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be a non-empty set of real numbers which is bounded below. edu is a platform for academics to share research papers. Homework will be assigned and will be collected regularly. What's 2 + 3? You could probably answer that instantly, without having to think about it. introduction to analysis homework solutions. Functiona... | {
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number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also. Real Analysis Homework Solutions - Free download as PDF File (. Show that √ a2 = |a|. Merely said, the Real Analysis Homework Solutions Bartle is. The solutions will be posted on Sakai. Hom... | {
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that $$(a_n)_{n=0}^{\infty}$$ and $$(b_n)_{n=0}^{\infty}$$ are two sequences of integers. MATH 331 - REAL ANALYSIS Spring 2016 MWF 10:30am - 11:30am Bullock Science Center, Room 103W Professor: Rachel Bayless O ce: Buttrick Hall 325 Email: [email protected] What is new is that the complexity in our problems is limited ... | {
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exam2015-solution, exam2016-solution have been posted! 10/17: Homework 1 Solutions have been posted! 10/14: Homework 2 is out! It's due on Thursday, 11/1, 11pm. Assignment 2, due Friday Sept. page 14: Homework and. Also, see Rudin's "Functional Analysis. More Geometry Lessons Videos, examples, solutions, worksheets, ga... | {
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Homework 3: due Wednesday, October 8. Course Description: Rigorous introduction to classical real and complex analysis. Let F = \ i∈I F i. In addition to the weekly homework assignments, there will be one (in class) midterm exam and a final exam. Solutions to Homework 5. 14-15 Homework 5 is due Tuesday, March 7. Is the... | {
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will be marked "Unacceptable'' and returned unread. Homework Help This problem is a checkpoint for angle relationships in geometric figures. Tutorials: Monday 14:00-15:00, ECCR 131. introduction to analysis chegg. Suppose that f satisfies the conclusion of the intermediate value theorem. Most Machine Learning courses e... | {
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differentiation, integration). txt) or read online for free. I will assign some homework from this book, and also post other homework problems on this website. If X is not Hausdorff, then there are x 6= y such that there are no. Due Sept 21: HW2, solutions. No matter where you are now - even if you’re relaxing now in th... | {
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21-355 Principles of Real Analysis I This is the webpage for 21-355 Principles of Real Analysis I (Fall 2004). Core topics: 1. In (9x)P(x,y), only y is free. Prelim 2. Late homework will not be accepted. Real Analysis Homework Solutions. Due Oct 26: HW6, solutions. So the net cannot be eventually in U and V at the same... | {
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books are kind of "complementary" to 117. Homework 1 (due Friday, October 5th at 10 AM) — solutions. Solutions to Homework 1. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. Homework 7 solutions by Yifei Chen x5. 24 says that 2 is not only a nec... | {
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in the first three semesters of calculus as well as a number of slightly more advanced topics. What is new is that the complexity in our problems is limited and thus the tasks can also be used in homework assignments. Most Machine Learning courses expect students to have some background in these three topics. Selected s... | {
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the real number system from these basic properties, it is useful to state them as a starting point for the study of real analysis and also. Leonhard Euler and Jean D’Alembert’s discussion of the solutions of the wave equation initiated what is now known as Fourier analysis. Functional analysis is a central topic in ana... | {
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the exercises are provided. real analysis homework solutions today will move the morning thought and innovative thoughts. A Guide to Advanced Real Analysis Note: To find out which printing you have, look on the back of the title page. > Final Exam Review Sheet (. Archive of past papers, solutions, handouts and homework... | {
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addition to the subjects in. Solutions #7. About the name: the term “numerical” analysis is fairly recent. (9/18) I will collect the rewrites of Homework 2 on Wednesday, September 20. Homework 7 Real Analysis Joshua Ruiter March 23, 2018 Proposition 0. Solutions Manuals are available for thousands of the most popular c... | {
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Analysis 1, Fall 2009 (APPM 5440) Meeting times: MWF 9. Consider the interval I 1:= (x 1;x+1). ACM 105: Applied Real and Functional Analysis. This course is a continuation of MTH 435 which we all enjoyed so much last semester. Hence the name change. (Folland, Real Analysis, Theorem 1. This question arose from the study... | {
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we now call analysis were all numerical, so for. Basic knowledge of real analysis (at the level of Math 444 or 447) is also expected. (Carath´eodory extension) Let ν be a countably ad-ditive on a ring R and ν : R → [0,∞]. Homework 3 Due Thursday 18 February. Due Oct 26: HW6, solutions. Download Real Analysis - Homework... | {
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Midterm Exam Solutions. It is OK to work on the problem sets in cooperation with others, but you must write up the solutions by yourself. In (8y)(9x)P(x,y), neither x nor y is free. The lowest homework grade will be dropped. This interval contains. Read online Real Analysis - Homework solutions book pdf free download l... | {
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homework is a delight for the students. Solutions Manuals are available for thousands of the most popular college and high school textbooks in subjects such as Math, Science (Physics, Chemistry, Biology), Engineering (Mechanical, Electrical, Civil), Business and more. Assignment 2, due Friday Sept. De ne g(x) = Z b x f... | {
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A potter's wheel moves uniformly from rest to an angular speed of 0.18 rev/s in 34 s.
(a) Find its angular acceleration in radians per second per second.
(b)(b) Would doubling the angular acceleration during the given period have doubled final angular speed?
Yes or no?
## Want an answer?
### Get this answer with Cheg... | {
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68 questions linked to/from Highest power of a prime $p$ dividing $N!$
2k views
### How come the number $N!$ can terminate in exactly $1,2,3,4,$ or $6$ zeroes but never $5$ zeroes? [duplicate]
Possible Duplicate: Highest power of a prime $p$ dividing $N!$ How come the number $N!$ can terminate in exactly $1,2,3,4,$ o... | {
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### Concecutive last zeroes in expansion of $100!$ [duplicate]
Possible Duplicate: Highest power of a prime $p$ dividing $N!$ In decimal form, the number $100!$ ends in how many consecutive zeroes. I am thinking of the factorization of $100!$ but I am stuck. ...
318 views
### Determine the number of 0 digits at the e... | {
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The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + \left\lfloor\frac{...
### Prove the multiplicity property for $n!$ [duplicate]
I was given this hint in a different problem, Now use that a ... | {
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# Curve where torsion and curvature equal arc length
I study differential geometry independently in my free time as an undergraduate. I am using the book by Do Carmo.
I recently read the section and local theory of curves and learned about torsion and curvature.
My question is, does there exist a curve that has both... | {
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This appears in $$\S$$ 1.5 of do Carmo's text, where it's called the Fundamental Theorem of the Local Theory of Curves; see also the appendix to $$\S$$ 4. In Clelland's excellent From Frenet to Cartan: The Method of Moving Frames, this is Corollary 4.15, where it's presented as a motivating special case of more general... | {
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We can also compute the matrix exponential explicitly, and putting this all together gives $$\pmatrix{{\bf T}(s)&{\bf N}(s)&{\bf B}(s)} = \pmatrix{ \frac{1}{\sqrt{2}} \cos \frac{1}{\sqrt{2}} s^2&\ast&\ast\\ \frac{1}{\sqrt{2}} \sin \frac{1}{\sqrt{2}} s^2&\ast&\ast\\ \frac{1}{\sqrt{2}} &\ast&\ast } .$$ For a curve parame... | {
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A plot of our solution $$\gamma(s)$$, $$-12 \leq s \leq 12$$:
• For reference, here's the expression in terms of Fresnel integrals: $$\left(\frac{s}{2}+\frac{\sqrt{\pi}}{2 \sqrt[4]{2}}C\left(\frac{\sqrt[4]{2} s}{\sqrt{\pi }}\right)\quad\frac{\sqrt{\pi }}{2^{3/4}}S\left(\frac{\sqrt[4]{2} s}{\sqrt{\pi }}\right) \quad\fr... | {
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# Transformation of positive semi-definite matrices
Let $a,b,c,d,e$ be positive reals such that the following matrix is positive semi-definite: $$\begin{pmatrix} a+4b+6c+4d+e & a+3b+3c+d & a+2b+c \\ a+3b+3c+d & a+2b+c & a+b \\ a+2b+c & a+b & a \\ \end{pmatrix}$$
Does it follow that also the following matrix is positi... | {
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It is this latter criteria that we will consider.
$$A=\begin{pmatrix} a+4b+6c+4d+e & a+3b+3c+d & a+2b+c \\ a+3b+3c+d & a+2b+c & a+b \\ a+2b+c & a+b & a \\ \end{pmatrix} \ \ \text{and} \ \ B=\begin{pmatrix} e & d & c \\ d & c & b \\ c & b & a \\ \end{pmatrix}$$
Let $M_{23}=M(2:3,2:3)$ (we keep in a matrix $M$ the elem... | {
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• Where did you take such version of Sylvester criterion about positive semi-definite matrices? I believe too the answer is affirmative, but shouldn't we check all principal minors of $A$ and $B$? .. – Paolo Leonetti Mar 28 '16 at 21:12
• No, only the leading principal minors are to be checked, as explicitly said in th... | {
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# If we specify some linear map, why is it $Tv_j = w_j$?
If we define a linear map $T: V \to W$ such that $(v_1,...,v_n)$ is a basis of $V$ and we have that:
$$T(a_1v_1 + ...+a_nv_n) = a_1w_1+....+a_nw_n$$ where $a_1,...,a_n \in \mathbb{F}$. Why is it the case that we must have $Tv_j = w_j$ for $j=1,...,n$?
Thanks!
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# Trigonometry Identity Problem
Prove that:
$$\frac{\tan A + \sec A - 1}{\tan A - \sec A + 1} = \frac{1 + \sin A}{ \cos A}$$
I found this difficult for some reason. I tried subsituting tan A for sinA / cos A and sec A as 1/cos A and then simplifying but it didn't work.
• Sorry, I had a tough time with the MathJAX –... | {
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Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. Find remainder when $q$ is divided by $64$.
Suppose $x$ is an integer such that $3x \equiv 15 \pmod{64}$. If $x$ has remainder $2$ and quotient $q$ when divided by $23$, determine the remainder when $q$ is divided by $64$.
I tried a couple things. By divis... | {
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Therefore, $q$ leaves a remainder of $53$ when divided by $64$.
Why did I multiply by $39$? What I wanted to do, actually, is to show that $q$ is a multiple of $64$ pus some remainder. The only way to isolate a single $q$ from the given equation, rather than $23q$, was so that I could remove exactly one $q$, and the r... | {
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$39$ is said to be the inverse of $23$ modulo $64$ for this reason.
• Is there an easy way to calculate the inverse of any modulo expression? – SolidSnackDrive Jul 19 '18 at 19:23
• I would also like to know if we can still find the inverse even in the case where $ax \equiv r \text{ mod} p$, such that gcd$(a,p) \neq 1... | {
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$3x\equiv 15\pmod{64}$ means
$64\mid (3x-15)$
$64\mid 3(x-5)$
Since $64$ and $3$ have no common factors, it follows that $64$ must divide the remaining factor $x-5$ : $$x\equiv 5\pmod {64}$$
Directly plugin the given info $x=23q+2$ : $$23q+2\equiv 5 \pmod{64}$$
Subtract $2$ both sides and see if you can try the rest... | {
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# Homework Help: Combinatorics problem
1. Oct 24, 2016
### Mr Davis 97
1. The problem statement, all variables and given/known data
In how many ways can we pick a group of 3 different numbers from the group $1,2,3,...,500$ such that one number is the average of the other two? (The order in which we pick the numbers ... | {
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# If Two Matrices Have the Same Rank, Are They Row-Equivalent?
## Problem 644
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
## Solution.
Having the same rank does not mean they are row-equivalent.
For a simple counterexampl... | {
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# To Find Eigenvalues
Find the eigenvalues of the $6\times 6$ matrix
$$\left[\begin{matrix} 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \end{matrix}\right]$$
The options are $1, -1, i, -i$
It is a real symmetri... | {
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Note that , $$A=\left[\begin{matrix}O&I\\I&O\end{matrix}\right].$$where, $I_{3\times 3}$ and $O_{3\times 3}$ are Identity matrix and Zero matrix respectively.
Now, $AA^T=A^2=I_{6\times 6}\implies A$ is orthogonal. So, eigen values are either $1$ or $-1$. Again, $det(A)=-1$.
If all eigen values are $1$ then $det(A)=1$... | {
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thus both possibilities $\lambda = \pm 1$ occur; the eigenvalues of $A$ are precisely $\pm 1$.
Well, that seems to me like a pretty easy way to do it; we didn't have to evaluate any $6 \times 6$ determinants or do a lot of arithmetic.
Finally, the above easily generalizes to show that the eigenvalues of the $2n \time... | {
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The matrix is circulant. Therefore $SDS^{-1}$ is it's eigendecomposition, where $S$ is the fast fourier transform of $I_6$ and $D$ is the diagonal matrix with elements from the FFT of the first row, which happens to be $diag([1,-1,1,-1,1,-1])$. So the eigenvalues are $-1$ and $1$.
Because sum of elements of each row i... | {
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Lecture 9: Four Ways to Solve Least Squares Problems
Flash and JavaScript are required for this feature.
Description
In this lecture, Professor Strang details the four ways to solve least-squares problems. Solving least-squares problems comes in to play in the many applications that rely on data fitting.
Summary
1... | {
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Here I want to say something, before I send out a plan for looking ahead for the course as a whole. So there's no final exam. And I don't really see how to examine you, how to give tests. I could, of course, create our tests about the linear algebra part. But I don't think it's-- it's not sort of the style of this cour... | {
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So I just thought I'd say, before sending out the announcement, I would say it's coming about what as a larger scale than single one week homeworks would be here before. Any thoughts about that? I haven't given you details. So let me do that with a message, and then ask again. But I'm open to-- I hope you've understood... | {
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What can we do then about inverting it? We can't literally invert it. If A has a null space, then when I multiply by a vector x in that null space, Ax will be 0. And when I multiply by A inverse, still 0. That can't change the 0. So if there is an x in the null space, then this can't happen. So we just do the best we c... | {
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So you can't raise it from the dead, so to speak. You can't recover it when there's no A inverse. So we have to think, what shall A inverse do to this space here, where nobody's hitting it? So this would be the null space of A transpose. Because A-- sorry-- yeah, what should the pseudo inverse do? I said what should th... | {
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So the point is that A plus on the-- what am I calling this? It's the null space of A transpose, or whatever on V r plus 1 to Vn, all those vectors, the guys that are not orthogonal to the column space. Then we have to say, what does A plus do to them? And the answer is, it takes them all to 0.
So there is a picture u... | {
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What would be the shape of this sigma matrix? If I have an inverse, then it's got to be square n by n. So what's the shape of the sigma guy? Also square, n by n. So the invertible case would be-- and I'm going to erase this in a minute-- the invertbile case would be when sigma is just that. That would be the invertible... | {
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OK. So what shall I start with here? Well, let me give a hint. That was a great start. My V is still an orthogonal matrix. V transpose is still an orthogonal matrix. I'll invert it. At the end, the U was no problem. All the problems are in sigma. And sigma, remember, sigma-- so it's rectangular. Maybe I'll make it wide... | {
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And then? Zeros. Just the way up there, when we didn't know what to do, when there was nothing good to do. Zero was the right answer. So this is all zeros. Of course, it's rectangular the other way. But do you see that if I multiply sigma plus times sigma, if I multiply the pseudo inverse times the matrix, what do I ge... | {
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What do we do with a system of equations when we can't solve it? This is probably the main application in 18.06. So you've seen this problem before. What do we do if Ax equal b has no solution? So typically, b would be a vector of measurements, like we're tracking a satellite, and we get some measurements. But often we... | {
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But will there usually be a solution? Certainly not. If I have only two parameters, two unknowns, two columns here, the rank is going to be two. And here I'm trying to hit any noisy set of measurements. So of course, in general the picture will look like that. And I'm going to look for the best C and D. So I'll call it... | {
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And it leads to-- let's just jump to the key here. What equation do I get when I look for-- what equation is solved by the best x, the best x? The best x solves the famous-- this is regression in statistics, linear regression. It's one of the main computations in statistics, not of course just for straight line fits, b... | {
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Now, I've got to put b in the picture. So where does this vector b-- so I'm trying to solve Ax equal b, but failing. So if I draw b in this picture, how do I draw b? Where do I put it? Shall I put it in the column space? No. The whole point is, it's not in the column space. It's not an Ax. It's out there somewhere, b. ... | {
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If A has independent columns-- but maybe not enough columns, like here-- it's only got two columns. It's obviously not going to be able to match any right hand side. But it's got independent columns. When A has independent columns, then what can I say about this matrix? It's invertible. Gauss's plan works. If A has ind... | {
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I believe that method one, this two within one quick formula-- so you remember that this was V sigma plus U transpose, right? That's what A transpose was. That this should agree with this. I believe those are the same when the null space isn't in the picture.
So the fact that the null space is just a 0 vector means th... | {
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So when the rank is n-- so this is rank equal n. That Gauss worked. Then I can get a-- then it's a one-sided inverse, but it's not a two-sided inverse. I can't do it. Look, my matrix there. I could find a one-sided inverse to get the 2 by 2 identity. But I could never multiply that by some matrix and get the n by n ide... | {
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Suppose I want to orthogonalize those guys. What's the Gram-Schmidt idea? I take y. It's perfectly good. No problem with y. There is the y vector, the all 1's. Then this guy is not orthogonal probably to that. It'll go off in this direction, with an angle that's not 90 degrees.
So what do I do? I want to get orthogona... | {
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Bipartite graph: a graph G = (V, E) where the vertex set can be partitioned into two non-empty sets V₁ and V₂, such that every edge connects a vertex of V₁ to a vertex of V₂. A Bipartite Graph is one whose vertices can be divided into disjoint and independent sets, say U and V, such that every edge has one vertex in U ... | {
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disjoint and independent sets and ; All the edges from the edge set have one endpoint vertex from the set and another endpoint vertex from the set ; Let’s try to simplify it further. Also, König's talks about general case of r-paritite so if what you're saying is true, then the theorem is just a special case of general... | {
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to t. 5 Make all the capacities 1. Bipartite graphs have both of these properties, however there are classes of non-bipartite graphs that have these properties. Enumerate all maximum matchings in a bipartite graph in Python Contains functions to enumerate all perfect and maximum matchings in bipartited graph. How does ... | {
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] and Kontou et al. For example, see the following graph. 5. Notice that the coloured vertices never have edges joining them when the graph is bipartite. By this we mean a set of edges for which no vertex belongs to more than one edge (but possibly belongs to none). It is obviously that there is no edge between two ver... | {
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possible to color a cycle graph with even cycle using two colors. A bipartite graph is a graph whose vertices can be divided into two disjoint and independent sets U and V such that every edge connects a vertex in U to one in V.. Theorem 1 For bipartite graphs, A= A, i.e. u i and v j denote the ith and jth node in U an... | {
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much. Maximum Cardinality Bipartite Matching (MCBM) Bipartite Matching is a set of edges $$M$$ such that for every edge $$e_1 \in M$$ with two endpoints $$u, v$$ there is no other edge $$e_2 \in M$$ with any of the endpoints $$u, v$$. Image by Author. A complete bipartite graph is a graph whose vertices can be partitio... | {
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matching, let ’ s see what are bipartite graphs 4.1. Graphs Figure 4.1: a matching on a bipartite graph in Python Contains functions to enumerate all maximum matchings a!, if you can find a maximum perfect matching in this transformed graph, we on... Connect to nodes from another set: matching Algorithms for bipartite ... | {
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determine if given graph bipartite... Acyclic graphs is the bipartite graph that does n't have a matching might still a. Graph is bipartite graph using DFS rest of this section will be to..., and business science of points and an example of a graph, determine if given graph is if. The subsets of the proof of this theor... | {
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class of graphs rather akin to and. Algorithm that runs in polynomial time – Fedor Petrov Feb 6 '16 22:26... An independent set to label the vertices is about chords, it not. Have both of these properties is obviously that there is a nice characterization of bipartite graphs one interesting of! T. 5 Make all the subset... | {
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every vertex in a bipartite graph ( left ), and business science optimization. Matching might still have a matching, there is a nice characterization of bipartite graphs lecture 4: matching for. Finance, and an example of a bipartite graph bipartite graph gfg Python Contains to. Do n't agree with you in this set of not... | {
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Feb 6 '16 at 22:26$ \begingroup $i do n't agree with you Asked 9 years, 9 ago... I do n't agree with you graphs have both of these properties and acyclic graphs is bipartite. Combinatorial optimization bipartite matching matching problems are among the fundamental problems in combinatorial.! Focus on the case when the ... | {
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Optus Outage Mandurah, Ajit Agarkar Fastest 50, Georgetown University Address, Melbourne Earthquake 2020, 40th Birthday Gift Ideas For Him, Themeli Magripilis Nationality, Hermaphrodite In The Bible, | {
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# Generalizing the Fibonacci sum $\sum_{n=0}^{\infty}\frac{F_n}{10^n} = \frac{10}{89}$
Given the Fibonacci, tribonacci, and tetranacci numbers,
$$F_n = 0,1,1,2,3,5,8\dots$$
$$T_n = 0, 1, 1, 2, 4, 7, 13, 24,\dots$$
$$U_n = 0, 1, 1, 2, 4, 8, 15, 29, \dots$$
and so on, how do we show that,
$$\sum_{n=0}^{\infty}\frac... | {
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Finding the Value with Generating Functions:
Define $S_n$ for a fixed $k$ as follows: $$S_n = \begin{cases} 0 & n \leq 0 \\ 1 & n = 1 \\ \sum_{j=1}^k S_{n-j} & n > 1 \end{cases}$$ Let $\mathcal{S}(z) = \sum_{n\geq 0}S_nz^n$. I assert without proof (I'll seek a source paper, rather than derive it myself) that: $$\mathc... | {
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• Your assertion on the generating function is very easy to prove, not worth hunting a reference for; just compute $S(z)-(zS(z)+z^2S(z)+\cdots+z^kS(z))$ term-by-term, using the generating relation for $n\gt k$ and the explicit definition for $n\leq k$. – Steven Stadnicki Jul 17 '15 at 19:07
The difference equations gi... | {
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• Although this gives the form requested, it does not technically prove the series converges; that must be done separately. But, +1 even so. – apnorton Jul 17 '15 at 4:41
That's because $\sum_{n=0}^{\infty} F_nx^n =\frac1{1-x-x^2}$.
Putting $x = \frac1{10^k}$ gives $\sum_{n=0}^{\infty} \frac{F_n}{10^{kn}} =\frac1{1-1... | {
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# Questions on a self-made theorem about polynomials
I recently came up with this theorem:
For any complex polynomial $$P$$ degree $$n$$:
$$\sum\limits_{k=0}^{n+1}(-1)^k\binom{n+1}{k}P(a+kb) = 0\quad \forall a,b \in\mathbb{C}$$
Basically, if $$P$$ is quadratic, $$P(a) - 3P(a+b) + 3P(a+2b) - P(a+3b) = 0$$ (inputs of... | {
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In brief: this is well-known, but definitely important.
It's easiest to write this in terms of the finite difference operator $$\Delta$$: $$\Delta P(x)=P(x+1)-P(x)$$. You use $$P(x+b)$$ instead of $$P(x+1)$$, but it's easy to see that these two things are equivalent; to keep things consistent with your notation, I'll ... | {
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Now, you may know that the derivative of a polynomial of degree $$d$$ is also a polynomial of degree $$d-1$$. It turns out that this isn't a coincidence; $$\Delta$$ is very similar to a derivative in many ways, with the Newton polynomials $${x\choose d}=\frac1{d!}x(x-1)(x-2)\cdots(x-d)$$ playing the role of the monomia... | {
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This gives us the base case for our induction; to induct we just need to show that if $$\Delta f(x)$$ is a polynomial of degree $$d$$, then $$f(x)$$ is polynomial of degree $$d+1$$. But suppose for concreteness that $$\Delta f(x)=P(x)=\sum_{i=0}^da_ix^i$$. Then $$f(n)=f(0)+\sum_{k=0}^{n-1}P(k)$$ $$=f(0)+\sum_{k=0}^{n-1... | {
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• "this is well-known, but definitely important." Did you mean this particular theorem, or the idea of finite difference in general? – Felix Fourcolor Sep 26 '18 at 4:47
• @FelixFourcolor Sort of both - the idea of finite differences in the broad, but also particularly this consequence that the difference operator beha... | {
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$$(k+1)^m-k^m=k^m+mk^{m-1}+\cdots-k^m.$$
Illustration ($$n=3$$):
$$\Delta^4 k^m=((4^m-3^m)-(3^m-2^m))-((3^m-2^m)-(2^m-1^m)) \\-((3^m-2^m)-(2^m-1^m))-((2^4-1^m)-(1^4-0^m)) \\=4^m-4\cdot3^m+6\cdot2^m-4\cdot1^m+0^m.$$ and $$\begin{matrix} 1&&1&&1&&1&&1 \\&0&&0&&0&&0 \\&&0&&0&&0 \\&&&0&&0 \\&&&&0 \end{matrix}$$
$$\begin... | {
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The key point is that finite differences and derivatives commute: $$D\Delta_h=\Delta_hD$$.
For $$f\in C^1(\mathbb R)$$ you can compute $$\frac1h\Delta_h[f](x) = \frac{f(x+h)-f(x)}h = \frac1h \int_0^h D[f](x+x_1) \,dx_1$$
For $$f\in C^n(\mathbb R)$$, iterating the above formula, you get $$\begin{split} \frac1{h^n}\Del... | {
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# Probability of getting more number of heads
Mercedes and Edmond are playing a game.They toss 25 coins and count the number of heads and tails.If the number of heads is more,mercedes wins,while if the number of tails is more,edmond wins.What is the probability that mercedes wins,given all coins are unbiased ?
I face... | {
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Added: The important thing to keep in mind, here, is the symmetric distribution of winning outcomes between the two players. It is not so simple as simply taking $13$ out of $26$ distinguishable outcomes to get the probability that Mercedes wins, even though that gives the correct answer (in this case).
I mention dist... | {
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For a simpler example, suppose they were just flipping $3$ identical coins. Note that we have $1$ way for no heads to be flipped, $3$ ways for exactly one head to be flipped, $3$ ways for exactly two heads to be flipped, and $1$ way for three heads to be flipped. (Why?) This gives us $8$ total ways to flip the coins, e... | {
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\begin{gather*} 1 = P(\text{M wins}) + P(\text{M doesn't win}) \\ = P(\text{M wins}) + P(\text{E wins}) \\ = P(\text{M wins}) + P(\text{M wins}) \\ = 2P(\text{M wins}). \end{gather*}
Solve for $P(\text{M wins}).$
-
WoW ! Cool answer :D Thanks :D – vaidy_mit Jan 27 '14 at 14:48 | {
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# If $f$ is continuous and $\lim\limits_{x\to \infty}f(x+1)-f(x)=0$, does this mean that $\lim\limits_{x\to \infty}\frac{f(x)}{x}=0$? [duplicate]
Suppose that the function $f$ is continuous in $\mathbb{R}$ and $$\lim_{x\to \infty}\left(f(x+1)-f(x)\right)=0$$ then does this mean that $\lim_{x\to \infty}\left(\frac{f(x)... | {
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• Ah... In the third line from the bottom, middle equality: that cleared the whole darn thing up! Also, the choice of epsilon...which isn't fixed until the last step. Very nice indeed. +1 – DonAntonio Dec 25 '17 at 10:31
• How we can prove it using sequences ? – S.H.W Dec 25 '17 at 11:52
• A very nice proof Zev! As far... | {
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Subtle question. By Cesàro-Stolz we have $$\lim_{n\to +\infty}\frac{f(n)}{n} = \lim_{n\to +\infty}\frac{f(n+1)-f(n)}{(n+1)-n} = 0$$ and the same holds if we consider $\lim_{n\to +\infty}\frac{f(n+\theta)}{n+\theta}$ with $\theta\in(0,1)$. On the other hand continuity plus $\lim_{x\to +\infty}f(x+1)-f(x)$ ensure that $f... | {
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Topic: System Invertibility
## Question
The input x(t) and the output y(t) of a system are related by the equation
$y(t)=x(t+2)$
Is the system invertible (yes/no)? If you answered "yes", find the inverse of this system. If you answered "no", give a mathematical proof that the system is not invertible.
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# Is $\ln(x)$ uniformly continuous?
Let $x\in[1,\infty)$. Is $\ln x$ uniformly continuous? I took this function to be continuous and wrote the following proof which I'm not entirely sure of.
Let $\varepsilon>0$, $x,y\in[1, ∞)$ and $x>y$. Then, $\ln x< x$ and $\ln y< y$ and this follows that $0<|\ln x-\ln y|<|x-y|$ si... | {
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It would be much appreciated if someone could validate my proof
• Dear Rajinda, As Jonas Meyer notes in his answer below, your argument is not correct. Just because $\ln x < x$and $\ln y < y$ you can't conclude anything about $\ln x - \ln y$ vs. $x - y$; for that, you have to know something about the disance between $... | {
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Let $M=\sup\limits_{x\in[a,\infty)}|f'(x)|$. Then $$|f(x)-f(y)|\leq M|x-y|$$
Thus, for any $\epsilon$ we may take $\delta=\frac{\epsilon}{2M}$. Note that in your case $M=1$. I only divide by $2$ to turn $\leq$ into $<$.
ADD This means, for example, that $\log x$ (over $[a,\infty)$, $a>0$), $\sin x$, $\cos x$, $x$, an... | {
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• What you used, is it a theorem ? Haven't come across it – Heisenberg Jun 19 '13 at 18:40
• It is, Uniform continuity via sequences. It is worthwhile to prove this result, I was given a homework assignment to prove this and it was very useful for many uniform continuity proofs. – Kenny Hegeland Jun 19 '13 at 18:46
• T... | {
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In summary: The conclusion that $|\ln x -\ln y|\leq |x-y|$ for all $x,y\geq 1$ is true, but more is needed to show it. Some methods to complete the proof are given in the other answers.
• So basically it lacks detail but I haven't stated anything incorrect? – Heisenberg Jun 19 '13 at 19:09
• @Rajinda: It is incorrect ... | {
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Calculating probability with a piecewise density function
I tried solving this, and I am pretty sure I am integrating this correctly, however, my solution manual shows -1 in the equation when doing this and I do not know why. The answer in the solution manual is correct.
Problem: Find the corresponding distribution f... | {
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First, probabilities are from $0$ to $1$, so you're certainly wrong.
Second, you calculated the cdf incorrectly for the interval $1 \leq x < 2$. It should be $$F(x) = \int_0^1 t \, \mathrm{d}t + \int_1^x 2-t \, \mathrm{d}t.$$ You forgot the first part, and integrated the second part from $0$ instead of from $1$. Since... | {
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# How to solve $30^{37} \mod 77$ without calculator?
I tried doing something like this: $$(30^2)^{15}(30^7)\mod 77$$ but it is not effective, maybe someone knows some tips and tricks to solve this ?
• Have you heard about the Chinese remainder theorem? It says that because $77 = 7\cdot 11$, where $7$ and $11$ are cop... | {
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