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# More than continuum many functions
Suppose we have a function $f_\alpha:\mathbb{N} \to o$ for each $\alpha<c^+$ (successor cardinal of $c=2^{\aleph_{0}}$), where $o$ is some ordinal. Show that there exists a set $S \subseteq c^+$ such that $|S|=c^+$ and for every $\alpha,\beta \in S$ where $\alpha<\beta$ we have tha... | {
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I need the following generalization of (a special case of) the Dushnik-Miller theorem: If the set $[c^+]^2$ of $2$-element subsets of $c^+$ is partitioned into countably many pieces $P_n$ ($n\in\omega$), then either there is a set $H\subseteq c^+$ of order-type $c^+$ with $[H]^2$ (the set of $2$-element subsetes of $H$... | {
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• I had the same idea (when I wrote coloring I was meaning partitioning), but must have missed that specific partition theorem because it looks like we did indeed learn it. Thanks! – ctlaltdefeat Jun 8 '13 at 21:25
• @user14111 An alternative to winning the lottery: Find the table of contents of the Handbook, go to the... | {
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If $S_0$ is stationary, then there are a stationary $S_1\subseteq S_0$ and an $\eta<\mathfrak{c}^+$ such that $\eta_\alpha=\eta$ for all $\alpha\in S_1$. There are only $\mathfrak{c}$ distinct possibilities for
$$\left\{\{\beta(\alpha,n,k):k<\ell(n)\}:n\in\omega\right\}\;,$$
so there is $S_2\subseteq S_1$ such that $... | {
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If $o=o'+1$ is a successor ordinal, we will isolate the maximal possible output $M$ (i.e. $o'$). We consider the partial functions $f'_\alpha$ where $M$ is replaced by $\bot$ (undefined). By hypothesis, there is $S'\subseteq c^+$ verifying all the requirements on the $f'_\alpha$'s.
Now for each $g:\mathbb N\to \{o',\{... | {
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# Matrix Multiplication Divide And Conquer | {
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Divide-and-Conquer Matrix Factorization Lester Mackeya Ameet Talwalkara Michael I. The Karatsuba algorithm provides a striking example of how the \Divide and Conquer" technique can achieve an asymptotic speedup over an ancient algorithm. Divide and conquer is a way to break complex problems into smaller problems that a... | {
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Search Selection Matrix Multiplication Convex Hull * * * * * * * * * Selection Find the kth smallest (largest) item in a list. •These are huge matrices, say n ≈50,000. Zima (SCS, UW) Module 4: Divide and Conquer Winter 20207/14. 7: Matrix multiplication using Strassen’s algorithm. Greedy Algorithms Idea: Find solution ... | {
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from the “ordinary” computation (although you. Matrix multiplication. Matrix multiplication is particularly easy to break into subproblems, because it can be performed blockwise. Given a n x n matrix where each of the rows and columns are sorted in ascending order, find the kth smallest element in the matrix. I want to... | {
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we looked at one way of parallelizing matrix multiplication. Perkalian Matrix dengan Divide and Conquer dan Algoritma Strassen Otniel and 13508108 Program Studi Teknik Informatika Sekolah Teknik Elektro dan Informatika Institut Teknologi Bandung, Jl. 8 When not to use D & C Multiplying matrices example • Given: two mat... | {
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in a field, or, more generally, in a ring or even a semiring. Integer Multiplication 3. More on Recurrence Relations. Divide and Conquer is a recursive problem-solving approach which break a problem into smaller subproblems, recursively solve the subproblems, and finally combines the solutions to the subproblems to sol... | {
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2020 - Lecture 13 : Recurrences and Divide and Conquer - PPT, Algorithms Notes | EduRev is made by best teachers of. It consists of rows and columns. Divide-and-Conquer Matrix Factorization Lester Mackeya Ameet Talwalkara Michael I. Shivakumar: Exploiting Geographical Location Information of Web Pages. analysis of a di... | {
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XY. Recipe for solving common divide-and-conquer recurrences: Terms. The problem is not actually to perform the multiplications, but merely to decide the sequence of the matrix multiplications involved. T1 - Error-free transformation of matrix multiplication with a posteriori validation. Divide-and-conquer. Reading: Ch... | {
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substitution method for solving recurrences 4. With divide-and-conquer multiplication, we split each of the numbers into two halves, each with n/2 digits. Greedy Algorithms Idea: Find solution by always making the choice that looks. Brief review of the tridiagonal DC method. Where the idea came from is unclear, however... | {
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the only explicit exam-ples in [9,3] describe Karatsuba multiplication. Introduction. Divide the problem into a number of sub-problems that are smaller instances of the same problem. – The above naturally leads to divide-and-conquer solution: ∗ Divide X and Y into 8 sub-matrices A, B, C, and D. The equation 4. 3 The D ... | {
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multiplication is easy to break into subproblems because it can be performed blockwise. Here the dimensions of matrices must be a power of 2. 1 of Introduction to Algorithms introduces Merge sort algorithm, Chapter 4 "Divide and conquer" introduces The maximum-subarray problem and Strassen's algorithm for matrix multip... | {
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shapes are 2 n × 2 n for some n. Strassens’s Matrix Multiplication • Strassen (1969) showed that 2x2 matrix multiplication can be accomplished in 7 multiplications and 18 additions or subtractions 𝑇𝑛= 7𝑇. On the left is the normal 4-by-4 matrix multiplication. Topic: Divide and Conquer 24 The Divide-and-Conquer way:... | {
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the size at the beginning, we. When working over the integers and taking into account the growth of coefficients, the general bound for matrix multiplication specialises to. This is because there is an overhead of dividing each time, copying, adding, etc. With divide-and-conquer multiplication, we split each of the numbe... | {
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Properties, Multiplication of large integers, Strassen‟s Matrix Multiplication. It uses divide and conquer strategy, and thus, divides the square matrix of size n to n/2. I've implemented the O(log_2 7) Strassen algorithm once (which should be really simple after implementing normal divide and conquer) and after benchm... | {
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inputs, and returns the sum of the results from those. The total time spent is then divided by the. We can treat each element as a row of the matrix. First example: matrix multiplication Matrix multiplication is one of the basic operations that you can do with matrices and a classic problem used in concurrent and paral... | {
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of Algorithms all about? Why do we care about speci cations and proving guarantees? The Karatsuba multiplication algorithm. Divide-and-Conquer algorithsm for matrix multiplication A = A11 A12 A21 A22 B = B11 B12 B21 B22 C = A×B = C11 C12 C21 C22 Formulas for C11,C12,C21,C22: C11 = A11B11 +A12B21 C12 = A11B12 +A12B22 C2... | {
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that has to compute on ‘n’ input the divide and conquer strategy suggest. delete() Creating a new Directory using File. Matrix multiplication. World's Best PowerPoint Templates - CrystalGraphics offers more PowerPoint templates than anyone else in the world, with over 4 million to choose from. Also, observe that divide... | {
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of two matrices using the naive method is O(n 3), whereas using the divide and conquer approach (ie. Contribute to saulmm/Divide-and-conquer development by creating an account on GitHub. Divide and Conquer: 992 24 632 408 1600 272 720 1232 512 0 512 384 460 17 405 497 Could someone tell me what I am doing wrong for div... | {
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different sizes many times. Architecture-efficient Strassen's Matrix Multiplication: A Case Study of Divide-and-Conquer Algorithms By Paul Pauca, Xiaobai Sun, Siddhartha Chatterjee and Alvin Lebeck Abstract. But the algorithm is not very practical, so I recommend either naive multiplication, which runs in $\mathcal{O}(... | {
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the subproblems into a global solution. Convex Hull algorithms (plus more on Mergesort, Quicksort, etc. The divide and conquer strategy •A first example : sorting a set S of values sort (S) = if |S| ≤ 1 then return S else divide (S, S1, S2) fusion (sort (S1), sort (S2)) end if fusion is linear is the size of its parame... | {
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than the original versions and within 2-60% of a high-performance hand crafted implementation. Our goal is to reduce this total time for multiplying two polynomials to using Divide and Conquer. 5 Strassen' matrix multiplication 2. Optimisation of Constant Matrix Multiplication Operation Hardware Using a Genetic Algorit... | {
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to choose from. [35:45] Naive (standard) algorithm for multiplying matrices. Strassens's Matrix Multiplication • Strassen (1969) showed that 2x2 matrix multiplication can be accomplished in 7 multiplications and 18 additions or subtractions 𝑇𝑛= 7𝑇. This document is highly rated by students and has been viewed 263 ti... | {
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its own recursive divide and conquer approach as defined by strassen’s methodology[9][10] to obtain partitioned matrix multiplication. Such systems are able to support a large volume of parallel communication of various patterns in constant time. its just an example of parallel programming. id Penghitungan matrix sanga... | {
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Conquer and Combine. January 2, 2013 January 3, 2013 saeediqbalkhattak How to multiply any two integer using divide & Conquer approach. And this is a super cool algorithm for two reasons. Split each matrix into 4 of size (n / 2) x (n / 2) 2. The multiply() method takes 3 matrices and their indexes and using the divide ... | {
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blockwise. For better under understanding lets see one more sorting technique called quick sort. Architecture-efficient Strassen's Matrix Multiplication: A Case Study of Divide-and-Conquer Algorithms By Paul Pauca, Xiaobai Sun, Siddhartha Chatterjee and Alvin Lebeck Abstract. Consider two matrices: Matrix A have n rows... | {
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and 18 additions or subtractions 𝑇𝑛= 7𝑇. Posts: 31,956; Joined: 06-March 08; Re: Divide and Conquer Matrix Multiplication. I assume from the question that the code has to cope with matrices of arbitrary size up to some reasonably sane limit. Description Presents the mathematical techniques used for the design and an... | {
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Illustration of listFiles() and list() method of F Find SubArray having Maximum Sum using Divide and CREATING A NEW FILE USING java. ; The median of a finite list of numbers can be found by arranging all the numbers from lowest value to highest value and picking the middle one. In the previous post, we discussed some a... | {
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be solved and then a method must be found to combine subsolutions into a solution of a whole. In this note, log will always mean log 2 (base-2 logarithm). The divide and conquer strategy •A first example : sorting a set S of values sort (S) = if |S| ≤ 1 then return S else divide (S, S1, S2) fusion (sort (S1), sort (S2)... | {
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# Toronto Math Forum
## MAT244--2019F => MAT244--Lectures & Home Assignments => Chapter 7 => Topic started by: nadia.chigmaroff on November 03, 2019, 12:04:50 PM
Title: Transforming a system of linear equations to a single higher-order equation
Post by: nadia.chigmaroff on November 03, 2019, 12:04:50 PM
Hi,
In the te... | {
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General criteria: System with constant coefficients could be reduced to a single equation iff each eigenspace is $1$-dimensional. To understand why we need to consider solutions to a homogeneous equation and to a homogeneous system.
For an equation one of the solutions is $t^{m-1} e^{kt}$ where $k$ is characteristic r... | {
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# Questions about torsion of a curve in $\mathbb{R}^3$ and analogues of torsion in higher dimensions
Suppose we have a curve $\alpha(s) : I \to \mathbb{R}^3$ parametrized by arc-length that has nowhere-vanishing second derivative, so that we are able to define the torsion $\tau(s)$ for every $s \in I$. It is clear to ... | {
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Arguably the simplest class of examples are those for which the curvature $\kappa$ and torsion $\tau$ are constants for each curve in the family. Any such curve is a helix (including the degenerate case of zero torsion, which gives a circle) and can be parameterized by $$\alpha(t) := (r \cos t, r \sin t, bt)$$ for some... | {
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So, to produce a family of helices $\color{#bf0000}{\alpha_m(t)}$ with constant prescribed curvature $\kappa$ and varying torsion $\tau$, we need only pick (nonconstant) functions $r(m), b(m)$ that satisfy $$\kappa = \frac{r(m)}{r(m)^2 + b(m)^2}$$ for any prescribed constant $\kappa > 0$. Rearranging shows that this eq... | {
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For a curve that is embedded in $\Bbb R^n$ is there a quantity indicating how far the curve is from being embedded in $\Bbb R^n$, and is this even useful? | {
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Yes, there are higher-order analogues of torsion for Euclidean spaces $\Bbb R^n$, $n > 3$. Recall that in in $\Bbb R^3$ (1) one can always choose (at least for curves with nonvanishing curvature) a unique adapted orthonormal frame $({\bf T}, {\bf N}, {\bf B})$ along a given smooth curve, and (2) derivatives of the curv... | {
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particular generic curves in $\Bbb R^n$ have, and are generically determined by, $n - 1$ curvature functions. (Note that this general pattern captures the $2$-dimensional case, too, in which there is only a single invariant, just the usual (signed) curvature.) | {
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• Maybe I've been working too long on this question, but, I think you want $r(u)/\kappa$ so you get back $\kappa = \kappa$ for the functions $r(u)$ and $c(u)$ you describe. Great answer. It's set me free to work on something else now. – James S. Cook May 28 '15 at 5:41
• Thanks, and I'm glad you found it useful. I've a... | {
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# Periodic Orbit property
A topological space $X$ satisfies "Periodic orbit property", briefly POP, if for every continuous map
$f:X \to X$, there exist a natural number $n$ and a point $x_{0}\in X$ such that $f^{n}(x_{0})=x_{0}$
Obviously fixed point property(FPP) implies POP.
For a natural number $n$,a topologica... | {
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-
Let $R_{\alpha}:S^1\to S^1$ be an irrational rotation, then $R^n_{\alpha}$ has no fixed point for every $n\geqslant 1$ – Juan Valdez Dec 24 '13 at 20:08
So $S^{1}$ is not a POP manifold. But I search for a manifold which is POP but is not n-POP for all $n\in \mathbb{N}$. – Ali Taghavi Dec 25 '13 at 8:00
What are ex... | {
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Proof of 2. We will need the following two observations.
Lemma 1: Let $f_0, f_1$ be linear operators acting on two finite-dimensional vector spaces $V_0, V_1$. If $\text{tr}(f_0^k) = \text{tr}(f_1^k)$ for $k$ between $1$ and $\text{max}(\dim V_0, \dim V_1)$, then $f_0$ and $f_1$ have the same nonzero eigenvalues with ... | {
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$$L(f^k) = \text{tr}(f_0^k) - \text{tr}(f_1^k).$$
By Lemma 2, the eigenvalues of $f_0$ and $f_1$ are all nonzero, so if $f_0$ and $f_1$ have the same nonzero eigenvalues then in particular $\dim V_0 = \dim V_1$. By the contrapositive of Lemma 1, if $\chi(X) = \dim V_0 - \dim V_1 \neq 0$, then there exists some $k$ bet... | {
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# Real VS Complex for integrals: $\int_0^\infty \frac{dx}{1 + x^3}$
The integral $$\int_0^\infty \frac{dx}{1 + x^4} = \frac{\pi}{2\sqrt2}$$ can be evaluated both by a complex method (residues) and by a real method (partial fraction decomposition). The complex method works also for the integral $$\int_0^\infty \frac{dx... | {
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Note that for $a > 0$, $$\int_0^N \frac{1}{x+a}\ dx = \ln(N+a) - \ln(a) = \ln(N) - \ln(a) + o(1)\ \text{as} \ N \to \infty$$ while \eqalign{\int_0^N \frac{x+a}{(x+a)^2 + b^2}\ dx &= \frac{1}{2} \left(\ln((N+a)^2+b^2) - \ln(a^2+b^2)\right)\cr &= \ln(N) - \ln(a^2+b^2) + o(1) \ \text{as} \ N \to \infty\cr} and (if $b > 0$... | {
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- | {
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##### Activity4.2.5
We will use Sage to find the eigenvalues and eigenvectors of a matrix. Let's begin with the matrix $$A = \left[\begin{array}{rr} 1 \amp 2 \\ 2 \amp 1 \\ \end{array}\right] \text{.}$$
1. We can find the characteristic polynomial of a matrix $$A$$ by writing A.charpoly('lam'). Notice that we have to... | {
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3. Let's now use Sage to determine the reduced row echelon form of $$A-I\text{:}$$
What result does Sage report for the reduced row echelon form? Why is this result not correct?
4. Because the arithmetic Sage performs with floating point entries is only approximate, we are not able to find the eigenspace $$E_1\text{.... | {
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# A closed form for $\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(\sqrt{x}\right)\ e^{-x\sqrt{2}}}{x}dx$
Let $\operatorname{erfi}(x)$ be the imaginary error function $$\operatorname{erfi}(x)=\frac{2}{\sqrt{\pi}}\int_0^xe^{z^2}dz.$$ Consider the integral $$I=\int_0^\infty\frac{\sin(x)\ \operatorname{erfi}\left(... | {
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\begin{align}\int_{0}^{\infty} dx \, \text{erfi}(\sqrt{x}) \, e^{-\sqrt{2} x} \, e^{i k x} &=\frac{2}{\sqrt{\pi}} \int_0^1 dt \, \int_{0}^{\infty} dx \, \sqrt{x} e^{x t^2} \, e^{-\sqrt{2} x} \, e^{i k x}\\ &= \frac{2}{\sqrt{\pi}} \int_0^1 dt \, \int_{0}^{\infty} dx \, \sqrt{x} e^{-(\sqrt{2}-t^2-i k) x}\\ &= \int_0^1 \f... | {
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or
$$\frac12 \log{\left [\frac{1+\sqrt{4-2 \sqrt{2}}+\sqrt{2 \left(1-\sqrt{2}+\sqrt{4-2 \sqrt{2}}\right)}}{1+\sqrt{4-2 \sqrt{2}}-\sqrt{2 \left(1-\sqrt{2}+\sqrt{4-2 \sqrt{2}}\right)}}\right ]} \approx 0.625774$$
• @achillehui: you are too kind. You're not bad yourself, by the way. – Ron Gordon Sep 22 '13 at 4:35
• @Ro... | {
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# Thread: probability of 52 card deck
1. ## probability of 52 card deck
I have a few simple probability questions regarding draws from a deck of cards.
1) If two cards are drawn face down, what is the probability that the second card is an ace?
2) If it is known the first card draw is an ace, how would that change t... | {
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I think it's: . $\frac{3}{51}$ . . . . Right!
Just reduce it to $\frac{1}{17}$
3) What is the probability that 2 randomly drawn cards are both aces?
I think it's: . $\frac{4}{52}\cdot\frac{3}{51} \:=\:\frac{1}{221}$ . . . . Yes!
4) If two cards are drawn from a deck,
how many different combinations of the two cards... | {
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1. ## velocity graph
sorry the image is rather small. It shows the velocity of an object between time 0 and 9.
-When does the object obtain its greatest speed? I think it obtains its greatest speed at t=8, since time vs. velocity shows acceleration. Is this right
-The object was at its origin at t=3. When does it re... | {
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The graph is below the axis to the right of t= 6 so the object is moving back to the right. The line passing through the points (5, 2) and (8, -4) is given by the equation
$y= \frac{-4- 2}{8- 5}= \frac{-6}{3}(t- 5)+ 2= -2(t- 5)+ 2= -2t+ 12$. If distance the object will have moved between t= 6 and t= x (which we would l... | {
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# Inspecting the function $f(x)=-x\sqrt{1-x^2}$
We are just wrapping up the first semester calculus with drawing graphs of functions. I sometimes feel like my reasoning is a bit shady when I am doing that, so I decided to ask you people from Math.SE.
I am supposed to draw a graph (and show my working) of the function... | {
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Second derivative \begin{align} \ f''(x) &=(\frac{2x^2-1}{\sqrt{1-x^2}})' \\ & = \frac{(2x^2-1)'\sqrt{1-x^2}-(2x^2-1)(\sqrt{1-x^2})'}{1-x^2} \\ & = \frac{4x\sqrt{1-x^2}-(2x^2-1)\frac{-x}{\sqrt{1-x^2}}}{1-x^2} \\ & = \frac{4x\frac{1-x^2}{\sqrt{1-x^2}}-(2x^2-1)\frac{-x}{\sqrt{1-x^2}}}{1-x^2} \\ & = \frac{\frac{-2x^3+3x}{... | {
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-
This looks very nice. I have only two suggestions: 1) ephasize throughout that you are only considering the interval $[0,1]$. e.g, "in the interval $[0,1]$, $f''(x)=0$ has only the solution $x=0$. 2) When examining the local mins and maxes, it would be easier and more rigorous to just note $f'\le0$ on the interval $[... | {
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(Just don't forget to graph the function too!)
-
Haha, thanks, I've edited the question, as I am supposed to do all the working out. I am more interested in not giving claims I haven't explicitly proven (or at least demonstrated enough), rather than correct answers (which I can, as you pointed out, check with software... | {
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# Math Help - centroid of a hemisphere
1. ## centroid of a hemisphere
I must find the z centroid of a hemisphere with radius a. It's base is on the x-y plane and its dome extends up the z axis. I am using the following equations to determine the centroid.
$\overline{z}=\frac{\int_V\tilde{z} dV}{\int_V dV}$
I am usi... | {
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"openwebmath_perplexity": 484.388693061674,
"openwebmath_score": 0.9029038548469543,
"... |
$= { \int_0^a \rho^3d\rho \int_0^{2\pi}d\theta \int_0^{\pi /2} \sin\phi \cos\phi d\phi \over \int_0^a \rho^2 d\rho \int_0^{2\pi}d\theta \int_0^{\pi /2} \sin\phi d\phi }$
$= { (a^4/4)(2\pi)(1/2) \over (a^3/3)(2\pi)(1)}={3a\over 8}$.
7. I also did it this way
$hemisphere = a^2=y^2+x^2$
$\tilde{z}=\frac{\int_{V}\tilde... | {
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"openwebmath_score": 0.9029038548469543,
"... |
# Documentation
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# diric
Dirichlet or periodic s... | {
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for any nonzero integer N. This function has period 2π for odd N and period 4π for even N. Its peak value is 1, and its minimum value is –1 for even N. The magnitude of the function is 1/N times the magnitude of the discrete-time Fourier transform of the N-point rectangular window. | {
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Did I prove this limit correctly?
Given $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to \infty}b_n = M$ implies that $\lim_{n \to \infty}2a_n + 3b_n = 2L + 3M$
Proof
Assume $\lim_{n \to \infty}a_n = L$ and $\lim_{n \to \infty}b_n = M$ and $\forall \epsilon > 0$, we have $\frac{\epsilon}{4}>0$ and $\frac{\epsilon}{6}>0... | {
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"openwebmath_score": 0.9540525674819946,
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Find the polynomial of the fifth degree with real coefficients such that…
Find the polynomial of the fifth degree with real coefficients such that the number 1 is a zero of the polynomial but to the second degree, the number $1+i$ is a zero but to the first degree and if divided by $(x+1)$ gives the remainder $10$, an... | {
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"openwebmath_score": 0.7958022356033325,
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If $1$ is a zero with multiplicity $2$, the polynomial is divisible by $(x-1)^2$.
If $1+\mathrm i\;$ is a complex root, as the polynomial has real coefficients, its conjugate is another root. Hence the polynomial is divisible by $$(x-1-\mathrm i)(x-1+\mathrm i)=x^2-2x+2.$$ Hence we have $$p(x)=(x-1)^2(x^2-2x+2)(ax+b)$$... | {
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"ta... |
# How can you find the cubed roots of $i$?
I am trying to figure out what the three possibilities of $z$ are such that
$$z^3=i$$
but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you solve this geometrically? Any help would be greatly appreciated.
-
## 6 Answers... | {
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Recalling basic trigonometry, the rectangular coordinates of $A$ are $\left(\cos\frac{\pi}{6}, \sin\frac{\pi}{6}\right)$ (the triangle OMA is rectangle at M). Thus, $A = \cos\frac{\pi}{6} + i \sin\frac{\pi}{6} = \frac{\sqrt{3}}{2} + i\frac{1}{2}$.
If you don't remember the values of $\cos\frac{\pi}{6}$ and $\sin\frac{... | {
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The first equation presents us with two cases:
I -- $\ a \ = \ 0 \$ :
$$a \ = \ 0 \ \ \Rightarrow \ \ b \ ( \ 0 \ - \ b^2 ) \ = \ -b^3 \ = \ 1 \ \ \Rightarrow \ \ b \ = \ -1 \ \ \Rightarrow \ \ z \ = \ 0 - i \ \ ;$$
II -- $\ a^2 \ - \ 3b^2 \ = \ 0$ :
$$a^2 \ = \ 3b^2 \ \ \Rightarrow \ \ b \ ( \ 3 \cdot [3b^2] \ - \... | {
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And we can stop there because this is a polynomial equation of degree 3, and the Fundamental Theorem of Algebra guarantees that it has at most 3 distinct roots. The solution set is thus $z \in \{ \frac{\sqrt{3}}{2} + \frac{1}{2} i, \frac{-\sqrt{3}}{2} + \frac{1}{2} i, -i \}$.
-
The answer of @Petaro is best, because ... | {
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# Please verify my induction proof.
I would like to show that the following statement is true by the principle of mathematical induction (I must only use induction, not other theorems to justify my answer)
If $n$ is odd natural number, then $n^3-n$ is divisible by 24.
My proof:
Base Case: For $n=1, n^3-n = 1-1 = 0$... | {
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"openwebmath_score": 0.5844190716743469,
"ta... |
Remark $\$ A slicker way to prove that equality is to note that $\,f(k) = \rm RHS-LHS$ is at most quadratic (cubic terms cancel), so to verify that it is zero it suffices to show that it has $3$ roots, e.g. verify $\,0 = f(1) = f(2) = f(3).\,$ But this looks like the base of an inductive proof! Indeed, if you study the... | {
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"openwebmath_score": 0.5844190716743469,
"ta... |
# not both $2^n-1,2^n+1$ can be prime.
I am trying to prove that not both integers $2^n-1,2^n+1$ can be prime for $n \not=2$. But I am not sure if my proof is correct or not:
Suppose both $2^n-1,2^n+1$ are prime, then $(2^n-1)(2^n+1)=4^n-1$ have 2 precisely two prim factors. Now $4^n-1=(4-1)(4^{n-1}+4^{n-2}+ \cdots +... | {
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"openwebmath_perplexity": 185.85185609349978,
"openwebmath_score": 0.903502345085144,
"tag... |
# Invertible skew-symmetric matrix
I'm working on a proof right now, and the question asks about an invertible skew-symmetric matrix. How is that possible? Isn't the diagonal of a skew-symmetric matrix always $0$, making the determinant $0$ and therefore the matrix is not invertible?
• Having vanishing diagonal entri... | {
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# Difference between revisions of "2000 AIME I Problems/Problem 15"
## Problem
A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and th... | {
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## Solution 2
To simplify matters, we want a power of $2$. Hence, we will add $48$ 'fake' cards which we must discard in our actual count. Using similar logic as Solution 1, we find that 1999 has position $1024$ in a $2048$ card stack, where the fake cards towards the front.
Let the fake cards have positions $1, 3, 5... | {
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With these recursions and the base case we found earlier, we calculate $a_{2000} = \boxed{927}$. To calculate this by hand, a helpful trick is finding that if $a_n=1$, then $a_{2n-1}=1$ as well. Once we find $a_{1537}=1$, the answer is just $1+(2000-1537)\cdot2$. - Frestho | {
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# sin
Symbolic sine function
## Description
example
sin(X) returns the sine function of X.
## Examples
### Sine Function for Numeric and Symbolic Arguments
Depending on its arguments, sin returns floating-point or exact symbolic results.
Compute the sine function for these numbers. Because these numbers are not... | {
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u = symunit;
syms x
sinf = sin(f)
sinf =
[ sin((pi*x)/180), sin(2)]
You can calculate sinf by substituting for x using subs and then using double or vpa.
## Input Arguments
collapse all
Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables,... | {
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# Is there anything special with a 3x3 matrix where the 3rd row is 0 0 1?
I'm coding using p5.js and I'm looking at this method https://p5js.org/reference/#/p5/applyMatrix
Using that method, I can multiply my current matrix with any matrix of the form:
$$\begin{pmatrix} a & c & e \\ b & d & f \\ 0 & 0 & 1 \\ \end{pm... | {
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"openwebmath_score": 0.6525617241859436,
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The integral $\int_0^8 \sqrt{x^4+4x^2}\,dx$
$\displaystyle \int_0^8 \sqrt{x^4+4x^2}\,dx$.
Alright, so I thought I had this figured out. Here's what I did:
1. I factor out an $x^2$ to get $\sqrt{x^2(x^2+4)}$.
2. I let $x = 2\tan(\theta)$, therefore the integrand is $\sqrt{4\tan^2(\theta) (4\tan^2(\theta) + 4)}$.
3. F... | {
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The main problem I spot with your development is that you forgot to change the $dx$ when you did the change of variable. (And you should be able to evaluate $\sec(\arctan a)$ as well; we'll get to that shortly).
So: you start with $$\int_0^8 \sqrt{x^4+4x^2}\,dx = \int_0^8 \sqrt{x^2(x^2+4)}\,dx.$$
Then you do the chan... | {
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Now, $\sec(0) = 1$. What about $\sec(\arctan(4))$?
Say $\psi$ is an angle with $\tan(\psi)=4$. Take a right triangle with this angle; by scaling, we may assume the opposite side has length $4$ and the adjacent side has length $1$. Then the hypotenuse has length $\sqrt{17}$, so the cosine of $\psi$ is $\frac{1}{\sqrt{1... | {
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Now try a $u$-substitution.
-
Or just spot an antiderivative directly. – Geoff Robinson Jul 29 '11 at 16:24
Alright, so that works and I got the right answer that way, but why didn't my trig substitution work? @Geoff: Well, it looks a lot like an arcsec.. if it were in the form of 1 / sqrt{x^4 + 4x^2}dx, then I'd kno... | {
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# Probability of urns
Four identical urns each contain 3 balls. In urn one, all three balls are black; urn two, 2 black 1 white; urn three, 1 black and 2 are white; urn four, all balls are whites.
One of the urn is picked at random, and a ball is chosen from the urn, which turns out to be white. What is the probabili... | {
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We know one urn has all black balls, a second has 2 black and 1 white, a third has 2 white and 1 black, and the final one has 3 white ones. Counting how many white balls and black balls there are and looking at their proportion relative to the total number of balls, we find that the probability of selecting a white or ... | {
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For $P(W)$ we have:
$P(W)=P(W|U_1)P(U_1)+P(W|U_2)P(U_2)+P(W|U_3)P(U_3)+P(W|U_4)P(U_4)$.
The first summand is 0, cause $P(W|U_1)=0$ since $U_1=(N,N,N)$. On the other hand, $P(W|U_2)=\frac{1}{3}$, $P(W|U_3)=\frac{2}{3}$ and $P(W|U_4)=1$, while $P(U_2)=P(U_3)=P(U_4)=\frac{1}{4}$.
Therefore have $P(W)=\frac{1}{3}\frac{1... | {
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In your case, $m=3$ (there's one urn with only white balls, and that contains 3 balls), and $n=6$ ($0+1+2+3$), thus the probability is $3/6=1/2$.
One could also make a "tree diagram" for this. There are four "branches", presumably equally probable, with each urn appearing with probability $\ \frac{1}{4} \$ from the ra... | {
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# Linear Program (LP) Tutorial¶
For instructions on how to run these tutorial notebooks, please see the README.
## Important Note¶
Please refer to mathematical program tutorial for constructing and solving a general optimization program in Drake.
## Linear Program¶
A linear program (LP) is a special type of optimi... | {
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If we call AddLinearCost again, the total cost stored in prog is the summation of all the costs. You can see that prog.linear_costs() will have two entries.
In [ ]:
cost2 = prog.AddLinearCost(2 * x[1] + 3)
print(f"number of linear cost objects: {len(prog.linear_costs())}")
If you know the coefficient of the linear c... | {
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"lm_q1q2_score": 0.843627927315521,
"lm_q2_score": 0.8558511396138365,
"openwebmath_perplexity": 2649.606721104745,
"openwebmath_score": 0.44954240322113037,
"tags": nu... |
# Call AddConstraint to add a bounding box constraint x[0] >= 1
print(f"number of bounding box constraint objects: {len(prog.bounding_box_constraints())}")
# Call AddLinearConstraint to add a bounding box constraint x[1] <= 2
print(f"number of bounding box constraint objects: {len(prog.bounding_box_constraints())}")
... | {
"domain": "jupyter.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180627184412,
"lm_q1q2_score": 0.843627927315521,
"lm_q2_score": 0.8558511396138365,
"openwebmath_perplexity": 2649.606721104745,
"openwebmath_score": 0.44954240322113037,
"tags": nu... |
In [ ]:
# Add a linear constraint 2x[0] + 3x[1] <= 2, 1 <= 4x[1] + 5y[2] <= 3.
# This is equivalent to lower <= A * [x;y[2]] <= upper with
# lower = [-inf, 1], upper = [2, 3], A = [[2, 3, 0], [0, 4, 5]].
A=[[2., 3., 0], [0., 4., 5.]],
lb=[-np.inf, 1],
ub=[2., 3.],
vars=np.hstack((x, y[2])))
print(linear_constraint)
I... | {
"domain": "jupyter.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180627184412,
"lm_q1q2_score": 0.843627927315521,
"lm_q2_score": 0.8558511396138365,
"openwebmath_perplexity": 2649.606721104745,
"openwebmath_score": 0.44954240322113037,
"tags": nu... |
In [ ]:
# Solve an optimization program
# min -3x[0] - x[1] - 5x[2] -x[3] + 2
# s.t 3x[0] + x[1] + 2x[2] = 30
# 2x[0] + x[1] + 3x[2] + x[3] >= 15
# 2x[1] + 3x[3] <= 25
# -100 <= x[0] + 2x[2] <= 40
# x[0], x[1], x[2], x[3] >= 0, x[1] <= 10
prog = MathematicalProgram()
# Declare x as decision variables.
x =... | {
"domain": "jupyter.org",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9857180627184412,
"lm_q1q2_score": 0.843627927315521,
"lm_q2_score": 0.8558511396138365,
"openwebmath_perplexity": 2649.606721104745,
"openwebmath_score": 0.44954240322113037,
"tags": nu... |
# Prove that the expected length $E [n_{h(k)}]$ of the list containing key $k$ is at most $1 + \alpha$
Theorem: Suppose that a hash function $$h$$ is chosen from a universal collection of hash functions and is used to hash n keys into a table $$T$$ of size $$m$$, using chaining to resolve collisions. If key $$k$$ is n... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9879462219033657,
"lm_q1q2_score": 0.8436198817866394,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 185.1419566804205,
"openwebmath_score": 0.8000425100326538,
"tag... |
So, have 2 cases here,
1. When $$k$$ is NOT table $$T$$: If $$k \notin T$$ , then $$n_{h(k)} = Y_k$$ and $$|{l : l \in T ~and~ l \ne k}| = n$$. Thus $$E [n_{h(k)}] = E[Y_k] \le n/m = \alpha$$, where $$h(k)$$ is the hash function that hashes $$k$$ to a slot. $$h(k)$$ is a simple division method.
2. When $$k$$ in table ... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9879462219033657,
"lm_q1q2_score": 0.8436198817866394,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 185.1419566804205,
"openwebmath_score": 0.8000425100326538,
"tag... |
Consider the case when $$k \in I$$. In this case, there are $$n-1$$ other keys in $$I$$. Whatever be the choice of hash function $$h$$, key $$k$$ always maps to $$S_{h}$$. Therefore, it accounts to a length of $$1$$ at $$S_h$$. To find the expected number of keys in $$I \setminus \{k\}$$ that maps to the same slot as $... | {
"domain": "stackexchange.com",
"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9879462219033657,
"lm_q1q2_score": 0.8436198817866394,
"lm_q2_score": 0.8539127566694178,
"openwebmath_perplexity": 185.1419566804205,
"openwebmath_score": 0.8000425100326538,
"tag... |
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