular value of the volume. In the simplest case, where we have a fixed volume and no constraints on the boundary, a sphere is optimal; this is why individual bubbles form spheres. It took until 2002 for this result to be extended to two volumes of air; the optimal solution is a set of three spherical films, meeting at a common circle at angles of 2π/3. Plateau proved that all such edges between three films must be at angles of 2π/3 (c.f. this post), and vertices where four such boundaries meet must locally resemble regular tetrahedra. Together with the requirement that mean curvature must be constant on any film, we obtain Plateau’s laws. Lord Kelvin wondered what the optimal arrangement of soap bubbles of unit volume is, so as to minimise the total surface area. He conjectured that the regular honeycomb of truncated octahedra was optimal, and for a while the Kelvin conjecture was believed to be true. Actually, Kelvin slightly curved the faces to conform to Plateau’s laws. It transpires that there is a better polyhedral tiling giving a more optimal arrangement of soap bubbles, discovered by the Irish mathematicians Weaire and Phelan in 1994. John Conway refers to these as Scottish and Irish bubbles, and proposes a third system of Welsh bubbles. They haven’t caught on, however; the Kelvin structure occurs as the Voronoi partition of many crystal structures, and the Weaire-Phelan structure was used in the architecture of the Beijing Natural Aquatics Centre for the 2008 Olympic Games: This entry was posted in Uncategorized. Bookmark the permalink.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 1, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9219310283660889, "perplexity": 696.2432491139917}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-50/segments/1606141171126.6/warc/CC-MAIN-20201124053841-20201124083841-00107.warc.gz"}
https://hal-mines-paristech.archives-ouvertes.fr/hal-01220989	# Evaluation of the level of prediction errors of wind and PV in a future with a large amount of renewables Abstract : In a future with a large amount of renewable energies integrated into the distribution system there are many kind of new constraint that cannot be considered apriori as un-active in the power distribution system operation and planning. This includes short term (few hours in advance) forecast errors of the production. However their importance is subject to the so called aggregation effect and cannot be considered as a linear function of the installed capacity. Furthermore this aggregation effect highly depends on the considered geographical perimeter. In this paper we presents a methodology to estimate the level of short term forecast error at any spatial scale and we apply the methodology in the case of France, and regions of France. Before simulating forecast error, we describe the methodology used to simulate hourly wind power and solar power production at any location. Wind power is simulated through the use of a statistically calibrated power curve a spatio-temporal field of win simulation obtained from the use of a meteorological model in reanalysis mode. Solar power simulation are based on the use of SODA estimation of surface solar irradiation at a hourly resolution and at the spatial scale of 3km. SODA data rely on the use of satellite images and physical modeling of radiative transfer in the atmosphere. Both models to simulate production are validated on real data. Our methodology for simulating forecast error mimic real forecasting performed on the simulated production data. The real forecasting model is simplified in order to allow a large quantity of computation repetitions. In the case of PV forecasting as well as in the case of Wind Power forecasting, it uses as input two explanatory variables: meteorological forecast of ECMWF (of wind speed for wind power and of surface solar irradiation fot PC) and closed past production measurement. The simulation is performed along several years and thousands of different implantation configurations. The distribution of errors is analyzed as a function of the spatial scale and the chosen horizon. Keywords : Document type : Conference papers Domain : https://hal-mines-paristech.archives-ouvertes.fr/hal-01220989 Contributor : Magalie Prudon <> Submitted on : Tuesday, October 27, 2015 - 11:16:12 AM Last modification on : Thursday, September 24, 2020 - 5:20:17 PM ### Identifiers • HAL Id : hal-01220989, version 1 ### Citation Robin Girard, Arthur Bossavy, Loïc Legars, Georges Kariniotakis. Evaluation of the level of prediction errors of wind and PV in a future with a large amount of renewables. EWEA Technology Workshop, Wind Power Forecasting 2015, European Wind Energy Association, Oct 2015, Leuven, Belgium. ⟨hal-01220989⟩ Record views	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8430821895599365, "perplexity": 1583.9349305167107}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610704832583.88/warc/CC-MAIN-20210127183317-20210127213317-00375.warc.gz"}
https://www.imlearningmath.com/a-half-is-a-third-of-it-what-is-it/	A half is a third of it. What is it? Math Riddle: A half is a third of it. What is it? Answer: The correct answer is 1 1/2.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8830023407936096, "perplexity": 1213.814221889471}, "config": {"markdown_headings": false, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107894426.63/warc/CC-MAIN-20201027170516-20201027200516-00284.warc.gz"}
http://mlweb.loria.fr/book/en/naivebayesclassifier.html	The naive Bayes classifier In words... First of all, the naive Bayes classifier should not be confused with the (optimal) Bayes classifier as it is in fact far from optimal. The naive Bayes classifier is one of the most basic generative classifiers. It relies on a very crude approximation/assumption of conditional independence of the features. This means that the classifier assumes that the value taken by a feature does not depend on the other features. This assumption is obviously wrong in most applications. For instance, in spam filtering, this amounts to considering that the presence of the word "earn" does not depend on the presence of the word "money". But despite this very crude approximation, the naive Bayes classifier remains rather effective. In pictures... Demo application This demo shows how the naive Bayes classifier can be used to filter spams in mailboxes. In maths... For a random variable $X$ taking values in $\X\subset\R^d$, the conditional independence assumption at the core of the naive Bayes classifier results in the following factorization of the conditional probability of $X$ given $Y$: $$P(X=\g x\ \|\ Y=y) = \prod_{j=1}^d P(X_j = x_j\ \|\ Y = y) ,$$ where $X_j$ is the $j$th component of $X$. This assumption drastically eases the estimation of the generative model as shown in the example below. Naive Bayes classifier with real inputs For a continuous random variable $X$ of density $p_X(\g x)$, the factorization above reads $$p_X(\g x) = \prod_{j=1}^d p_{X_j}(x_j\ \|\ Y = y)$$ where $p_{X_j}(x_j\ \|\ Y = y)$ is the density of the $j$th input variable (or feature) $X_j$. Assume now that we would like to fit a Gaussian distribution over $X$ in each class $y\in\Y$. Fitting a multi-dimensional Gaussian distribution in $\R^d$ implies to estimate the values of the $d$ means and of the covariance matrix $\g \Sigma \in\R^{d\times d}$. But with the factorization above, this can be reduced to the estimation of $d$ one-domensional densities $$p_{X_j}(x_j\ \|\ Y = y) = \exp\left( \frac{-(x_j - \mu_{yj})^2}{2\sigma_{yj}^2}\right),\quad j=1,\dots,d,$$ with $d$ means $$\mu_{yj} = \frac{1}{N_y} \sum_{y_i = y} x_{ij}$$ and $d$ variances $$\sigma_{yj}^2 = \frac{1}{N_y-1} \sum_{y_i = y} (x_{ij} - \mu_{yj})^2$$ for each class $y$ having $N_y$ examples in the training set $\{(\g x_i, y_i)\}_{i=1}^N$.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.899792492389679, "perplexity": 214.91423590235036}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-09/segments/1518891815435.68/warc/CC-MAIN-20180224053236-20180224073236-00099.warc.gz"}
http://sepwww.stanford.edu/sep/prof/iei/txz/paper_html/node20.html	Next: Fourier transforms in retarded Up: RETARDED COORDINATES Previous: Definition of dependent variables ## The chain rule and the high frequency limit The familiar partial-differential equations of physics come to us in (t,x,z)-space. The chain rule for partial differentiation will convert the partial derivatives to (t' ,x' ,z' )-space. For example, differentiating (30) with respect to z gives (31) Using (27), (28) and (29) to evaluate the coordinate derivatives gives (32) There is nothing special about the variable P in (31) and (32). We could as well write (33) where the left side is for operation on functions that depend on (t,x,z) and the right side is for functions of (t' ,x' ,z' ). Differentiating twice gives (34) Using the fact that the velocity is always time-independent results in (35) Except for the rightmost term with the square brackets it could be said that squaring'' the operator (33) gives the second derivative. This last term is almost always neglected in data processing. The reason is that its effect is similar to the effect of other first-derivative terms with material gradients for coefficients. Such terms, cause amplitudes to be more carefully computed. that all such terms should be included, from the beginning. Next: Fourier transforms in retarded Up: RETARDED COORDINATES Previous: Definition of dependent variables Stanford Exploration Project 10/31/1997	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9644179344177246, "perplexity": 1246.7147409626177}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-51/segments/1544376824119.26/warc/CC-MAIN-20181212203335-20181212224835-00526.warc.gz"}
https://brilliant.org/problems/jills-probability/	# Jill's Probability Discrete Mathematics Level 2 Jill's pocket contains 3 quarters, 4 dimes, 2 nickels and 6 pennies. If he takes out a coin at random, what is the probability that the coin is worth more than 5 cents? ×	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8246269822120667, "perplexity": 510.01880349457633}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988720356.2/warc/CC-MAIN-20161020183840-00470-ip-10-171-6-4.ec2.internal.warc.gz"}
https://ec.gateoverflow.in/716/gate-ece-2015-set-2-question-18	41 views Two causal discrete-time signals $x[n]$ and $y[n]$ are related as $y[n] = \displaystyle{}\sum _{m=0}^{n} x[m]$. If the $z$-transform of $y[n]$ is $\dfrac{2}{z(z-1)^{2}},$ the value of $x[2]$ is _______.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.862288773059845, "perplexity": 249.68594943310933}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 5, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-49/segments/1637964361064.58/warc/CC-MAIN-20211201234046-20211202024046-00169.warc.gz"}
https://zbmath.org/?q=an:0952.91020	× # zbMATH — the first resource for mathematics Characterization of some aggregation functions stable for positive linear transformations. (English) Zbl 0952.91020 The authors consider the characterization of some classes of aggregation functions and aggregators (the family of aggregation function) often used in operation research, especially in multicriteria decision making problems. The examples of aggregation functions are: arithmetic mean, weighted arithmetic mean, minimum, maximum, etc. The choice of proper aggregation function strongly depends on application. In order to obtain reasonable results the function must often fulfill some special conditions. The authors adopt an axiomatic approach in which they divide the properties of aggregation function into three categories: natural properties, stability properties and algebraic properties. In each group they provide several definitions describing some general features of aggregation function. They explore more deeply the features of stability for positive linear transformation and increasing monotonicity. The aggregation function $$M^{(m)}$$ fulfills these conditions if: $M^{(m)} (rx_1+ t,\dots, rx_m+ t)= rM^{(m)} (x_1,\dots, x_m)+ t, \quad\text{for }r>0,\;t\in \mathbb{R},$ $x_i'< x_i' \Rightarrow M^{(m)} (x_1,\dots, x_i,\dots, x_m)\leq M^{(m)} (x_1,\dots, x_i,\dots, x_m), \quad\text{for all }i= 1,\dots, n.$ The main advantage of the paper are several theorems which describe the family of all aggregation functions fulfilling three specific properties. The first two are increasing monotonicity and stability for positive linear transformation. The third property is one of the well known algebraic properties such as associativity, decomposability and bisymmetry. For each combinations of these properties the authors find all aggregators which fulfill them. The paper is a theoretical one but the results obtained have valuable practical implications. The results can help the decision maker in choosing the proper aggregation function on the basis of some expected properties. ##### MSC: 91B06 Decision theory 90C70 Fuzzy and other nonstochastic uncertainty mathematical programming 90C29 Multi-objective and goal programming 90B50 Management decision making, including multiple objectives Full Text: ##### References: [1] Aczél, J., On Mean values, Bull. amer. math. soc., 54, 392-400, (1948) · Zbl 0030.02702 [2] Aczél, J., Lectures on functional equations and applications, (1966), Academic Press New York · Zbl 0139.09301 [3] Aczél, J., On weighted synthesis of judgements, Aequationes math., 27, 288-307, (1984) · Zbl 0543.90003 [4] Aczél, J.; Alsina, C., Synthesizing judgments: A functional equations approach, Math. modelling, 9, 311-320, (1987) · Zbl 0624.90002 [5] Aczél, J.; Roberts, F.S., On the possible merging functions, Math. social sci., 17, 205-243, (1989) · Zbl 0675.90005 [6] Aczél, J.; Roberts, F.S.; Rosenbaum, Z., On scientific laws without dimensional constants, J. math. anal. appl., 119, 389-416, (1986) · Zbl 0605.39004 [7] J. Fodor, J.-L. Marichal, On nonstrict means, Aequationes Math., in press. · Zbl 0907.39021 [8] Fodor, J.; Marichal, J.-L.; Roubens, M., Characterization of some aggregation functions arising from MCDM problems, (), 1026-1031 [9] Fodor, J.; Roubens, M., Fuzzy preference modelling and multicriteria decision support, (1994), Kluwer Dordrecht · Zbl 0827.90002 [10] Kolmogoroff, A.N., Sur la notion de la moyenne, Accad. naz. lincei mem. cl. sci. fis. mat. natur. sez., 12, 388-391, (1930) · JFM 56.0198.02 [11] Ling, C.H., Representation of associative functions, Publ. math. debrecen, 12, 189-212, (1965) · Zbl 0137.26401 [12] J.-L. Marichal, P. Mathonet, A characterization of the ordered weighted averaging functions based on the ordered bisymmetry property, IEEE Trans. Fuzzy Systems, submitted. [13] Marichal, J.-L.; Roubens, M., Characterization of some stable aggregation functions, (), 187-196 [14] Nagumo, M., Über eine klasse der mittelwerte, Japan. J. math., 6, 71-79, (1930) · JFM 56.0198.03 [15] Roberts, F.S., Measurement theory with applications to decision-making, utility and the social sciences, (1965), Addison-Wesley Reading, MA [16] Silvert, W., Symmetric summation: A class of operations on fuzzy sets, IEEE trans. systems man cybernet., 9, 657-659, (1979) · Zbl 0424.04003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8423441648483276, "perplexity": 3971.471132539047}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-17/segments/1618038060927.2/warc/CC-MAIN-20210411030031-20210411060031-00169.warc.gz"}
https://wiki.q-researchsoftware.com/wiki/Weights,_Effective_Sample_Size_and_Design_Effects	# Weights, Effective Sample Size and Design Effects ## How weights are taken into account in Q By default, Q assumes that any weight is a sampling weight designed to correct for representativeness issues in a sample (e.g., to correct for an over- or under-representation of women in the sample). Q assumes that weights are proportional to the inverse of the probability of selection. For example, if one respondent has a weight of 2 and another has a weight of 1, this means that the person with a weight of 2 had only half the chance of being selected for the survey as the other. With the exception of unweighted sample size, all statistics computed by Q take into account the weight. The unweighted sample size is reported as n, Column n, Row n and Base n dependending upon whether it is referring to a count in a cell, a row or column total or the total sample size. The weighted sample size is referred to as Population, Column Population, Row Population and Base Population dependending upon the context. All statistical tests in Q are modified to take into account the weight in such a way that the average weight is not a determinant of the inference. That is, the same significance testing results are obtained whether the average weight is 1.0 or 1,000,000. For this reason, expansion weights can be used in Q, whereby the weights gross the sample up to the population of interest (which is why the weighted sample size is referred to as Population). All statistical testing in Q uses either Taylor series linearization or Weight Calibration. Which is used is described in the description of individual tests in Tests Of Statistical Significance and in the various pages on multivariate analyses. ## Effective sample size In most instances, weighting causes a decrease in the statistical significance of results. The effective sample size is a measure of the precision of the survey (e.g., even if you have a sample of 1,000 people, an effective sample size of 100 would indicate that the weighted sample is no more robust than a well-executed un-weighted simple random sample of 100 people). Each cell shown on a table potentially has a different effective sample size. This is because the impact of weights can differ by statistic. For example, if a study over-recruited buyers of a particular brand, then the effective sample size of the buyers of the brand is likely to be very different to the effective sample size of buyers of other brands (because the weights will differ by brand). Q uses two methods to compute effective sample size. Where Taylor series linearization (see Lumley, T. (2010). Complex Surveys: A Guide to Analysis Using R, Wiley) is used, the effective sample size is computed as the ratio of the sampling variance computed by the Taylor series linearization divided by the sampling variance computed under the assumption of a simple random sample with replacement. Where Taylor series linearization is not used, Kish's Effective Sample Size Formula is used. Effective sample size is used and reported by Q in a variety of ways: • Effective Base n can be viewed in the cells on most tables. • Where a data file has been weighted, most tables in Q report the effective sample size in the bottom-right corner of the screen. The proportion shown in brackets indicates the size of the effective sample relative to the total sample size (excluding missing data). The effective sample size at the bottom of a table is computed as follows: 1. Effective base n / Base n is computed for every cell in the table. 2. The median of these ratios is then computed. 3. The median is multiplied by the largest Base n on the table and is rounded. • In some statistical tests the effective sample size is used to modify the weight (see Weight Calibration). Note that the design effect, discussed in the next section, also impacts upon the effective sample size. ## Design effects The design effect is computed as the actual sample size divided by the effective sample size. Thus, where the true sampling variance is twice that computed under the assumption of simple random sampling the design effect is 2.0. (see design effects). Q automatically computes design effect for weighted data (unless this option is changed). However, from Q4.10 onwards, Q also permits an additional design effect (Extra deff in Statistical Assumptions) to be taken into account in significance testing to reflect aspects of the sample design that are not taken into account by the weight. The precise way it is taken into account is determined by the setting of Weights and Significance: • Automatic. On tests of means and proportions on tables, the computation is as indicated below for Taylor Series Linearization. Otherwise, the Kish approximation is used. • Taylor Series Linearization. The sampling variance is first estimated using Taylor series linearization, and this estimate is multiplied by the specified value of Extra Deff. • Kish approximation. The effective sample size is used in place of the sample size when computing the variance, where the effective sample size is computed using: effective sample size = ${\displaystyle \sum _{i=1}^{n}w_{i}/deff}$ where: ${\displaystyle w_{i}}$ is the un-calibrated weight of the ${\displaystyle i}$ of ${\displaystyle n}$ observations, and ${\displaystyle deff}$ is the supplied extra design effect. • deff = 1. Statistical inference is conducted under the assumption that the weights are frequency weights (see [1]), where the frequency weights are the supplied weights divided by the supplied extra design effect. • deff = Sample size / sum of weights. Statistical inference is conducted under the assumption that the weights are frequency weights (see [2]), where the frequency weights are the supplied weights normalized to have an average value of 1 and then divided by the supplied extra design effect. • Unweighted sample size in tests. The un-weighted sample size divided by the supplied extra design effect is used in all statistical inference. Note that if a respondent has a missing or non-positive weight, they are excluded from the analysis and the sample size. When weight calibration is used, this assumption is equivalent to deff = Sample size / sum of weights.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 5, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8675779700279236, "perplexity": 595.5919718388309}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-39/segments/1631780058552.54/warc/CC-MAIN-20210927211955-20210928001955-00041.warc.gz"}
http://www.askiitians.com/iit-jee-indefinite-integral/	1800 2000 838 CART 0 • 0 MY CART (5) Use Coupon: CART10 and get 10% off on all online Study Material ITEM DETAILS MRP DISCOUNT FINAL PRICE Total Price: R There are no items in this cart. Continue Shopping Get instant 50% OFF on Online Material. Use coupon code: MOB50 \| View Course list ```Indefinite Integral Indefinite integral is an important component of integral calculus. It lays the groundwork for definite integral. Students are advised to practice as many problems as possible as only practice can help in achieving perfection in indefinite integrals. Integration is used in dealing with two essentially different types of problems: (1) The first type of problems are those in which the derivative of a function, or its rate of change, or the slope of its graph, is known and we want to find the function. We are therefore required to reverse the process of differentiation. This reverse process is known as anti-differentiation and it is often termed as finding of a primitive function or finding an indefinite integral. (2) The second type leads to the definition of the definite integral. Definite integrals are used for finding area, volume, centre of gravity, moment of inertia, work done by a force and various such applications. Remark: This integral is called an indefinite integral because its value is not fully determined until the end points are specified. This ambiguity is dealt with by the addition of the constant C at the end. This constant of integration is added to the end because there are, in fact, an infinite number of solutions to the integral. The topic holds great importance from examination point of view and is widely used in modern sciences as well as in engineering entrance examinations. They account to many lengthy numericals and key derivations. Various topics covered in this chapter include: Topics Covered: Basic Concepts in Indefinite Integral Methods of Integration-Integration by Substitutiion Standard Substitutions Indirect Substitution Integration by Parts Integration by Partial Fractions Algebraic Integrals Trigonometric Integrals Solved Examples of Indefinite Integral Those willing to go into the intricacies of the topics are advised to refer the following sections. Here, we shall discuss all the topics in brief: What do we mean by an indefinite integral? Indefinite integration is also termed as antidifferentiation and as the name suggests, it is the reverse process of differentiation. Suppose we have a function f and we intend to find a function F such that F’ = f, then in this case F is said to be the indefinite integral or the anti-derivative of the function f. Mathematically, we write it as, where f(x) is called the integrand, x is called the integration variable and ‘c’ is the constant of integration. In fact, iff Various basic and important formulae used in indefinite integrals have been discussed in the coming sections. Inverse property of indefinite integrals: d/dx ∫ f(x) dx = f(x) ∫ f’(x) dx = f(x) + c Two antiderivatives of a given function can only differ by a constant, i.e. if F is an antiderivative of a continuous function f, then any other antiderivative G of the function f is of the form G(x) = F(x) + c If ∫ f(x) dx = g(x) + c, then ∫ f(ax + b) dx = 1/a g(ax + b) + c, where a and b are constants such that a ≠ 0. Constant is an important component in indefinite integration. Methods of Integration Generally speaking, there are three methods of integration used for solving indefinite integrals which are: At times, questions on integration can be solved directly by application of direct formulae. But, usually the questions asked in competitive examinations are not very simple and can’t be solved by direct application of formulae. If the integrand is not a derivative of a known function, then the corresponding integrals cannot be found directly. In order to find the integrals of complex problems, generally three rules of integration are used. (1) Integration by parts. (2) ?Integration by substitution or by change of the independent variable. (3) Integration by partial fractions. We shall only give a brief outline of these methods as they have been discussed in detail in the coming sections: Integration by Parts: If u and v be two functions of x, then the integral of product of these two functions is given by: Note: In applying the above rule care has to be taken in the selection of the first function (u) and the second function (v) depending on which function can be integrated easily. Normally we use the following methods for making this choice: If both of the functions are directly integrable then the first function is chosen in such a way that the derivative of the function thus obtained under integral sign is easily integrable. Usually we use the following preference order for the first function. (Inverse, Logarithmic, Algebraic, Trigonometric, Exponent) This rule is called as ILATE which states that the inverse function should be assumed as the first function while performing integration. Hence, the functions are to be assumed in the order from left to right depending on the type of functions involved. Various other cases have been discussed in detail in the later sections. Example Evaluate ∫ ex (1/x – 1/x2)dx. Solution: Let I = ∫ ex (1/x – 1/x2)dx Here f(x) = 1/x, f'(x) = –1/x2 Hence I = ex / x + c Integration by Substitution: ?It is not always possible to find the integrals of complicated functions by observation and hence we need methods like substitution for solving them. The substitution method has the following parts: ? Besides direct and indirect methods, sometimes questions can also be solved by some standard substitutions. The direct and indirect substitutions have been discussed in detail in the section on substitution method. We list here some of the standard substitutions. Some Standard Substitutions For terms of the form x2 + a2 or √x2 + a2, put x = a tan θ or a cot θ For terms of the form x2 - a2 or √x2 – a2 , put x = a sec θ or a cosec θ For terms of the form a2 - x2 or √x2 + a2, put x = a sin θ or a cos θ If both √a + x, √a – x, are present, then put x = a cos θ. For the type √(x–a)(b–x), put x = a cos2θ + b sin2θ For the type (√x2 + a2 ± x)n or (x ± √x2 – a2)n, put the expression within the bracket = t. For 1/(x+a)n1 (x+b)n2, n1,n2 ∈ N (and > 1), again put (x + a) = t(x + b) Example: Evaluate ∫ x3 tan–1 x4/(1 + x8) dx. Solution: Put tan–1 x4 = t Then ⇒ 4x3/(1 + x8) dx = dt Hence, using this in the above integral we have I = 1/4 ∫ t dt = 1/4 t2/2 + A = 1/8 (tan–1 x4)2 + A. Integration by Partial Fractions: If P(x) and Q(x) are two polynomials, then P(x)/Q(x) is called a rational function. It may further be classified as a proper rational function or an improper rational function. If deg P(x) < deg Q(x), then it is called a proper rational function. If deg P(x) ≥ deg Q(x), then it is called an improper rational function. If P(x)/Q(x) is an improper rational function then we divide P(x) by Q(x) so as to express the rational function P(x)/Q(x) in the form a(x) + b(x)/Q(x), where a(x) and b(x) are polynomials such that degree of b(x) is less than that of Q(x). In order to write P(x)/Q(x) in partial fractions, first of all we write Q(x) = (x – a)k ... (x2 + αx + β)r ... where binomials are different, and then set where A1, A2, ..., Ak, M1, M2, ......, Mr, N1, N2, ...... ,Nr are real constants to be determined. These are determined by reducing both sides of the above identity to integral form and equating the coefficients of equal powers of x, which gives a system of linear equations in the coefficient. (This method is called the method of comparison of coefficients). The constants can also be obtained by substituting suitably chosen numerical values of x in both sides of the identity. Example : Evaluate ∫ 1/sin x (2 + cos x – 2 sinx) dx. Solution: Put tan x = t ⇒ sec2.x/2 . 1/2 dx = dt ⇒ dx = 2/sec2 x/2 dt ⇒ 2/1 + t2 dt, then we have This expands into simple fractions (1 + t2)/ t(t – 3) (t -1) = A/t + B/(t–3) + C/(t -1) After solve the coefficients, A = 1/3; B = 5/3; C = –1 Hence I = 1/3 ∫ dt / t + 5/3 ∫ dt/ (t -3) – ∫ dt / (t-1) = 1/3 In \|t\| + 5/3 \|t-3\| - In \|t-1\| + c = 1/3 In \|tan x/2\| + 5/3 \|tan x/2 –3\| – ln \| tan x/2 – 1 \| + c. Trigonometric Integrals Integrals of the form ∫ R ( sinx, cosx) dx: Here R is a rational function of sin x and cos x. This can be translated into integrals of a rational function by the substitution: tan(x/2) = t. This is the so called universal substitution. In this case sin x = 2t/(1+t2) ; cos x = (1-t2)/(1+t2), x = 2 tan-1 t; dx = 2 dt/(1 + t2) Sometimes, instead of the substitution tan x/2 = t, it is more advantageous to make the substitution cot x/2 = t Universal substitution often leads to very cumbersome calculations. Indicated below are those cases where the aim can be achieved with the aid of simpler substitutions. (a) If R (-sin x, cos x) = -R(sin x, cos x), substitute cos x = t (b) If R (sin x, -cos x) = -R(sin x, cos x), substitute sin x = t (c) If R (-sin x, - cos x) = R(sin x, cos x), substitute tan x = t Various other cases of substitution in case of trigonometric functions have been discussed in detail in the later sections. Example: Evaluate ∫ √sin x cos x dx. Solution: Put sin x = t Then cos dx = dt Substituting these values in the given question we have, ∫ t1/2 = 2/3 t3/2 + c = 2/3 (sin x)3/2 + c Some Basic Properties of Indefinite Integral ∫ k f(x) dx = k ∫ f(x) dx, where k is any number. Hence, a constant can be taken out of the integral sign. ∫ - f(x) dx = - ∫ f(x), this actually follows from the above property only for k = -1. ∫ (f(x) ± g(x)) dx = ∫ f(x) dx ± ∫ g(x)) dx, this property can be extended to any number of functions. The same cannot be said for multiplication and division of functions i.e. .∫ f(x) / g(x) dx ≠ ∫ f(x) dx / ∫ g(x) dx and similarly ∫ f(x) . g(x) dx ≠ ∫ f(x) dx . ∫ g(x)) dx ? Q1. We need to integrate an inverse and a logarithmic function. Then the order of functions should be: (a) inverse function as the first function and logarithmic as the second. (b) the functions can be chosen in any order. (c) logarithmic function should be chosen as the first and inverse as the second. (d) can’t say Q2. Indefinite integration is also termed as (a) irregular integration (b) anti differentiation (c) none of these (d) reverse integration Q3. While using integration by parts, the sequence of the functions is decided by the rule (a) TIALE (b) IATE (c) ILETA (d) ILATE Q4. f(x)/g(x) is said to be a proper rational function if (a) deg f(x) > deg g(x) (b) deg f(x) = deg g(x) (c) deg f(x) < deg g(x) (d) deg f(x) ≠ deg g(x) Q5. Two integrals of the same function (a) are exactly the same (b) are entirely different (c) differ only by a constant (d) may or may not be same Q5 Q5 Q5 Q5 Q5 (a) (b) (d) (c) (c) Related Resources: For getting an idea of the type of questions asked, refer the previous year papers.? You may wish to refer definite integral.	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9747283458709717, "perplexity": 876.8329837121549}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-40/segments/1443738087479.95/warc/CC-MAIN-20151001222127-00103-ip-10-137-6-227.ec2.internal.warc.gz"}
https://euclid.math.temple.edu/events/seminars/analysis/	# Analysis Seminar Current contact: Gerardo Mendoza The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract. • Monday January 23, 2017 at 14:40, Wachman 617 Scott Armstrong, NYU I will describe recent developments in the quantitative homogenization of linear elliptic equations in divergence form, emphasizing some ideas arising from the calculus of variations and the role played by a new elliptic regularity theory for equations with random coefficients. • Monday January 30, 2017 at 14:40, Wachman 617 Irina Mitrea In this talk I will discuss well-posdness results for the Dirichlet problem for second-order, homogeneous, elliptic systems, with constant complex coefficients, in the upper half space, with boundary data from Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. A key tool in this analysis is establishing boundedness of the Hardy-Littlewood maximal operator on appropriate Köthe function spaces. This is joint work with Dorina Mitrea, Marius Mitrea and Jose Maria Martell. • Monday February 20, 2017 at 14:40, Wachman 617 Guy David, Courant Institute, New York University Since the work of Cheeger, many non-smooth metric measure spaces are now known to support a differentiable structure for Lipschitz functions. The talk will discuss this structure on metric measure spaces with quantitative topological control: specifically, spaces whose blowups are topological planes. We show that any differentiable structure on such a space is at most 2-dimensional, and furthermore that if it is 2-dimensional the space is 2-rectifiable. This is partial progress on a question of Kleiner and Schioppa, and is joint work with Bruce Kleiner. • Monday February 27, 2017 at 14:40, Wachman 617 TBA Federico Tournier, University of La Plata and IAM, Argentina • Monday March 20, 2017 at 14:40, Wachman 617 TBA Charles Epstein, University of Pennsylvania • Monday April 17, 2017 at 14:40, Wachman 617 TBA Yannick Sire, John Hopkins • Monday May 1, 2017 at 14:40, Wachman 617 TBA Tadele Mengesha, University of Tennessee, Knoxville	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8451824188232422, "perplexity": 2535.438799507981}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2017-09/segments/1487501171932.64/warc/CC-MAIN-20170219104611-00278-ip-10-171-10-108.ec2.internal.warc.gz"}
http://solvemathproblems.org/pre-algebra-help/	## Pre-Algebra Calculator Study for Tests/Quizzes The more you practice, the better prepared you’ll be. The calculator will give you extra problems to try. Look for the topic you wish to review in the Examples section of the calculator. A problem will show in the box. (If it’s hard to understand, that’s because it’s written in calculator notation. Click the Show button to view the problem in the standard mathematical format.) Select Topic tells the calculator what to do with the problem; just make sure this matches the directions for the type of problem you want to practice. Solve the problem and check your answer using the calculator. If you answer correctly, you’ll know you’re on the right track. If not, you’ll need to go back over your work to find your mistakes. If you can’t find it, ask a parent or friend for help or sign up for Mathway. Mathway will show you how to work the problem step-by-step so that you’ll be able to find your mistake and see how to solve the problem correctly. How to Use the Calculator 1. You can enter the problem by using an example or by using the symbols. 2. Click the Show button next to Math Format. This will show your problem in the normal mathematical format that you’re used to. If it looks wrong and you need more help, click the ? Box next to the Enter Problem field. 3. The Select Topic dropdown will fill in with the most common type of problem, but if you want the calculator to do something different, just choose the correct option from the dropdown. 5. If you made a mistake, you need to figure out what you did wrong. To see the steps, sign up for Mathway. Using an example: Scroll through the topics to find the type of problems you want to check or practice. This will provide an example in the calculator so that you can see how it is formatted. You can then change the numbers or variables to fit the problem you are trying to check. Using the Symbols Parenthesis – They indicate multiplication or that the operation inside should be done first. Brackets – Use brackets if you need a parenthesis within parenthesis – The brackets go on the outside as seen in this example: [3 + 2(10 -1)] ÷ 7. Absolute Value – The absolute value tells how far away a number is from zero. It’s always the same number but positive. For example,\|3\| is 3 and \|-3\| is also 3. Fractions – Type the numerator and denominator inside the parenthesis that will come up. To create a mixed number, delete the parenthesis and put a space between the whole number and the numerator of the fraction. For example, for 2¼ type 2 1/4. Exponents – Type the base before the ^ symbol and the exponent in parenthesis. For example, 5^(2) for 52. Remember that the exponent tells how many times the base is multiplied by itself. Square Roots – Type the radicand (the number inside the square root symbol) inside the parenthesis. Square roots find what number times itself equals the radicand. For example, the square root of 49 is 7 because 7 * 7 = 49. Other Roots – Type the index after the √ symbol and the radicand inside the parenthesis. For example, use √3:(8) for $\sqrt[3]{8}$ . Remember that a different index means that the answer must be multiplied by itself that many times to equal the radicand. In our $\sqrt[3]{8}$ example, 2 * 2 * 2 = 8, so 2 would be the answer because 2 times itself 3 (the index) times is 8. Note: If no index is given, it is assumed to be two and is just called a square root. Coordinates – Type a coordinate as you normally would – such as (1,5). Point/Slope – Use this button if you know one coordinate and the slope (m) of a line. You can then either find the equation of the line or see the graph of the line. Type the coordinate inside the parenthesis and the slope after the m= . For example, a line with slope of ½ and the coordinate (3,5) would look like this: (3,5),m=1/2. Scientific Notation – Inside the brackets, type the number then the exponent. For example, 2.6 x 10 $^8$ would look like this: sci[2.6,8]. Remember that negative exponents are used for numbers less than zero. Greater Than or Equal To – If you need to use just the greater than sign ( > ), simply type it using your keyboard. (Hit shift then the period.) Less Than or Equal To – If you need to use just the less than sign ( < ), simply type it using your keyboard. (Hit shift then the comma.) Rectangle – In the brackets, type the length and width of the rectangle. The calculator can then give you the perimeter or area. Remember that perimeter is the distance around the outside of the figure (like a fence around a yard) and that area is the amount of surface the figure covers (like carpeting in a room.) Circle – Type the radius in the brackets. Radius is the distance from the center to the outside. If you’re given the diameter (the distance all the way across the circle), divide it by two to find the radius. The calculator can then produce the area or circumference of the circle. Circumference means the distance around the outside of the circle. The formula for the area of a circle is $A = \pi r$ ² and the circumference is $C = 2\pi r$ . Triangle – Type the base and the height in the brackets. Remember that the height of a triangle makes a right angle with the base – it is not one of the sides unless you have a right triangle. The area of a triangle is found by using the formula A = ½bh. Parallelogram – A parallelogram has two pairs of parallel sides. Type the base and the height in the brackets. Remember that the height of the parallelogram makes a right angle with the base. The formula for the area of a parallelogram is A = bh. Trapezoid – A trapezoid has only one set of parallel sides. In the brackets, type one of the bases then the height then the other base. *Notice that the height must go in the middle. The bases are the two parallel sides of the trapezoid and the height makes a right angle with both bases. The formula for the area of a trapezoid is $A = \frac{1}{2}(b_{1}+b_{2})h$ . Rectangular Prisms – Rectangular prism is the formal name for a box. Type the length, width, and height in the brackets. You can then find the volume or the surface area. Volume is the amount of space inside a figure (like the amount of water that will fit into a pool). Surface area is the area of all of the outside surfaces (like how much wrapping paper it would take to wrap the box). Cylinder – Type the height first then the radius. Remember that the radius is the distance from the center of the circle to the outside of the circle. The formula for the volume of a cylinder is $V = \pi r^{2}h$ . Cone – Type the height first then the radius. Remember that the radius is the distance from the center of the circle to the outside of the circle. The formula for the volume of a cone is $V = \frac{1}{3} \pi r^2$ . Rectangular pyramid – Type the height first then the length and width of the base. The formula for the volume of a rectangular pyramid is $V = \frac{1}{3} lwh$ . Sphere – Type the radius of the sphere inside the brackets. Remember that the radius is the distance from the center of the sphere to the outside. If you are given the diameter (the distance all the way across), you must divide it by 2 to find the radius. The formula for the volume of a sphere is $V = \frac{4}{3} \pi r^2$ . The surface area can be found using the formula $V = \frac{4}{3} \pi r^2$ . Division sign – For multiplication, use the asterisk button on your keyboard. (Hit shift then 8). Pi – Pi is a unique number that describes the relationship between the circumference and the diameter of every circle. Pi is approximately equal to 3.14. Math Study Tips • Pay attention in class. Listening and asking questions in class is really important. If you miss the instruction your teacher gives, you’ll have to figure it out on your own later. Not only will this take a lot more time, but it’ll also be more difficult to learn by yourself. So avoid the temptation to write notes to your friends or doodle during class and listen up! You’ll be glad you did. • • Take your homework seriously. Math homework is designed to give you a chance to practice what you learned in class. Your tests and quizzes will contain problems very similar to the ones on your homework, so if you know how to do your homework, you’ll know how to work them on the test. However, if you don’t know how to do the homework, you’re going to be in trouble. So do your very best. If you run into problems you can’t figure out, ask someone for help or write down a question to ask the teacher when you go over the homework in class the next day. • • Learn from your mistakes. You are going to make mistakes in math – it’s part of the process. What’s important is that you learn from your mistakes so that you don’t keep making them over and over. Whenever you get a problem wrong, you need to go back and figure out where you messed up. If you can’t find your error, ask a friend, parent, or teacher for help. Or, if your parents sign you up for Mathway, you’ll be able to see the steps for each problem and can find your error from there.	{"extraction_info": {"found_math": true, "script_math_tex": 11, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 11, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8192141652107239, "perplexity": 289.705618399376}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2015-18/segments/1429246647589.15/warc/CC-MAIN-20150417045727-00139-ip-10-235-10-82.ec2.internal.warc.gz"}
http://fashionphotographycourse.com/how-to-amx/page.php?7fd468=fundamental-theorem-of-calculus-derivative-of-integral	- The integral has a variable as an upper limit rather than a constant. continuous over that interval, because this is continuous for all x's, and so we meet this first Question 6: Are anti-derivatives and integrals the same? Our mission is to provide a free, world-class education to anyone, anywhere. Second fundamental, I'll Compute the derivative of the integral of f(x) from x=0 to x=3: As expected, the definite integral with constant limits produces a number as an answer, and so the derivative of the integral is zero. pretty straight forward. is just going to be equal to our inner function f theorem of calculus tells us that if our lowercase f, if lowercase f is continuous In this section we present the fundamental theorem of calculus. The second part states that the indefinite integral of a function can be used to calculate any definite integral, \int_a^b f(x)\,dx = F(b) - … our original question, what is g prime of 27 going to be equal to? seems to cause students great difficulty. The second fundamental both sides of this equation. evaluated at x instead of t is going to become lowercase f of x. Practice: Finding derivative with fundamental theorem of calculus This is the currently selected item. This description in words is certainly true without any further interpretation for indefinite integrals: if F(x) is an antiderivative of f(x), then: Example 1: Let f(x) = x3 + cos(x). The Fundamental Theorem of Calculus The single most important tool used to evaluate integrals is called “The Fundamental Theo-rem of Calculus”. To log in and use all the features of Khan Academy, please enable JavaScript in your browser. it's actually very, very useful and even in the future, and to three, and we're done. The great beauty of the conclusion of the fundamental theorem of calculus is that it is true even if we can't (easily, or at all) compute the integral in terms of functions we know! So the left-hand side, First, we must make a definition. In general, you can skip the multiplication sign, so 5x is equivalent to 5x. How Part 1 of the Fundamental Theorem of Calculus defines the integral. Fundamental theorem of calculus. integral like this, and you'll learn it in the future. fundamental theorem of calculus. Let f(x, t) be a function such that both f(x, t) and its partial derivative f x (x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ t ≤ b(x), x 0 ≤ x ≤ x 1.Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x 0 ≤ x ≤ x 1. Another way of stating the conclusion of the fundamental theorem of calculus is: The conclusion of the fundamental theorem of calculus can be loosely expressed in words as: "the derivative of an integral of a function is that original function", or "differentiation undoes the result of integration". To understand the power of this theorem, imagine also that you are not allowed to look out of the window of the car, so that you have no direct evidence of how far the car has tra… abbreviate a little bit, theorem of calculus. I'll write it right over here. The Fundamental Theorem of Calculus Three Different Quantities The Whole as Sum of Partial Changes The Indefinite Integral as Antiderivative The FTC and the Chain Rule The Indefinite Integral and the Net Change Indefinite Integrals and Anti-derivatives A Table of Common Anti-derivatives The Net Change Theorem The NCT and Public Policy Substitution Note the important fact about function notation: f(x) is the same exact formula as f(t), except that x has replaced t everywhere. It bridges the concept of an antiderivative with the area problem. If you're behind a web filter, please make sure that the domains .kastatic.org and .kasandbox.org are unblocked. respect to x of g of x, that's just going to be g prime of x, but what is the right-hand And so we can go back to The Area under a Curve and between Two Curves The area under the graph of the function f (x) between the vertical lines x = a, x = b (Figure 2) is given by the formula S … try to think about it, and I'll give you a little bit of a hint. The fact that this theorem is called fundamental means that it has great significance. A function F(x) is called an antiderivative of a function f (x) if f (x) is the derivative of F(x); that is, if F'(x) = f (x).The antiderivative of a function f (x) is not unique, since adding a constant to a function does not change the value of its derivative: This calculus video tutorial explains the concept of the fundamental theorem of calculus part 1 and part 2. we have the function g of x, and it is equal to the In the 1 -dimensional case this is the fundamental theorem of calculus for n = 1 and we can take higher derivatives after applying the fundamental theorem. Let’s now use the second anti-derivative to evaluate this definite integral. In Example 4 we went to the trouble (which was not difficult in this case) of computing the integral and then the derivative, but we didn't need to. might be some cryptic thing "that you might not use too often." One way to write the Fundamental Theorem of Calculus (7.2.1) is: ∫ a b f ′ (x) d x = f (b) − f (a). Fundamental Theorem: Let ∫x a f (t)dt ∫ a x f (t) d t be a definite integral with lower and upper limit. definite integral like this, and so this just tells us, The derivative with $1 per month helps!! definite integral from 19 to x of the cube root of t dt. Proof of the First Fundamental Theorem of Calculus The ﬁrst fundamental theorem says that the integral of the derivative is the function; or, more precisely, that it’s the diﬀerence between two outputs of that function. (The function defined by integrating sin(t)/t from t=0 to t=x is called Si(x); approximate values of Si(x) must be determined by numerical methods that estimate values of this integral. The result of Preview Activity 5.2 is not particular to the function $$f (t) = 4 − 2t$$, nor to the choice of “1” as the lower bound in the integral that defines the function $$A$$. Introduction. Khan Academy is a 501(c)(3) nonprofit organization. a Fundamental Theorem of Calculus tells us how to find the derivative of the integral from to of a certain functio AP® is a registered trademark of the College Board, which has not reviewed this resource. Question 5: State the fundamental theorem of calculus part 2? The first thing to notice about the fundamental theorem of calculus is that the variable of differentiation appears as the upper limit of integration in the integral: Think about it for a moment. Donate or volunteer today! So the derivative is again zero. Imagine for example using a stopwatch to mark-off tiny increments of time as a car travels down a highway. The conclusion of the fundamental theorem of calculus can be loosely expressed in words as: "the derivative of an integral of a function is that original function", or "differentiation undoes the result of … Now, I know when you first saw this, you thought that, "Hey, this Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. Fundamental Theorem of Calculus Part 1: Integrals and Antiderivatives As mentioned earlier, the Fundamental Theorem of Calculus is an extremely powerful theorem that establishes the relationship between differentiation and integration, and gives us a way to evaluate definite integrals without using Riemann sums or calculating areas. hey, look, the derivative with respect to x of all of this business, first we have to check$ \displaystyle y = \int^{3x + 2}_1 \frac{t}{1 + t^3} \,dt \$ Integrals Show Instructions. out what g prime of x is, and then evaluate that at 27, and the best way that I some of you might already know, there's multiple ways to try to think about a definite If an antiderivative is needed in such a case, it can be defined by an integral. Theorem: (First Fundamental Theorem of Calculus) If f is continuous and b F = f, then f(x) dx = F (b) − F (a). But this can be extremely simplifying, especially if you have a hairy Think about the second then the derivative of F(x) is F'(x) = f(x) for every x in the interval I. What is that equal to? The theorem already told us to expect f(x) = 3x2 as the answer. Conic Sections (Reminder: this is one example, which is not enough to prove the general statement that the derivative of an indefinite integral is the original function - it just shows that the statement works for this one example.). ), When the lower limit of the integral is the variable of differentiation, When one limit or the other is a function of the variable of differentiation, When both limits involve the variable of differentiation. The fundamental theorem of calculus explains how to find definite integrals of functions that have indefinite integrals. The fundamental theorem of calculus (FTC) establishes the connection between derivatives and integrals, two of the main concepts in calculus. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. :) https://www.patreon.com/patrickjmt !! First, actually compute the definite integral and take its derivative. It also tells us the answer to the problem at the top of the page, without even trying to compute the nasty integral. is, what is g prime of 27? We work it both ways. Some of the confusion seems to come from the notation used in the statement of the theorem. The result is completely different if we switch t and x in the integral (but still differentiate the result of the integral with respect to x). So we wanna figure out what g prime, we could try to figure Using the Fundamental Theorem of Calculus to evaluate this integral with the first anti-derivatives gives, ∫2 0x2 + 1dx = (1 3x3 + x)\|2 0 = 1 3(2)3 + 2 − (1 3(0)3 + 0) = 14 3 Much easier than using the definition wasn’t it? Finding derivative with fundamental theorem of calculus: chain rule And what I'm curious about finding or trying to figure out Furthermore, it states that if F is defined by the integral (anti-derivative). The fundamental theorem of calculus and accumulation functions, Functions defined by definite integrals (accumulation functions), Practice: Functions defined by definite integrals (accumulation functions), Finding derivative with fundamental theorem of calculus, Practice: Finding derivative with fundamental theorem of calculus, Finding derivative with fundamental theorem of calculus: chain rule, Practice: Finding derivative with fundamental theorem of calculus: chain rule, Interpreting the behavior of accumulation functions involving area. To be concrete, say V x is the cube [ 0, x] k. There are several key things to notice in this integral. derivative with respect to x of all of this business. That is, to compute the integral of a derivative f ′ we need only compute the values of f at the endpoints. Well, that's where the First, it states that the indefinite integral of a function can be reversed by differentiation, \int_a^b f(t)\, dt = F(b)-F(a). Using the fundamental theorem of calculus to find the derivative (with respect to x) of an integral like. This makes sense because if we are taking the derivative of the integrand with respect to x, … (3 votes) See 1 more reply Compute the derivative of the integral of f(x) from x=0 to x=t: Even though the upper limit is the variable t, as far as the differentiation with respect to x is concerned, t behaves as a constant. General form: Differentiation under the integral sign Theorem. function replacing t with x. work on this together. It also gives us an efficient way to evaluate definite integrals. Here are two examples of derivatives of such integrals. This theorem of calculus is considered fundamental because it shows that definite integration and differentiation are essentially inverses of each other. ∫ V x F (x 1,..., x k) d V where V x is some k -dimensional volume dependent on x. Unless the variable x appears in either (or both) of the limits of integration, the result of the definite integral will not involve x, and so the derivative of that definite integral will be zero. We'll try to clear up the confusion. condition or our major condition, and so then we can just say, all right, then the derivative of all of this is just going to be this inner About; One of the first things to notice about the fundamental theorem of calculus is that the variable of differentiation appears as the upper limit of integration in the integral. The Second Fundamental Theorem of Calculus. You da real mvps! Here's the fundamental theorem of calculus: Theorem If f is a function that is continuous on an open interval I, if a is any point in the interval I, and if the function F is defined by. The calculator will evaluate the definite (i.e. we'll take the derivative with respect to x of g of x, and the right-hand side, the It converts any table of derivatives into a table of integrals and vice versa. Within the theorem the second fundamental theorem of calculus, depicts the connection between the derivative and the integral— the two main concepts in calculus. the integral is called an indefinite integral, which represents a class of functions (the antiderivative) whose derivative is the integrand. that our inner function, which would be analogous So we're going to get the cube root, instead of the cube root of t, you're gonna get the cube root of x. Well, it's going to be equal Answer: As per the fundamental theorem of calculus part 2 states that it holds for ∫a continuous function on an open interval Ι and a any point in I. The fundamental theorem of calculus states that the integral of a function f over the interval [a, b] can be calculated by finding an antiderivative F of f: ∫ = − (). side going to be equal to? Well, no matter what x is, this is going to be Calculus tells us that the derivative of the definite integral from to of ƒ () is ƒ (), provided that ƒ is conti The Second Fundamental Theorem of Calculus shows that integration can be reversed by differentiation. on the interval from a to x, so I'll write it this way, on the closed interval from a to x, then the derivative of our capital f of x, so capital F prime of x F(x) = integral from x to pi squareroot(1+sec(3t)) dt By the fundamental theorem of calculus, the derivative of Si(x) is sin(x)/x. The fundamental theorem of calculus relates the evaluation of definite integrals to indefinite integrals. second fundamental theorem of calculus is useful. Compute the derivative of the integral of f(t) from t=0 to t=x: This example is in the form of the conclusion of the fundamental theorem of calculus. The fundamental theorem of calculus has two separate parts. Thus, the two parts of the fundamental theorem of calculus say that differentiation and integration are inverse processes. Pause this video and So let's take the derivative - The variable is an upper limit (not a lower limit) and the lower limit is still a constant. Imagine also looking at the car's speedometer as it travels, so that at every moment you know the velocity of the car. with bounds) integral, including improper, with steps shown. Example 3: Let f(x) = 3x2. (Sometimes this theorem is called the second fundamental theorem of calculus.). All right, now let's The theorem says that provided the problem matches the correct form exactly, we can just write down the answer. to our lowercase f here, is this continuous on the Example 5: Compute the derivative (with respect to x) of the integral: To make sure you understand the derivative of a definite integral, figure out the answer to the following problem before you roll over the expression to see the answer: Notes: (a) the answer is valid for any x > 0; the function sin(t)/t is not differentiable (or even continuous) at t = 0, since it is not even defined at t = 0; (b) this problem cannot be solved by first finding an antiderivative involving familiar functions, since there isn't such an antiderivative. The value of the definite integral is found using an antiderivative of … Stokes' theorem is a vast generalization of this theorem in the following sense. Lesson 16.3: The Fundamental Theorem of Calculus : ... Notice the difference between the derivative of the integral, , and the value of the integral The chain rule is used to determine the derivative of the definite integral. This math video tutorial provides a basic introduction into the fundamental theorem of calculus part 1. Now the fundamental theorem of calculus is about definite integrals, and for a definite integral we need to be careful to understand exactly what the theorem says and how it is used. It tells us, let's say we have definite integral from a, sum constant a to x of Something similar is true for line integrals of a certain form. interval from 19 to x? of both sides of that equation. The fundamental theorem of calculus (FTC) is the formula that relates the derivative to the integral and provides us with a method for evaluating definite integrals. Example 4: Let f(t) = 3t2. If you're seeing this message, it means we're having trouble loading external resources on our website. can think about doing that is by taking the derivative of The Fundamental Theorem of Calculus. - [Instructor] Let's say that Well, we're gonna see that Second, notice that the answer is exactly what the theorem says it should be! Example 2: Let f(x) = ex -2. to the cube root of 27, which is of course equal Now, the left-hand side is some function capital F of x, and it's equal to the Thanks to all of you who support me on Patreon. lowercase f of t dt. Derivatives Derivative Applications Limits Integrals Integral Applications Riemann Sum Series ODE Multivariable Calculus Laplace Transform Taylor/Maclaurin Series Fourier Series Functions Line Equations Functions Arithmetic & Comp. Suppose that f(x) is continuous on an interval [a, b]. The (indefinite) integral of f(x) is, so we see that the derivative of the (indefinite) integral of this function f(x) is f(x). 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Variable is an upper limit rather than a constant example 2: Let f ( ). Us an efficient way to evaluate this definite integral table of integrals and vice versa, what g... Furthermore, it can be reversed by differentiation that it has great significance, the left-hand side is straight... Also tells us the answer it travels, so that at every moment know... To evaluate this definite integral State the fundamental theorem of calculus ( FTC ) the. The car trademark of the fundamental theorem of calculus relates the evaluation definite. 4: Let f ( x ) = 3x2 as the answer ′. Where the second fundamental, I'll abbreviate a little bit, theorem of calculus is useful this theorem the. Case, it means we 're having trouble loading external resources on our website all features... The function to expect f ( x ) = ex -2 resources on our website by. The lower limit ) and the lower limit ) and the lower limit is still constant! Bounds ) integral, including improper, with steps shown it has great significance for. That this theorem in the following sense world-class education to anyone,.... Calculus Part 1 of the function it bridges the concept of the fundamental theorem of calculus, derivative! You 're behind a web filter, please enable JavaScript in your browser -2! On Patreon the car give you a little bit of a certain form figure... Integration and differentiation are essentially inverses of each other to notice in this.! Having trouble loading external resources on our website integrals and vice versa, including improper, with steps shown area. = 3x2 an upper limit rather than a constant using a stopwatch to mark-off tiny increments of time a! Can be reversed by differentiation message, it can be reversed by differentiation continuous on an [! About finding or trying to figure out is, to compute the integral! To find the derivative of Si ( x ) /x defined by the integral has a variable as upper... Is pretty straight forward the velocity of the car 's speedometer as it travels, so that at moment... A constant How Part 1 of the fundamental theorem of calculus to find the derivative of the car to! A free, world-class education to anyone, anywhere examples of derivatives of such integrals Let ’ s use... = 3x2 as the answer can just write down the answer integral, including improper, with shown. There are several key things to notice in this integral the multiplication sign, that. Integrals of a hint well, that 's where the second fundamental theorem of calculus to find the derivative both! Correct form exactly, we can go back to our original question, what is g of! Second, notice that the domains .kastatic.org and .kasandbox.org are unblocked fundamental because it shows that can... Without even trying to compute the definite integral and take its derivative actually compute the integral sign theorem a! A 501 ( c ) ( 3 ) nonprofit organization message, it can be reversed differentiation! All right, now let's work on this together speedometer as it travels, so that at every moment know... Trouble loading external resources on our website calculus relates the evaluation of definite integrals to indefinite integrals 501 c... Web filter, please make sure that the domains .kastatic.org and .kasandbox.org are.! Down a highway limit rather than a constant our website a stopwatch to tiny! Derivative f ′ we need only compute the integral sign theorem the car to mark-off tiny increments of time a! You 're behind a web filter, please enable JavaScript in your browser can go back to our original,... Lower limit ) and the lower limit is still a constant fundamental theorem of calculus derivative of integral is an upper (! Indefinite integrals be equal to ′ we need only compute the definite integral and take its derivative of. 501 ( c ) ( 3 ) nonprofit organization the features of Khan Academy is a vast generalization this. The endpoints fundamental theorem of calculus derivative of integral give you a little bit, theorem of calculus Part 1 and Part 2 first actually! Not a lower limit ) and the lower limit is still a constant the domains .kastatic.org and.kasandbox.org... A little bit of a certain form, I'll abbreviate a little bit theorem... Mission is to provide a free, world-class education to anyone, anywhere is! Is defined by the fundamental theorem of calculus Part 2 relates the evaluation of integrals! Still a constant integration and differentiation are essentially inverses of each other a trademark! A free, world-class education to anyone, anywhere, theorem of calculus relates the evaluation of integrals... Integration and differentiation are essentially inverses of each other tiny increments of time as a car travels down a.! Of Khan Academy, please make sure that the answer the main concepts in calculus... Reversed by differentiation the values of f at the endpoints it should be 'm curious about finding trying... Nasty integral examples of derivatives into a table of derivatives of such integrals limit rather than a constant Part. At the endpoints can go back to our original question, what is g prime of 27 going be! That it has great significance ) See 1 more reply How Part 1 of the concepts... Of you who support me on Patreon the lower limit is still a.! Tutorial explains the concept of an antiderivative with the area problem Let 's the. Behind a web filter, please enable JavaScript in your browser page, without even trying to compute the integral. X 's speedometer as it travels, so 5x is equivalent to . ) and the lower limit ) and the lower limit ) and the limit... As the answer is exactly what the theorem already told us to expect f ( ). Let f ( x ) is continuous on an interval [ a, b ] efficient. Shows that integration can be defined by an integral to come from the used! Calculus has two separate parts you can skip the multiplication sign, so 5x is to!, it can be reversed by differentiation this message, it means we 're having trouble loading resources. Every moment you know the velocity of the theorem a variable as an upper limit rather than constant... The velocity of the fundamental theorem of calculus to find the derivative of both of! Each other = 3t2 is continuous on an interval [ a, ]. Tells us the answer enable JavaScript in your browser theorem says that provided the problem at top... Already told us to expect f ( x ) = 3x2 as the answer it also tells us the to. Both sides of that equation the lower limit is still a constant bridges the concept of the fundamental theorem calculus! Me on Patreon this integral as an upper limit ( not a lower limit ) and the lower limit still... Line integrals of a derivative f ′ we need only compute the values of at. Si ( x ) = 3x2 ) nonprofit organization the domains .kastatic.org and.kasandbox.org. Of Khan Academy is a vast generalization of this theorem is called fundamental means that it great! It travels, so that at every moment you know the velocity of the Board. Explains the concept of the theorem if f is defined by the fundamental of! 'M curious about finding or trying to figure out is, what g! Where the second fundamental, I'll abbreviate a little bit of a hint exactly... A car travels down a highway interval [ a, b ] Sometimes this theorem in the statement of fundamental... It shows that integration can be defined by an integral equal to second, notice that the to. Provided the problem matches the correct form exactly, we can go back to our question. Javascript in your browser velocity of the fundamental theorem of calculus is fundamental... That is, what is g prime of 27 a free, world-class education to anyone,.... Defines the integral of a hint Let f ( x ) is continuous on an interval [ a b. The definite integral 're behind a web filter, please enable JavaScript in your.! Main concepts in calculus. ) into a table of integrals and vice versa derivatives and integrals the same sin... The page, without even trying to figure out is, to compute the values of f at top... Let ’ s now use the second fundamental theorem of calculus to find the derivative fundamental theorem of calculus derivative of integral! The fact that this theorem in the following sense x ` defines the integral ( anti-derivative ) the form! Evaluation of definite integrals I 'm curious about finding or trying to compute the nasty integral tells... What Is The Largest Private College In North Carolina, Properties Of Real Numbers Quiz Pdf, How To Install 12x12 Wall Tile, Arcmap Layout View, Sephora Pro Powder Brush, Wireless Remote Switch Box, Starbucks Green Tea Frappuccino Price, Coloration Crossword Clue, Alice And Olivia Shirts, Sedum Plug Plants, Iams Proactive Health Smart Puppy Large Breed,	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.985245943069458, "perplexity": 349.29920807188165}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-10/segments/1614178357641.32/warc/CC-MAIN-20210226115116-20210226145116-00101.warc.gz"}
http://math.stackexchange.com/questions/296147/the-relation-between-arbitrary-measure-space-and-the-lebesgue-integral/296159	# The relation between arbitrary measure space and the Lebesgue integral Let $(X, \mathcal F, \mu)$ be a measure space and $f\in M^+(X,\mu)$ (the measurable non-negative functions), and $t>0$. Now let $$S_f(t)=\{x\in X:f(x)>t\} \quad \Psi_f(t)=\mu(S_f(t))$$ Prove that $$\int_Xfd\mu=\int_0^\infty \Psi_f(t)dt$$ I guess the non-trivial part is to prove it for simple functions. The rest follows from B. Levy's theorem. Namely if $\{f_n\}_{n=0}^\infty$ a monotone increasing sequence of measurable functions, then it has a limit $$f:f\in M^+ \land \lim_{n\to \infty}\int_Xf_nd\mu=\int_Xfd\mu$$ - – Martin Feb 6 '13 at 10:13 Using Tonelli's/Fubini's theorem, we have \begin{align} \int_0^\infty \Psi_f(t)\,\mathrm dt&=\int_0^\infty\int_X 1_{S_f}(t)\,\mathrm \mu(\mathrm dx)\,\mathrm dt=\int_X\,\int_0^\infty 1_{S_f}(t)\,\mathrm dt\,\mu(\mathrm dx)\\ &=\int_X\int_0^\infty 1_{[0,f(x)]}\,\mathrm dt\,\mathrm \mu(\mathrm dx)=\int_X f(x)\,\mu(\mathrm dx), \end{align} which is justified by the assumption that $f$ is non-negative. However, you need your measure space to be $\sigma$-finite in order for Tonelli/Fubini to work. Yes, thank you. $1_Y$ must be the characteristic function of Y, but what do you mean by $\mu(dx)$? This is the first time I see this notation. It must be what I mean by $d\mu$, must it not? – Student Feb 6 '13 at 13:36 Yes, $\mathrm d\mu$ and $\mu(\mathrm d x)$ is the same. I just prefer writing $\mu(\mathrm dx)$ whenever I write $f(x)$. For example $$\int f(x)\,\mu(\mathrm dx)\quad \text{vs.}\quad\int f(x)\,\mathrm d\mu$$ – Stefan Hansen Feb 6 '13 at 16:03	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 1, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9973738789558411, "perplexity": 489.8013443380379}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-07/segments/1454701962902.70/warc/CC-MAIN-20160205195242-00090-ip-10-236-182-209.ec2.internal.warc.gz"}
https://brilliant.org/problems/problem-of-squares/	# Problem of Squares Geometry Level pending ABCD is a square in which AC is the diagonal. Any line segment XY cuts AB at X and BC at Y such that XYllAC. If length of Square is 8cm and length of BY is 2cm, then value of AX+BY ? ×	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8669505715370178, "perplexity": 2007.2775847381959}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-44/segments/1476988719465.22/warc/CC-MAIN-20161020183839-00463-ip-10-171-6-4.ec2.internal.warc.gz"}
http://mathonline.wikidot.com/applying-lebesgue-s-dominated-convergence-theorem-2	Applying Lebesgue's Dominated Convergence Theorem 2 # Applying Lebesgue's Dominated Convergence Theorem 2 Recall from the Lebesgue's Dominated Convergence Theorem page that if: • 1) $(f_n(x))_{n=1}^{\infty}$ is a sequence of Lebesgue integrable functions on $I$. • 2) $(f_n(x))_{n=1}^{\infty}$ converges to a limit function $f$ almost everywhere on $I$. • 3) There exists a Lebesgue integrable function $g$ on $I$ such that $\mid f_n(x) \mid \leq g(x)$ almost everywhere on $I$ and for all $n \in \mathbb{N}$ (dominating criterion). Then we can conclude that: • a) $f$ is Lebesgue integrable on $I$. • b) $\displaystyle{\int_I f(x) \: dx = \int_I \lim_{n \to \infty} f_n(x) \: dx = \lim_{n \to \infty} \int_I f_n(x) \: dx}$. We will now look at applying Lebesgue's dominated convergence theorem. ## Example 1 Use Lebesgue's dominated convergence theorem to show that the Lebesgue integral $\displaystyle{\int_I \frac{x \ln x}{ln(x) \ln (1 + x)} \: dx}$ exists, where $I = [0, 1]$. Notice that the function $f(x) = \ln (x) \ln (1 + x)$ is continuous on all of $I$ except for $x = 0$. Define a sequence of functions $(f_n(x))_{n=1}^{\infty}$ where $f_n$ is defined for all $n \in \mathbb{N}$ by: (1) \begin{align} \quad f_n(x) = \left\{\begin{matrix} 0 & 0 \leq x < \frac{1}{n}\\ \ln(x) \ln (1 + x) & \frac{1}{n} \leq x \leq 1 \end{matrix}\right. \end{align} Notice that $0$ is Lebesgue integrable on $\left [ 0, \frac{1}{n} \right )$ for all $n \in \mathbb{N}$ and $\ln(x) \ln (1 + x)$ is Lebesgue integrable on $\left [ \frac{1}{n}, 1 \right ]$ for all $n \in \mathbb{N}$. So, by the theorems presented on the Additivity of Lebesgue Integrals on Subintervals of General Intervals page we have that $f_n$ is Lebesgue integrable for all $n \in \mathbb{N}$. Moreover, it's not hard to see that $(f_n(x))_{n=1}^{\infty}$ converges to $f$ almost everywhere on $I$ (everywhere except at $x = 0$). Using L'Hospital's rule twice gives us that: (2) \begin{align} \quad \lim_{x \to 0} [\ln(x) \ln (1 + x)] = \lim_{x \to 0} \frac{\ln (x)}{\frac{1}{\ln(1 + x)}} = \lim_{x \to 0} \frac{\frac{1}{x}}{\frac{-\frac{1}{1 + x}}{\ln^2 (1 + x)}} = \lim_{x \to 0} \frac{-(1 + x)\ln^2 (1 + x)}{x} = \lim_{x \to 0} \left [-\ln^2 (1 + x) + -2 \ln(1 + x) \right ] = 0 \end{align} So $f(x) = \ln (x) \ln (x+1)$ is bounded on $[0, 1]$, i.e., there exists some $M \in \mathbb{R}$, $M > 0$ such that $\mid f(x) \mid \leq M$ for all $x \in [0, 1]$. Let $g(x) = M$. Then $g$ is Lebesgue integrable on $I$. From the way each $f_n$ is defined, we also see that for all $x \in [0, 1]$ and for all $n \in \mathbb{N}$ we have that: (3) \begin{align} \quad \mid f_n(x) \mid \leq M = g(x) \end{align} So by Lebesgue's dominated convergence theorem we can conclude that $\displaystyle{\int_I \ln (x) \ln(1 + x) \: dx}$ exists. A graph of $f$ is given below.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 3, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9998181462287903, "perplexity": 145.5978579535934}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-39/segments/1537267159165.63/warc/CC-MAIN-20180923075529-20180923095929-00014.warc.gz"}
https://ir.cwi.nl/pub/13868	A stable computational scheme for the conical function $P^{\mu}_{-1/2+i\tau}(x)$ for $x>-1$, real $\tau$, and $\mu\le 0$ or $\mu\in\mathbb{N}$ is presented. The scheme combines uniform asymptotic expansions for large $\|\mu\|$ with the application of the three-term recurrence relation on the $\mu$ index in the direction of decreasing $\|\mu\|$ when $x>0$. When $x<0$, the conditioning of recursion is the opposite, and conical functions can be computed in the direction of increasing $\|\mu\|$. Gil, A, Segura, J, & Temme, N.M. (2009). Computing the conical function $P^{\mu}_{-1/2+i\tau}(x)$. SIAM Journal on Scientific Computing, 31(3), 1716–1741.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9854015707969666, "perplexity": 268.56274293677365}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-04/segments/1547583831770.96/warc/CC-MAIN-20190122074945-20190122100945-00163.warc.gz"}
http://zims-en.kiwix.campusafrica.gos.orange.com/wikipedia_en_all_nopic/A/Co-NP	# co-NP In computational complexity theory, co-NP is a complexity class. A decision problem X is a member of co-NP if and only if its complement X is in the complexity class NP. In simple terms, co-NP is the class of problems for which there is a polynomial-time algorithm that can verify "no" instances (sometimes called counterexamples) given the appropriate certificate. Equivalently, co-NP is the set of decision problems where the "no" instances can be accepted in polynomial time by a non-deterministic Turing machine. Unsolved problem in computer science:${\displaystyle {\textsf {NP}}\ {\overset {?}{=}}\ {\textsf {co-NP}}}$ (more unsolved problems in computer science) An example of an NP-complete problem is the circuit satisfiability problem: given a Boolean circuit, is there a possible input for which the circuit outputs true? To give a proof of a "yes" instance, one must specify a valid input to the circuit such that the output will be true. The complementary problem asks: "given a Boolean circuit, do all possible inputs to the circuit output false?". Even if someone claims to know the answer to the complementary question for a given set, we do not know how to verify their solution efficiently. To give a proof of the "yes" instance, we may give all possible inputs to the circuit and saying it outputs true. However, verifying their claim is just as inefficient as checking all possible inputs in the first place. Because of this dilemma, we do not know if the circuit satisfiability problem is in co-NP. Note however that the complementary question itself is in co-NP, since the original question is in NP. ## Relationship to other classes P, the class of polynomial time solvable problems, is a subset of both NP and co-NP. P is thought to be a strict subset in both cases (and demonstrably cannot be strict in one case and not strict in the other). NP and co-NP are also thought to be unequal.[1] If so, then no NP-complete problem can be in co-NP and no co-NP-complete problem can be in NP.[2] This can be shown as follows. Suppose there exists an NP-complete problem X that is in co-NP. Since all problems in NP can be reduced to X, it follows that for every problem in NP we can construct a non-deterministic Turing machine that decides its complement in polynomial time, i.e., NP co-NP. From this it follows that the set of complements of the problems in NP is a subset of the set of complements of the problems in co-NP, i.e., co-NP NP. Thus co-NP = NP. The proof that no co-NP-complete problem can be in NP if NP co-NP is symmetrical. If a problem can be shown to be in both NP and co-NP, that is generally accepted as strong evidence that the problem is probably not NP-complete (since otherwise NP = co-NP). An example of a problem that is known to belong to both NP and co-NP (but not known to be in P) is integer factorization: given positive integers m and n determine if m has a factor less than n and greater than one. Membership in NP is clear; if m does have such a factor then the factor itself is a certificate. Membership in co-NP is also straightforward: one can just list the prime factors of m, all greater or equal to n, which the verifier can confirm to be valid by multiplication and the AKS primality test. It is presently not known whether there is a polynomial-time algorithm for factorization, equivalently that integer factorization is in P, and hence this example is interesting as one of the most natural problems known to be in NP and co-NP but not known to be in P. ## References 1. Hopcroft, John E. (2000). Introduction to Automata Theory, Languages, and Computation (2nd Edition). Boston: Addison-Wesley. ISBN 0-201-44124-1. Chap. 11. 2. Goldreich, Oded (2010). P, NP, and NP-completeness: The Basics of Computational Complexity. Cambridge University Press. p. 155. ISBN 9781139490092.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 1, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8961041569709778, "perplexity": 310.35247885561904}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-04/segments/1610703519395.23/warc/CC-MAIN-20210119135001-20210119165001-00349.warc.gz"}
http://www.bellcurves.com/gmat-practice-questions/functions/2013-DEC-02/question	An "alpha series" is defined by the recursive rule: An = (An - 1)k, where k is a constant. If the 1st term of a particular "alpha series" is 64 and the 25th term is 192, what is the 9th term? (A) (B) (C) (D) (E) • (646) 414-1586 CONNECT WITH US	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8448790311813354, "perplexity": 967.666201570637}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-42/segments/1413507454577.44/warc/CC-MAIN-20141017005734-00225-ip-10-16-133-185.ec2.internal.warc.gz"}
http://math.stackexchange.com/questions/143872/computing-the-trace-and-determinant-of-ab-given-eigenvalues-of-a-and-an-ex	# Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$ Let $A$ be $4\times 4$ matrix with real entries such that $-1$, $1$, $2$, and $-2$ are its eigenvalues. If $B = A^4 - 5A^2+5I$, where $I$ denotes $4\times 4$ identity matrix, then what would be determinant and trace of matrix $A+B$? - Please state all conditions in the title, lest it causes confusion, thanks. P.S. I did not downvote, and knew not why it is voted, but I might surmise... – awllower May 11 '12 at 15:41 Dear sir there is limited word alloted for title. Thanks for pointing my mistakes. I will take care of that in near future. Being a new member i am not much aware of rules and regulations. But soon i wll get awre of all the rules. – srijan May 11 '12 at 16:16 Since $A$ is $4\times 4$ and its eigenvalues are $2$, $-2$, $1$, and $-1$, the minimal and characteristic polynomials of $A$ agree and are both equal to $$(t-1)(t+1)(t-2)(t+2) = (t^2-1)(t^2-4) = t^4 - 5t^2 + 4.$$ In particular, by the Cayley-Hamilton Theorem, $$A^4 - 5A^2 + 4I = 0,$$ and therefore $$B+A = A^4 - 5A^2 + 5I + A = (A^4-5A^2+4I) + (A+I) = A+I.$$ Now notice that $\lambda$ is an eigenvalue of $A$ if and only if $\alpha\lambda+\beta$ is an eigenvalue of $\alpha A+\beta I$, to conclude that the eigenvalues of $B+A=A+I$ are $0$, $-1$, $2$, and $3$. Therefore, the trace is $0-1+2+3 = 4$, and the determinant is $0$ (since $A+I$ is not invertible, or since the determinant is the product of the eigenvalues). - Sincerely thanks to you for your help and wonderfull way of tackling this problem. – srijan May 11 '12 at 16:54 Pick one of the eigenvectors of $A$, call it $\bf v$, with corresponding eigenvalue $\lambda$. Can you work out $(A+B){\bf v}$? From that, can you work out the determinant and trace? - Why my question is being down voted? I beg pardon if i did something wrong to this community. – srijan May 11 '12 at 13:22 Thanks for being lenient. – srijan May 11 '12 at 13:25 I didn't get you sir. – srijan May 11 '12 at 13:28 OK, let's start at the beginning. What does it mean to say that 2 is an eigenvalue of $A$? – Gerry Myerson May 11 '12 at 13:36 That mean there will exist at least one non zero vector, call it $v$, with corresponding eigenvalue $\lambda$ such that $Av = \lambda v$ – srijan May 11 '12 at 13:42 Forgive me if I mistake what Gerry is trying to say, but I believe it is this: 1. If $v$ is an eigenvector of $A$, then $v$ is an eigenvector of $A+B$ (what is the associated eigenvalue, then?). Use the formula for $B$, and note that $A^n(v) = \lambda^nv$. 2. From the associated eigenvalues you get for $A+B$, you should be able to explicitly state the characteristic polynomial for $A+B$. 3. For a 4x4 matrix, the trace is the negative of the coefficient of the cubic term in the characteristic polynomial, and the determinant is the constant term. - Maybe the first point in your answer should make clear what the eigenvalues of A+B look like? And that directly solves the problem, right? – awllower May 11 '12 at 15:39 Start by finding the eigenvalues of $B$, using the eigenvalues of $A$. Find the eigenvalues of $A+B$ using this. Then recall that the determinant here will be the product of the eigenvalues of $A+B$ and the trace will be the negative of the sum of the eigenvalues of $A+B$. -	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.977655291557312, "perplexity": 171.6818487732061}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-18/segments/1461860121737.31/warc/CC-MAIN-20160428161521-00073-ip-10-239-7-51.ec2.internal.warc.gz"}
https://stats.stackexchange.com/questions/374361/can-we-apply-kl-divergence-to-the-probability-distributions-on-different-domains	# Can we apply KL divergence to the probability distributions on different domains? When I was reading the original paper of t-SNE, I had an question whether or not we can apply KL divergence to the discrete probability distributions on different domains. In the paper, they measure the dissimilarity between two discrete (conditional) distributions on high dimensional domains and low dimensional domains by KL divergence. However, according to the wikipedia Kullback–Leibler divergence entry, KL divergence for discrete probability distribution is defined on the same probability space. This implies the same sample space must be used for the probability distribution. Can we apply KL divergence to the probability distributions on different domains?	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9629247784614563, "perplexity": 272.5691583186011}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2022-33/segments/1659882573163.7/warc/CC-MAIN-20220818033705-20220818063705-00258.warc.gz"}
http://math.stackexchange.com/questions/142987/prove-the-map-is-null-homotopy	# Prove the map is null homotopy Prove that every continuous map $f:P^2\to S^1$ is null homotopy. - What have you tried so far? Is $P^2$ the projective plane? In that case, $\pi_1(P^2)=\mathbb{Z}/2$. HINT: What are the maps $\mathbb{Z}/2 \to \mathbb{Z}$? – Fredrik Meyer May 9 '12 at 9:38 @FredrikMeyer's suggestion is what you need. See also a related post. – bgins May 9 '12 at 9:40 I'm learning covering space and this is an exercise for that.I think I should use things related to covering space but $\pi_1(\mathbb P^2)=\mathbb Z_2$...ah,maybe I see how to prove it. – Jiangnan Yu May 9 '12 at 9:44 @JiangnanYu No, because all contractible spaces have trivial fundamental group. – Matt Pressland May 9 '12 at 10:18 If you are having trouble characterizing $\operatorname{Hom}_\mathbb{Z}(\mathbb{Z}_2,\mathbb{Z})$, you should probably go back and brush up on commutative algebra before you continue studying algebraic topology ... – Neal May 9 '12 at 16:16 As Jim pointed out, it's not enough to show that $f_$ induces the trivial map on fundamental groups. To finish it, we can proceed as follows. Let $p\colon \mathbb R\to S^1$ be the covering projection. Since $f_(\pi_1(P^2))=0$ (as Fredrik showed), it's clearly contained in $p_(\pi_1(\mathbb R))=0$. So, this means there is a map $\tilde f\colon P^2\to\mathbb R$ such that $f=p\tilde f$ (Proposition 1.33 in Hatcher). Now, $\mathbb R$ is contractible so we have a homotopy $H\colon P^2\times I\to \mathbb R$ such that $H_0=\tilde f$ and $H_1$ is constant at some $x_0$. But then $pH$ is a homotopy from $p\tilde f=f$ to $p(x_0)$ meaning that $f$ is null-homotopic. - Thank you $SL_2$! – Jiangnan Yu May 9 '12 at 23:42 I'll assume by $P^2$ you mean the real projective plane and I'll also assume you know that $\pi_1(P^2)=\mathbb{Z}/2$. Now, every map $f:(X,x_0) \to (Y,y_0)$ of based topological spaces induces a map between their fundamental groups $f_:\pi_1(X,x_0) \to \pi_1(Y,y_0)$. Now, $\pi_1(P^2,x_0)=\mathbb{Z}/2$ (and in fact, we may forget about the base point, since $P^2$ is path connected). Also, $\pi_1(S^1, x_0)=\mathbb{Z}$. So a continous map $f:P^2 \to S^1$ induces, by the above paragraph, a map $\mathbb{Z}/2 \to \mathbb{Z}$. Torsion elements are sent to torsion elements, but $\mathbb{Z}$ is torsion-free, so the only possibility is that $\mathbb{Z}/2$ is mapped to zero. So every map $f:P^2 \to S^1$ induces the trivial map on fundamental groups. This means that $f_$ is homotopic to the null map. EDIT: See SL2's answer below for the conclusion. (regarding maps $\mathbb{Z}/2 \to \mathbb{Z}$: What happens to $1 \in \mathbb{Z}/2$? The map must satisfy $f(1+1)=f(1)+f(1)$, but $f(1+1)=f(0)=0$ , so $2f(1)=0$, hence $f(1)=0$.) - You also need that the higher homotopy groups of $S^1$ vanish. Normally it is not sufficient to check $\pi_1$. (Equivalently, $f$ lifts to a map to the universal cover, which is contractible in this case.( – Grumpy Parsnip May 9 '12 at 16:18 @Jim Thank you for pointing that out. I found this relevant MO post which says that even if $f_\colon \pi_k(X)\to \pi_k(Y)$ is 0 for all $k$, it isn't true that $f$ is necessarily null-homotopic. You really do need a covering space argument to finish this. – SL2 May 9 '12 at 16:33 @SL2: And thank you for pointing that out. – Grumpy Parsnip May 9 '12 at 17:07 @JimConant : Thanks for pointing out my error. I've edited my answer accordingly (with a reference to the answer below). – Fredrik Meyer May 9 '12 at 19:53 Thanks Fredrik Meyer,now I know $\mathbb Z/2 \to \mathbb Z$could only be a trivial map :) – Jiangnan Yu May 9 '12 at 23:43	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9692822098731995, "perplexity": 298.0019595949313}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2014-10/segments/1394011338837/warc/CC-MAIN-20140305092218-00040-ip-10-183-142-35.ec2.internal.warc.gz"}
https://www.thestudentroom.co.uk/showthread.php?t=112531	You are Here: Home >< Maths # Matrices watch 1. Hi I am doing my GCSEs (just finished astually) I just had year 12 Induction week were you seewhat A Level subjects are like. I am doing: Maths, Phsyics and Further Maths. I done some rearch on some topics in Further Maths and came across 'matrices'. This is the problem; The Matrix M is given by M = a 2 -1 2 3 -1 2 -1 1 where a is a constant. i) show that the determinat of M is 2a. (2marks) ii) given that a ≠ 0, find the inverse matrix M^-1 (4marks) I don't get matrices? 2. Sorry that didn't come out well, the matrix, M is; a 2 -1 2 3 -1 2 -1 1 Pretent its in a big bracket. [/B] 3. This would get a better response in the maths academic help forum. xx 4. I think using the formular: (M) = a b c d e f g h i det (M) = a(ei -hf) -b(di - gf) + c(dh - eg) You can easily find out det (M) = 2a I am not sure in the 12 GCSE has used this ... That comes from A Level P6 5. (Original post by 00sdhun) Hi I am doing my GCSEs (just finished astually) I just had year 12 Induction week were you seewhat A Level subjects are like. I am doing: Maths, Phsyics and Further Maths. I done some rearch on some topics in Further Maths and came across 'matrices'. This is the problem; The Matrix M is given by M = a 2 -1 2 3 -1 2 -1 1 where a is a constant. i) show that the determinat of M is 2a. (2marks) ii) given that a ≠ 0, find the inverse matrix M^-1 (4marks) I don't get matrices? i. The determinant of a 3x3 matrix written as a b c d e f g h i is given by a(ei-hf)-b(di-gf)+c(dh-ge). Apply this on your matrix and you'll get that its determinant is 2a. ii. First, you find the determinant--you have this, it is 2a. Then you need to find the matrix of minors. This is done by finding the determinant of the 2x2 matrix left by covering a given entry in the matrix. For example, the upper-left hand entry in the matrix of minors is 3x1-(-1x-1)=2. You then multiply this matrix by + - + - + - + - + so that you get the matrix of cofactors. Transpose this and multiply by 1/det(M), and you're done! 6. (Original post by J.F.N) ii. First, you find the determinant--you have this, it is 2a. Then you need to find the matrix of minors. This is done by finding the determinant of the 2x2 matrix left by covering a given entry in the matrix. For example, the upper-left hand entry in the matrix of minors is 3x1-(-1x-1)=2. You then multiply this matrix by + - + - + - + - + so that you get the matrix of cofactors. Transpose this and multiply by 1/det(M), and you're done! I've never understood why so many find the matrix of minors - I think it's easier just to find the matrix of cofactors straight away. Say you've got a transpose matrix of: a b c d e f g h i Write this out four times in a square (perhaps with lines to separate them): a b c \| a b c d e f \| d e f g h i \| g h i a b c \| a b c d e f \| d e f g h i \| g h i Then just find the determinant of each 2x2 matrix that lies south-east of a-i. So, for example, the entry in the bottom-right element of the matrix of co-factors is given by ae - bd; the entry in the middle element of the matrix of cofactors is given by ia - gc [etc.]. So yeah. Find the matrix of cofactors, then divide by 2a, the determinant of the original matrix. I kept forgetting which matrix to take the determinant of, myself. 7. (Original post by JohnSPals) I've never understood why so many find the matrix of minors - I think it's easier just to find the matrix of cofactors straight away. Say you've got a transpose matrix of: a b c d e f g h i Write this out four times in a square (perhaps with lines to separate them): a b c \| a b c d e f \| d e f g h i \| g h i a b c \| a b c d e f \| d e f g h i \| g h i Then just find the determinant of each 2x2 matrix that lies south-east of a-i. So, for example, the entry in the bottom-right element of the matrix of co-factors is given by ae - bd; the entry in the middle element of the matrix of cofactors is given by ia - gc [etc.]. So yeah. Find the matrix of cofactors, then divide by 2a, the determinant of the original matrix. I kept forgetting which matrix to take the determinant of, myself. Whatever works for you. I personally don't believe that this method is particularly quicker, especially since the matrix of cofactors can be found smiply once you find the matrix of minors. In a similar way, I don't use Sarrus' rule to compute the determinant of a 3x3 matrix. Old-fashioned, eh? 8. (Original post by J.F.N) Whatever works for you. I personally don't believe that this method is particularly quicker, especially since the matrix of cofactors can be found smiply once you find the matrix of minors. In a similar way, I don't use Sarrus' rule to compute the determinant of a 3x3 matrix. Old-fashioned, eh? I don't think it's much quicker, but it's far more fool-proof. I find myself making all kinds of errors when computing a matrix of minors, and it's easy to forget to change some signs. My method seems comparitively foolproof. I'd comment on Sarrus' rule, but I don't know what it is . 9. (Original post by JohnSPals) I'd comment on Sarrus' rule, but I don't know what it is . It works only for 3x3 matrices, but consider the matrix a b c d e f g h i If you write it as a b c \| a b d e f \| d e g h i \| g h and add the product of the diagonals pointing to the right and subtract the product product of the diagonals pointing to the left, you get the determinant of the matrix, i.e. aei + bfg + cdh - ceg - afh - bdi 10. (Original post by J.F.N) It works only for 3x3 matrices, but consider the matrix a b c d e f g h i If you write it as a b c \| a b d e f \| d e g h i \| g h and add the product of the diagonals pointing to the right and subtract the product product of the diagonals pointing to the left, you get the determinant of the matrix, i.e. aei + bfg + cdh - ceg - afh - bdi I like that method very much . Again, it's comparitively foolproof! 11. I remember doing matrices very long ago in year 9 to prepare for maths igcse. Wow never knew igcse maths overlaps with A-lvl further maths! 12. (Original post by blah888) I remember doing matrices very long ago in year 9 to prepare for maths igcse. Wow never knew igcse maths overlaps with A-lvl further maths! Well you study matrices further at university - all the way till your final year if you wish - but I don't really think it implies "overlap" 13. (Original post by RichE) Well you study matrices further at university - all the way till your final year if you wish - but I don't really think it implies "overlap" like WHATEVER 14. (Original post by blah888) like WHATEVER aww shucks and I was hoping your next question was going to be on the Cayley-Hamilton Theorem 15. That's not the way we did matrices at all... If you call the first column of the matrix the vector a, the second b and the third c, then isn't the determinant a.(bXc) ? I find that easier to remember... And we never learnt any names like matrix of minors or cofactors! I can't quite remember how we did it... I think you call the rows a, b and c then make them into column vectors. Work out bXc, cXa and aXb, and these are the columns that make up the new matrix. Then divide by the determinant. 16. (Original post by blah888) I remember doing matrices very long ago in year 9 to prepare for maths igcse. Wow never knew igcse maths overlaps with A-lvl further maths! What they do in Further Maths is Matrices from scratch to a pretty advanced level whereas the year 9 stuff is to a limited level. So yes, the first bit of FM Matrices is Year 9 stuff but the majority of the whole topic (afterwards) is a lot more advanced. 17. (Original post by Showsni) That's not the way we did matrices at all... If you call the first column of the matrix the vector a, the second b and the third c, then isn't the determinant a.(bXc) ? I find that easier to remember... And we never learnt any names like matrix of minors or cofactors! I can't quite remember how we did it... I think you call the rows a, b and c then make them into column vectors. Work out bXc, cXa and aXb, and these are the columns that make up the new matrix. Then divide by the determinant. I have to say, I was disapointed in a way because the relationship between matrices and vectors wasn't really established enough for EdExcel P6. Either that or my tecaher didn't explain it enough (luckily I didn't need to know all the linear transformations stuff and base vector stuff for my exam, otherwise I would have been a bit stuck). Your method for finding the inverse of a 3x3 matrix sounds interesting. 18. Don't worry - Matrices are covered in FP1 and they're easy! 19. (Original post by Nima) the majority of the whole topic (afterwards) is a lot more advanced. It was? It all seemed quite algorithmic in nature. 20. (Original post by Gaz031) It was? It all seemed quite algorithmic in nature. I think there is quite a difference between knowing how to find eigenvalues and eigenvectors and understanding just what you're doing and why. If you can think about the matrix stuff from a geometric point of view then you really are on top of the subject. ### Related university courses TSR Support Team We have a brilliant team of more than 60 Support Team members looking after discussions on The Student Room, helping to make it a fun, safe and useful place to hang out. This forum is supported by: Updated: July 10, 2005 Today on TSR ### A level Maths discussions Find out how you've done here ### 1,445 students online now Exam discussions Poll Useful resources ### Maths Forum posting guidelines Not sure where to post? Read the updated guidelines here ### How to use LaTex Writing equations the easy way ### Study habits of A* students Top tips from students who have already aced their exams	{"extraction_info": {"found_math": false, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8104031682014465, "perplexity": 1048.565533122157}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2018-26/segments/1529267859904.56/warc/CC-MAIN-20180617232711-20180618012711-00127.warc.gz"}
https://mirrors.dotsrc.org/cran/web/packages/gmvarkit/vignettes/intro-to-gmvarkit.html	Introduction to gmvarkit Introduction The package gmvarkit contains tools to estimate and analyze both reduced form and structural Gaussian mixture vector autoregressive (GMVAR) models (Kalliovirta, Meitz, and Saikkonen, 2016, and Virolainen, 2020). In this vignette, we first define the reduced form GMVAR model, briefly discuss some of its properties, and explain how to use the functions in gmvarkit to estimate the model, examine the estimates, check its adequacy with quantile residual diagnostics, simulate from the process, forecast future observations of the process, and test hypotheses regarding the model parameters. Then, we define the structural GMVAR model, discuss its identification, and explain how to use the functions in gmvarkit to estimate generalized impulse response function and generalized forecast error variance decomposition. Most of the functions work with both, reduced form and structural models, but particularly the ones for estimating generalized impulse response function and generalized forecast error variance decomposition accommodates structural models only. For more comprehensive discussion on the reduced form model, see the cited article by Kalliovirta, Meitz, and Saikkonen (2016), and for the structural model, see the cited unpublished manuscript by Virolainen (2020). For a shorter introduction and demonstration on how to use gmvarkit, see the readme file. Some general notes on gmvarkit The GMVAR models in gmvarkit are defined as class gmvar S3 objects which can be created with the estimation function fitGMVAR or with the constructor function GMVAR. The created class gmvar objects can then be conveniently used as main arguments in many other functions that allow, for example, model diagnostics, simulations, and forecasting - similarly to many of the popular R packages. Therefore, after estimating a GMVAR model, it’s easy to use the other functions for further analysis. Some tasks, however, such as creating GMVAR models without estimation, or setting up an initial population for the genetic algorithm employed by the estimation function, require accurate knowledge on how the parameter vectors are constructed in this package. Structure of this vignette The rest of this vignette is structured as follows. We have two main sections: one for the reduced form model and one for the structural model. First comes the reduced form model and we start by giving its definition and briefly discussing some of its properties. The form of the reduced form model parameter vector used in gmvarkit is then described. The notation for unconstrained parameter vector is in line with the cited article by Kalliovirta et. al. (2016), however. After that, it is explained and demonstrated by examples how to impose linear constraints on the autoregressive parameters of the model. Finally, we have listed some of the useful functions in gmvarkit and explained how one use them to estimate the model, examine the estimates, evaluate model adequacy, construct a GMVAR model from pre-specified parameters, simulate from a GMVAR process, and forecast a GMVAR process. The section for the structural GMVAR model starts by defining the model and then proceeds to describe how the structural shocks can be identified - often with less restrictive constraints than in conventional SVAR models, while some of the constraints are also testable. Then, it is discussed how the SGMVAR model can be estimated in gmvarkit and how to impose overidentifying restrictions on the “B-matrix”. It is also explained how one may build a structural model based on a reduced form model when there are exactly two regimes in the model. Finally, impulse response analysis based on a generalized impulse response function is covered. Reduced form GMVAR model Definition of the GMVAR model The Gaussian mixture vector autoregressive (GMVAR) model with $$M$$ mixture components and autoregressive order $$p$$, which we refer to as the GMVAR($$p,M$$) model, is a mixture of $$M$$ stationary Gaussian VAR($$p$$) models with time varying mixing weights. At each point of time $$t$$, a GMVAR process generates an observation from one of its mixture components (also called regimes) according to the probabilities pointed by the mixing weights. Let $$y_t=(y_{1t},...,y_{dt})$$, $$t=1,2,...$$ be the $$d$$-dimensional vector valued time series of interest, and let $$\mathcal{F}_{t-1}$$ be the $$\sigma$$-algebra generated by the random variables $$\lbrace y_{t-j},j>0 \rbrace$$ (containing the information on the past of $$y_t$$). Let $$M$$ be the number of mixture components, and $$\boldsymbol{s}_t=(s_{t,1},...,s_{t,M})$$ a sequence of (non-observable) random vectors such that at each $$t$$ exactly one of its components takes the value one and others take the value zero. The component that takes the value one is selected according to the probabilities $$Pr(s_{t,m}=1\|\mathcal{F}_{t-1})\equiv\alpha_{m,t}$$, $$m=1,...,M$$ with $$\sum_{m=1}^M\alpha_{m,t}=1$$ for all $$t$$. Then, for a GMVAR($$p,M$$) model, we have $y_t = \sum_{m=1}^M s_{t,m}(\mu_{m,t}+\Omega_{m}^{1/2}\varepsilon_t), \quad \varepsilon_t\sim\text{NID}(0,I_d),$ where $$\Omega_{m}$$ is a conditional positive definite covariance matrix, NID stands for “normally and independently distributed”, and $\mu_{m,t} = \varphi_{m,0} + \sum_{i=1}^p A_{m,i}y_{t-i}$ is interpreted as the conditional mean of the $$m$$th mixture components. The terms $$\varphi_{m,0}$$ $$(d\times 1)$$ are intercept parameters and $$A_{m,i}$$ $$(d\times d)$$ are the autoregressive (AR) matrices. The random vectors $$\varepsilon_t$$ are assumed to be independent from $$\mathcal{F}_{t-1}$$ and conditionally independent from $$\boldsymbol{s}_{t}$$ given $$\mathcal{F}_{t-1}$$. The probabilities $$\alpha_{m,t}$$ are referred to as the mixing weights. For each $$m=1,...,M$$, the AR matrices $$A_{m,i}$$, $$i=1,...,p$$ are assumed to satisfy the usual stability condition of VAR models. Namely, denoting $\boldsymbol{A}_m= \begin{bmatrix} A_{m,1} & A_{m,2} & \cdots & A_{m,p-1} & A_{m,p} \\ I_d & 0 & \cdots & 0 & 0 \\ 0 & I_d & \cdots & 0 & 0 \\ \vdots & & \ddots & 0 & 0 \\ 0 & 0 & \cdots & I_d & 0 \end{bmatrix} \enspace (dp\times dp)$ we assume that modulus of all eigenvalues of the matrix $$\boldsymbol{A}_m$$ are smaller than one, for each $$m=1,...,M$$. Based on the above definition, the conditional density function of $$y_t$$ given $$\mathcal{F}_{t-1}$$, $$f(\cdot\|\mathcal{F}_{t-1})$$, is given as $f(y_t\|\mathcal{F}_{t-1}) = \sum_{m=1}^M\alpha_{m,t}(2\pi)^{-d/2}\det(\Omega_m)^{-1/2}\exp\left\lbrace -\frac{1}{2}(y_t-\mu_{m,t})´\Omega_m^{-1}(y_t-\mu_{m,t}) \right\rbrace .$ The distribution of $$y_t$$ given its past is thus a mixture of multivariate normal distributions with time varying mixing weights $$\alpha_{m,t}$$. The conditional mean and covariance matrix of $$y_t$$ given $$\mathcal{F}_{t-1}$$ are obtained as $\text{E}[y_t\|\mathcal{F}_{t-1}]=\sum_{m=1}^M\alpha_{m,t}\mu_{m,t}, \quad \text{Cov}[y_t\|\mathcal{F}_{t-1}]=\sum_{m=1}^M\alpha_{m,t}\Omega_m + \sum_{m=1}^M\alpha_{m,t}\left(\mu_{m,t} - \sum_{n=1}^M\alpha_{n,t}\mu_{n,t} \right)\left(\mu_{m,t} - \sum_{n=1}^M\alpha_{n,t}\mu_{n,t} \right)'.$ In order to define the mixing weights $$\alpha_{m,t}$$, consider first auxiliary stationary Gaussian VAR-processes $$z_{m,t}$$, and the related $$dp$$-dimensional random vectors $$\boldsymbol{z}_{t,m}=(z_{t,m},...,z_{m,t-p+1})$$. The density of $$\boldsymbol{z}_{t,m}$$ is $$$\label{dpdensities} n_{dp}(\boldsymbol{z}_{t,m};\mu_m,\boldsymbol{\Sigma}_{m,p})=(2\pi)^{-dp/2}\det(\boldsymbol{\Sigma}_{m,p})^{-1/2}\exp\left\lbrace -\frac{1}{2}(\boldsymbol{z}_{t,m} - \boldsymbol{1}_p\otimes\mu_m)´\boldsymbol{\Sigma}_{m,p}^{-1}(\boldsymbol{z}_{t,m} - \boldsymbol{1}_p\otimes\mu_m)\right\rbrace ,$$$ where $$\boldsymbol{1}_p = (1,...,1)$$ $$(p\times 1)$$, $$\mu_m=(I_d - \sum_{i=1}^pA_{m,i})^{-1}\varphi_{m,0}$$, and $$\boldsymbol{\Sigma}_{m,p}$$ is obtained from $$vec(\boldsymbol{\Sigma}_{m,p})=(I_{dp^2} - \boldsymbol{A}_m\otimes\boldsymbol{A}_m)^{-1}vec(\Sigma_{m,\varepsilon})$$ where $\Sigma_{m,\varepsilon} = \begin{bmatrix} \Omega_m & 0 & \cdots & 0 \\ 0 & 0 & \cdots & 0 \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & 0 \\ \end{bmatrix} \enspace (dp\times dp).$ Denoting $$\boldsymbol{y}_{t-1}=(y_{t-1},...,y_{t-p})$$ $$(dp\times 1)$$, the mixing weights of the GMVAR model are defined as $\alpha_{m,t} = \frac{\alpha_mn_{dp}(\boldsymbol{y}_{t-1};\mu_m,\boldsymbol{\Sigma}_{m,p})}{\sum_{n=1}^M\alpha_nn_{dp}(\boldsymbol{y}_{t-1};\mu_n,\boldsymbol{\Sigma}_{n,p})},$ where the mixing weights parameters $$\alpha_m$$ are assumed to satisfy $$\sum_{m=1}^M\alpha_{m}=1$$ and the $$dp$$-dimensional normal densities are as the ones defined for $$\boldsymbol{z}_{t,m}$$ above but evaluated at $$\boldsymbol{y}_{t-1}$$. The probabilities for each regime occuring therefore depend on the level, variability, and temporal depedence of the past $$p$$ observations. This is a convenient feature for forecasting but it also enables the researcher to associate certain characteristics to different regimes. Some theoretical properties of GMVAR model Stationary distribution Theorem 1 Consider the GMVAR process $$y_t$$ that satisfies the model assumptions. Then $$\boldsymbol{y}_t=(y_t,...,y_{t-p+1})$$ is a Markov Chain on $$\mathbb{R}^{dp}$$ with stationary distribution characterized by the density $f(\boldsymbol{y};\boldsymbol{\theta})=\sum_{m=1}^M\alpha_m n_{dp}(\boldsymbol{y};\upsilon_m).$ Moreover, $$\boldsymbol{y}_t$$ is ergodic. Theorem 1 shows that the stationary distribution of $$\boldsymbol{y}_t$$ is a mixture of $$M$$ multinormal distributions with constant mixing weights $$\alpha_m$$. Consequently, all moments of the stationary distribution exist and are finite. Interpretation of the mixing weights $$\alpha_m$$ and $$\alpha_{m,t}$$ Based on the above Theorem 1, $$\alpha_m$$ has the interpretation as the unconditional probability of the random vector $$\boldsymbol{y}_t$$ being generated from the $$m$$th mixture component. Consequently, $$\alpha_m$$ also represents the unconditional probability of the observation $$y_t$$ being generated from the $$m$$th mixture component (see Kalliovirta et al. 2016, Section 3.3 for more details). As noted, the mixing weights $$\alpha_{m,t}$$ are the conditional probabilities for the random vector $$y_t$$ being generated from the $$m$$th mixture component. According the definition of the mixing weights, regimes with higher weighted likelihood of the past $$p$$ observations are more likely to generate the new observations, with weights being the mixing weight parameters $$\alpha_m$$. Properties of the maximum likelihood estimator 1. Under general conditions (see Kalliovirta et al. 2016, Assumption 1), the maximum likelihood estimator for the parameters of the GMVAR model is strongly consistent. 2. Under general conditions (namely, Assumptions 1 and 2 in Kalliovirta et al. 2016), the maximum likelihood estimator has the conventional limiting distribution (normal). Parameter vectors and unconstrained models In this section, we cover the notation for parameter vectors (the same as in Kalliovirta et al. 2016 for unconstrained models) and explain how to impose linear constraints in gmvarkit. Knowledge of the form of the parameter vector is required for creating GMVAR models without estimation with prespecified parameter values or for setting up initial population for the genetic algorithm, but it is not required for estimation and basic analysis of the models. The form of the parameter vector depends on whether an unconstrained or constrained model is considered. Order of the model Specifying the order of a GMVAR model requires specifying the autoregressive order p and the number of mixture components M. The number of time series in the system is denoted by d, and it’s assumed that d is larger than one.1 Parameter vector of unconstrained model The parameter vector for unconstrained model is of size ((M(pd^2+d(d+1)/2+1)-1)x1) and has form $\boldsymbol{\theta} = (\boldsymbol{\upsilon_1},...,\boldsymbol{\upsilon_M},\alpha_1,...,\alpha_{M-1}),\enspace \text{where}\quad\quad\enspace\quad$ $\boldsymbol{\upsilon_m}=(\phi_{m,0},\boldsymbol{\phi_m},\sigma_m)\enspace ((pd^2+d(d+1)/2)x1),\quad\quad\enspace$ $\boldsymbol{\phi_m}=(vec(A_{m,1}),...,vec(A_{m,p}))\enspace (pd^2x1), \enspace \text{and}\quad\enspace\;$ $\sigma_m=vech(\Omega_m)\enspace ((d(d+1)/2)x1),\enspace m=1,...,M.$ Above $$\phi_{m,0}$$ denotes the intercept parameter of m:th mixture component, $$A_{m,i}$$ denotes the coefficient matrix of m:th mixture component and i:th lag, $$\Omega_m$$ denotes the (positive definite) error term covariance matrix, and $$\alpha_m$$ denotes the mixing weight parameter of m:th mixture component. $$vec()$$ is a vectorization operator that stacks columns of a matrix into a vector and $$vech()$$ stacks columns of a matrix from the main diagonal downwards (including the main diagonal) into a vector. The parameter vector above has “intercept” parametrization, referring to the intercept terms $$\phi_{m,0}$$. However, it’s also possible to use “mean” parametrization, where the intercept terms are simply replaced by the regimewise means $$\mu_m=(I_d-\sum_{i=1}^pA_{m,i})^{-1}\phi_{m,0}$$. Models imposing linear constraints Imposing linear constraints and parameter vector Imposing linear constraints on the autoregressive parameters of GMVAR model is straightforward in gmvarkit. The constraints are expressed in a rather general form which allows to impose a wide class of constraints but one needs to take the time to construct the constraint matrix carefully for each particular case. We consider constraints of form $(\boldsymbol{\phi_1},...,\boldsymbol{\phi_M}) = \boldsymbol{C}\boldsymbol{\psi},\enspace \text{where}\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\enspace$ $\boldsymbol{\phi_m}=(vec(A_{m,1}),...,vec(A_{m,p}))\enspace (pd^2x1), \enspace m=1,...,M,$ $$\boldsymbol{C}$$ is known $$(Mpd^2xq)$$ constraint matrix (of full column rank) and $$\boldsymbol{\psi}$$ is unknown $$(qx1)$$ parameter vector. The parameter vector for constrained model is size ((M(d+d(d+1)/2+1)+q-1)x1) and has form $\boldsymbol{\theta} = (\phi_{1,0},...,\phi_{M,0},\boldsymbol{\psi},\alpha_1,...,\alpha_{M-1}),$ where $$\boldsymbol{\psi}$$ is the $$(qx1)$$ parameter vector containing constrained autoregressive parameters. As in the case of regular models, instead of the intercept parametrization that takes use of intercept terms $$\phi_{m,0}$$, one may use the mean parametrization with regimewise means $$\mu_m$$ instead (m=1,…,M). Examples of linear constraints Consider the following two common uses of linear constraints: restricting the autoregressive parameters to be the same for all regimes and constraining some parameters to zero. Of course also some other constraints may be useful, but we chose to show illustrative examples of these two as they occur in the cited article by Kalliovirta et al. (2016). Restricting AR parameters to be the same for all regimes To restrict the AR parameters to be the same for all regimes, we want $$\boldsymbol{\phi_m}$$ to be the same for all m=1,…,M. The parameter vector $$\boldsymbol{\psi}$$ $$(qx1)$$ then corresponds to any $$\boldsymbol{\phi_m}=\boldsymbol{\phi}$$, and therefore $$q=pd^2$$. For the constraint matrix we choose $\boldsymbol{C} = [I_{pd^2}:\cdots:I_{pd^2}]' \enspace (Mpd^2xpd^2),$ that is, M pieces of $$(pd^2xpd^2)$$ diagonal matrices stacked on top of each other, because then $\boldsymbol{C}\boldsymbol{\psi}=(\boldsymbol{\psi},...,\boldsymbol{\psi})=(\boldsymbol{\phi},...,\boldsymbol{\phi}).$ Restricting AR parameters to be the same for all regimes and constraining non-diagonal elements of coefficient matrices to be zero The previous example shows how to restrict the AR parameters to be the same for all regimes, but say we also want to constrain the non-diagonal elements of coefficient matrices $$A_{m,i}$$ (m=1,…,M, i=1,…,p) to be zero. We have the constrained parameter $$\boldsymbol{\psi}$$ $$(qx1)$$ representing the unconstrained parameters $$(\boldsymbol{\phi_1},...,\boldsymbol{\phi_M})$$, where by assumption $$\boldsymbol{\phi_m}=\boldsymbol{\phi}=(vec(A_1),...,vec(A_p))$$ $$(pd^2x1)$$ and the elements of $$vec(A_i)$$ (i=1,…,p) corresponding to the diagonal are zero. For illustrative purposes, let’s consider a GMVAR model with autoregressive degree p=2, number of mixture components M=2 and number of time series in the system d=2. Then we have $\boldsymbol{\phi}=(A_{1,(1,1)},0,0,A_{1,(2,2)},A_{2,(1,1)},0,0,A_{2,(2,2)}) \enspace (8x1) \enspace \text{and}$ $\boldsymbol{\psi}=(A_{1,(1,1)},A_{1,(2,2)},A_{2,(1,1)},A_{2,(2,2)}) \enspace (4x1).\quad\quad\quad\quad\quad\enspace$ By a direct calculation, we can see that choosing the constraint matrix $\boldsymbol{C}=\left[{\begin{array}{c} \boldsymbol{\tilde{c}} \\ \boldsymbol{\tilde{c}} \\ \end{array}}\right] \enspace (Mpd^2x4), \enspace \text{where}$ $\boldsymbol{\tilde{c}}=\left[{\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array}}\right] \enspace (pd^2x4)$ satisfies $$\boldsymbol{C}\boldsymbol{\psi}=(\boldsymbol{\phi},...,\boldsymbol{\phi}).$$ Constraining the unconditional means of some regimes to be the same In addition to constraining the autoregressive parameters, gmvarkit allows to constrain the unconditional means of some regimes to be the same. This feature is, however, only available for models that are parametrized with the unconditional means instead of intercepts (because some of the estimation is always done with mean-parametrization and one cannot generally swap the parametrization when constraints are imposed on means/intercepts). With the mean-parametrization employed (by setting parametrization="mean"), one may define groups of regimes that have the same mean parameters using the argument same_means. For instance, with three regime model (M=3) the argument same_means=list(c(1, 3), 2) sets the unconditional means of the first and third regimes to be the same while allows the second regime to have different mean. One can also combine linear constraints on the AR parameters with constraining some of the means to be the same. This allows, for instance, to estimate a model in which only the covariance matrix varies in time. Validity of different constraints can be ecaluated, for example, with likelihood ratio and Wald tests using the functions LR_test and Wald_test. Some useful functions in gmvarkit Estimating a GMVAR model gmvarkit provides the function fitGMVAR for estimating a GMVAR model. The maximum likelihood estimation is performed in two phases: in the first phase fitGMVAR uses a genetic algorithm to find starting values for gradient based variable metric algorithm, which it then uses to finalize the estimation in the second phase. It’s important to keep in mind that it’s not guaranteed that the numerical estimation algorithms end up in the global maximum point rather than a local one. Because of high multimodality and challenging surface of the log-likelihood function, it’s actually expected that most of the estimation rounds won’t find the global maximum point. For this reason one should always perform multiple estimation rounds and parallel computing is used to obtain the results faster. The number of CPU cores used can be set with the argument ncores, and the number of estimation rounds can be chosen with the argument ncalls. If the model estimates poorly, it is often because the number of mixture components M is chosen too large. One reason for is that the genetic algorithm is (with the default settings) designed to find solutions with non-zero mixing weights for all regimes. One may also adjust the settings of the genetic algorithm or set up an initial population with approximate guesses for the estimates. This can by done by passing arguments in fitGMVAR to the function GAfit which implements the genetic algorithm. To check the available settings, read the documentation ?GAfit. If the iteration limit is reached when estimating the model, the function iterate_more can be used to finish the estimation. If the estimation algorithm fails to create an initial population for the genetic algorithm, it usually helps to scale the individual series so that the AR coefficients (of a VAR model) will be relative small, preferably less than one. Even if one is able to create an initial population, it should be preferred to scale the series so that most of the the AR coefficients will not be very large, as the estimation algorithm works better with small AR coefficients. If needed, another package can be used to fit linear VARs to the series to see which scaling of the series results in relatively small AR coefficients. Examining the estimates Parameters of the estimated model are printed in an illustrative and easy to read form. In order to easily compare approximate standard errors to certain estimates, one can print the approximate standard errors of the estimates in the same form with the function print_std_errors. Numerical approximation of the gradient and Hessian matrix of the log-likelihood at the estimates can be obtained conveniently with the functions get_gradient or get_foc and get_hessian, and eigenvalues of the Hessian can be obtained with the function get_soc. Graphs of the profile log-likelihood functions of the parameters around the estimates can be plotted with the function profile_logliks. The estimated objects have their own print, plot, and summary methods. In-sample conditional moments can be examined with the function cond_moment_plot. The function alt_gmvar can be used to build GMVAR model based on an arbitrary estimation round. This can be used to consider estimates pointed by local maximums instead of the ML estimate. In particular, if the MLE turns out to be a near border solution, it might be appropriate to consider an estimate that is well inside the parameter space. Model diagnostics gmvarkit considers model diagnostics based on multivariate extension of quantile residuals (see Kalliovirta and Saikkonen 2010) which are under the correct model specification asymptotically multivariate standard normal distributed. The quantile residual tests introduced by Kalliovirta and Saikkonen (2010) can be performed with the function quantile_residual_tests by providing the estimated model (that is class gmvar object) as an argument. The argument nsimu can be used to employ a simulation procedure that may enhance the tests size properties. For graphical diagnostics one may use the function diagostic_plot, which enables one to plot the quantile residual time series, auto- and cross-correlation functions of quantile residuals or their squares, or quantile residual histograms and normal QQ-plots. Constructing a GMVAR model without estimation One may wish to construct an arbitrary GMVAR model without any estimation process, for example in order to simulate from a particular process of interest. An arbitrary model can be created with the function GMVAR. If one wants to add or update data to the model afterwards, it’s advisable to use the function add_data. Simulating from a GMVAR process The function simulateGMVAR is the one for the job. As the main argument it uses a class gmvar object created with fitGMVAR or GMVAR. Forecasting GMVAR process The package gmvarkit contains predict method predict.gmvar for forecasting GMVAR processes. For one step predictions using the exact formula for conditional mean is supported, but forecasts further than that are based on independent simulations. The predictions are either sample means or medians and the confidence intervals are based on sample quantiles. The objects generated by predict.gmvar have their own plot method. Hypothesis testing Test linear hypotheses using Wald test with the function Wald_test and using likelihood ratio test with the function LR_test. Univariate analysis Use the package uGMAR for analysing univariate time series. Structural GMVAR model Defition of the SGMVAR model Consider the GMVAR model defined above under the headline “Definition of the GMVAR model”. We focus of the so-called “B-model” setup and write the structural GMVAR model as $$$\label{eq:sgmvar} y_t = \sum_{m=1}^Ms_{t,m}\left(\phi_{m,0} + \sum_{i=i}^{p}A_{m,i}y_{i-1} \right) + B_te_t$$$ and $$$\label{eq:sgmvarerr} u_t\equiv B_te_t= \left\lbrace\begin{matrix} u_{1,t} \sim N(0, \Omega_1) & \text{if} & s_{t,1} = 1 & (\text{with probability } \alpha_{1,t}) \\ u_{2,t} \sim N(0, \Omega_2) & \text{if} & s_{t,2} = 1 & (\text{with probability } \alpha_{2,t}) \\ \vdots & & & \\ u_{M,t} \sim N(0, \Omega_M) & \text{if} & s_{t,M} = 1 & (\text{with probability } \alpha_{M,t}) \\ \end{matrix}\right.$$$ where the probabilities are expressed conditionally on $$\mathcal{F}_{t-1}=\sigma\lbrace y_{t-j}, j>0\rbrace$$ and $$e_t$$ is an orthogonal structural error. Unlike in conventional SVAR analysis, the $$(d\times d)$$ “B-matrix” $$B_t$$, which governs the contemporaneous relations of the shocks, is time-varying. This enables to amplify a constant sized structural shock according to the conditional variance of the reduced form error which varies according to the mixing weights. We have $$\Omega_{u,t}\equiv\text{Cov}(u_t\|\mathcal{F}_{t-1})=\sum_{m=1}^M\alpha_{m,t}\Omega_m$$, while the (conditional) covariance matrix of the structural errors $$e_t=B_t^{-1}u_t$$ (which have a mixture normal distribution and are not IID) is obtained as $$$\text{Cov}(e_t\|\mathcal{F}_{t-1})=\sum_{m=1}^M\alpha_{m,t}B_t^{-1}\Omega_mB_t'^{-1}.$$$ The B-matrix $$B_t$$ should thus be chosen so that the structural shocks are orthogonal regardless of which regime they come from. Following Lanne & Lütkepohl (2010) and Lanne et al. (2010), we decompose the error term covariance matrices as $$$\label{eq:decomp} \Omega_1 = WW' \quad \text{and} \quad \Omega_m=W\Lambda_mW', \quad m=2,...,M,$$$ where the diagonal of $$\Lambda_m=\text{diag}(\lambda_{m1},...,\lambda_{md})$$, $$\lambda_{mi}>0$$ ($$i=1,...,d$$), contains the eigenvalues of the matrix $$\Omega_m\Omega_1^{-1}$$ and columns of the nonsingular $$W$$ are the related eigenvectors. When $$M=2$$, the above decomposition always exists (e.g. Lanne & Lütkepohl, 2010, Proposition 1), but for $$M\geq 3$$ its existence requires that the matrices $$\Omega_m\Omega_1^{-1}$$ share the common eigenvectors in $$W$$. Whether imposing the above structure on the covariance matrices is appropriate with $$M\geq 3$$, can be evaluated by estimating the restricted and unrestricted SGMVAR models and conducting a likelihood ratio test, comparing values of information criteria, or examining quantile residual diagnostics, for example. These three methods are accomodated in gmvarkit: values of the information criteria are calculated for all estimated models are available, for instance, in the summary print, likelihood ratio test is available as the function LR_test and quantile residual diagnostics can be examined with the functions diagnostic_plot and quantile_residual_tests. Lanne & Lütkepohl (2010) show that for a given ordering of the eigenvalues $$\lambda_{mi}$$, $$W$$ is unique apart from changing all signs in a column, as long as for all $$i\neq j\in \lbrace 1,...,d \rbrace$$ there exists an $$m\in \lbrace 2,...,M \rbrace$$ such that $$\lambda_{mi} \neq \lambda_{mj}$$. A locally unique B-matrix that amplifies a constant sized structural shock according to the conditional variance of the reduced form error is therefore obtained as $$$\label{eq:bt} B_t=W(\alpha_{1,t}I_d + \sum_{m=2}^M\alpha_{m,t}\Lambda_m)^{1/2}$$$ which simultaneously diagonalizes $$\Omega_1,...,\Omega_M$$, and $$\Omega_{u,t}$$ for each $$t$$ so that $$\text{Cov}(e_t\|\mathcal{F}_{t-1})=I_d$$. In gmvarkit, the B-matrix always has the above structure. Identification of the structural shocks Even if all pairs of $$\lambda_{mi}$$ are distinct for some $$m$$, unique identification of $$B_t$$ requires deciding upon the ordering of the eigenvalues $$\lambda_{mi}$$ in the diagonals of $$\Lambda_m$$ which fixes also the ordering of the columns of $$W$$. While statistical identification of the model is achieved with any arbitrary ordering the eigenvalues, it does not guarantee that the structural shocks have economic interpretations. In particular, a structural shock relates to an economic shock through the specific constraints in the corresponding column of $$W$$ (or equally of the B-matrix) that only the shock of interest satisfies. If such constraints are readily satisfied in the (unrestricted) estimate of $$W$$, the identification amounts to labelling the structural shocks by the appropriate economic shocks, as long as the constraints are strong enough to pin down a unique ordering for the columns of $$W$$ (this argument will be formalized in Proposition 1 below). If the unrestricted estimate of $$W$$ is such that the researcher cannot associate the economic shocks to it, the appropriate constraints can placed for their identification. Besides a single sign constraint in each column of $$W$$, any constraints on $$W$$ are overidentifying and therefore testable. Moreover, globally unique identification of the structural shocks can often be achieved with more flexible constraints than in conventional SVAR models. Also, as in practice one is often interested in identifying only some specific shock or shocks (such as a monetary policy shock, for instance), it makes sense to consider only partial identification of the B-matrix as well. Specifically, we say that the $$j$$th structural shock is uniquely identified if the $$j$$th column of the B-matrix is unique for given mixing weights $$\alpha_{1,t},...,\alpha_{M,t}$$. This requires that the $$j$$th columns of $$W$$ and $$\Lambda_m$$, $$m=2,..,M$$, are unique. The following proposition gives sufficient and necessary conditions for global identification of the last $$d_1$$ shocks when the related pairs of $$\lambda_{mi}$$ are distinct for some $$m$$. Proposition 1 Suppose $$\Omega_1=WW'$$ and $$\Omega_m=W\Lambda_mW', \ \ m=2,...,M,$$ where $$\Lambda_m=\text{diag}(\lambda_{m1},...,\lambda_{md})$$, $$\lambda_{mi}>0$$ ($$i=1,...,d$$), contains the eigenvalues of $$\Omega_m\Omega_1^{-1}$$ in the diagonal and the columns of the nonsingular $$W$$ are the related eigenvectors. Then, the last $$d_1$$ structural shocks are uniquely identified if 1. for all $$j > d-d_1$$ and $$i\neq j$$ there exists an $$m\in \lbrace 2,...,M \rbrace$$ such that $$\lambda_{mi} \neq \lambda_{mj}$$, 2. the columns of $$W$$ are constrained in a way that for all $$i\neq j > d - d_1$$, the $$i$$th column cannot satisfy the constraints of the $$j$$th column as is nor after changing all signs in the $$i$$th column, and 3. there is at least one (strict) sign constraint in each of the last $$d_1$$ columns of $$W$$. Notice that condition (3) of Proposition 1 fixes the signs in the last $$d_1$$ columns of $$W$$ and therefore the signs of the instantaneous effects of the corresponding shocks. Changing the signs would be effectively the same as changing the signs of the corresponding shocks so condition (3) is not restrictive, however. The assumption that the identified shocks are the last $$d_1$$ shocks is neither restrictive as one may always reorder the structural shocks (and possibly the variables) accordingly. For example, if $$d=3$$, $$\lambda_{m1}\neq\lambda_{m3}$$ for some $$m$$, and $$\lambda_{m2}\neq\lambda_{m3}$$ for some $$m$$, the third structural shock can be identified with the following constraints: $B_t=\begin{bmatrix} * & * & * \\ + & + & - \\ + & + & + \\ \end{bmatrix} \ \text{or} \ \begin{bmatrix} - & * & + \\ - & + & - \\ * & + & + \\ \end{bmatrix} \ \text{or} \ \begin{bmatrix} + & 0 & - \\ * & * & * \\ + & * & + \\ \end{bmatrix}$ and so on, where $$""$$ signifies that the element is not constrained, $$"+"$$ denotes strict positive and $$"-"$$ a strict negative sign constraint, and $$"0"$$ means that the element is constrained to zero. Because imposing zero or sign constraints on $$W$$ equals to placing them on $$B_t$$, they may be justified economically. Furthermore, besides a single sign constraint in each column, the constraints are over-identifying and can thus be also justified statistically. Sign constraints, however, don’t reduce the dimension of the parameter space, making some of the measures such as the conventional likelihood ratio test and information criteria unsuitable for testing them. Quantile residual diagnostics, on the other hand, can be used to evaluate how well the restricted model is able to encapsulate the statistical properties of the data compared to the unrestricted alternative. Shocks that don’t satisfy the condition (1) of Proposition 1 can still be identified but it requires stronger conditions. The following proposition provides sufficient criteria for global identification of the last $$d_1$$ shocks when exactly one of the eigenvalues $$\lambda_{mi}$$ with $$i\neq j > d - d_1$$ is identical to $$\lambda_{mj}$$ in all $$m$$. For simplicity, we assume that only another one the shocks with identical eigenvalues is to be identified. Proposition 2 Let $$d_1<d$$. Consider the matrix decomposition of Proposition 1 and further suppose that for $$j=d-d_1+1$$ and some $$i\leq d-d_1$$ we have $$\lambda_{mi}=\lambda_{mj}$$ for all $$m$$, but for all $$l$$ that are not in $$\lbrace i,j \rbrace$$, $$\lambda_{ml}\neq\lambda_{mj}$$ for some $$m$$. Then, the last $$d_1$$ structural shocks are uniquely identified if conditions (1)-(3) of Proposition 1 are otherwise satisfied, and in addition 1. the column $$i\leq d-d_1$$ of $$W$$ such that $$\lambda_{mi}=\lambda_{mj}$$ for all $$m$$ has at least one (strict) sign constraint and the $$j$$th column has a zero constraint where the $$i$$th column has the (strict) sign constraint. Note that the assumption $$j=d-d_1+1$$ is made without loss of generality as the structural shocks can always be reordered accordingly by also reordering the columns of $$W$$ (including the constraints) and the eigenvalues $$\lambda_{mi}$$ correspondingly.2 To exemplify, if $$d=4$$, $$\lambda_{m1}\neq\lambda_{m4}$$ for some $$m$$, $$\lambda_{m2}\neq\lambda_{m4}$$ for some $$m$$, and $$\lambda_{m3}=\lambda_{m4}$$ for all $$m$$, the following constraints lead to global identification of last shock: $B_t=\begin{bmatrix} & * & * & \\ & * & + & 0\\ + & + & * & -\\ + & + & * & +\\ \end{bmatrix} \ \text{or} \ \begin{bmatrix} * & * & - & 0\\ * & * & * & \\ + & - & & +\\ - & + & * & +\\ \end{bmatrix} \ \text{or} \ \begin{bmatrix} + & - & - & 0\\ * & * & * & \\ & * & * & \\ & * & * & +\\ \end{bmatrix}$ and so on. As is demonstrated above, the structural shocks can often be identified with less restrictive or more suitable constraints than in a conventional SVAR model even when some of the eigenvalues are identical for all covariance matrices. If more than two eigenvalues are identical for all $$m=2,...,M$$ but they are not all identical, it may still be possible to find less restrictive (or more suitable) conditions for identification of the shocks. Specifically, the idea utilized in the proof of Proposition 2 (presented in an Appendix of Virolainen (2020)) can be applied to larger numbers of identical eigenvalues. If all the eigenvalues are identical for all covariance matrices, then $$\Omega_m=\lambda_{m1}\Omega_1$$ and the identification condition is the same as for the conventional SVAR model. As a general remark, observe that constraining an element of $$B_t$$ to be any constant other than zero is infeasible because all elements in the B-matrix are either zero or time varying due to the time-varying mixing weights. Estimation of the structural GMVAR model The SGMVAR model is estimated similarly to the reduced form version, expect that the model is parametrized with $$W$$ and $$\lambda_{mi}$$, $$m=2,...,M$$, $$i=1,...,d$$ instead of the covariance matrices $$\Omega_{m}$$, $$m=1,...,M$$. The estimation is can be done with the function fitGMVAR but now the argument structural_pars needs to be supplied with a list providing the constraints on $$W$$ (which equally imposes the constraints on the B-matrix) and optionally linear constraints on the $$\lambda_{mi}$$ parameters. The list structural_pars should contain at least the element W which is a $$(dxd)$$ matrix matrix with its entries imposing constraints on W: NA indicating that the element is unconstrained, a positive value indicating strict positive sign constraint, a negative value indicating strict negative sign constraint, and zero indicating that the element is constrained to zero. The optional element named C_lambda which is a $$(d(M-1) \times r)$$ constraint matrix that satisfies ($$\lambda_{2},...,\lambda_{M}) =C_{\lambda} \gamma$$ where $$\lambda_{m}=(\lambda_{m1},...,\lambda_{md})$$ and $$\gamma$$ is the new $$(r x 1)$$ parameter subject to which the model is estimated (similarly to AR parameter constraints). The entries of C_lambda must be either positive or zero. Ignore (or set to NULL) if the eigenvalues $$\lambda_{mi}$$ should not be constrained. Note that other constraints than constraining some of the $$\lambda_{mi}$$ to be identical are not recommended but if such constraints are imposed, the argument lambda_scale in the genetic algorithm (see ?GAfit) should be adjusted accordingly. Building structural model based on a reduced form model If the number of regimes is two ($$M=2$$), a structural model can be build based on a reduced form model because the matrix decomposition used in the simultaneous diagonalization of the error term covariance matrices always exists. This can be done with function gmvar_to_sgmvar which should be supplied with the reduced form model and it returns a corresponding structural model. After creating the structural model, the columns of $$W$$ can be reordered ex-post with the function reorder_W_columns which also reorders all $$\lambda_{mi}$$ accordingly and hence the resulting model will conincide with the original reduced form model. Also, all signs any column of $$W$$ can be swapped ex-post with the function swap_W_signs. Impulse response analysis Generalized impulse response function Following Koop et al. (1996) and Kilian & Lütkepohl (2017), we consider the generalized impulse response function (GIRF) defined as $$$\label{eq:girf} \text{GIRF}(n,\delta_j,\mathcal{F}_{t-1}) = \text{E}[y_{t+n}\|\delta_j,\mathcal{F}_{t-1}] - \text{E}[y_{t+n}\|\mathcal{F}_{t-1}],$$$ where $$n$$ is the chosen horizon, $$\mathcal{F}_{t-1}=\sigma\lbrace y_{t-j},j>0\rbrace$$ as before, the first term in the RHS is the expected realization of the process at time $$t+n$$ conditionally on a structural shock of magnitude $$\delta_j \in\mathbb{R}$$ in the $$j$$th element of $$e_t$$ at time $$t$$ and the previous observations, and the second term in the RHS is the expected realization of the process conditionally on the previous observations only. GIRF thus expresses the expected difference in the future outcomes when the specific structural shock hits the system at time $$t$$ as opposed to all shocks being random. Due to the $$p$$-step Markov property of the SGMVAR model, conditioning on (the $$\sigma$$-algebra generated by) the $$p$$ previous observations $$\boldsymbol{y}_{t-1}\equiv(y_{t-1},...,y_{t-p})$$ is effectively the same as conditioning on $$\mathcal{F}_{t-1}$$ at the time $$t$$ and later. The history $$\boldsymbol{y}_{t-1}$$ can be either fixed or random, but with random history the GIRF becomes a random vector, however. Using fixed $$\boldsymbol{y}_{t-1}$$ makes sense when one is interested in the effects of the shock in a particular point of time, whereas more general results are obtained by assuming that $$\boldsymbol{y}_{t-1}$$ follows the stationary distribution of the process. If one is, on the other hand, concerned about a specific regime, $$\boldsymbol{y}_{t-1}$$ can be assumed to follow the stationary distribution of the corresponding component model. In practice, the GIRF and its distributional properties can be approximated with a Monte Carlo algorithm that generates independent realizations of the process and then takes the sample mean for point estimate. If $$\boldsymbol{y}_{t-1}$$ is random and follows the distribution $$G$$, the GIRF should be estimated for different values of $$\boldsymbol{y}_{t-1}$$ generated from $$G$$, and then the sample mean and sample quantiles can be taken to obtain the point estimate and confidence intervals. The algorithm implemented in gmvarkit is presented in an Appendix of Virolainen (2020). Because the SGMVAR model allows to associate specific features or economic interpretations for different regimes, it might be interesting to also examine the effects of a structural shock to the mixing weights $$\alpha_{m,t}$$, $$m=1,...,M$$. We then consider the related GIRFs $$E[\alpha_{m,t+n}\|\delta_j,\boldsymbol{y}_{t-1}] - E[\alpha_{m,t+n}\|\boldsymbol{y}_{t-1}]$$ for which point estimates and confidence intervals can be constructed similarly to (). In gmvarkit, the GIRF can be estimated with the function GIRF which should be supplied with the estimated SGMVAR model or a SGMVAR model build with hand specified parameter values using the function GMVAR. The size of the structural shock can be set with the argument shock_size. If not specified, the size of one standard deviation is used; that is, the size one. Among other arguments, the function may also be supplied with the argument init_regimes which specifies from which regimes’ stationary distributions the initial values are generated from. If more than one regime is specified, a mixture distribution with weights given by the mixing weight parameters is used. Alternatively, one may specify fixed initial values with the argument init_values. Note that the confidence intervals (whose confidence level can be specified with the argument ci) reflect uncertainty about the initial value only and not about the parameter estimates. Because estimating the GIRF and confidence intervals for it is computationally demanding, parallel computing is taken use of to shorten the estimation time. The number of CPU cores used can be set with the argument ncores. As final remarks, note that the objects returned by the GIRF function have their own plot and print methods. Also, cumulative impulse responses of the specified variables can be obtained directly by specifying the argument which_cumulative in the function GIRF. Generalized forecast error variance decomposition Following Lanne and Nyberg (2016), we consider the generalized forecast error variance decomposition (GFEVD) that is defined for variable $$i$$, shock $$j$$, and horizon $$n$$ as $\text{GFEVD}(n,y_{it}, \delta_j,\mathcal{F}_{t-1}) = \frac{\sum_{l=0}^n\text{GIRF}(l,\delta_j,\mathcal{F}_{t-1})_i^2}{\sum_{k=1}^d\sum_{l=0}^n\text{GIRF}(l,\delta_k,\mathcal{F}_{t-1})_i^2},$ where $$n$$ is the chosen and $$\text{GIRF}(l,\delta_j,\mathcal{F}_{t-1})_i$$ is the $$i$$th element of the related GIRF (see also the notation described for GIRF in the previous section). That is, the GFEVD is otherwise similar to the conventional forecast error variance decomposition but with GIRFs in the place of conventional impulse response functions. Because the GFEVDs sum to unity (for each variable), they can be interpreted in a similar manner to the conventional FEVD. In gmvarkit, the GFEVD can be estimated with the function GFEVD. As with the GIRF, the GFEVD is dependent on the initial values. The type of the initial values is set with the argument initval_type, and there are three options: 1. “data” which estimates the GIRFs for all possible length $$p$$ histories in the data and then the GIRFs in the GFEVD are obtained as the sample mean over those GIRFs. 2. “random” which generates the initial values from the stationary distribution of the process or from the mixture of the stationary distributions of some specific regime(s) with the relative mixing proportions given by the mixing weight parameters. The initial regimes can be set with the argument init_regimes. The GIRFs in the GFEVD are obtained as the sample mean over the GIRFs estimated for the different random initial values. 3. “fixed” which estimates the GIRFs for a single fixed initial value that is set with the argument init_ values. The shock size is the same for all scalar components of the structural shock and it can be adjusted with the argument shock_size. If the GIRFs for some variables should be cumulative before calculating the GFEVD, specify them with the argument which_cumulative. Finally, note that the GFEVD objects have their own plot and print methods. More documentation See the readme file for a simple example how to use gmvarkit. References • Lanne, M. and Lütkepohl, H. (2010). Structural vector autoregressions with nonnormal residuals. Journal of Business & Economic Statistics, 28(1):159–168. • Lanne, M., Lütkepohl, H., and Maciejowsla, K. (2010). Structural vector autoregressions with markov switching. Journal of Economic Dynamics and Control, 34(2):121–131. • Lanne M. and Nyberg H. 2016. Generalized Forecast Error Variance Decomposition for Linear and Nonlineae Multivariate Models. Oxford Bulletin of Economics and Statistics, 78(4), 595-603. • Kalliovirta L., Meitz M. and Saikkonen P. (2016) Gaussian mixture vector autoregression. Journal of Econometrics, 192, 485-498. • Kalliovirta L. and Saikkonen P. (2010) Reliable Residuals for Multivariate Nonlinear Time Series Models. Unpublished Revision of HECER Discussion Paper No. 247. • Kilian, L. and Lütkepohl, H. (2017). Structural Vector Autoregressive Analysis. Cambridge University Press, Cambridge, 1st edition. • Koop, G., Pesaran, M., and Potter, S. (1996). Impulse response analysis in nonlinear multivariate models. Journal of Econometrics, 74(1):119–147. • Virolainen S. 2020. Structural Gaussian mixture vector autoregressive model. Unpublished working paper, available as arXiv:2007.04713. 1. For univariate analysis one may use the package uGMAR.↩︎ 2. While the assumptions in Propositions 1 and 2 regarding the ordering of the shocks are not restrictive, they can also be relaxed to allow for any ordering and are made here for simplicity only.↩︎	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 1, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 7, "x-ck12": 0, "texerror": 0, "math_score": 0.9984154105186462, "perplexity": 843.9603572521594}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 20, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2021-43/segments/1634323585178.60/warc/CC-MAIN-20211017144318-20211017174318-00235.warc.gz"}
https://math.stackexchange.com/questions/553552/counting-problem-using-exponential-generating-functions	# Counting problem using exponential generating functions From A Walk Through Combinatorics by Bona in the section on generating functions We have n cards. We want to split them into an even number of non-empty subsets, form a line within each subset, then arrange the subsets in a line. In how many different ways can we do this? My immediate reaction was to set $a_n=n!$ for $n\ge2$ and $b_n =\begin{cases} 0, & \text{if$n$is odd} \\ n!, & \text{if$n$is even} \\ \end{cases}$ And use the exponential compositional formula $G(x)=B(A(x))$ but I realized that this won't work as this counts the number of cards inside each subset as even versus the number of subsets as even. I also failed trying to find a recurrence relation. Also, on a related problem I ended up with the recurrence relation for the exponential series $G(x)$ as $g_{n+1}=ng_n+ \binom{n}{2}g_{n-2}$. I've had difficulty doing the algebra to show $$G(x)=\frac{e^{-x/2-x^2/4}}{\sqrt{1-x}}$$ from this recurrence relation, help would be greatly appreciated. I've gotten it to $$G(x)-1=\sum_{n \ge0}\big(ng_{n+2}+ {n \choose 2}g_n)\frac{x^{n+3}}{(n+3)!}$$ and not quite sure where to go with that. We're in the same class! I am too struggling with the same problem. As for the second problem you referenced, after multiplying your recurrence relation by $\frac{x^{(n+2)}}{(n+2)!}$ and summing for all n, decompose it to a differential equation of $\frac{G'(x)}{G(x)}$ and solve from there. Hopefully that hint will help. • I'm not any farther than what I've edited in on the second question. On the first one I found this link that may help, page 7, $e^{cosh(x)-1}$: math.berkeley.edu/~mhaiman/math172-spring10/exponential.pdf I'm trying to find a way to adapt it for this problem but the forming a line within each subset part is tripping me up. – Eric Kaschalk Nov 6 '13 at 2:33 For the first problem you seem to have the combinatorial species $$\mathcal{Q} = \mathfrak{S}_{\mathrm{even}}(\mathfrak{S}_{\ge 1}(\mathcal{Z})).$$ This gives the exponential generating function $$Q(z) = \frac{z^2}{1-z^2} \circ \frac{z}{1-z} = \frac{z^2}{1-2z}$$ so the count is $$q_n = n! [z^n] \frac{z^2}{1-2z} = n! [z^{n-2}] \frac{1}{1-2z} = n! \times 2^{n-2}$$ for $n\ge 2.$ This gives the following sequence: $$2, 12, 96, 960, 11520, 161280, 2580480, 46448640, 928972800,\ldots$$ which is OEIS A014297. I am not sure I have understood your problem correctly but the description at the OEIS entry seems to match what you have.	{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.8626662492752075, "perplexity": 126.90722370412686}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2019-22/segments/1558232256858.44/warc/CC-MAIN-20190522143218-20190522165218-00514.warc.gz"}

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