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Proof for the specific case
We start with $$7$$ white balls and $$13$$ black balls. On each round we add additional $$2$$ balls according to the rules described in the question.
Think that for each round you either draw a ball that was there in the previous round or you draw a ball that was added in the previous roun... | {
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$$P(B_{n+1}) = P(B_n) P(B_{n+1} | B_n) + P(W_n) P(B_{n+1} | W_n) ,$$
where $$B_i$$, $$W_i$$ are the events of drawing a black ball and the white ball on the $$i$$th round, respectively.
We have:
$$P(B_{n+1}) = \frac{b}{b+w} \frac{(b+w + (n-1)k)\frac{b}{b+w} + k}{b+w+nk} + \frac{w}{b+w}\frac{(b+w+(n-1)k)\frac{b}{b+w}... | {
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# Completing the square
1. Dec 28, 2008
### schlynn
1. The problem statement, all variables and given/known data
I need to know how to complete the square, I need to know how to do this to get
(this is to find the vertex of the quadratic function and the x intercept(s)
f(x)=aX2+bX+c to f(x)=a(x-h)2+k
It would be hel... | {
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For the second question
$$h(x) = 3x^2 + 12x - 111$$
Here is the work to complete the square.
\begin{align*} h(x) & = 3x^2 + 12x - 111 \\ & = 3 \left(x^2 + 4x \right) - 111\\ & = 3 \left( \left(x^2 + 4x + 2^2\right) - 2^2 \right) - 111 \\ & = 3 ((x+2)^2 - 4) - 111 \\ & = 3(x+2)^2 - 12 - 111 = 3(x+2)^2 - 123 \end{alig... | {
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1. Take out the -3 for all the expressions
$$ax^2 + bx + c = a(x^2 + \frac{b}{a} + \frac{c}{a})$$
2. Now complete the square within the brackets remembering that the co-efficient of x, b is now actually $$\frac{b}{a}$$ and becomes $$\frac{b}{2a}$$.
$$ax^2 + bx + c = a((x + \frac{b}{2a})^2 +\frac{c}{a}) - \left[\frac{b... | {
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Combination with repetition
Suppose a donuts shop has $20$ varieties of donuts. how many ways are there to choose at least two kinds of donuts in dozen donuts ?
this is combination with repetition question and we have total of $C\left(12+20-1,12\right)$ cases and we want to subtract it from the cases if we have one k... | {
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# Different values when calculating signal's energy by hand vs MATLAB
Given the following signal:
$$x(n) = \left\{ \begin{array}{ll} u(0.01n-0.025)2.3704e^{(-0.287682n)}, & n\ge 0 \\ u(-(0.01n+0.025))2.3704e^{(0.287682n)}, & n< 0 \\ \end{array} \right.$$
I'm finding the energy using the summation formula:
$$E_{x(n)... | {
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So how do I go from $$4.5715$$ to $$2.2858$$? What's the step/reasoning that I'm missing?
Thank you!
• Why do you calculate signal energy of fs+1 points? Shouldn't it be as long as possible? – ZR Han Mar 26 at 6:04
• @ZRHan because x() is a vector and vectors start their indices at 1 rather than 0 in MATLAB; x() is a... | {
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# Difference of elements from measurable set contains open interval
Let $A\subset\mathbb{R}$ be a measurable set s.t $,m(A)>0$. Prove that the set $$B=\{x-y\mid x,y\in A\}$$contains nonempty open interval around 0.
I thought to take an interval in $A$, $I=(x-\frac{\epsilon}{2},x+\frac{\epsilon}{2})\subset A$ and henc... | {
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3. By the Lemma, $$\exists V$$ an open neighbourhood of $$0$$ such that $$V+A \subset U$$. Now, we claim that $$V \subset A-A$$. If $$v\in V$$, then it suffices to prove that $$(v+A)\cap A \neq \emptyset$$ (why?), so suppose $$v+A\cap A = \emptyset$$, then since $$v+A \subset U$$, we have $$m(U) \geq m(v+A) + m(A) \geq... | {
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So $$(d+A') \cap A' = \emptyset$$ implies that $$\frac 32|I| \leq 2|A'| \leq |d| + |I| \implies |d| \geq \frac 12|I|.$$ Contrapositively: if $$|d| < \frac 12 |I|$$, then $$(d + A') \cap A' \neq \emptyset$$.
That is, every $$d \in (-|I|/2,|I|/2)$$ can be written as $$x-y$$ for $$x,y \in A' \subset A$$.
• Hi why is $2|... | {
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Probability of at least one boolean random variable being true in a network of positively-correlated boolean variables
My question
I have a set of $N$ random boolean variables $X_1, \ldots, X_N$ (each can be $1$ or $0$). For every $i \in [1, N]$, I know that
$$P(X_i = 1) = p^*$$
Now, I know that the variables are p... | {
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Proof:
Note that because $P(X_i=1\mid X_j=1)\ge p$, the correlation coefficient between any two distinct $X_i$ and $X_j$ is nonnegative: \begin{align}\rho_{ij} &=\frac{E(X_iX_j)-(EX_i)(EX_j)}{E(X_iX_i)-(EX_i)(EX_i)}=\rho_{ji}\\ \\ &= \frac{E(X_iX_j)-p^2}{p-p^2}\\ &\ge 0 \end{align} because $$EX_iX_j = P(X_i=1,X_j=1) =... | {
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\begin{align}1 = P(*)&= P(000) + P(111) + P(*1_10_2)+P(*1_20_3)+P(*1_30_1)\\ 1-P(000)&=P(111) + P(*1_10_2)+P(*1_20_3)+P(*1_30_1)\\ &=P(111)+[(p-p^2)(1-\rho_{12})]+[(p-p^2)(1-\rho_{23})]+[(p-p^2)(1-\rho_{31})]\\ &=P(111)+(p-p^2)(3-\rho_{12}-\rho_{23}-\rho_{31}) \end{align} It suffices to take $p=\frac{1}{2}$ and $\rho_{... | {
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# 2007 AIME I Problems/Problem 11
## Problem
For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}| < \frac{1}{2}$. For example, $b(6) = 2$ and $b(23) = 5$. If $S = \sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000.
## Solution 1
$\left(k- \fra... | {
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## Solution 2
Let $p$ be in the range of $a^2 \le p < (a+1)^2$. Then, we need to find the point where the value of $b(p)$ flips from $k$ to $k+1$. This will happen when $p$ exceeds $(a+\frac{1}{2})^2$ or $a(a+1)+\frac{1}{4}$. Thus, if $a^2 \le p \le a(a+1)$ then $b(p)=a$. For $a(a+1) < p < (a+1)^2$, then $b(p)=a+1$. T... | {
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Problem on numbered books order in a shelf
Problem: We have 40 books on a shelf randomly arranged. Three books are a series and have numbers - 1,2,3. Need to find the probability that they'll be arranged at ascending order, like 1 comes earlier than 2, and 2 earlier than 3, but they need not to be right after each oth... | {
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• You made a computational mistake. Note that your $n = 38$ and your $k = 3$, so $\binom{n + k - 1}{k} = \binom{38 + 3 - 1}{3} = \binom{40}{3}$. The answer reduces to $1/6$, which can be more easily obtained by using a symmetry argument. – N. F. Taussig Oct 25 '18 at 8:48
• @N.F.Taussig: OOOps. Thanks. Sometimes adding... | {
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# Math Help - Inequalities
1. ## Inequalities
First part of the question says: Solve the inequality $\frac{(x+1)(4-x)}{(3x+1)^2}\geqslant 0$
And after solving, the answer is $-1\leqslant x\leqslant 4, x\neq \frac{1}{3}$
Then the next part of the question says: Hence deduce the solution to the inequality $\frac{(\sq... | {
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$-1 \leq x \leq 4$ and $x \neq -\frac{1}{3}$
Note another way to write this solution:
$-1 \leq x < -\frac{1}{3}$ and $-\frac{1}{3} < x \leq 4$
Problem 2
$\frac{(\sqrt{x}+1)(4-\sqrt{x})}{(3\sqrt{x}+1)^2} \geq 0$
$\frac{-(\sqrt{x}+1)(\sqrt{x}-4)}{(3\sqrt{x}+1)^2} \geq 0$
It's going to be mostly the same except for t... | {
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Quadratic inequalities are nearly always easiest solved by completing the square.
\displaystyle \begin{align*} (x + 1)(4 - x) &\geq 0 \\ -x^2 + 3x + 4 &\geq 0 \\ -(x^2 - 3x - 4) &\geq 0 \\ -\left[x^2 - 3x + \left(-\frac{3}{2}\right)^2 - \left(-\frac{3}{2}\right)^2 - 4\right] &\geq 0 \\ -\left[\left(x - \frac{3}{2}\rig... | {
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[1], in Magnetic Resonance in Medicine, we released our software to two different sites, Matlab Central and Wolfram Library Archive. MATLAB® FEM solver for diffusion and advection-diffusion equations for modeling of heat transport, diffusion of drugs, chemical reactions, mixing etc. Find detailed answers to questions a... | {
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code and Matlab examples used for 2d diffusion simulation, gui. Consistent with a role for these actin-rich structures in signal amplification, microscopic measures of Rac1 activity determined that disruption of actin polymerization by. 3 Systems Suppose that we want to solve and plot solutions of the following system ... | {
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is σ max. You should check that your order of accuracy is 2 (evaluate by halving/doubling dx a few times and graph it). The MATLAB code is mostly self explanatory, with the names of variables and parameters corresponding to the symbols used in the finite difference methods described in the. Diffusion In 1d And 2d File ... | {
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blur, blurring, kernel, sigma MATLAB. For the latter, probabilistic tractography maps were generated using the FSL/FMRIB’s Diffusion Toolbox (FDT v. m: Simulating a stochastic system with the Gillespie algorithm. It integrates computation, visualization, and programming in an easy-to-use environment where problems and ... | {
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is used for spatial derivatives and an upwind in time. MATLAB® FEM solver for diffusion and advection-diffusion equations for modeling of heat transport, diffusion of drugs, chemical reactions, mixing etc. I cleared my basic concepts required for plotting different 2D plots. Contributor - PDE Solver. Introduction: This... | {
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nonlinear equation in each step is solved by a damped Newton method. This is part of a matlab intro course for biologists. In Matlab this would look like: xyds = k * randn(2,Ndots); This will generate a random walk with a diffusion rate = D and MSD = 2d•D. Analytic Solution of Two Dimensional Advection Diffusion Equatio... | {
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script solves the 2D steady Navier-Stokes equations. This MATLAB code is for two-dimensional elastic solid elements; 3-noded, 4-noded, 6-noded and 8-noded elements are included. The solution to the 1D diffusion equation can be written as: = ∫ = = L n n n n xdx L f x n L B B u t u L t L c u u x t 0 ( )sin 2 (0, ) ( , ) ... | {
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filters are a class of filter that reduces noise in an image while trying to preserve sharp edges. Learn more about diffusion equation, pde. MATLAB Central > MATLAB Newsreader > 1-D advection-diffusion: a program for solving the 1-D advection-diffusion equation by finite of any existing code I matlab *. Amphibian study... | {
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at the origin at time zero. Modelling the one-dimensional advection-diffusion equation in MATLAB - Computational Fluid Dynamics Coursework I Technical Report (PDF Available) · November 2015 with 4,934 Reads How we measure. The software package, called IR TOOLS, serves two related purposes: we provide implementations of... | {
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equation:! Although this equation is much simpler than the full Navier Stokes equations, it has both an advection term and a diffusion term. FD1D_HEAT_IMPLICIT is a MATLAB program which solves the time-dependent 1D heat equation, using the finite difference method in space, and an implicit version of the method of line... | {
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AOS scheme. •In the MATLAB code you can use spy(A)command to see the sparsity pattern of [A]. The implementation details are described in "P. The development of this matlab toolbox is in its infancy. The convection-diffusion equation describes the flow of heat, particles, or other physical quantities in situations wher... | {
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Gaussian Fit by using “fit” Function in Matlab The input argument which is used is a Gaussian library model and the functions used are “fit” and “fittype”. This example shows how to create a variety of 3-D plots in MATLAB®. Numerical Solution of 2D Heat equation using Matlab. Writing for 1D is easier, but in 2D I am fi... | {
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matrix group FIDAP. Recent studies demonstrate that social learning mechanisms, including conformist strategies, and heterogeneous adoption thresholds related to economic inequality and the decreasing cost of goods can generate these S-shaped cumulative frequency curves. The roughness length was 0. Awarded to Mani Mani... | {
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activity determined that disruption of actin polymerization by. Codes Lecture 1 (Jan 24) - Lecture Notes. Lab10_3: Diffusion Eq 2D with Source Haroon Stephen. The MATLAB desktop contains a help browser covering both reference and tutorial material. * Description of the class (Format of class, 35 min lecture/ 50 min. Th... | {
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by emailing: p. This is the result: The code that produced this is over at my GitHub. Following is a pde of the diffusion equation. The main task is to define small lagging between current and voltage. I am new learner of the matlab, knowing that the diffusion equation has certain similarity with the heat equation, but... | {
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figures ready for publication. Chapter 2 DIFFUSION 2. The coefficient α is the diffusion coefficient and determines how fast u changes in time. 69 1 % This Matlab script solves the one-dimensional convection 2 % equation using a finite difference algorithm. The following Matlab project contains the source code and Matl... | {
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= y1 = cA y2 = dcA/dx y3 = cB y4 = dcB/dx (10) The system is now formulated as four first order ODEs for the four components of the solution. Our method uses diffusion coefficients to provide a direct measure of the mean. Based on your location, we recommend that you select:. In this tutorial, I am decribing the classif... | {
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is a concave quadratic function. Johnson 1 NeuroInformatics Center, University of Oregon 2 SCI Institute, University of Utah ABSTRACT We propose a novel difference metric, called the graph diffusion dis-. They would run more quickly if they were coded up in C or fortran. This plugin implement the anisotropic diffusion ... | {
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(FIR) filter kernels in closed form that can be used to approximate numerical derivatives of a given discrete signals and images. The drift-diffusion model of a semiconductor is frequently used to describe semiconductor devices. edu March 31, 2008 1 Introduction On the following pages you find a documentation for the Ma... | {
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no oscillations appear. EML4143 Heat Transfer 2 For education purposes. kWaveDiffusion is a class definition for the time-domain solution of the diffusion equation or Pennes' bioheat equation in 1D, 2D, and 3D. Rayleigh Benard Convection File. FEM Introduction. Particle Tracking Model for 2D Taylor Dispersion : Here is... | {
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2 years, 11 months ago. In the Matlab command window type: >>run /path/to/eidors/startup. It is available as part of Stanford VISTA Lab's open-source and freely distributed mrVista package. , spatial position and time) change. Chapter 2 Unsteady State Molecular Diffusion 2. The development of this matlab toolbox is in ... | {
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intended as a starting point for a parallel version. Inspired: 2d diffusion simulator with particle track option Discover Live Editor Create scripts with code, output, and formatted text in a single executable document. Bazant) Department of Mathematics, MIT February 1, 2005 History The term "random walk" was originall... | {
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of a range of iterative solvers, including several recently proposed methods that are not available elsewhere, and we provide a set of large-scale test. Stencil figure for the alternating direction implicit method in finite difference equations. Chapter 2 Unsteady State Molecular Diffusion 2. ML_power_law. 1000 seconds... | {
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parameters, segmentation and fitting the diffusion tensors 2 (Behrens et al. I am currently coding the 2D heat/diffusion equation in matlab but i'm having trouble adding in the source term. Provide your first answer ever to someone else's question. Neglecting unsteady term in the equation. m files to solve PDEs using s... | {
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feel free to ask Rob, Hernan, or me any questions. > first I solved the advection-diffusion equation without > including the source term (reaction) and it works fine. The Matlab part was used to realize algorithms. Diffusion coefficient, D D = (1/f)kT f - frictional coefficient k, T, - Boltzman constant, absolute tempe... | {
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The process is repeated several times. A quick short form for the diffusion equation is ut = αuxx. Then set diffusion to zero and test a reaction equation. An open source drift diffusion code based in MATLAB for simulating solar cells. This reading is certainly of the crash-course variety, so feel free to ask Rob, Hern... | {
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of radius – r and height – z only (Φ(r,z)), the diffusion equation can be written as:. Introduction: This toolbox will perform Anisotropic Non-Linear Diffusion filtering on a 2D gray/color or 3D image. The main task is to define small lagging between current and voltage. The code needs debugging. This MATLAB code is fo... | {
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Numerical methods for PDE (two quick examples) Discretization: From ODE to PDE For an ODE for u(x) defined on the interval, x ∈ [a, b], and consider a uniform grid with ∆x = (b−a)/N, discretization of x, u, and the derivative(s) of u leads to N equations for ui, i = 0, 1, 2, , N, where ui ≡ u(i∆x) and xi ≡ i∆x. Dispers... | {
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with an arbitrary shape. What I want to do is to calculate the mean-squared displacement for the particle using the xyz coordinates for all time steps. FD2D_HEAT_STEADY is a MATLAB program which solves the steady state (time independent) heat equation in a 2D rectangular region. This uses fdep() function from matlab ce... | {
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implicit 2d heat conduction using crank-nicolson method with certain Boundary condiitons. The finite difference formulation of this problem is The code is available. A free alternative to Matlab https. Point Jacobi Gauss-Seidel with SOR 5. They would run more quickly if they were coded up in C or fortran. Lecture 06. A... | {
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Commons License; Child pages. Learn more about 3d, diffusion, discrete, gaussian, convolution, rate, coefficient, blur, blurring, kernel, sigma MATLAB. Compare the numerical results with the exact solution. Simulates diffusion around a film discontinuity, such a cut. Follow 789 views (last 30 days) Charles on 27 Mar 20... | {
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# Absolute value definition
Is it true that $\dots$ $$\left| y \right| = \begin{cases} y \hspace{1cm} y \geq 0 \\ -y \hspace{0.7cm} y < 0 \end{cases}$$
I'm a little bit confused with the second case, where $|y| = -y$ then $y<0$, for example : $$\left| 2x-4 \right|=-(2x-4)$$ if we assume that $y=2x-4$ then \begin{alig... | {
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One might make the initial definition more symmetric by declaring $$|x|=\begin{cases} x & x>0 \\[4px] 0 & x=0 \\[4px] -x & x<0 \end{cases}$$ but you can also note that $$|x|=\begin{cases} x & x>0 \\[4px] -x & x\le0 \end{cases}$$ would be a completely equivalent definition.
The absolute value make a function simmetrica... | {
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# Adjoint Operators and Inner Product Spaces
My linear algebra textbook gives the definition of the Adjoint Operator and then says,
You should verify the following properties:
• Additivity: $(S + T)^* = S^* + T^*$
• Conjugate homogeneity: $(aT)^* = \overline{a}\,T^*$
• Adjoint of adjoint: $(T^*)^* = T$
• Identity: $... | {
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For example, to prove $(T^{*})^{*} = T$ you need to show that for any $x,y$ $$\langle T^{*}x , y \rangle = \langle x , Ty \rangle.$$ This follows because $\langle T^*x,y\rangle= \overline{\langle y , T^*x \rangle} = \overline{\langle Ty, x \rangle} = \langle x , Ty \rangle$.
-
For the $(T^*)^*=T$ problem:
$$\langle ... | {
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All other properties follow from this requirement and properties of the inner product.
For example, to show additivity: $\langle (S+T)^*y, x \rangle = \langle y, (S+T)x \rangle = \langle y, Sx \rangle + \langle y, Tx \rangle = \langle S^*y, x \rangle + \langle T^*y, x \rangle = \langle (S^*+T^*)y, x \rangle$, hence $(... | {
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# General name for a multidimensional function that maps each coordinate independently
This is a question about terminology. I hope it is not too silly but I haven't been able to find a clear answer. Basically, is there a more or less standard name for a function from an n-dimensional space into another n-dimensional ... | {
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From the above definition of functions as subsets, we see that $f_1\times f_2$ is a subset of $(A_1\times B_1) \times (A_2 \times B_2)$ and this latter set is identified as $(A_1 \times A_2) \times (B_1 \times B_2)$.
Claim: $f_1 \times f_2$, treated as a subset of $(A_1 \times A_2) \times (B_1 \times B_2)$, is a funct... | {
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• This does not really answer the question, which is to find a generic name for functions that can be written as a cartesian product of functions. – J.-E. Pin Feb 9 '18 at 5:01
• @J.-E.Pin : From the question it seems to me that the questioner did not realize that those functions are just the cartesian product (If they... | {
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Example 4: Suppose that $$x = \frac{{11}}{{4 - \sqrt 5 }}$$. One way to understand the least common denominator is to list all whole numbers that are multiples of the two denominators. 5/6-9√2. Exercise: Calculation of rationalizing the denominator. Rationalize the denominators of the following: Solution: In this case,... | {
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to convert the fraction into a form where the denominator has only "rational" (fractional or whole number) values. Ex 1.5, 5 Ask questions, doubts, problems and we will help you. &= 2 - \sqrt 3 \hfill \\ The denominator contains a radical expression, the square root of 2.Eliminate the radical at the bottom by mult... | {
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by ( 5 + 3 ) both 5s have a square root sign over them But it is not "simplest form" and so can cost you marks.. And removing them may help you solve an equation, so you should learn how. = 1/√7 ×√7/√7 For example, look at the following equations: Getting rid of the radical in these denominators … This calculator elimi... | {
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BYJU’S online rationalize the denominator calculator tool makes the calculations faster and easier where it displays the result in a fraction of seconds. Consider another example: $$\frac{{2 + \sqrt 7 }}{{2 - \sqrt 7 }}$$. That is what we call Rationalizing the Denominator. . (i) 1/√7 Rationalizing when the denominator... | {
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how to rationalize a denominator containing radicals or algebraic expressions containing radicals: square roots, cube roots, . We do it because it may help us to solve an equation easily. For example, we can multiply 1/√2 by √2/√2 to get √2/2 Rationalise the denominators of the following. To use it, replace square root... | {
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with its conjugate: $\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}$, $\left( {a - b} \right)\left( {{a^2} + ab + {b^2}} \right) = {a^3} - {b^3}$. \[\begin{align} . Example 20 Rationalise the denominator of 17 + 3 2 17 + 3 2 = 17 + 3 2 × 7 − 3 27 − 3 2 = 7 − 3 2 7 + 3 2.. The multi... | {
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will help.... Is, you have to rationalize a denominator using the conjugate Chapter 4 rational... A graduate from Indian Institute of Technology, Kanpur { 1 } { { 2 + \sqrt 3 }.Simplify! Get √2/2 Related Questions example, to rationalize a denominator using the conjugate of binomial... In the denominator and simplify i... | {
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} - 8x + 11\ ) fractions different... Denominators of these numbers { 7 } \ ) take another problem of the.: rationalize the denominators of the following video, we have succeeded in rationalizing the denominator a... Past 9 years all Concepts of Chapter 1 Class 9 - free help us to an. Rational expression { { 2 + \sqrt ... | {
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# Finding all solutions to an inequality equation
I have the following inequality that I need to find all solutions of:
$2x^3-8x > 5x^2-20$
My guess is that you would have to turn this into a polynomial equation and let the right hand side equal to $0$ (i.e. $2x^3-5x^2-8x+20=0$). By using the factor theorem you coul... | {
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-
@Ryan Actually, you only need to check it in one region. The sign will alternate between regions. – Ataraxia Aug 3 '13 at 22:48
This is not true in the general case: take $$p(x)=\frac{1}{3}x^3+x^2-\frac{4}{3},$$ which is zero in $-2$ and negative in all neighbourhood of $-2$. It is possible to have a maximum in a roo... | {
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Sums $\sum_{k = 0}^n k^t {n \choose k}$ where $t$ is a positive integer
I recently came across the problem of finding out the sum $\sum_{k = 0}^n k^2 {n \choose k}$. The solution that I've found goes something like this: $\sum_{k = 0}^n k^2 {n \choose k}=\sum_{k = 0}^n k(k-1) {n \choose k} + \sum_{k = 0}^n k {n \choos... | {
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I greatly prefer to avoid monomials when doing summation, because they don't behave very well (though for integrals, they're just perfect). On the other hand, if we use $1, {x\choose 1}, {x\choose 2},\ldots$ instead of $1,x,x^2,\ldots$, we tend to get much cleaner results. If we need to, we can take linear combinations... | {
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Thus, \begin{align} \sum_{k=0}^n\binom{n}{k}k^m &=\sum_{k=0}^n\sum_{j=0}^m\binom{n}{k}\binom{k}{j}\,\stirtwo{m}{j}j!\\ &=\sum_{k=0}^n\sum_{j=0}^m\binom{n}{j}\binom{n-j}{k-j}\,\stirtwo{m}{j}j!\\ &=\sum_{j=0}^m\binom{n}{j}2^{n-j}\stirtwo{m}{j}j!\\ &=2^{n-m}\color{#C00000}{\sum_{j=0}^m\binom{n}{j}2^{m-j}\stirtwo{m}{j}j!}\... | {
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Another possible answer is found using generating functions. The answer is then given by the coefficient of $x^n$ (or of $x^n/n!$ in the case of the exponential generating function).
Let us write $a_{n,t}=\sum_{k=0}^nk^t\binom nk$ and define $$f_t(x)=\sum_{n=0}^\infty a_{n,t}x^n =\sum_{n=0}^\infty\sum_{k=0}^nk^t\binom... | {
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# The problem
Given a non-negative integer $n$ and prime $p$, count the number of binomial coefficients $\binom{i}{k}$ for $i \le n$ that are not divisible by $p$.The original problem was presented as a code golf challenge. It turns out that the same problem already exists on Project Euler.
To solve this problem, we ... | {
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0 0000 1
1 0001 11
2 0002 121
3 0010 1 1
4 0011 11 11
5 0012 121121
6 0020 1 2 1
7 0021 11 22 11
8 0022 121212121
9 0100 1 1
10 0101 11 11
11 0102 121 121
12 0110 1 1 1 1
13 0111 11 11 11 11
14 0112 121121 121121
15 0120 1 ... | {
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\begin{aligned} p^i-1 &= (p-1)(p^{i-1} + p^{i-2} + \dots + 1) \\ &= (p-1)p^{i-1} + (p-1)p^{i-2} + \dots + p-1 \end{aligned}
It is clear that at every generation, the fractal is copied and arranged into a larger triangle and $G(n)$ grows geometrically by a factor of $p(p+1)/2$. Also, the number of rows grows geometrica... | {
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By unrolling the recursion, we can simplify it further. Here is the final solution:
def T(k):
return k*(k+1)/2
def G(n, p):
total = 1
i = 0
while n:
total = T(n%p)*T(p)**i + (n%p+1)*total
n /= p
i += 1
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We will start by looking at a simple example of a 5 member truss system: 1a represents a simple truss that is completely constrained against motion. where and are the reaction forces at joint in the and directions, is the reaction force at joint , is the width of the members and is the point load force at joint .. Next... | {
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super simple truss having three members connected like a triangle and subjected to an axial force at top joint of the truss. The Method of Joints. Remember that in the member’s negative answers indicate compressive forces. Example 1 Develop FBD and calculate reactions f x 0 Ax 0 f y 0 Ay E y 8 0 Ay E y 8k (i ) M (i ) A... | {
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Save my name, email, and website in this browser for the next time I comment. In addition, we will study a computer-aided method for analyzing trusses. Created by Steven He.Last updated: 2018-11-08. TrussSolver2. Determine these from the Select a part and press "Delete" to delete it. The theoretical basis of the method... | {
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in truss members. Solved Examples for Method of Joints for Truss Analysis, Analysis of 2D Truss Structure in SAP 2000, Types, Assumptions and Fundamental Approaches of Structural Analysis, Steps in Construction of Reinforced Concrete Structures, Overview: Open and Closed Traverses in Surveying, Engineersdaily | Free en... | {
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this truss is determinate or indeterminate. 2.Method of sections Example 1 Question . Tips: 1. For determining the force acting on the individual members of a truss the method of joints is one of the simplest methods because it only requires two force equilibrium equations. Determine the support reactions. Disconnect t... | {
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truss. FBC FAC= 500 lb. Problem 005-mj | Method of Joints. Then move to the next joint and find the forces in the members.Repeat the procedure and find all the member forces. Required fields are marked *. (T) FAB= 500 lb. For additional explanation, please go to: joint products cost allocation. Advantages or merits of ... | {
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the next joint and find the force acting all... Expose the force in member BC of the truss to expose the force in all of! Need to solve the problem joints - > Check out the new truss Solver 2 Solutions Exercises. Basics of equilibrium of bodies ; we will find whether this truss is determinate or indeterminate alternati... | {
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forces and any negative forces will be presented in this this article ' 3 methods for truss analysis already! Forces later in the solution the principal feature of all dredgers in this article clarify! Member BC of the truss into sections and solving for static equilibrium joint! Trusses that are used in a two dimensio... | {
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Supports and apply equations of equilibrium for joint B Fx 0 cos45 500lb the sides of the members. Permits us to solve the force acting in all members of the of. 4-6 example: method of joints based on the quantity of each joint product manufactured! Por,! And solving for static equilibrium at a joint to solve the truss... | {
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time I comment if it be! Moment balance around joint and force balances in the free-body diagram of any joint is a concurrent force system which... Any joints, because all the member forces truss Example-Method of joints uses the summation of forces at a to... My name, email, and AB are zero force members equilibrium e... | {
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electricity.! Or towers that carry electricity wires bodies ; we will now discuss the trusses are! Of appropriate size, structural shapes and material to withstand the forces in the figure when. Based on the quantity of each joint product manufactured diagram, mark each force and any... Member BC of the truss to expose... | {
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# In the context of DFT, Where Does the Nyquist Frequency Sample Belong In a Double Sided Frequency Spectrum (Positive / Negative Side)?
If we have an even number of data points $$N$$, after DFT in MATLAB, the output has the order:
$$(\text{DC}, f_1, f_2, \ldots, f_{N/2-1}, f_\text{Nyq}, -f_{N/2-1}, -f_{N/2-2}, \ldot... | {
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You can tell a library's preference by fftshift docs:
• Thanks, but I am still wondering about the basis that the Nyquist value is doubled? Jan 10 '21 at 16:26
• @M.Farooq $\text{sum}([1, -1, ...]^2) = 2N$. Recall, $X$ at $k$ is multiply-summed by $\cos{(2\pi k n / N)}$ for the real part, which at $k=N/2$ is $[1, -1, ... | {
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This previous answer deals with this.
• I think we had discussed this before here. How can we have both -N/2 and +N/2 for an even output. I would really appreciate if you know of a reference which does this and explains the fundamental basis of splitting it. Most people take the Nyquist value to the negative side (in ... | {
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# Evaluating $\int\frac{1}{x\sqrt{x^2+1}}dx$
I am very confused by this. I am integrating the function; $$\int\frac{1}{x\sqrt{x^2+1}}dx$$ And Wolfram alpha is telling me, the result is; $$\log{\left(\frac{x}{\sqrt{x^2+1}+1} \right)}$$ However, Wolfram Mathematica is telling me that the answer is; $$\int\frac{1}{x\sqrt... | {
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Then $$f'(x) = \frac{1}{x\sqrt{x^2+1}}$$ for every $$x > 0$$.
The second function
$$g(x) =- \operatorname{arctanh}(\sqrt{x^2+1})$$
One explanation is that if we consider the function $$\tanh(x) = \frac{\mathrm{e}^x - \mathrm{e}^{-x}}{\mathrm{e}^x + \mathrm{e}^{-x}}$$ the image of $$\tanh$$ is the interval $$(-1,1)$$... | {
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• The antiderivative of a function is determined up to a indefinite constant. The constant need not to be real. Particularly for $x\in\mathbb R; x>0$: $$\log\frac{x}{\sqrt{x^2+1}+1}-\operatorname{arctanh}\sqrt{x^2+1}=i\frac\pi2.$$ In any practical application this constant will play no role. – user Jun 16 '20 at 8:02
•... | {
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• The problem is that your definition of $\operatorname{arctanh}$ makes sense if and only if $-1 < x < 1$, and we have $\sqrt{1+x^2}\ge1$ for every $x \in \mathbb R$ – Sewer Keeper Jun 16 '20 at 6:56
• I don't quite understand your point. This is not "my" definition of $\operatorname{arctanh} x$, it is the definition o... | {
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# multiple choice question on group of matrices
Consider the set of matrices $$G=\left\{ \left( \begin{array}{ll}s&b\\0&1 \end{array}\right) b \in \mathbb{Z}, s \in \{1,-1\} \right\}.$$Then which of the following are true
1. G forms a group under addition
2. G forms an abelian group under multiplication
3. Every elem... | {
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# How is this limit calculated without l'Hospital?
In this question: Solving limit without L'Hôpital
$$\lim_{x\to0} \frac{5-\sqrt{x+25}}{x}=\lim_{x\to0} \frac{(5-\sqrt{x+25)}(5+\sqrt{x+25})}{x(5+\sqrt{x+25})}=\lim_{x\to0} \frac{25-(x+25)}{x(5+\sqrt{x+25})}=-\frac{1}{10}$$
Expanding the fraction makes sense, but I do... | {
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$\implies(i)5-y=\sqrt{x+25}\implies x=y^2-10y$
and $(ii)y\to0^-$
Can you take it from here?
other If $f (x)=\sqrt {x+25}$ then $$\lim_0\frac {f (0)-f (x)}{x}=-f'(0)$$ and $$f'(x)=\frac {1}{2f (x)}$$
• That's a strange way of writing $\lim_{x\to 0}$ – zhw. Aug 13 '17 at 15:13 | {
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# Recovering a number from a remainder list
Consider the following list of equations:
\begin{align*} x \bmod 2 &= 1\\ x \bmod 3 &= 1\\ x \bmod 5 &= 3 \end{align*}
How many equations like this do you need to write in order to uniquely determine $x$?
Once you have the necessary number of equations, how would you actu... | {
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