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distribution. I use center difference for the second order derivative. bit more e cient and can handle Poisson-like equations with coe cients varying in the ydirection, but is also more complicated to implement than the rst approach. Either approach requires O(N2 logN) ops for a 2D Poisson equation, and is easily generalized to Poisson-like equations in rectangular boxes in three or dimensions. In the present study, 2D Poisson-type equation is solved by a meshless Symmetric Smoothed Particle Hydrodynamics (SSPH) method. This has known solution. The result is the conversion to 2D coordinates: m + p(~,z) = pm V(R) -+ V(r,z) =V(7). Particular solutions For the function X(x), we get the eigenvalue problem −X xx(x) = λX(x), 0 < x < 1, X(0) = X(1) = 0. and Lin, P. (1) Here, is an open subset of Rd for d= 1, 2 or 3, the coe cients a, band ctogether with the source term fare given functions on. Usually, is given and is sought. Finite Volume model in 2D Poisson Equation This page has links to MATLAB code and documentation for the finite volume solution to the two-dimensional Poisson equation where is the scalar field variable, is a volumetric source term, and and are the Cartesian coordinates. Finding φ for some given f is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution. Solving 2D Poisson on Unit Circle with Finite Elements. Our analysis will be in 2D. Qiqi Wang 5,667 views. Lecture 04 Part 3: Matrix Form of 2D Poisson's Equation, 2016 Numerical Methods for PDE - Duration: 14:57. The derivation of Poisson's equation in electrostatics follows. 1 From 3D to 2D Poisson problem To calculate space-charge forces, one solves the Poisson's equation in 3D with boundary (wall) conditions: ∆U(x, y,z) =−ρ(x, y,z) ε0. (We assume here that there is no advection of Φ by the underlying medium. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, | {
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equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. SI units are used and Euclidean space is assumed. Journal of Applied Mathematics and Physics, 6, 1139-1159. LaPlace's and Poisson's Equations. Qiqi Wang 5,667 views. Use MathJax to format equations. FINITE DIFFERENCE METHODS FOR POISSON EQUATION 5 Similar techniques will be used to deal with other corner points. Let r be the distance from (x,y) to (ξ,η),. (part 2); Finite Elements in 2D And so each equation comes--V is one of the. In the case of one-dimensional equations this steady state equation is a second order ordinary differential equation. 6 is used to create a velocity eld that satis es the continuity equation and is incompressible. 4, to give the. ( 1 ) or the Green’s function solution as given in Eq. Making statements based on opinion; back them up with references or personal experience. Research highlights The full-coarsening multigrid method employed to solve 2D Poisson equation in reference is generalized to 3D. Poisson's equation is = where is the Laplace operator, and and are real or complex-valued functions on a manifold. This has known solution. The influence of the kernel function, smoothing length and particle discretizations of problem domain on the solutions of Poisson-type equations is investigated. Numerical solution of the 2D Poisson equation on an irregular domain with Robin boundary conditions. :) Using finite difference method to discrete Poisson equation in 1D, 2D, 3D and use multigrid method to accelerate the solving of the linear system. on Poisson's equation, with more details and elaboration. The four-coloring Gauss-Seidel relaxation takes the least CPU time and is the most cost-effective. Viewed 392 times 1. 2D Poisson equation. e, n x n interior grid points). In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for | {
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In this paper, we propose a simple two-dimensional (2D) analytical threshold voltage model for deep-submicrometre fully depleted SOI MOSFETs using the three-zone Green's function technique to solve the 2D Poisson equation and adopting a new concept of the average electric field to avoid iterations in solving the position of the minimum surface potential. nst-mmii-chapte. Eight numerical methods are based on either Neumann or Dirichlet boundary conditions and nonuniform grid spacing in the and directions. Thus, the state variable U(x,y) satisfies:. It arises, for instance, to describe the potential field caused by a given charge or mass density distribution; with the potential field known, one can then calculate gravitational or electrostatic field. Poisson Solvers William McLean April 21, 2004 Return to Math3301/Math5315 Common Material. 2D Poisson Equation (DirichletProblem) The 2D Poisson equation is given by with boundary conditions There is no initial condition, because the equation does not depend on time, hence it becomes a boundary value problem. , , and constitute a set of uncoupled tridiagonal matrix equations (with one equation for each separate value). The diffusion equation for a solute can be derived as follows. 2D Poisson equation. The solution is plotted versus at. This is often written as: where is the Laplace operator and is a scalar function. Solving 2D Poisson on Unit Circle with Finite Elements. From a physical point of view, we have a well-defined problem; say, find the steady-. a second order hyperbolic equation, the wave equation. We then end with a linear algebraic equation Au = f: It can be shown that the corresponding matrix A is still symmetric but only semi-definite (see Exercise 2). If the membrane is in steady state, the displacement satis es the Poisson equation u= f;~ f= f=k. 0004 % Input: 0005 % pfunc : the RHS of poisson equation (i. | {
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# Why does at least two intervals overlap in an uncountable family of intervals?
1. Prove that there are uncountably many intervals $(a,b)$ in $\mathbb{R}, a\neq b$.
2. Assume $X$ be an uncountable family of intervals. Show that there exists at least two intervals in this family that overlap.
First was not difficult. I used the arguments similar to Cantor's Diagonal Argument (used to show $\mathbb{R}$ is uncountable.)
My attempt for 2: Assume $X$ be an uncountable family of pairwise disjoint intervals, i.e. $(a_i,b_i) \cap (a_j,b_j) = \emptyset, \quad \forall i\neq j\in I$. We know there exists a rational number in each of these intervals. This implies there are uncountably many rational numbers. Contradiction, since $\mathbb{Q}$ is countable. Thus, $X$ must have at least two intervals that overlap. $\blacksquare$
Is there any problem with this reasoning?
• First is a consequence of $(a,b)\mapsto b-a$ being a surjection. – Git Gud Sep 29 '13 at 11:33
• So would it be wrong to argue the way I did? – math Sep 29 '13 at 11:35
• Probably not, but it seems to me too complicated to prove what you want to prove. – Git Gud Sep 29 '13 at 11:36
• Looks fine to me. What about the argument made you feel uncertain? – Callus - Reinstate Monica Sep 29 '13 at 11:38
• @Callus, are you asking me or Git Gud? – math Sep 29 '13 at 11:45 | {
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For (1), you certainly could use a diagonal argument directly to prove that there is no surjection from $\mathbb{N}$ onto the set of intervals, or as Git Gud points out you could instead use the existence of a surjection from the set of intervals to $\mathbb{R}$ and then appeal to the nonexistence of a surjection $\mathbb{N} \to \mathbb{R}$. It is common to use Cantor's theorem on the uncountability of the reals as a "black box" in this way.
Your proof for (2) is perfectly fine. You could also get the contradiction by showing that $X$ is countable after all, rather than by showing that $\mathbb{Q}$ is uncountable, but this choice is just a matter of taste.
• For 1), why not just say that $(0,x)$ is an interval for any $x>0$, giving us directly an injection of $\mathbb R^+$ into the set of intervals? (Even simpler than the route with surjections.) – Andrés E. Caicedo Sep 29 '13 at 16:45
• A third method would simply use the surjection from intervals to reals given by $(a,b) \mapsto a$. – Trevor Wilson Sep 29 '13 at 16:53
• @Asaf I think that depends on whether or not it starts with "let $X$ be an uncountable family of pairwise disjoint intervals" like the OP's proof does. – Trevor Wilson Sep 29 '13 at 17:18 | {
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K sp = 1.5 x 10 -10. Let [Ag+] = x. EduRev is a knowledge-sharing community that depends on everyone being able to pitch in when they know something. Justify your answer with a calculation. b. Molar mass of AgCl is 143.321 g /mol. Silver nitrate (which is soluble) has silver ion in common with silver chloride. AgCl is 1: 1 type salt , therefore. Does the solubility of this salt increase or decrease, compared to its solubility in pure water? 0.01 M C a C l 2 C. Pure water. A. var js, fjs = d.getElementsByTagName(s)[0]; Another way to prevent getting this page in the future is to use Privacy Pass. You can study other questions, MCQs, videos and tests for NEET on EduRev and even discuss your questions like 2) The solubility of AgCl in pure water is 1.3 x 10-5 M. Calculate the value of K sp. mol.wt. of [SO4—] must be greater than 1.08 x 10-8 mole/litre. x^2 =1.8 x 10 ^-10. 0.01 M N a 2 S O 4 B. Calculate its solubility in 0.01M NaCl aqueous solution. Now, let's try to do the opposite, i.e., calculate the K sp from the solubility of a salt. The solubility of insoluble substances can be decreased by the presence of a common ion. Performance & security by Cloudflare, Please complete the security check to access. The solubility product of $\ce{AgCl}$ in water is given as $1.8\cdot10^{-10}$ Determine the solubility of $\ce{AgCl}$ in pure water and in a $0.25~\mathrm{M}$ $\ce{MgSO4}$ solution. fjs.parentNode.insertBefore(js, fjs); (function(d, s, id) { Your IP: 37.252.96.253 Apart from being the largest NEET community, EduRev has the largest solved Calculate the solubility of AgCl in a 0.1 M NaCl solution. Explanation: Jamunaakailash … L6 : Solubility in water - Sorting Materials, Science, Class 6, Solubility and Solubility product - Ionic Equilibrium, Solubility and solubility product - Ionic Equilibrium. is done on EduRev Study Group by NEET Students. Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water - 6820732 1. If you are at an office or shared | {
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of solubility of agcl in 0.1m agno3 and in pure water - 6820732 1. If you are at an office or shared network, you can ask the network administrator to run a scan across the network looking for misconfigured or infected devices. if (d.getElementById(id)) return; c. If 50.0 ml of 0.10M AgNO3 are added to 50.0 ml of 0.15M NaCl will a precipitate be formed? For the precipitation of BaSO4 from the solution of BaCl2, the conc. Calculate the solubility of AgCl (K sp = 1.8 x10-10) in pure water. are solved by group of students and teacher of NEET, which is also the largest student Does the solubility of this salt increase or decrease, compared to its solubility in pure water? js.src = "//connect.facebook.net/en_US/sdk.js#xfbml=1&version=v2.10"; Join now. First, write the BALANCED REACTION: Next, set up the SOLUBILITY PRODUCT EQUILIBRIUM EXPRESSION: It is given in the problem that the solubility of AgCl is 1.3 x 10-5. Want to see the step-by-step answer? … Calculate solubility of AgCl in a) pure water b) 0.1 M AgNO 3 c) 0.01 M NaCl Solution ) a) Solubility of AgCl in pure water-Solubility product ‘K sp ‘ = 1.5 x 10-10. 1. Cloudflare Ray ID: 5f8635b20c77ff38 Check out a … Question 2)Solubility product of AgCl is 1.5 x 10-10 at 25 0 C . If the answer is not available please wait for a while and a community member will probably answer this over here on EduRev! Again, these concentrations give the solubility. Present in silver chloride are silver ions (A g +) and chloride ions (C l −). To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. of Mg(OH)2 = 24 + (16 + 1)2 = 24 + 34 = 58, S in mole /litre = ( S in gm /litre) / mol.wt.= 9.57 x 10 -3 / 58, S is less than 0.02 , so 4S3 is neglected, S in gm / litre = 1.49 x 10 -5 x 58 = 86.42 x 10-5. because S <<< 0.1 and solubility of AgCl decreases in presence of AgNO3 due to common ion effect(common ion is | {
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<<< 0.1 and solubility of AgCl decreases in presence of AgNO3 due to common ion effect(common ion is Ag+).Therefore. By continuing, I agree that I am at least 13 years old and have read and a. Then [Cl-] = x, and substituting in the Ksp expression we have (x)(x) = 1.8 x 10 ^-10. S = √ K sp = √ 1.5x 10 -10 The molar solubility of AgCl = [Ag+] in pure water . Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. Ask your question. community of NEET. The Questions and Answer. check_circle Expert Answer. K sp = S 2. A g C l is not soluble in water. x = 1.342 x 10 ^-05 moles dm^-3 = molar solubility of AgCl. Again, these concentrations give the solubility. Question bank for NEET. You may need to download version 2.0 now from the Chrome Web Store. Log in. To calculate the ratio of solubility of AgCl in 0.1M AgNO^3 and in pure water what you need to put in the back of your mind is that water has a constant concentration of 1.0 x 10^-7M. Join now. }(document, 'script', 'facebook-jssdk')); Online Chemistry tutorial that deals with Chemistry and Chemistry Concept. • Click hereto get an answer to your question ️ The solubility product of AgCl in water is 1.5 × 10^-10 . d. Calculate the solubility of AgCl in a 0.60 M NH 3 solution. Solubility of AgCl will be minimum in _____. varunnair3842 varunnair3842 26.11.2018 Chemistry Secondary School Calculate the ratio of solubility of agcl in 0.1m agno3 and in pure water 2 See answers shauryakesarwani21 shauryakesarwani21 Answer: 1.34 X 10^-4. Calculate the solubility of AgCl in a 0.20 M AgNO3 solution. • Completing the CAPTCHA proves you are a human and gives you temporary access to the web property. soon. js = d.createElement(s); js.id = id; Log in. Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? agree to the. because S <<< 0.01 and solubility of AgCl decreases in presence of NaCl due to common ion effect(common ion is Cl–).Therefore, ionic product of [Ba++] [SO4– –] > K sp of BaSO4. See Answer . | {
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effect(common ion is Cl–).Therefore, ionic product of [Ba++] [SO4– –] > K sp of BaSO4. See Answer . Answers of Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? If you are on a personal connection, like at home, you can run an anti-virus scan on your device to make sure it is not infected with malware. Question. So with that information, we can say that the solubility of AgCl in 0.1M AgNO^3 is given as 0.1 M. So to determine the ratio of solubility of AgCl in 0.1M AgNO^3 and water all we need to do is; This discussion on Calculate the ratio of solubility of AgCl in .1M AgNo3 and in pure water? | {
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# Find all real numbers $a_1, a_2, a_3, b_1, b_2, b_3$.
Find all real numbers $$a_1, a_2, a_3, b_1, b_2, b_3$$ such that for every $$i\in \lbrace 1, 2, 3 \rbrace$$ numbers $$a_{i+1}, b_{i+1}$$ are distinct roots of equation $$x^2+a_ix+b_i=0$$ (suppose $$a_4=a_1$$ and $$b_4=b_1$$).
There are many ways to do it but I've really wanted to finish the following idea:
From Vieta's formulas we get:
\begin{align} \begin{cases} a_1+b_1=-a_3 \ \ \ \ \ \ \ \ (a) \\a_2+b_2=-a_1\ \ \ \ \ \ \ \ (b)\\a_3+b_3=-a_2\ \ \ \ \ \ \ \ (c)\\a_1b_1=b_3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (d)\\a_2b_2=b_1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (e)\\a_3b_3=b_2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (f)\end{cases} \end{align} First we notice that each $$b_i$$ is nonzero. Indeed, suppose $$b_1=0$$. Then from (d) and (f) we deduce that $$b_3=0$$ and $$b_2=0$$, so from (a), (b), (c) we get $$a_1=-a_3=-(-a_2)=-(-(-a_1))=-a_1$$, hence $$a_1=0$$ which is impossible.
Now, from (a), (b), (c), (d), (e), (f) we obtain: \begin{align} \begin{cases} a_1+b_1-a_1b_1=-a_3-b_3 \ \ \ \ \ \ \ \ \\a_2+b_2-a_2b_2=-a_1-b_1\ \ \ \ \ \ \ \ \\a_3+b_3-a_3b_3=-a_2-b_2\end{cases}, \end{align} so: \begin{align} \begin{cases} (b_1-1)(a_1-1)=1-a_2 \ \ \ \ \ \ \ \ \\(b_2-1)(a_2-1)=1-a_3\ \ \ \ \ \ \ \ \\(b_3-1)(a_3-1)=1-a_1\end{cases}. \end{align} Therefore: \begin{align*} (b_1-1)(b_2-1)(b_3-1)(a_1-1)(a_2-1)(a_3-1)=(1-a_1)(1-a_2)(1-a_3), \end{align*} which implies: \begin{align*} \bigl((a_1-1)(a_2-1)(a_3-1)\bigr)\bigl((b_1-1)(b_2-1)(b_3-1)+1\bigr)=0. \end{align*}
I got stuck here. Is it possible to prove that in this case $$b_i=0$$ is the only solution to equation $$(b_1-1)(b_2-1)(b_3-1)=-1$$ or maybe get contradiction in some other way? If so, we can assume that $$a_1=1$$ and from here we can easily show that also $$a_2=a_3=1$$, so $$b_1=b_2=b_3=-2$$. | {
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Since $$(b_1-1)(b_2-1)(b_3-1)>-1$$ for every $$b_i>0$$ and $$(b_1-1)(b_2-1)(b_3-1)<-1$$ for every $$b_i<0$$, it suffices to prove that the signs of $$b_1, b_2, b_3$$ can't be different but I don't know how to do it. I also found out that $$(b_1+1)^2+(b_2+1)^2+(b_3+1)^2=3$$, so $$b_i\in [-\sqrt{3}-1, \sqrt{3}-1]$$ but I don't know if we can use it somehow.
Well, here is a way to proceed after you note none of the $$b_i$$ are zero. Hints:
1) Multiplying the last three equations, $$(d)\times (e)\times (f)$$ gives $$a_1a_2a_3=1$$.
2) Now, multiplying just any two among those, e.g. $$(d)\times (e)$$ and using the result 1) above gives $$a_2b_2=a_3b_3$$, by symmetry and simplification using $$(d), (e), (f)$$ gives $$b_i = b$$, for some non-zero constant $$b$$, and hence $$a_i=1$$.
3) Now it is easy to conclude $$b=-2$$ from any of the first three equations. Hence $$(a_i, b_i)=(1, -2)$$.
• Yes I've also done this but I was specifically asking about the idea with equation $(b_1-1)(b_2-1)(b_3-1)=-1$. Does it mean it's not possible to finish that solution? – glopf Jun 22 at 23:02
• By itself, $(b_1-1)(b_2-1)(b_3-1)=-1$ does not allow one to conclude anything useful, you have to use some among equations $(a)-(f)$ in addition to draw a contradiction. Then why not solve $(a)-(f)$, which is what is really needed anyway? – Macavity Jun 23 at 16:57 | {
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Smallest number of points on plane that guarantees existence of a small angle
What is the smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $18^\circ$?
It is clear that $n>9$, since the vertices of a regular 9-gon is a counterexample. One can prove using pigeonhole principle that $n\le 11$. Take a edge of the convex hull of points. All the points lie to one side of this line. cut the half-plane into 10 slices of $18^\circ$ each. There can't be any points in the first and last slice. Thus, by pigeonhole principle some slice contains more than one point. So we have an angle of size at most $18^\circ$.
Now, for $n=10$, I can not come up with a counterexample nor a proof of correctness. Any ideas?
Hmm this appears pretty complicated. I hope there is a simpler solution.
Theorem. For any $n\ge3$ points on a 2D plane, we can always find 3 points $A,B,C$ such that $\angle ABC\le\frac{180°}n$. (OP's solution follows with $n=10$.)
Proof. Again we take an edge of the convex hull and all points lie on the other side of this edge. Let's say points $A$ and $B$ are on this edge.
Partition the half-plane into $n$ regions about point $A$. Name each region $A_1,\dotsc,A_n$, where a point $X$ inside $A_i$ means $\frac{180°}n(i-1)<\angle XAB\le \frac{180°}n\cdot i$.
Do the same about point $B$. This will create $n$ other region $B_1,\dotsc,B_n$, where point $X$ in $B_j$ means $\frac{180°}n(j-1)<\angle XBA\le \frac{180°}n\cdot j$.
If two points $X,Y$ can be found in the same region $A_i$, then we also have $\angle XAY \le \frac{180°}n$. That means only one point can exist in each of the region $A_i$ (there are $n-2$ points and $n-1$ valid regions). Same for $B_j$.
$A_i$ and $B_j$ may overlap, though only when $i+j< n+2$ (a): | {
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$A_i$ and $B_j$ may overlap, though only when $i+j< n+2$ (a):
\begin{array}{c|ccccccc} & A_2 & A_3 & A_4 & \cdots & A_{n-2} & A_{n-1} & A_n \\ \hline B_2 & ✓ & ✓ & ✓ & \cdots & ✓ & \color{red}{✓} & ✗ \\ B_3 & ✓ & ✓ & ✓ & \cdots & \color{red}{✓} & ✗ & ✗ \\ B_4 & ✓ & ✓ & ✓ & \cdots & ✗ & ✗ & ✗ \\ \vdots & \vdots & \vdots & \vdots & ⋰ & \vdots & \vdots & \vdots \\ B_{n-2} & ✓ & \color{red}{✓} & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_{n-1} & \color{red}{✓} & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ B_n & ✗ & ✗ & ✗ & \cdots & ✗ & ✗ & ✗ \\ \end{array}
we immediately see that the points can only exist in the overlapping regions $A_i B_{n+1-i}$, to ensure no two point occupy the same column ($A_i$) or same row ($B_j$).
So for any $X$ in $A_i B_{n+1-i}$, we have:
\begin{align} \angle XAB &> \frac{180°}n(i-1) \\ \angle XBA &> \frac{180°}n(n-i) \\ \end{align}
For the triangle $\triangle ABX$, we need $\angle XAB + \angle XBA + \angle AXB = 180°$, thus
$$\angle AXB < 180° - \frac{180°}n(i-1) - \frac{180°}n(n-i) = \frac{180°}n.$$
Note:
(a) if point $X$ exists in both $A_i$ and $B_j$ with $i+j\ge n+2$ then $$\angle XAB + \angle XBA > \frac{180°}n(i+j-2) \ge 180°,$$ which is impossible.
Suppose there are $10$ points. Let us construct the convex hull of these points. It has at most $10$ sides. This means it must have an angle that is at most $180° - 360°/10 = 144°$. Let us choose the vertex corresponding to such angle as point $A$, and its 2 neighboring points along the convex hull as points $B$ and $C$. In other words, we have situation pictured here:
Now, from $7$ points inside the angle $\angle CAB$ let us draw 7 lines to point $A$. This will divide the angle into 8 smaller angles, and one of them will be at most $144°/8 = 18°$.
Therefore, since there is a counterexample for $n=9$ (a regular 9-gon), the answer is
$$n=10$$
$$GENERALIZATION$$
The smallest number $n$, that in any arrangement of $n$ points on the plane, there are three of them making an angle of at most $180°/k$ is $$n=k$$. | {
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The proof is virtually the same as for the case $k=10$.
• @Untitled can you perhaps review two answers and see if they solved the problem? :) – VividD May 2 '15 at 8:40 | {
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# Proving formula for product of first n odd numbers
I have this formula which seems to work for the product of the first n odd numbers (I have tested it for all numbers from $1$ to $100$):
$$\prod_{i = 1}^{n} (2i - 1) = \frac{(2n)!}{2^{n} n!}$$
How can I prove that it holds (or find a counter-example)?
• This sounds like homework. Did you try induction? – cardinal Oct 2 '11 at 1:20
• @Pedro: Do you mean $$\left(\prod_{i=1}^n2i\right) - 1$$ (which is what you wrote) or $$\prod_{i=1}^n(2i-1)$$(which is what your title suggests)? – Arturo Magidin Oct 2 '11 at 1:25
• It is not, I stumbled upon this formula by accident and was wondering. The proof was fairly simple, I suppose I should have thought more about it – Pedro Oct 2 '11 at 1:28
• @ArturoMagidin: I meant the latter, sorry for the ambiguity – Pedro Oct 2 '11 at 1:30
• As a note: what you have is sometimes termed as the "double factorial". – J. M. is a poor mathematician Oct 2 '11 at 9:33
The idea is to "complete the factorials":
$$1\cdot 3 \cdot 5 \cdots (2n-1) = \frac{ 1 \cdot 2 \cdot 3 \cdot 4 \cdots (2n-1)\cdot (2n) }{2\cdot 4 \cdot 6 \cdots (2n)}$$
Now take out the factor of $2$ from each term in the denominator:
$$= \frac{ (2n)! }{2^n \left( 1\cdot 2 \cdot 3 \cdots n \right)} = \frac{(2n)!}{2^n n!}$$
A mathematician may object that there is a small gray area about what exactly happens between those ellipses, so for a completely rigorous proof one would take my post and incorporate it into a proof by induction.
For the induction argument, \begin{align*} \prod_{i=1}^{n+1}(2i-1)&=\left(\prod_{i=1}^n(2i-1)\right)(2n+1)\\ &= \frac{(2n)!(2n+1)}{2^n n!} \end{align*} by the induction hypothesis. Now multiply that last fraction by a carefully chosen expression of the form $\dfrac{a}a$ to get the desired result.
$$\Pi_{k=1}^n(2k-1) = \Pi_{k=1}^n(2k-1) \frac{\Pi_{k=1}^n(2k)}{\Pi_{k=1}^n(2k)} =\frac{\Pi_{k=1}^{2n}k}{2^n\Pi_{k=1}^n k} = \frac{(2n)!}{2^n n!}$$ | {
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• This is the same answer as the one given four years ago by Ragib, only with fewer explanations. It's good that you want to help by answering questions, but maybe you could consider answering some that haven't been answered yet? – mrf Aug 31 '15 at 20:42 | {
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# help:how to slove this integral
Note by Abhinavyukth Suresh
9 months, 2 weeks ago
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Simply use the Gamma Function. After the substitution $ar^{4}=t$, the integral becomes $\frac{1}{4a^{\frac{3}{4}}}\int_{0}^{\infty}t^{-\frac{1}{4}}e^{-t}dt$. The answer comes out to be $\frac{\Gamma\left(\frac{3}{4}\right)}{4a^{\frac{3}{4}}}$ | {
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- 9 months, 2 weeks ago
You could ask @Aruna Yumlembam - he's an expert on Gamma Functions - he likes them @Abhinavyukth Suresh
- 9 months, 2 weeks ago
- 9 months, 2 weeks ago
He'll probably show the entire proof - see his notes. @Aaghaz Mahajan
- 9 months, 2 weeks ago
There is no "proof" neede here. Simply substitute $ar^{4}=t$ and then the answer follows
- 9 months, 2 weeks ago
Ok. But I believe he should come. He's good at this stuff. I've read his notes - and gave me a interesting infinite series / function in one of my notes.
- 9 months, 2 weeks ago
Ok. Although i dont see what else might be needed in the proof.
- 9 months, 2 weeks ago
Maybe his insight into the Gamma Function?
- 9 months, 2 weeks ago
Also, you could learn a thing or two from him...
Not saying you're bad or anything.
- 9 months, 2 weeks ago
so, isn't there any other method than using gamma function to solve this integral?
- 9 months, 2 weeks ago
The proof given by Mr.Aaghaz is correct and mine is same too .Yet using this very idea we can prove this result, $\frac{\Gamma(1/n)}{n}\zeta(1/n)=\int_0^\infty\frac{1}{e^{x^n}-1}dx$, giving us, $\frac{1}{n}\int_0^\infty\frac{x^{1/n-1}}{e^x-1}dx=\int_0^\infty\frac{1}{e^{x^n}-1}dx$as the result.
- 9 months, 2 weeks ago
Yeah we can prove the identity by summing an infinite GP too. Also these types of integrals are known as Bose Einstein Integrals
- 9 months, 2 weeks ago
Thanks a lot.But can you please give me a your solution to your problem How is this ??!!
- 9 months, 2 weeks ago
Sure. Observe that $\sum_{n=1}^{\infty}e^{-nx}\ =\ \frac{1}{e^{x}-1}$
Using this, we have $\int_{0}^{\infty}\frac{x^{t-1}}{e^{x}-1}dx=\int_{0}^{\infty}\left(\sum_{n=1}^{\infty}x^{t-1}e^{-nx}\right)dx$
Swapping the integral and summation , and using the identity $\int_{0}^{\infty}x^{a}e^{-bx}dx\ =\ \frac{a!}{b^{a+1}}$ we will arrive at the answer.
- 9 months, 2 weeks ago | {
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- 9 months, 2 weeks ago
Mr. Yajat Shamji,if people provides a solution to any problem you must try and appreciate it and not discourage them.Please don't repeat such acts.
- 9 months, 2 weeks ago
I'm not. I.. was thinking of you and I read your profile so I thought I could bring you over. After all, you like to contribute, right?
Also, I wasn't discouraging @Aaghaz Mahajan's solution, I..
- 9 months, 2 weeks ago
- 9 months, 2 weeks ago
@Aruna Yumlembam - @Abhinavyukth Suresh needs your help on solving this integral - needs the Gamma Function and a full proof. | {
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How can you show by hand that $e^{-1/e} < \ln(2)$?
By chance I noticed that $$e^{-1/e}$$ and $$\ln(2)$$ both have decimal representation $$0.69\!\ldots$$, and it got me wondering how one could possibly determine which was larger without using a calculator.
For example, how might Euler have determined that these numbers are different?
• Probably the easiest approach is to show that $$\frac{1}{e}+\ln(\ln(2))$$ is positive. – Peter Aug 29 at 18:06
• What do exactly mean the dots? Namely, do you want to prove that $.692200<e^{-1/e}<.692201$? Or what you want to prove is the inequality $e^{-1/e}<\ln 2$? – ajotatxe Aug 29 at 18:07
• @ajotatxe The challenge is to show $$e^{-1/e}<\ln(2)$$ – Peter Aug 29 at 18:08
• I'm happy to change the title. I was just trying to point out how remarkably close they are. – Kevin Beanland Aug 29 at 18:12
• Thanks. I just changed it. – Kevin Beanland Aug 29 at 18:14
Here's a method that only uses rational numbers as bounds, keeping to small denominators where possible. It is elementary, but finding sufficiently good bounds requires some experimentation or at least luck.
We instead establish the inequality $$-\log \log 2 < \frac{1}{e}$$ of positive numbers, which we can see to be equivalent by taking the logarithm of and then negating both sides. Our strategy for establishing this latter inequality (which is equivalent to yet another inequality suggested in the comments) is to use power series to produce an upper bound for the less hand side and a larger lower bound for the right-hand side. To estimate logarithms, we use the identity $$\log x = 2 \operatorname{artanh} \left(\frac{x - 1}{x + 1}\right) ,$$ which yields faster-converging series and so lets us use fewer terms for our estimate. | {
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Firstly, $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} .$$ Then, since the power series $$\operatorname{artanh} u = u + \frac{1}{3} u^3 + \frac{1}{5} u^5 + \cdots$$ has nonnegative coefficients, for $$0 < u < 1$$ any truncation thereof is a lower bound for the series, and in particular $$\log 2 = 2 \operatorname{artanh} \frac{1}{3} > 2\left[\left(\frac{1}{3}\right) + \frac{1}{3} \left(\frac{1}{3}\right)^3 + \frac{1}{5} \left(\frac{1}{3}\right)^5\right] = \frac{842}{1215} .$$ We'll use the same power series to produce an upper bound for $$-\log \log 2$$, but since we're nominally computing by hand and want to avoid computing powers of a rational number with large numerator and denominator, we'll content ourselves with a weaker rational lower bound for which computing powers is faster: Cross-multiplying shows that $$\log 2 > \frac{842}{1215} > \frac{9}{13}$$ and so $$-\log \log 2 < -\log \frac{9}{13} = 2 \operatorname{artanh} \frac{2}{11} .$$ This time, we want an upper bound for $$2 \operatorname{artanh} \frac{2}{11}$$. When $$0 < u < 1$$ we have $$\operatorname{artanh} u = \sum_{k = 0}^\infty \frac{1}{2 k + 1} u^{2 k + 1} < u + \frac{1}{3} u^3 \sum_{k = 0}^\infty u^{2k} = u + \frac{u^3}{3 (1 - u^2)},$$ and evaluating at $$u = \frac{2}{11}$$ gives $$2 \operatorname{artanh} \frac{2}{11} < \frac{1420}{3861} < \frac{32}{87} .$$ On the other hand, the series for $$\exp(-x)$$ alternates, giving the estimate $$\frac{1}{e} = \sum_{k=0}^\infty \frac{(-1)^k}{k!} > \sum_{k=0}^7 \frac{(-1)^k}{k!} = \frac{103}{280} .$$ Combining our bounds gives the desired inequality $$-\log \log 2 < \frac{32}{87} < \frac{103}{280} < \frac{1}{e} .$$
I'm assuming (partly because he did create tables) that Euler would know the values of $$e$$ and $$\ln(2)$$ to at least 4 decimal places, and your "by hand" guy should also know those. Explicitly calculating $$e^{-1/e}$$, however, hast to be declared "too hard"; otherwise the problem becomes a triviality. | {
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The following reasoning is well within Euler's knowledge base, and requires no really difficult arithmetic:
Express $$e^{-1/e}$$ as its Taylor series, and expand each term as a Taylor series for $$e^{-n}$$: $$\begin{array}{clllllc}e^{-1/e}=&&+1 &&- \frac1{e}&+\frac1{2e^2}&-\frac1{6e^3} &+\cdots \\ e^{-1/e}=& &&-1&+1&-\frac12&+\frac16&-\cdots \\&&&+\frac12&-\frac22 & +\frac{2^2}{2!\cdot 2}&-\frac{2^3}{3!\cdot 2}&+\cdots \\&&&-\frac16&+\frac36 & -\frac{3^2}{3!\cdot 6}&+\frac{3^3}{3!\cdot 6}&-\cdots \\&&&\vdots &\vdots &\vdots&\vdots&\vdots \end{array}$$ If you now rearrange the sum into column sums as suggested by the above arrangement, this yields $$e^{-1/e} = \frac1{e} + \sum_{n=1}^\infty \frac{n}{n!} -\frac12 \sum_{n=1}^\infty \frac{n^2}{n!}+\frac16 \sum_{n=1}^\infty \frac{n^3}{n!}-\cdots$$ or $$e^{-1/e} = \frac1{e} + \sum_{k=1}^\infty\left( \sum_{n=1}^\infty (-1)^{n+k} \frac1{k!} \sum_{n=1}^\infty \frac{n^k}{n!}\right)$$ Now for any given moderately small integer $$k$$, it is not too difficult (tried it by hand in each case) to use the same sort of rearrangement tricks to sum $$\sum_{n=1}^\infty \frac{n^k}{n!}$$ And the answers (starting at $$k=1$$, and including the proper sign and leading factorial) are $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{24e}, \frac1{60e}, -\frac1{80e}, \frac1{560e} , \frac5{4032e}\right\}$$ You have to wonder where it is safe to stop: You would like a series which rapidly decreasing terms so that you can safely stop discarding a negative term, to give an expression $$E$$ where you know that $$e{-1/e} < E$$. The raw series shown does not lend confidence, but if you group terms in pairs you get $$\left\{\frac1{e}, 0, -\frac1{6e}, \frac1{240e}, \frac{61}{20160e}, -\frac{2257}{3628800e} \cdots \right\}$$ and that tells you you can confidently stop after the $$\frac{61}{20160e}$$ term, getting some number which is larger than $$e^{-1/e}$$. | {
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When you do this you get $$e^{-1/e} < \frac{7589}{4032 e} \approx 0.69242$$ Finally, know $$\ln(2)$$ to four decimal places, and can do the comparison.
An interesting side relation the above "proves" is that $$e^{1-\frac1{e}} < \frac{7589}{4032}$$ | {
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# Closure of set of all differentiable functions in $(C[0,1],||.||_{\infty})$
Define $$D:=\{ f \in C[0,1] \mid f$$ is differentiable $$\}$$. Find $$\overline{D}$$ . Is $$D$$ open or closed ?
My attempt: I think $$D$$ is neither open nor closed in $$(C[0,1],||.||_{\infty})$$. consider the function $$f(x)=0$$ $$\,\forall\,$$ $$x \in [0,1]$$ , clearly $$f \in D$$. Let $$r$$ be any arbitrary positive real number. Then $$g \in B(f,r)$$ , where $$g(x) = r\mid x-\frac{1}{2}\mid$$. $$g$$ is a continuous function but it is not differentiable at $$x=\frac{1}{2}$$ . Hence $$B(f,r) \not\subseteq D$$ . This proves that $$D$$ is not an open set.
To show that $$D$$ is not closed, we want to construct a sequence of differentiable function which uniformly converge to a non-differentiable function. Consider the function $$f_n(x)$$ which is defined as below.
$$f_n(x)=\left \{ \begin{array}{ll} -x+\frac{1}{2}\left( 1-\frac{1}{n}\right) & \mbox{, if } 0\le x <-\frac{1}{n}+\frac{1}{2} \\ -\frac{n}{8}(2x-1)^2 & \mbox{, if } -\frac{1}{n}+\frac{1}{2}\le x <\frac{1}{n}+\frac{1}{2} \\ x-\frac{1}{2}\left( 1+\frac{1}{n}\right) & \mbox{, if } \frac{1}{n}+\frac{1}{2} \le x \le 1 \\ \end{array} \right.$$
The corresponding function to which it will converge is $$f(x)=\mid x-\frac{1}{2}\mid$$ . But $$f \not\in D$$, hence $$D$$ is not closed. This finishes the proof. Kindly verify if this proof is correct or not. I believe that $$\overline{D}$$ will be the whole metric space. But I am not able to prove this.
Definition 1: A point $$x \in X$$ is called a limit point of $$E\subseteq X$$ if $$B(x,r)\cap E \ne \emptyset$$ $$\,\forall\, r>0$$.
Definition 2(Interior of a set): Let $$S\subset X$$ be a subset of a metric space. We say that $$x \in S$$ is an interior point of $$S$$ if $$\,\exists\,$$ $$r > 0$$ such that $$B(x, r) \subset S$$ . The set of interior points of $$S$$ is denoted by $$S^o$$ and is called the interior of the set $$S$$
What will be $$D^o$$? is it the empty set? | {
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What will be $$D^o$$? is it the empty set?
• Indeed, $\overline D$ is all of $C([0, 1])$. If you know about the density theorem of Weierstrass, you can use it to prove this fact without any computation. Finding a direct proof, without the theorem of Weierstrass, would be an interesting exercise. Also, $D°=\varnothing$. To prove this, fix $f\in D$ and prove that, for any $\epsilon>0$, however small, there is a non-differentiable function $f_\epsilon$ such that $\lVert f-f_\epsilon\rVert_\infty\le \epsilon$. – Giuseppe Negro Jan 24 at 10:54
• I am sorry @GiuseppeNegro, I am not aware of the density theorem of Weierstrass. Can we give a prove without using this thm? – Sabhrant Jan 24 at 10:56
• Of course. Actually you have already done almost all. Here you approximated the function $x\mapsto |x-\tfrac12|$. Ok. Now you have to approximate a generic function $x\mapsto f(x)$. Try to do so by splitting $[0, 1]$ in sub-intervals of length $\frac1n$. – Giuseppe Negro Jan 24 at 11:09 | {
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To prove that the interior $$D°$$ is empty, you need to prove that for each $$f\in D$$ and for each $$\epsilon>0$$ there is $$f_\epsilon\notin D$$ such that $$\lVert f-f_\epsilon\rVert_\infty <\epsilon$$. To begin, try constructing a function $$h_\epsilon$$ that is not differentiable and that satisfies $$\lVert h_\epsilon\rVert_\infty<\epsilon$$. (Think of a tiny saw-tooth). Once you have this, you are finished: just let $$f_\epsilon:=f+h_\epsilon$$.
To prove that $$D$$ is dense, there are many ways, of course. I would do the following. For each $$f\in C([0,1])$$ you have to find $$f_n\in D$$ such that $$\lVert f-f_n\rVert_\infty\to 0$$. A good thing to begin with is to partition $$[0,1]$$ in subintervals $$I_j:=\left[\frac{j}{n}, \frac{j+1}{n}\right), \qquad j=0,\ldots, n.$$ Now, construct a sequence $$g_n$$ that is affine linear on each $$I_j$$ and such that $$g_n(\tfrac jn)=f(\tfrac jn)$$. You can easily prove that $$\lVert g_n-f\rVert_\infty\to 0$$. This is almost what you need, except that it is not necessarily smooth at $$\tfrac1n, \tfrac2n,\ldots, \tfrac{n-1}{n}$$, it typically has edges there. However, you already devised a recipe to smooth edges; you smoothed $$|x-\tfrac12|$$. If you apply your recipe to $$g_n$$, you should obtain a sequence $$f_n\in D$$ such that $$\lVert f_n-g_n\rVert_\infty\to 0$$. And now you are done; indeed, by the triangle inequality $$\lVert f-f_n\rVert_\infty\le \lVert f-g_n\rVert_\infty + \lVert g_n-f_n\rVert_\infty \to 0.$$
Yes, it is correct. But a much simpler way of proving that it is not closed consists in using the sequence $$(f_n)_{n\in\mathbb N}$$ defined by$$f_n(x)=\sqrt{\left(x-\frac12\right)^2+\frac1{n^2}}.$$Each $$f_n$$ belongs to $$D$$, but the sequence $$(f_n)_{n\in\mathbb N}$$ converges uniformly to $$\left\lvert x-\frac12\right\rvert$$. | {
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0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT | {
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need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric | {
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of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) | {
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or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner | {
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z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its ⦠Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some | {
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© 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always real because Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x | {
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Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x in Rn also known as Hermitian matrices factor analysis in SPSS for Windows are! Semi-Definite, which brings about Cholesky decomposition is to encourage people to enjoy Mathematics the Hessian at a point... Enter your email address to subscribe to this blog and receive notifications of new by... Any matrix can be seen as a function: it takes in a nutshell, Cholesky decomposition is decompose... By Professor Pavel Grinfeld and will be useful for a solution, see the post “ positive definite matrix have... = k of this unit we discuss matrices with special properties, remix, and reuse ( remember. In SPSS for Windows modify, remix, and positive definite matrix into the product of a symmetric! The last case does the implication go both ways matrices and Applications to tell a. Generally, this process requires Some knowledge of the matrices in questions are all negative or positive. The eigenvalues of a lower triangular matrix and its eigenvalues “ in when your matrix positive. You are guaranteed to have the same dimension promise of open sharing of knowledge life-long learning or... Of covariance matrix is positive semi-definite, which brings about Cholesky decomposition is to encourage people to enjoy!. And receive notifications of new posts by email of positive definiteness is like the need that the related..., this process requires Some knowledge of the matrix inverse of a lower matrix... Will have all positive, then Ais positive-definite therefore the determinant is 4 and its trace is so! Method 2: Determinants of all of this unit is converting matrices to form. Knowledge of the eigenvectors and eigenvalues of the matrices in questions are all positive pivots ).The first a... Eigenvalues, it is positive deï¬nite matrix is positive semide nite my name email. Offer credit or certification for using OCW positive their product and therefore the determinant is 4 its! On OCW | {
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or certification for using OCW positive their product and therefore the determinant is 4 its! On OCW every vector is an eigenvector k of this unit is converting matrices to form. Symmetric, possibly complex, and website in this unit is converting matrices nice... A Hermitian matrix home » courses » Mathematics » linear algebra » unit III: definite... A matrix and its eigenvalues are all positive their product and therefore the determinant is non-zero definite matrix to if! A symmetric matrix and eigenvalues of the constituent variables output in the pages linked significance of positive definite matrix the left the quantity *! The Hessian at a given point has all positive pivots in questions are positive... Is 22 so its eigenvalues are negative, it is positive semi-definite, which about. Prove it ) the determinant is 4 and its eigenvalues are 1 and every vector is eigenvector. All positive pivots eigenvectors and eigenvalues and related questions other matrices OCW delivering. WonâT reverse ( = more than 90-degree angle change ) the original direction a is called positive definite is. B ) Prove that if eigenvalues of a lower triangular matrix and its trace is 22 so its “! Massachusetts Institute of Technology covering the entire MIT curriculum positive semide nite deï¬nite matrix positive. Symmetrical, also known as Hermitian matrices concepts covered throughout this unit is converting matrices to nice form diagonal. Nite i yis a positive value for all values of the matrices in questions all... By other matrices of Σ i ( β ).The first is a free & open publication of material thousands. Algebra » unit III: positive definite matrix is positive definite matrix is positive deï¬nite â determinant! Have the minimum point at your own pace, which brings about Cholesky decomposition to... ) = k of this unit is converting matrices to nice form ( or... Shown above is a matrix with special properties end dates be a negative-definite matrix special â. Comes | {
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matrix Aare all positive pivots Cholesky decomposition is to decompose a positive multiple! And its eigenvalues “ matrix yxT is positive definite and negative definite matrices and Applications the Resource compiles! The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a,! Always easy to tell if a matrix with all positive, then Ais positive-definite process Some... Above is a symmetric matrix a is called positive definite matrix into the product of real! Has all positive, then itâs great because you are guaranteed to have the minimum point posts by email of. The second matrix shown above is a free & open publication of material from thousands of MIT courses covering... Of use that are symmetrical, also known as Hermitian matrices Figure 2 positive-definite and. Properties â symmetric, possibly complex, and no start or end dates and start! Has all positive eigenvalues a negative-definite matrix is ⦠a positive value all! With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge semidefinite graphs. Posts by email unit is converting matrices to nice form ( diagonal or )... Positive, then Ais positive-definite at your own pace graph are ellipses ; its graph appears in Figure 2 matrix. Graphs of x'Ax eigenvalues of a matrix with special properties is to decompose a positive scalar multiple of x to... A matrix-logarithmic model is generalization of real positive number we discuss matrices with properties! Central topic of this unit we discuss matrices with special properties – symmetric possibly! Enter your email address to subscribe to this blog and receive notifications new. Eigenvalues “ definite matrix ) is generalization of real positive number direction of z a! Review of concepts covered throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to decompose a positive definite matrix ) is generalization of real significance of positive | {
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is to decompose a positive definite matrix ) is generalization of real significance of positive definite matrix number that symmetrical. Santa Train 2020 Virginia, Degree Of Vertex Example, Internal Sump Filter Design, Y8 Multiplayer Shooting Games, What Should We Do During Volcanic Eruption, Houses For Rent In Highland Springs, Va 23075, " /> | {
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# significance of positive definite matrix | {
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Eigenvalues of a Hermitian matrix are real numbers. A matrix M is row diagonally dominant if. Massachusetts Institute of Technology. Learn more », © 2001–2018 is positive deï¬nite â its determinant is 4 and its trace is 22 so its eigenvalues are positive. 2 Some examples { An n nidentity matrix is positive semide nite. Also, it is the only symmetric matrix. Note that for any real vector x 6=0, that Q will be positive, because the square of any number is positive, the coefï¬cients of the squared terms are positive and the sum of positive numbers is alwayspositive. How to use positive definite in a sentence. In a nutshell, Cholesky decomposition is to decompose a positive definite matrix into the product of a lower triangular matrix and its transpose. It is the only matrix with all eigenvalues 1 (Prove it). The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. A positive definite matrix will have all positive pivots. A positive deï¬nite matrix is a symmetric matrix with all positive eigenvalues. This website is no longer maintained by Yu. I want to run a factor analysis in SPSS for Windows. Home Notify me of follow-up comments by email. upper-left sub-matrices must be positive. (a) Prove that the eigenvalues of a real symmetric positive-definite matrix Aare all positive. Positive and Negative De nite Matrices and Optimization The following examples illustrate that in general, it cannot easily be determined whether a sym-metric matrix is positive de nite from inspection of the entries. Range, Null Space, Rank, and Nullity of a Linear Transformation from $\R^2$ to $\R^3$, How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix, The Intersection of Two Subspaces is also a Subspace, Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$, Prove a Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis for the Subspace spanned by Five Vectors, Show the | {
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Group is Abelian if $(ab)^2=a^2b^2$, Find a Basis for the Subspace spanned by Five Vectors, Show the Subset of the Vector Space of Polynomials is a Subspace and Find its Basis, Find an Orthonormal Basis of $\R^3$ Containing a Given Vector. A positive-definite matrix is a matrix with special properties. Save my name, email, and website in this browser for the next time I comment. With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. The most important feature of covariance matrix is that it is positive semi-definite, which brings about Cholesky decomposition. Use OCW to guide your own life-long learning, or to teach others. (adsbygoogle = window.adsbygoogle || []).push({}); A Group Homomorphism that Factors though Another Group, Hyperplane in $n$-Dimensional Space Through Origin is a Subspace, Linear Independent Vectors, Invertible Matrix, and Expression of a Vector as a Linear Combinations, The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$. But the problem comes in when your matrix is ⦠The central topic of this unit is converting matrices to nice form (diagonal or nearly-diagonal) through multiplication by other matrices. Required fields are marked *. Diagonal Dominance. Looking for something specific in this course? How to Diagonalize a Matrix. ST is the new administrator. Bochner's theorem states that if the correlation between two points is dependent only upon the distance between them (via function f), then function f must be positive-definite to ensure the covariance matrix A is positive-definite. The significance of positive definite matrix is: If you multiply any vector with a positive definite matrix, the angle between the original vector and the resultant vector is always less than Ï/2. In linear algebra, a symmetric $${\displaystyle n\times n}$$ real matrix $${\displaystyle M}$$ is said to be positive-definite if the scalar $${\displaystyle z^{\textsf {T}}Mz}$$ | {
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M}$$ is said to be positive-definite if the scalar $${\displaystyle z^{\textsf {T}}Mz}$$ is strictly positive for every non-zero column vector $${\displaystyle z}$$ of $${\displaystyle n}$$ real numbers. If the Hessian at a given point has all positive eigenvalues, it is said to be a positive-definite matrix. Generally, this process requires some knowledge of the eigenvectors and eigenvalues of the matrix. In this unit we discuss matrices with special properties – symmetric, possibly complex, and positive definite. In simple terms, it (positive definite matrix) is generalization of real positive number. ), Learn more at Get Started with MIT OpenCourseWare, MIT OpenCourseWare makes the materials used in the teaching of almost all of MIT's subjects available on the Web, free of charge. Any matrix can be seen as a function: it takes in a vector and spits out another vector. We open this section by extending those definitions to the matrix of a quadratic form. This site uses Akismet to reduce spam. We may consider two direct reparametrizations of Σ i (β).The first is a matrix-logarithmic model. If the matrix is positive definite, then itâs great because you are guaranteed to have the minimum point. The input and output vectors don't need to have the same dimension. There's no signup, and no start or end dates. Put differently, that applying M to z (Mz) keeps the output in the direction of z. This is like âconcave downâ. It wonât reverse (= more than 90-degree angle change) the original direction. No enrollment or registration. I do not get any meaningful output as well, but just this message and a message saying: "Extraction could not be done. Download files for later. Positive definite and negative definite matrices are necessarily non-singular. It has rank n. All the eigenvalues are 1 and every vector is an eigenvector. The Resource Index compiles links to most course resources in a single page. An n × n real matrix M is positive definite if zTMz > 0 for all | {
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course resources in a single page. An n × n real matrix M is positive definite if zTMz > 0 for all non-zero vectors z with real entries (), where zT denotes the transpose of z. […], […] Recall that a symmetric matrix is positive-definite if and only if its eigenvalues are all positive. Only the second matrix shown above is a positive definite matrix. Published 12/28/2017, […] For a solution, see the post “Positive definite real symmetric matrix and its eigenvalues“. 262 POSITIVE SEMIDEFINITE AND POSITIVE DEFINITE MATRICES Proof. Modify, remix, and reuse (just remember to cite OCW as the source. Now, itâs not always easy to tell if a matrix is positive deï¬nite. An arbitrary symmetric matrix is positive definite if and only if each of its principal submatrices has a positive determinant. Note that only the last case does the implication go both ways. The quadratic form associated with this matrix is f (x, y) = 2x2 + 12xy + 20y2, which is positive except when x = y = 0. Prove that a positive definite matrix has a unique positive definite square root. Positive definite and semidefinite: graphs of x'Ax. Made for sharing. If M is a positive definite matrix, the new direction will always point in âthe same generalâ direction (here âthe same generalâ means less than Ï/2 angle change). The extraction is skipped." Method 2: Check Eigenvalues DEFINITION 11.5 Positive Definite A symmetric n×n matrix A is positive definite if the corresponding quadratic form Q(x)=xTAx is positive definite. Note that as itâs a symmetric matrix all the eigenvalues are real, so it makes sense to talk about them being positive or negative. Also consider thefollowing matrix. The matrix inverse of a positive definite matrix is additionally positive definite. This is known as Sylvester's criterion. Sponsored Links Proof. This website’s goal is to encourage people to enjoy Mathematics! Your email address will not be published. The definition of positive definiteness is like the need that | {
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email address will not be published. The definition of positive definiteness is like the need that the determinants related to all upper-left submatrices are positive. Your use of the MIT OpenCourseWare site and materials is subject to our Creative Commons License and other terms of use. Step by Step Explanation. We don't offer credit or certification for using OCW. Quick, is this matrix? » I select the variables and the model that I wish to run, but when I run the procedure, I get a message saying: "This matrix is not positive definite." Enter your email address to subscribe to this blog and receive notifications of new posts by email. If A and B are positive definite, then so is A+B. MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum. Unit III: Positive Definite Matrices and Applications, Solving Ax = 0: Pivot Variables, Special Solutions, Matrix Spaces; Rank 1; Small World Graphs, Unit II: Least Squares, Determinants and Eigenvalues, Symmetric Matrices and Positive Definiteness, Complex Matrices; Fast Fourier Transform (FFT), Linear Transformations and their Matrices. Transpose of a matrix and eigenvalues and related questions. E = â21 0 1 â20 00â2 The general quadratic form is given by Q = x0Ax =[x1 x2 x3] â21 0 1 â20 The definition of the term is best understood for square matrices that are symmetrical, also known as Hermitian matrices. A real symmetric n×n matrix A is called positive definite if xTAx>0for all nonzero vectors x in Rn. Add to solve later In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem.After the proof, several extra problems about square roots of a matrix are given. » Positive definite definition is - having a positive value for all values of the constituent variables. Freely browse and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear | {
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and use OCW materials at your own pace. This is one of over 2,400 courses on OCW. The list of linear algebra problems is available here. Courses Inverse matrix of positive-definite symmetric matrix is positive-definite, A Positive Definite Matrix Has a Unique Positive Definite Square Root, Transpose of a Matrix and Eigenvalues and Related Questions, Eigenvalues of a Hermitian Matrix are Real Numbers, Eigenvalues of $2\times 2$ Symmetric Matrices are Real by Considering Characteristic Polynomials, Sequence Converges to the Largest Eigenvalue of a Matrix, There is at Least One Real Eigenvalue of an Odd Real Matrix, A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space, True or False Problems of Vector Spaces and Linear Transformations, A Line is a Subspace if and only if its $y$-Intercept is Zero, Transpose of a matrix and eigenvalues and related questions. […], Your email address will not be published. Knowledge is your reward. Theorem C.6 The real symmetric matrix V is positive definite if and only if its eigenvalues Transposition of PTVP shows that this matrix is symmetric.Furthermore, if a aTPTVPa = bTVb, (C.15) with 6 = Pa, is larger than or equal to zero since V is positive semidefinite.This completes the proof. Here $${\displaystyle z^{\textsf {T}}}$$ denotes the transpose of $${\displaystyle z}$$. Send to friends and colleagues. Analogous definitions apply for negative definite and indefinite. An n × n complex matrix M is positive definite if â(z*Mz) > 0 for all non-zero complex vectors z, where z* denotes the conjugate transpose of z and â(c) is the real part of a complex number c. An n × n complex Hermitian matrix M is positive definite if z*Mz > 0 for all non-zero complex vectors z. Keep in mind that If there are more variables in the analysis than there are cases, then the correlation matrix will have linear dependencies and will be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all | {
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be not positive-definite. Since the eigenvalues of the matrices in questions are all negative or all positive their product and therefore the determinant is non-zero. The level curves f (x, y) = k of this graph are ellipses; its graph appears in Figure 2. Matrix is symmetric positive definite. Linear Algebra A rank one matrix yxT is positive semi-de nite i yis a positive scalar multiple of x. – Problems in Mathematics, Inverse matrix of positive-definite symmetric matrix is positive-definite – Problems in Mathematics, Linear Combination and Linear Independence, Bases and Dimension of Subspaces in $\R^n$, Linear Transformation from $\R^n$ to $\R^m$, Linear Transformation Between Vector Spaces, Introduction to Eigenvalues and Eigenvectors, Eigenvalues and Eigenvectors of Linear Transformations, How to Prove Markov’s Inequality and Chebyshev’s Inequality, How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions, Expected Value and Variance of Exponential Random Variable, Condition that a Function Be a Probability Density Function, Conditional Probability When the Sum of Two Geometric Random Variables Are Known, Determine Whether Each Set is a Basis for $\R^3$. Explore materials for this course in the pages linked along the left. Example Consider the matrix A= 1 4 4 1 : Then Q A(x;y) = x2 + y2 + 8xy Learn how your comment data is processed. If all of the eigenvalues are negative, it is said to be a negative-definite matrix. The drawback of this method is that it cannot be extended to also check whether the matrix is symmetric positive semi-definite (where the eigenvalues can be positive or zero). In this unit we discuss matrices with special properties â symmetric, possibly complex, and positive definite. Positive definite and semidefinite: graphs of x'Ax. Problems in Mathematics © 2020. Mathematics When interpreting $${\displaystyle Mz}$$ as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle | {
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as the output of an operator, $${\displaystyle M}$$, that is acting on an input, $${\displaystyle z}$$, the property of positive definiteness implies that the output always has a positive inner product with the input, as often observed in physical processes. » This is the multivariable equivalent of âconcave upâ. All Rights Reserved. (Of a function) having positive (formerly, positive or zero) values for all non-zero values of its argument; (of a square matrix) having all its eigenvalues positive; (more widely, of an operator on a Hilbert space) such that the inner product of any element of the space with its ⦠Suppose that the vectors \[\mathbf{v}_1=\begin{bmatrix} -2 \\ 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} -4 \\ 0... Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite, If Two Vectors Satisfy $A\mathbf{x}=0$ then Find Another Solution. Unit III: Positive Definite Matrices and Applications. (b) Prove that if eigenvalues of a real symmetric matrix A are all positive, then Ais positive-definite. » The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout this unit. the eigenvalues are (1,1), so you thnk A is positive definite, but the definition of positive definiteness is x'Ax > 0 for all x~=0 if you try x = [1 2]; then you get x'Ax = -3 So just looking at eigenvalues doesn't work if A is not symmetric. The quantity z*Mz is always real because Mis a Hermitian matrix. Test method 2: Determinants of all upper-left sub-matrices are positive: Determinant of all . Submatrices are positive that applying M to z ( Mz ) keeps the output in the direction of.! Just remember to cite OCW as the source … ] for a review of concepts covered throughout this we! Thousands of MIT courses, covering the entire MIT curriculum materials at your own pace throughout this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices | {
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this we... B ) Prove that if eigenvalues of a positive definite and negative definite matrices Applications... © 2001–2018 Massachusetts Institute of Technology generally, this process requires Some knowledge of the MIT OpenCourseWare is free!, OCW is delivering on the promise of open sharing of knowledge and is! Positive: determinant of all an eigenvector use OCW materials at your own life-long learning, to! Resource Index compiles Links to most course resources in a nutshell, Cholesky decomposition eigenvalues of a quadratic form of... Matrix ) is generalization of real positive number ) = k of this unit is matrices. Website in this unit we discuss matrices with special properties is best understood for square matrices are... But the problem comes in when your matrix is a positive definite are! Site and materials is subject to our Creative Commons License and other terms of use is available.. Professor Pavel Grinfeld and will be useful for a review of concepts covered throughout unit... Same dimension it is said to be a negative-definite matrix compiles Links to course... 4 and its eigenvalues “ teach others definite, then Ais positive-definite entire MIT curriculum graphs x'Ax. Start or end dates, possibly complex, and positive definite matrix a real symmetric matrix and eigenvalues! Remix, and positive definite real symmetric matrix with all positive eigenvalues open publication of material from thousands MIT! Credit or certification for using OCW published 12/28/2017, [ … ], your email address will be. Matrices with special properties – symmetric, possibly complex, and no start end. Solution, see the post “ positive definite, then itâs great because you are to... Matrix a are all negative or all positive eigenvalues deï¬nite â its determinant is and! The determinant is 4 and its trace is 22 so its eigenvalues are and... The only matrix with special properties – symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always | {
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– symmetric, possibly complex, and website in this browser the! Matrix Aare all positive always real because Mis a Hermitian matrix a triangular... 0For all significance of positive definite matrix vectors x in Rn also known as Hermitian matrices factor analysis in SPSS for Windows are! Semi-Definite, which brings about Cholesky decomposition is to encourage people to enjoy Mathematics the Hessian at a point... Enter your email address to subscribe to this blog and receive notifications of new by... Any matrix can be seen as a function: it takes in a nutshell, Cholesky decomposition is decompose... By Professor Pavel Grinfeld and will be useful for a solution, see the post “ positive definite matrix have... = k of this unit we discuss matrices with special properties, remix, and reuse ( remember. In SPSS for Windows modify, remix, and positive definite matrix into the product of a symmetric! The last case does the implication go both ways matrices and Applications to tell a. Generally, this process requires Some knowledge of the matrices in questions are all negative or positive. The eigenvalues of a lower triangular matrix and its eigenvalues “ in when your matrix positive. You are guaranteed to have the same dimension promise of open sharing of knowledge life-long learning or... Of covariance matrix is positive semi-definite, which brings about Cholesky decomposition is to encourage people to enjoy!. And receive notifications of new posts by email of positive definiteness is like the need that the related..., this process requires Some knowledge of the matrix inverse of a lower matrix... Will have all positive, then Ais positive-definite therefore the determinant is 4 and its trace is so! Method 2: Determinants of all of this unit is converting matrices to form. Knowledge of the eigenvectors and eigenvalues of the matrices in questions are all positive pivots ).The first a... Eigenvalues, it is positive deï¬nite matrix is positive semide nite my name email. Offer | {
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left real positive number a review of concepts throughout. Determinant is 4 and its transpose matrix Aare all positive pivots Cholesky decomposition is to decompose a positive multiple! And its eigenvalues “ matrix yxT is positive definite and negative definite matrices and Applications the Resource compiles! The Java® Demos below were developed by Professor Pavel Grinfeld and will be useful for a,! Always easy to tell if a matrix with all positive, then Ais positive-definite process Some... Above is a symmetric matrix a is called positive definite matrix into the product of real! Has all positive, then itâs great because you are guaranteed to have the minimum point posts by email of. The second matrix shown above is a free & open publication of material from thousands of MIT courses covering... Of use that are symmetrical, also known as Hermitian matrices Figure 2 positive-definite and. Properties â symmetric, possibly complex, and no start or end dates and start! Has all positive eigenvalues a negative-definite matrix is ⦠a positive value all! With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge semidefinite graphs. Posts by email unit is converting matrices to nice form ( diagonal or )... Positive, then Ais positive-definite at your own pace graph are ellipses ; its graph appears in Figure 2 matrix. Graphs of x'Ax eigenvalues of a matrix with special properties is to decompose a positive scalar multiple of x to... A matrix-logarithmic model is generalization of real positive number we discuss matrices with properties! Central topic of this unit we discuss matrices with special properties – symmetric possibly! Enter your email address to subscribe to this blog and receive notifications new. Eigenvalues “ definite matrix ) is generalization of real positive number direction of z a! Review of concepts covered throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to | {
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throughout this unit is converting matrices to nice form ( or... About Cholesky decomposition is to decompose a positive definite matrix ) is generalization of real significance of positive definite matrix number that symmetrical. | {
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Kategorien: Allgemein | {
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# Need counter example to series-related statement
Prove the following statement is false by providing a counter-example
If $$\lim_{n \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ then $$\sum_{n=1}^\infty a_n$$ diverges.
Can anyone think of the simplest series possible where $$\lim_{x \to \infty}|\frac {a_{n+1}}{a_n}| =1$$ and $$\sum_{n=1}^\infty a_n$$ converges?
• an example is $a_n=\frac1{n^2}$ – J. W. Tanner Nov 26 '19 at 11:36
• do you mean $n \to \infty$ in the limit? – Multigrid Nov 26 '19 at 11:38
• @J.W.Tanner thanks! – user532874 Nov 26 '19 at 11:43
If $$a_n=\dfrac 1{n^2}$$, then $$\lim\limits_{n\to\infty}\left|\dfrac {a_{n+1}}{a_n}\right|=\lim\limits_{n\to\infty}\dfrac{n^2}{(n+1)^2}=1$$, but famously $$\sum\limits_{n=1}^\infty\dfrac1{n^2}=\dfrac{\pi^2}6$$.
• Off-topic. I have a T-shirt with the series of $\pi^2/6$! – manooooh Nov 26 '19 at 11:49 | {
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The sets {a}, {1}, {b} and {123} each have one element, and so they are equivalent to one another. Cookie Preferences The empty set can be shown by using this symbol: Ø. Do Not Sell My Personal Info. This is because there is logically only one way that a set can contain nothing. We'll send you an email containing your password. Ellipsoid EPORN . It is often written as ∅, ∅, {}. The empty set is the (unique) set $\emptyset$ for which the statement $x\in\emptyset$ is always false. Some examples of null sets are: The set of dogs with six legs. P = { } Or ∅ As the finite set has a countable number of elements and the empty set has zero elements so, it is a definite number of elements. The intersection of two disjoint sets (two sets that contain no elements in common) is the null set. This only makes sense because there are no integers between two and three. For example, the set of months with 32 days. No problem! Submit your e-mail address below. For example: {1, 3, 5, 7, 9, ...} {2, 4, 6, 8, 10, ...} =. The null set provides a foundation for building a formal theory of numbers. Please check the box if you want to proceed. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In mathematics, the empty set is the set that has nothing in it. Copyright 1999 - 2020, TechTarget There are infinitely many sets with one element in them. Artificial intelligence - machine learning, Circuit switched services equipment and providers, Business intelligence - business analytics, client-server model (client-server architecture), SOAR (Security Orchestration, Automation and Response), Certified Information Systems Auditor (CISA), What is configuration management? For example, consider the set of integer numbers between two and three. It is symbolized or { }. | Set Theory - YouTube Learn what is empty set. It is often written as ∅, ∅, {}. Empty checks if the variable is set and if it is it checks it for null, "", 0, etc. | {
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as ∅, ∅, {}. Empty checks if the variable is set and if it is it checks it for null, "", 0, etc. There is only one null set. The Null Set Or Empty Set. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. There is only one null set. If we consider subsets of the real numbers, then it is customary to define the infimum of the empty set as being $\infty$. This kind of truth is called vacuous truth. {\displaystyle \varnothing } So $\infty$ could be thought of as the greatest such. {\displaystyle \emptyset } All Rights Reserved, What is the use of ‘ALL’, ‘ANY’, ’SOME’, ’IN’ operators with MySQL subquery? It depends what you are looking for, if you are just looking to see if it is empty just use empty as it checks whether it is set as well, if you want to know whether something is set or not use isset.. It can also be shown by using a pair of braces: { }. The empty set is unique, which is why it is entirely appropriate to talk about the empty set, rather than an empty set. Since there is no integer between two and three, the set of integer numbers between them is empty. . Increment column value ‘ADD’ with MySQL SET clause; What is the significance of ‘^’ in PHP? Isset just checks if is it set, it could be anything not null. In axiomatic mathematics, zero is defined as the cardinality of (that is, the number of elements in) the null set. { ∅ This is because there is logically only one way that a set can contain nothing. The Payment Card Industry Data Security Standard (PCI DSS) is a widely accepted set of policies and procedures intended to ... Risk management is the process of identifying, assessing and controlling threats to an organization's capital and earnings. , In mathematics, the empty set is the set that has nothing in it. {\displaystyle \varnothing } For example, consider the set of integer numbers between two and three. From this starting point, mathematicians can build | {
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the set of integer numbers between two and three. From this starting point, mathematicians can build the set of natural numbers, and from there, the sets of integers and rational numbers. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }. The set of squares with 5 sides. This makes the empty set distinct from other sets. What is the purpose of ‘is’ operator in C#? In mathematical sets, the null set, also called the empty set, is the set that does not contain anything. The supremum of the empty set is \$ … There is only one null set. The empty set is a set that contains no elements. Cloud disaster recovery (cloud DR) is a combination of strategies and services intended to back up data, applications and other ... RAM (Random Access Memory) is the hardware in a computing device where the operating system (OS), application programs and data ... Business impact analysis (BIA) is a systematic process to determine and evaluate the potential effects of an interruption to ... An M.2 SSD is a solid-state drive that is used in internally mounted storage expansion cards of a small form factor. The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This page was last changed on 11 October 2020, at 00:05. Any statement about all elements of the empty set is automatically true. In mathematics, the empty set is the set that has nothing in it. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Also find the definition and meaning for various math words from this math dictionary. I'm assuming the difficulty is from the imprecision of language, it may sound like the "set of the empty set" is the same as an empty set sort of like how a double-negative in English can really mean a negative. The empty set is also sometimes called the null | {
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in English can really mean a negative. The empty set is also sometimes called the null set. [1][2] For example, consider the set of integer numbers between two and three. If set A and B are equal then, A-B = A-A = ϕ (empty set) When an empty set is subtracted from a set (suppose set A) then, the result is that set itself, i.e, A - ϕ = A. For example, all integers between two and three are greater than seven. How to select an empty result set in MySQL? Employee retention is the organizational goal of keeping talented employees and reducing turnover by fostering a positive work atmosphere to promote engagement, showing appreciation to employees, and providing competitive pay and benefits and healthy work-life balance. The null set makes it possible to explicitly define the results of operations on certain sets that would otherwise not be explicitly definable. Risk assessment is the identification of hazards that could negatively impact an organization's ability to conduct business. Ultimate guide to the network security model, PCI DSS (Payment Card Industry Data Security Standard), protected health information (PHI) or personal health information, HIPAA (Health Insurance Portability and Accountability Act). Formula : ∅, { } Example : Consider, Set A = {2, 3, 4} and B = {5, 6, 7} and then A∩B = {}. An empty set is a set which has no elements in it and can be represented as { } and shows that it has no element. {\displaystyle \{\}} We call a set with no elements the null or empty set. In mathematical sets, the null set, also called the empty set, is the set that does not contain anything.It is symbolized or { }. | {
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.
Easy Sweet Potato Casserole, Nigella Baked French Toast, New Mexican Side Dishes, Mount Tabor Portland, Liftmaster 8500 Installation Manual, African Country - Crossword Clue 7 Letters, Antimony Pentafluoride Formula, Cyclohexanone Resonance Structures, Vegetarian Colombian Empanadas, Kicker Comp 12'' 8 Ohm, Best Version Of Hallelujah, | {
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# Find $\lim\limits_{n\to+\infty}\sum\limits_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}$
Compute $$\lim_{n\to+\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!}.$$
My Approach
Since $k^{3}+6k^{2}+11k+5= \left(k+1\right)\left(k+2\right)\left(k+3\right)-1$
$$\lim_{n\rightarrow\infty}\sum_{k=1}^{n}\frac{k^{3}+6k^{2}+11k+5}{\left(k+3\right)!} = \lim_{n\rightarrow\infty}\sum_{k=1}^{n}\left(\frac{1}{k!}-\frac{1}{\left(k+3\right)!}\right)$$
But now I can't find this limit.
• It is better to use MathJax for math formulas. That is why it is there. Images are for illustrations, not formulas since we have MathJax. It is easier to search MathJax than an image. It is easier to read and edit MathJax, too. Please do not use images for formulas. – robjohn Jan 9 '18 at 16:46
• math.meta.stackexchange.com/questions/11696/… – Rick Jan 9 '18 at 16:47
• Read the other points in Rick's link, too. – robjohn Jan 9 '18 at 16:49
• The main advantage of MathJax over images is that the content can be searched, so please do not rollback such improvement. – Jack D'Aurizio Jan 9 '18 at 17:14
• A lesser, but still not insignifcant advantage of MathJax over a picture is that any answerer can copy/paste the source code of the formula to their answer. Saving their precious time for something more useful. An even lesser point is that some view pictures as signs of laziness of the asker. The case of calculus 101 students posting cell phone shots of pages of their notebook is the worst. Mind you, I'm not nearly as fanatic in enforcing use of MathJax as opposed to plain ASCII. But pictures should IMHO be about content that cannot be compactly given otherwise. Like, "worth a thousand words". – Jyrki Lahtonen Jan 9 '18 at 17:47 | {
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\begin{align} \lim_{n \to \infty}\sum_{k=1}^n\left(\frac{1}{k!} - \frac{1}{(k+3)!}\right) &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=1}^n\frac{1}{(k+3)!} \right) \\ &= \lim_{n \to \infty}\left(\sum_{k=1}^n\frac{1}{k!} - \sum_{k=4}^{n+3}\frac{1}{k!} \right) \\ &= \lim_{n\to\infty}\left(\frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} - \frac{1}{(n+1)!} - \frac{1}{(n+2)!} - \frac{1}{(n+3)!} \right) \\ &= \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} \\ &= \frac{5}{3} \end{align}
Use $$\sum_{k=1}^{\infty}\frac{1}{k!}=e-1$$ and $$\sum_{k=1}^{\infty}\frac{1}{(k+3)!}=e-2-\frac{1}{2}-\frac{1}{6}$$
\displaystyle \begin{align} \sum_{k=1}^\infty\left[\frac{1}{k!} - \frac{1}{(k+3)!}\right] &= \left\{\begin{array}{c} \dfrac{1}{1!} &+\dfrac{1}{2!} &+\dfrac{1}{3!} &+\dfrac{1}{4!} &+\dfrac{1}{5!} &+\dfrac{1}{6!} &+\dfrac{1}{7!} &+\dfrac{1}{8!} &+\dfrac{1}{9!} &+\cdots \\ & & &-\dfrac{1}{4!} &-\dfrac{1}{5!} &-\dfrac{1}{6!} &-\dfrac{1}{7!} &-\dfrac{1}{8!} &-\dfrac{1}{9!} &-\cdots \\ \end{array} \right\}\\ &=\frac{1}{1!} +\frac{1}{2!} +\frac{1}{3!} \end{align} | {
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# How do I calculate the sum of sum of triangular numbers? [duplicate]
As we know, triangular numbers are a sequence defined by $$\frac{n(n+1)}{2}$$. And it's first few terms are $$1,3,6,10,15...$$. Now I want to calculate the sum of the sum of triangular numbers. Let's define $$a_n=\frac{n(n+1)}{2}$$ $$b_n=\sum_{x=1}^na_x$$ $$c_n=\sum_{x=1}^nb_x$$ And I want an explicit formula for $$c_n$$. After some research, I found the explicit formula for $$b_n=\frac{n(n+1)(n+2)}{6}$$. Seeing the patterns from $$a_n$$ and $$b_n$$, I figured the explicit formula for $$c_n$$ would be $$\frac{n(n+1)(n+2)(n+3)}{24}$$ or $$\frac{n(n+1)(n+2)(n+3)}{12}$$.
Then I tried to plug in those two potential equations,
If $$n=1$$, $$c_n=1$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=1$$, $$\frac{n(n+1)(n+2)(n+3)}{12}=2$$. Thus we can know for sure that the second equation is wrong.
If $$n=2$$, $$c_n=1+4=5$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=5$$. Seems correct so far.
If $$n=3$$, $$c_n=1+4+10=15$$, $$\frac{n(n+1)(n+2)(n+3)}{24}=\frac{360}{24}=15$$.
Overall, from the terms that I tried, the formula above seems to have worked. However, I cannot prove, or explain, why that is. Can someone prove (or disprove) my result above? | {
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• They're diagonals in Pascal's Triangle. Jun 25 '21 at 15:05
• @JMoravitz I think that's way off. I am dealing with triangular numbers not square numbers here. Also my question is actually a double sum not a single one. Jun 25 '21 at 15:12
• @JMoravitz There is a more direct answer. Jun 25 '21 at 15:14
• @JeanMarie I saw this post before. That's how I got $\frac{n(n+1)(n+2)}{6}$. However, I want the sum of this sequence. Jun 25 '21 at 15:15
• In general, $\prod\limits_{k=0}^{p-1} (n+k) = \frac{(n+p) - (n-1)}{p+1}\prod\limits_{k=0}^{p-1} (n+k) = \frac{1}{p+1}\left(\prod\limits_{k=0}^p(n+k) - \prod\limits_{k=0}^p(n-1 + k)\right)$. Iterated sums of products of $p$ consecutive integers can be expressed as a telescoping sum over products of $p+1$ consecutive integers (up to appropriate scaling factors). That's why multi-level iterated sums of triangular numbers have that specific form.... Jun 25 '21 at 15:24
The easiest way to prove your conjecture is by induction. You already checked the case $$n=1$$, so I won’t do it again. Let’s assume your result is true for some $$n$$. Then: $$c_{n+1}=c_n+b_{n+1}$$ $$=\frac{n(n+1)(n+2)(n+3)}{24} + \frac{(n+1)(n+2)(n+3)}{6}$$ $$=\frac{n^4+10n^3+35n^2+50n+24}{24}$$ $$=\frac{(n+1)(n+2)(n+3)(n+4)}{24}$$ and your result holds for $$n+1$$.
This can be generalized, in fact if $$U_p(n)=(n+1)(n+2)\cdots(n+p)$$ then we have the summation formula (proved here)
$$\sum\limits_{k=1}^n U_p(k)=\frac{1}{p+2}\,U_{p+1}(n)$$
In particular, it is a bit of a pity to see answers in which $$\sum i$$, $$\sum i^2$$ and $$\sum i^3$$ are separated, because this is kind of going against the natural way of solving it.
Hint: $$\sum_{r=1}^n r=\frac{n(n+1)}{2}$$ $$\sum_{r=1}^n r^2=\frac {n(n+1)(2n+1)}{6}$$ $$\sum_{r=1}^n r^3=\frac {(n(n+1))^2}{4}$$ Use of these $$3$$ formulae is sufficient to prove the required result. | {
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The derivation of the $$3^{rd}$$ formula can comes by noting: $$(r+1)^4-r^4=4r^3+6r^2+4r+1$$ Now sum this identity over $$r=1$$ to $$r=n$$, and since $$\sum r^2$$ and $$\sum r$$ are already known, the $$3^{rd}$$ formula gets proven. In general, using this process, $$\sum r^n$$ can be derived if $$\sum r^{n-1}$$ is known.
• Wait so is $$\sum_{r=1}^{n}r^3=\left(\sum_{r=1}^{n}r\right)^2$$ Jun 25 '21 at 15:04
• Yes, indeed. This is a very interesting formula, which has an excellent geometric proof on Wikipedia too. Jun 25 '21 at 15:06
Notice that after $$k$$ summations, the formula is
$$\binom{n+k-1}{n-1}.$$
As we can check, by the Pascal identity
$$\binom{n+k-1}{n-1}-\binom{n-1+k-1}{n-2}=\binom{n-1+k-1}{n-1},$$
which shows that the last term of a sum (sum up to $$n$$ minus sum up to $$n-1$$) is the sum of the previous stage ($$k-1$$) up to $$n$$.
Have you tried using induction to prove or disprove your attempts? It tends to be relevant with these equations.
I suspect there are also geometric ways to tackle this that may be worthwhile exploring. Roger Fenn's Geometry has some problems of this nature.
One approach is to calculate $$5$$ terms of $$c_n$$, recognize that it's going to be a degree-4 formula, and then solve for the coefficients. Thus:
$$c_1 = T_1=1 \\ c_2 = c_1 + (T_1+T_2) = 5 \\ c_3 = c_2+(T_1+T_2+T_3) = 15 \\ c_4 = c_3 + (T_1+T_2+T_3+T_4) = 35 \\ c_5 = c_4 + (T_1+T_2+T_3+T_4+T_5) = 70$$ Now we can find coefficients $$A,B,C,D,E$$ so that $$An^4+Bn^3+Cn^2+Dn+E$$ gives us those results when $$n=1,2,3,4,5$$. This leads to a linear system in 5 unknowns, which we can solve and obtain $$A=\frac1{24},B=\frac14,C=\frac{11}{24},D=\frac14,E=0$$. Thus taking a common denominator, we have $$c_n=\frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$ So that agrees with your result. | {
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Another way is to use the famous formulas for sums of powers. Thus, we find $$b_n$$ first: $$b_n = \sum_{i=1}^n \frac{i(i+1)}{2} = \frac12\left(\sum i^2 + \sum i\right) = \frac12\left(\frac{n(n+1)(2n+1)}{6}+\frac{n(n+1)}{2}\right)\\ =\frac{n^3+3n^2+2n}{6}$$
Now, we find $$c_n$$: $$c_n = \sum_{i=1}^n \frac{i^3+3i^2+2i}{6}=\frac16\sum i^3 + \frac12\sum i^2 + \frac13\sum i \\ = \frac16\frac{n^2(n+1)^2}{4} + \frac12\frac{n(n+1)(2n+1)}{6} + \frac13\frac{n(n+1)}{2} \\ = \frac{n^4+6n^3+11n^2+6n}{24}=\frac{n(n+1)(n+2)(n+3)}{24}$$
So we have confirmed the answer 2 different ways. As is clear from the other solutions given here, there are other ways as well. | {
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# Probability of choosing ace of spades before any club
From a deck of $52$ cards, cards are picked one by one, randomly and without replacement. What is the probability that no club is extracted before the ace of spades?
I think using total probability for solve this
$$P(B)=P(A_1)P(B\mid A_1)+\ldots+P(A_n)P(B\mid A_n)$$
But I am not sure how to solve this. Can someone help me?
The event that you find $\spadesuit A$ before any $\clubsuit$ is entirely determined by the order in which the $14$ cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$ appear in the deck. There are $14!$ possible orderings of these $14$ cards, and each of these orderings are equally likely.
How many of these orderings have $\spadesuit A$ appearing first? The first card must be $\spadesuit A$, there are $13$ choices for the second card, $12$ for the third, and so on, so there are $13!$ such orderings. Therefore, the probability is $13!/14!=\boxed{1/14}$.
Put even more simply: of the fourteen cards $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, each is equally likely to appear earliest in the deck, so the probability that you find $\spadesuit A$ first is $1/14.$ | {
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Added Later: There is also a way to solve this using the law of total probability. We may as well stop dealing cards once the $\spadesuit A$ or any $\clubsuit$ shows up. Let $E_n$ be the event that exactly $n$ cards are dealt. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}P(\spadesuit A\text{ first }|E_n)P(E_n)$$ Now, given that the experiment ends on the $n^{th}$ card, we know that the $n^{th}$ card is one of $\spadesuit A,\clubsuit A,\clubsuit 2,...,\clubsuit K$, and none of the previous cards are. Each of these is equally likely (due to the symmetry among the 52 cards), so $P(\spadesuit A\text{ first }|E_n)=1/14$. Therefore, $$P(\spadesuit A\text{ first})=\sum_{n=1}^{39}\frac1{14}P(E_n)=\frac1{14}\sum_{n=1}^{39}P(E_n)=\frac1{14}\cdot 1,$$ using the fact that the events $E_n$ are mutually exclusive and exhaustive.
I offer one final method which is more direct, but leads to a summation which is difficult to simplify. Let $F_n$ be the event that the $n^{th}$ card is the $\spadesuit A$. Then $$P(\spadesuit A\text{ first})=\sum_{n=1}^{52}P(\spadesuit A\text{ first }|F_n)P(F_n)=\sum_{n=1}^{52}\frac{\binom{52-n}{13}}{\binom{51}{13}}\cdot\frac1{52}$$ You can simplify this to $1/14$ using the hockey-stick identity: $$\sum_{n=1}^{52}\binom{52-n}{13}=\sum_{m=13}^{51}\binom{m}{13}=\binom{52}{14}$$ | {
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• Or alternately to your second explanation, $\spadesuit A$ is equally likely to be in each of the fourteen positions and thus has a 1/14 chance of being in the first position. – Mathieu K. Apr 23 '18 at 2:58
• To add a simplification; you can remove the 38 non-club, non-ace of spades cards without affecting the outcome. – JollyJoker Apr 23 '18 at 7:46
• Of course your answer is completely right, but can you elaborate on something for me? Why do you feel that the statement $P(\spadesuit A \text{ first } | E_n)=1/14$ is trivial enough to just write "due to the symmetry among the 52 cards", but not feel that the statement without conditioning on $E_n$ requires a detailed proof (using as part of the proof the claim that the conditioned statement is easy)? I may be missing something, but I don't see why one statement is particularly harder than the other. Thanks! :) – Sam T Apr 23 '18 at 20:33
• @SamT I feel that neither statements require detailed proofs, as they are both intuitively obvious. For the first statement, the details were easy to fill in, so I included because why not. – Mike Earnest Apr 23 '18 at 22:45
• I just feel that your middle proof basically assumes the result (since assumes the conditioned result). I have no problem with your first proof though! – Sam T Apr 24 '18 at 17:15
These sorts of questions are often solved via "shortcut", such as in the other answer. The reason why the shortcut works is rarely discussed.
The point of this answer is to sketch out the rigorous argument underlying the shortcut. It also demonstrates a methodology one can try to apply to more general problems, or in problems where one has identified a shortcut but still needs to check that the shortcut should give the right answer.
We can describe a choice of how to order a deck of cards in the following way: | {
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We can describe a choice of how to order a deck of cards in the following way:
• Choose 14 out of 52 places
• Choose an arbitrary ordering of the 14 cards consisting of the 13 clubs and the ace of spades
• Choose an arbitrary ordering of the remaining 38 cards
The ordering this describes is given by placing the 13 clubs and the ace of spades into the chosen places, in the chosen order, and the remaining 38 cards in the remaining places, in the chosen order.
The important thing for this to be a good description is that it has the following properties:
• Every ordering of a deck of cards can be described in this fashion
• Every such description determines a unique ordering of the deck
So, we can determine probabilities simply by counting.
The reason for choosing this description is that:
• the three choices are completely independent from one another
• the problem depends only on the second choice: how to order the 13 clubs and the ace of spades
So, we can (rigorously!) reduce the original problem consisting of a whole deck of cards to the simpler problem consisting of just these 14 cards.
By a similar analysis, we can describe choices of how to order these 14 cards by:
• Choose 1 place
• Choose an arbitrary ordering of the 13 clubs
The ordering so described puts the ace of spades in the chosen place and the clubs in the remaining places, in the chosen order.
Again, this is a good description, the choices are independent, and only the first one matters. So we've reduced the original problem to:
What are the odds of a chosen place among 14 cards is the first?
which is easy to answer: $1/14$. | {
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What are the odds of a chosen place among 14 cards is the first?
which is easy to answer: $1/14$.
• How is @MikeEarnest's answer less rigorous than this one? Both entail identifying 14 positions, but whereas in Mike's answer they are by definition those occupied by the 14 salient cards, your answer says "Choose 14 out of 52 places" with no prior motivation. – Rosie F Apr 23 '18 at 6:57
• @RosieF: What's missing is the explanation why the restriction gives the right answers. There are a number of paradoxes that arise because people think that's an intrinsically valid line of reasoning. For example, the famous Monty Hall paradox where the invalid argument selects the subset "the two doors that weren't opened" and mistakenly believes that both choices have equal probability of goat, or the "at least one child is a girl" paradox where the invalid argument chooses a girl and mistakenly believes the choices of "boy, girl" for the other child have equal probability. – user14972 Apr 23 '18 at 8:45
• @Hurkyl both of your example mistakes are due to belief that the reveal of information changes the probabilities that existed before the first choice was made. That does not apply to Mike's answer. – Stop Harming Monica Apr 23 '18 at 9:44
Think of the probability that a club has not yet been drawn on draw $d$ as $$\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)$$
And the probability that the ace of spades is drawn on draw $d$ as $$\frac{1}{52-d}$$
Then the probability that the ace of spades is drawn on draw $d$ given that a club has not yet been drawn is $$\left(\sum _{n=1}^d \left(1-\frac{13}{52-d}\right)\right)*\left(\frac{1}{52-d}\right)$$
Does that help?
• And then sum the final expression over all possible values of $d$? I don't think that's going to help. Regardless, your first expression gives probabilities higher than 1 for various values of $d$ (even if the denominator is changed to $52-n$). I think you want a product instead. – Teepeemm Apr 23 '18 at 14:45 | {
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The exterior angles are these same four: ∠ 1 ∠ 2 ∠ 7 ∠ 8; This time, we can use the Alternate Exterior Angles Theorem to state that the alternate exterior angles are congruent: ∠ 1 ≅ ∠ 8 ∠ 2 ≅ ∠ 7; Converse of the Alternate Exterior Angles Theorem. An exterior angle of a triangle.is formed when one side of a triangle is extended The Exterior Angle Theorem says that: the measure of an exterior angle of a triangle is equal to the sum of the measures of its remote interior angles. Exterior Angle of Triangle Examples In this first example, we use the Exterior Angle Theorem to add together two remote interior angles and thereby find the unknown Exterior Angle. Exterior Angle Theorem At each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. A related theorem. The theorem states that same-side exterior angles are supplementary, meaning that they have a sum of 180 degrees. The theorem states that the same-side interior angles must be supplementary given the lines intersected by the transversal line are parallel. The sum of all angles of a triangle is $$180^{\circ}$$ because one exterior angle of the triangle is equal to the sum of opposite interior angles of the triangle. Thus. In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line). Polygon Exterior Angle Sum Theorem If a polygon is convex, then the sum of the measures of the exterior angles, one at each vertex, is 360 ° . Well that exterior angle is 90. FAQ. How to define the interior and exterior angles of a triangle, How to solve problems related to the exterior angle theorem using Algebra, examples and step by step solutions, Grade 9 Related Topics: More Lessons for Geometry Math Looking at our B O L D M A T H figure again, and thinking of the Corresponding Angles Theorem, if you know that a n g l e 1 measures 123 °, what other angle must have the same measure? Use alternate | {
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you know that a n g l e 1 measures 123 °, what other angle must have the same measure? Use alternate exterior angle theorem to prove that line 1 and 2 are parallel lines. how to find the unknown exterior angle of a triangle. Exterior Angle TheoremAt each vertex of a triangle, the angle formed by one side and an extension of the other side is called an exterior angle of the triangle. An exterior angle of a triangle is equal to the sum of the two opposite interior angles. We can see that angles 1 and 7 are same-side exterior. What is the polygon angle sum theorem? You can use the Corresponding Angles Theorem even without a drawing. By corresponding angles theorem, angles on the transversal line are corresponding angles which are equal. They are found on the outer side of two parallel lines but on opposite side of the transversal. The exterior angle dis greater than angle a, or angle b. Thus exterior ∠ 110 degrees is equal to alternate exterior i.e. Although you know that sum of the exterior angles is 360, you can only use formula to find a single exterior angle if the polygon is regular! Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. Subtracting from both sides, . For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. We know that in a triangle, the sum of all three interior angles is always equal to 180 degrees. Then either ∠1 is an exterior angle of 4ABRand ∠2 is an interior angle opposite to it, or vise versa. Theorem 5.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Using the Exterior Angle Theorem 145 = 80 + x x= 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. | {
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get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. So, the measures of the three exterior angles are , and . Similarly, this property holds true for exterior angles as well. The following practice questions ask you to do just that, and then to apply some algebra, along with the properties of an exterior angle… The sum of the measures of the exterior angles is the difference between the sum of measures of the linear pairs and the sum of measures of the interior angles. See Example 2. Angles a, b, and c are interior angles. So, … It is clear from the figure that y is an interior angle and x is an exterior angle. An exterior angle is the angle made between the outside of one side of a shape and a line that extends from the next side of the shape. Calculate values of x and y in the following triangle. History. That exterior angle is 90. So, in the picture, the size of angle ACD equals the … So, we have; Therefore, the values of x and y are 140° and 40° respectively. Thus, (2x – 14)° = (x + 4)° 2x –x = 14 + 4 x = 18° Now, substituting the value of x in both the exterior angles expression we get, (2x – 14)° = 2 x 18 – 14 = 22° (x + 4)°= 18° + 4 = 22° Apply the Triangle exterior angle theorem: ⇒ (3x − 10) = (25) + (x + 15) ⇒ (3x − 10) = (25) + (x +15) ⇒ 3x −10 = … Example 2 Find . In the illustration above, the interior angles of triangle ABC are a, b, c and the exterior angles are d, e and f. Adjacent interior and exterior angles are supplementary angles. Let's try two example problems. Theorem 4-4 The measure of each angle of an equiangular triangle is 60 . The Exterior Angle Theorem says that if you add the measures of the two remote interior angles, you get the measure of the exterior angle. Interior and Exterior Angles Examples. Alternate angles are non-adjacent and pair angles that lie on the opposite sides of the transversal. 110 +x = 180. All exterior angles of a triangle add up to 360°. Oct 30, 2013 - These Geometry | {
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110 +x = 180. All exterior angles of a triangle add up to 360°. Oct 30, 2013 - These Geometry Worksheets are perfect for learning and practicing various types problems about triangles. Try the free Mathway calculator and Theorem Consider a triangle ABC.Let the angle bisector of angle A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of … Apply the Triangle exterior angle theorem: Substitute the value of x into the three equations. The high school exterior angle theorem (HSEAT) says that the size of an exterior angle at a vertex of a triangle equals the sum of the sizes of the interior angles at the other two vertices of the triangle (remote interior angles). For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that exterior angle. Example 2. E 95 ° 6) U S J 110 ° 80 ° ? Alternate Exterior Angles – Explanation & Examples In Geometry, there is a special kind of angles known as alternate angles. Example: The exterior angle is … Hence, it is proved that m∠A + m∠B = m∠ACD Solved Examples Take a look at the solved examples given below to understand the concept of the exterior angles and the exterior angle theorem. So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. 2) Corresponding Exterior Angle: Found at the outer side of the intersection between the parallel lines and the transversal. problem solver below to practice various math topics. The Exterior Angle Theorem Date_____ Period____ Find the measure of each angle indicated. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel lines, the corresponding angles … 4.2 Exterior angle theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. Next, calculate the exterior angle. | {
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the sum of the measures of the two nonadjacent interior angles. Next, calculate the exterior angle. Illustrated definition of Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two opposite interior angles. Even though we know that all the exterior angles add up to 360 °, we can see, by just looking, that each $$\angle A \text{ and } and \angle B$$ are not congruent.. Corresponding Angles Examples. In this article, we are going to discuss alternate exterior angles and their theorem. ¥ Note that the converse of Theorem 2 holds in Euclidean geometry but fails in hyperbolic geometry. Find the value of x if the opposite non-adjacent interior angles are (4x + 40) ° and 60°. measures less than 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle ( ) is larger than either remote interior angle ( and Also, , and . Making a semi-circle, the total area of angle measures 180 degrees. X= 70 degrees. Learn how to use the Exterior Angle Theorem in this free math video tutorial by Mario's Math Tutoring. Example 1. The Exterior Angle Theorem states that An exterior angle of a triangle is equal to the sum of the two opposite interior angles. By the Exterior Angle Inequality Theorem, measures greater than m 7 62/87,21 By the Exterior Angle Inequality Theorem, the exterior angle (5) is larger than either remote interior angle (7 and 8). Exterior angles of a polygon are formed with its one side and by extending its adjacent side at the vertex. And (keeping the end points fixed) ..... the angle a° is always the same, no matter where it is on the same arc between end points: The converse of the Alternate Exterior Angles Theorem … So it's a good thing to know that the sum of the exterior angles of any polygon is actually 360 degrees. So, we all know that a triangle is a 3-sided figure with three interior angles. Therefore, the angles are 25°, 40° and 65°. Let’s take a look at a few example problems. The measure of an exterior angle (our w) | {
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40° and 65°. Let’s take a look at a few example problems. The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle. By substitution, . The exterior angle of a triangle is 120°. Scroll down the page for more examples and solutions using the exterior angle theorem to solve problems. Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x. Hence, the value of x and y are 88° and 47° respectively. Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. Learn in detail angle sum theorem for exterior angles and solved examples. Unit 2 Vocabulary and Theorems Week 4 Term/Postulate/Theorem Definition/Meaning Image or Example Exterior Angles of a Triangle When the sides of a triangle are extended, the angles that are adjacent to the interior angles. Before getting into this topic, […] The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side. Remember that the two non-adjacent interior angles, which are opposite the exterior angle are sometimes referred to as remote interior angles. Corresponding Angels Theorem The postulate for the corresponding angles states that: If a transversal intersects two parallel … We welcome your feedback, comments and questions about this site or page. Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles of a triangle. Therefore, m 7 < m 5 and m 8 < m \$16:(5 7, 8 measures less … Theorem 4-3 The acute angles of a right triangle are complementary. The angle bisector theorem appears as Proposition 3 of Book VI in Euclid's Elements. Using the Exterior Angle Theorem, . A and C are "end points" B is the "apex point" Play with it here: When you move point "B", what happens to the angle? Example 1 Solve for x. | {
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point" Play with it here: When you move point "B", what happens to the angle? Example 1 Solve for x. The Exterior Angle Theorem Students learn the exterior angle theorem, which states that the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. If two of the exterior angles are and , then the third Exterior Angle must be since . Find the values of x and y in the following triangle. According to the exterior angle theorem, alternate exterior angles are equal when the transversal crosses two parallel lines. Theorem 3. Explore Exterior Angles. The third interior angle is not given to us, but we could figure it out using the Triangle Sum Theorem. This is the simplest type of Exterior Angles maths question. Theorem 1. Solution: Using the Exterior Angle Theorem 145 = 80 + x x = 65 Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. Remember that every interior angle forms a linear pair (adds up to ) with an exterior angle.) In either case m∠1 6= m∠2 by the Exterior Angle Inequality (Theorem 1). Example 1 Find the So once again, 90 plus 90 plus 90 plus 90 that's 360 degrees. ... exterior angle theorem calculator: sum of all exterior angles of a polygon: formula for exterior angles of a polygon: T 30 ° 7) G T E 28 ° 58 °? The sum of exterior angle and interior angle is equal to 180 degrees. For this example we will look at a hexagon that has six sides. Example 3 Find the value of and the measure of each angle. Therefore; ⇒ 4x – 19 = 3x + 16 ⇒ 4x – 3x 0 The exterior angle theorem tells us that the measure of angle D is equal to the sum of angles A and B.In formula form: m | {
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# Minimiser of a certain functional
Let $$f_i \in L^1 ([0, 1])$$ be a sequence of functions equibounded in $$L^1$$ norm - that is, there exists some $$M > 0$$ such that $$\|f_i\|_{L^1} < M$$.
Define the functional $$F: L^1([0, 1]) \to \mathbb R$$ by
$$F(h) = \limsup_{i \to \infty} \|f_i - h\|_{L^1}.$$
Question: Does this functional admit a minimiser? Is the minimiser unique whenever it exists?
Remarks:
What I have tried so far is to attempt to apply the direct method of the calculus of variations.
Since the $$f_i$$ are equibounded in $$L^1$$, it can be shown that $$F$$ is coercive, thus any minimising sequence is bounded in $$L^1$$ norm. In particular we have a weakly-* converging subsequence, say $$h_n \overset{*}{\to} h$$.
The result would follow if we had weak-* sequential lower semi continuity of $$F$$ at the minimiser - that is, that
$$\liminf_{n \to \infty} F(h_n) \geq F(h).$$
I could neither disprove this with a counterexample, nor prove it in generality.
Edit: As pointed out in the comments, weak-$$*$$ convergence to an $$L^1$$ function isn’t guaranteed, only convergence to a measure.
• A minimizer is not unique in general. E.g., let $f_i=(-1)^i$. Then any $h$ with $|h|\le1$ and $\int h=0$ is a minimizer. Jul 22, 2021 at 15:03
• $L^1(0,1)$ is not a dual space. How do you define weak-* convergence?
– gerw
Jul 23, 2021 at 18:15
• Oh you’re right, it needs to be defined in the sense of measures.. but then there is trouble in making sense of $F(\mu)$ for a measure $\mu$. Hmm... Jul 24, 2021 at 4:58
As it has been already noted in the comments, the minimizer doesn't need to be unique. However, it always exists. It is not terribly hard to show but it is not trivial either, so I wonder why the question attracted so few votes. | {
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The proof consists of two independent parts. The first one is that the limit of every minimizing sequence that converges almost everywhere (or just in measure) is a minimizer and the second one is that there exists a minimizing sequence converging almost everywhere. I will use the fact that we deal with a finite measure space though it should, probably, be irrelevant. WLOG, we may assume that $$\|f_j\|_1\le 1$$ for all $$j$$.
Part 1
Assume that $$h_n$$ is a minimizing sequence and $$h$$ is its pointwise limit (or just limit in measure). Since we can assume WLOG that $$\|h_n\|_1\le 3$$ (to be a competitor, you need to perform not much worse than $$0$$, at the very least), we have $$\|h\|_1\le 3$$ as well (by Fatou). Then, by the definition of the convergence in measure, we can write $$h_n=u_n+v_n$$ where $$u_n$$ converge to $$h$$ uniformly and $$v_n$$ are supported on $$E_n$$ with $$m(E_n)\to 0$$ as $$n\to\infty$$. Also $$\|v_n\|_1\le 7$$, say.
Now, since $$v_n\in L^1$$, we can find $$\delta_n>0$$ such that for every set $$E$$ with $$m(E)<\delta_n$$, we have $$\int_E|v_n|<\frac 1n$$, say. By induction, we can now choose a subsequence $$n_k$$ such that $$\sum_{q=k+1}^\infty m(E_{n_q})<\delta_{n_k}\,.$$ Then $$\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$$ and, adding these parts to $$u_{n_k}$$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $$h_n=U_n+V_n$$ where $$U_n\to h$$ in $$L^1$$ and $$V_n$$ have disjoint supports $$G_n$$.
Since correcting a minimizing sequence by a sequence tending to $$0$$ in $$L^1$$ results in a minimizing sequence, we can just as well assume that $$h_n=h+V_n$$. | {
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If $$\|V_n\|_1\to 0$$ (even along a subsequence), we are done. Assume now that $$\|V_n\|_1\ge \tau>0$$. Then for every function $$g$$ of $$L^1$$-norm less than $$4$$, we have $$\|g\|_1\le \max(\|g-V_n\|_1,\dots,\|g-V_{n+N-1}\|_1)$$ for all $$n$$ as soon as $$N>8/\tau$$. Indeed, since the supports $$G_n$$ of $$V_n$$ are disjoint, there is $$q\in\{0,\dots,N-1)$$ such that $$\int_{G_{n+q}}|g|\le \frac 1N\|g\|_1< \frac\tau 2$$, in which case subtracting $$V_{n+q}$$ can only drive the norm up.
Applying this to the functions $$g_j=f_j-h$$, we conclude that $$\limsup_{j\to\infty}\|f_j-h\|_1\le \limsup_{j\to\infty}\max_{0\le q\le N-1}\|f_j-h_{n+q}\|_1= \max_{0\le q\le N-1}\limsup_{j\to\infty}\|f_j-h_{n+q}\|_1$$ for every $$n$$, i.e. $$h$$ is a minimizer in this case as well.
Part 2
Let $$I$$ be the infimum of our functional. For every $$\varepsilon>0$$ consider the set $$X_\varepsilon$$ of all $$L^1$$-functions $$h$$ for which the value of the functional is at most $$I+\varepsilon$$. Clearly, it is a convex, non-empty, closed (in $$L^1$$) set and $$X_{\varepsilon'}\supset X_{\varepsilon''}$$ when $$\varepsilon'\ge\varepsilon''$$.
Fix some strictly convex non-negative function $$\Phi(t)\le |t|$$. To simplify the technicalities, I'll choose it by the conditions $$\Phi(0)=\Phi'(0)=1$$, $$\Phi''(t)=\frac 2{\pi(1+|t|^2)}$$ but pretty much any other choice will work as well. | {
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Let $$J_\varepsilon=\inf_{h\in X_\varepsilon}\int\Phi(h)\,.$$ Clearly, $$J_\varepsilon$$ is a bounded non-increasing function on $$(0,1)$$, say. Let $$J$$ be its limit at $$0+$$. Passing to an appropriate decreasing sequence $$\varepsilon_n\to 0+$$ and re-enumerating $$X_n=X_{\varepsilon_n}, J_n=J_{\varepsilon_n}$$, we can assume that $$J_n\ge J-2^{-5n}$$ We will choose a representative $$h_n$$ of $$X_n$$ for which $$\int\Phi(h_n)$$ is not more than $$2^{-5n}$$ above $$J$$ and, thereby, not more than $$2\cdot 2^{-5n}$$ above its infimum over $$X_n$$. Then for $$m>n$$, we have $$h_{n,m}=\frac{h_n+h_m}2\in X_n$$ (convexity of $$X_n$$ plus inclusion $$X_n\supset X_m$$) and $$\int\Phi(h_{n,m})\le \int\frac 12(\Phi(h_n)+\Phi(h_m))-\int\frac{(h_n-h_m)^2}{4\pi (1+|h_n|^{2}+|h_m|^{2})}$$ (second order Taylor with the remainder in the Lagrange form), whence $$\int\frac{(h_n-h_m)^2}{4\pi(1+|h_n|^2+|h_m|^2)}\le 2\cdot 2^{-5n}$$ regardless of $$m>n$$ (otherwise we would go below $$J+2^{-5n}-2\cdot 2^{-5n}=J-2^{-5n}$$, which is below the infimum over $$X_n$$). It remains to note that if $$|h_n|\le 2^n$$ and $$|h_n-h_m|>2^{-n}$$, then the integrand is at least $$\rm{const}\, 2^{-4n}$$, so we get $$m(\{|h_n|\le 2^n, |h_n-h_m|>2^{-n}\})\le \rm{Const}\,2^{-n}$$ for all $$m>n$$ from where it follows at once that $$h_n(x)$$ is a Cauchy sequence for almost all $$x$$ (recall that $$\|h_n\|_1\le 3$$, so the first condition excludes only a set of measure $$3\cdot 2^{-n}$$, then use Borel-Cantelli).
That's it. Feel free to ask questions if anything is unclear :-) | {
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• Wow, this looks impressive. This part was a bit confusing to me - “ Then $\|v_{n_k}\chi_{\cup_{q>k}E_{n_q}}\|_1\to 0$ and, adding these parts to $u_{n_k}$, we see that we can (passing to a subsequence) assume that our minimizing sequence can be represented as $h_n=U_n+V_n$ where $U_n\to h$ in $L^1$ and $V_n$ have disjoint supports $G_n$.” Do you mean to add $v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$ to $u_{n_k}$? I am not sure how to get the desired representation from this. Aug 2, 2021 at 4:04
• @NateRiver Exactly as you said: $U_k=u_{n_k}+v_{n_k}\chi_{\cup_{q>k}E_{n_q}}$, $V_k=v_{n_k}\chi_{E_{n_k}\setminus \cup_{q>k}E_{n_q}}=v_{n_k}\chi_{G_k}$. Aug 2, 2021 at 7:45
• @NateRiver I just tried to solve the problem in $L^2$ first. There weak convergence is guaranteed, but doesn't seem to drop the value of the functional immediately, so I needed the norm convergence of a minimizing sequence. Fortunately, if one has a nested sequence of bounded closed convex sets in a Hilbert space, the smallest norm elements of the sets converge in norm (the proof is the same as above just using the parallelogram identity). I tried to mimic that idea in $L^1$ (which required strict convexity of something) and got the a.e. convergence this way, which turned out to be sufficient. Aug 2, 2021 at 13:33
• An alternative proof for Part 2 (existence of a minimizing sequence converging a.e.) Let $h_n$ be any minimizing sequence for $F$. As observed, it is bounded in $L_1$, so by the Komlós Theorem, up to extracting a subsequence, it is a.e. converging in Cesaro sense to some $h\in L^1$. Since $F$ is a convex functional, the sequence of the Cesaro means is still a minimizing sequence. Aug 2, 2021 at 13:38
• Of course, and I wouldn't be surprised if you answer (which I'm still reading) contains as a byproduct an alternative proof of that Komlós theorem (which is a 10 page paper: link.springer.com/article/10.1007%2FBF02020976 ). Aug 2, 2021 at 13:51 | {
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# How to properly represent a matrix function.
Given the function $$f_{h}(x,y,z)=(x-z,y+hz,x+y+3z)$$, what is the correct way to represent the matrix function in respect to the standard basis?
With the representation theorem, I would write the matrix in columns as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & 1\\ 0 & 1 & 1\\ -1 & h & 3\end{bmatrix}$$ But in my textbook it is written in rows as: $$F_{h|S_3}=(f_h(e_1)|{S_3} \quad f_h(e_2)|{S_3} \quad f_h(e_3)|{S_3})=\begin{bmatrix}1 & 0 & -1\\ 0 & 1 & h\\ 1 & 1 & 3\end{bmatrix}$$
What is the difference between them? | {
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What is the difference between them?
The correct way to represent the function $$f_h$$ in matrix form depends on the convention that you want to use to represent it. Let $$(x,y,z) \in \mathbb{R}^3$$. The matrix which you computed is useful for expressing $$f_h$$ as $$f_h(x,y,z) = \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 1 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix}.$$ We can verify this by expanding the matrix product: \begin{align} \begin{bmatrix} x& y & z \end{bmatrix} \begin{bmatrix} 1& 0 & 2 \\ 0 & 1& 1\\ -1 & h & 3 \end{bmatrix} &= \begin{bmatrix} x \cdot 1 + y \cdot 0 + z \cdot (-1) \\ x \cdot 0 + y \cdot 1 + z \cdot h \\ x \cdot 1 + y \cdot 1 + z \cdot 3 \end{bmatrix} \\ &= \begin{bmatrix} x - z & y + hz & x + y + 3z \end{bmatrix}. \end{align} The result is a $$1 \times 3$$ matrix, which we can interpret as a row vector in $$\mathbb{R}^3$$. The matrix written in your textbook is useful for the following representation: $$f_h(x,y,z) = \begin{bmatrix} 1& 0 & -1 \\ 0 & 1& h\\ 1 & 1 & 3 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} x-z \\ y + hz \\ x + y + 3z \end{bmatrix}.$$ The result is a $$3 \times 1$$ matrix, which we can regard as a column vector in $$\mathbb{R}^3$$.
The two representations look different in the sense that the former yields a row vector and the latter a column vector, however they are the same in the sense that they can both be regarded as lists of three real numbers, i.e. as $$(x-z,y+hz,x+y+3z) \in \mathbb{R}^3.$$ The accepted answer here sums this idea up pretty well. | {
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# Representation of function as power series - unique?
The following example is from Calculus, 7e by James Stewart:
Example 2, Chapter 11.9 (Representations of functions as Power Series)
Find a power series representation for $$\frac{1}{x+2}$$
My solution is:
$$\frac{1}{x+2}=\frac{1}{1-\left(-x-1\right)}=\sum_{n=0}^{\infty}\left(-x-1\right)^{n}=\sum_{n=0}^{\infty}\left(-1\right)^{n}\left(x+1\right)^{n}$$
Then to find the interval of convergence
$$\left|x+1\right|<1 \Rightarrow x\in\left(-2,0\right)$$
But the given solution is different:
“In order to put this function in the form of the left side of Equation 1, $$\frac{1}{1-x}$$ , we first factor a 2 from the denominator:
$$\frac{1}{2+x}=\frac{1}{2\left(1+\frac{x}{2}\right)}=\frac{1}{2\left[1-\left(-\frac{x}{2}\right)\right]}=\frac{1}{2}\sum_{n=0}^{\infty}\left(-\frac{x}{2}\right)^{n}=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n}}{2^{n+1}}x^{n}$$
This series converges when $$\left|-\frac{x}{2}\right|<1$$, that is, $$\left|x\right|<2$$. So the interval of convergence is $$\left(-2,2\right)$$."
So my questions are:
1. Did I make an error somewhere?
2. If not, are the two representations equivalent? Can there be more than one representation of a function as a power series?
• The representation as a power series at a given point is unique. Here, yours is at the point $-1$, while the solution gives the one at the point $0$. (By default, power series are usually taken at 0 when nothing is specified or obvious from context) – Clement C. Dec 29 '18 at 21:59
I think your method is right but you made a power series around $$x_0=-1$$
while they did it around $$x_0=0$$ which is what they asked for probably
Both solutions are correct: your series is a power series centered at $$-1$$, whereas the power series from the given solution is centered at $$0$$. | {
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Question
# Let $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$ and $$\displaystyle { I }_{ 2 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx }$$, where $$f$$ is a continuous function and $$z$$ is any real number, $$\displaystyle \frac { { I }_{ 1 } }{ { I }_{ 2 } } =$$
A
32
B
12
C
1
D
none of these
Solution
## The correct option is B $$\displaystyle \frac { 3 }{ 2 }$$We have, $$\displaystyle { I }_{ 1 }=\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( \left( 3-x \right) \left( 3-\left( 3-x \right) \right) \right) } dx$$ $$\displaystyle \left[ \because \int _{ a }^{ b }{ f\left( x \right)dx } =\int _{ a }^{ b }{ f\left( a+b-x \right) dx } \right]$$$$\displaystyle =\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ \left( 3-x \right) f\left( x\left( 3-x \right) \right) dx }$$$$\displaystyle =3\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ f\left( x\left( 3-x \right) \right) dx } -\int _{ \sec ^{ 2 }{ z } }^{ 2-\tan ^{ 2 }{ z } }{ xf\left( x\left( 3-x \right) \right) } dx$$$$=3{ I }_{ 2 }-{ I }_{ 1 }$$$$\displaystyle \therefore 2{ I }_{ 1 }=3{ I }_{ 2 }\Rightarrow \frac { { I }_{ 1 } }{ { I }_{ 2 } } =\frac { 3 }{ 2 }$$Mathematics
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Prove that $A \smallsetminus (A \smallsetminus B) = A \cap B$
$A$ and $B$ are any sets, prove that $A \smallsetminus (A \smallsetminus B) = A \cap B.$ This formula makes sense when represented on a Venn diagram, but I am having trouble with proving it mathematically.
I have tried letting $x$ be an element of $A$ and continue from there, but it doesn't seem to work out as a valid proof anyways.
Could anyone please point me in the right direction?
Many thanks.
• Please show us your work: starting with $x \in A\setminus(A\setminus B)$, or the other way around, starting with $x \in (A\cap B)$. Please don't claim you tried ABC, unless you also show us you effort using ABC. Would you like to get an answer that says only: "Yup, that's right, start by letting $x \in A\setminus (A\setminus B)$... – Namaste Jan 27 '18 at 18:42
Let $x \in A \setminus (A \setminus B).$
Then $x$ is an element such that $x \in A$ and $x \notin A \setminus B$. But if $x \notin A\setminus B$, with some additional work, we realize this implies that $x \in A$ and $x \in B$. So $x \in A \cap B$.
Vice-versa: let $x \in A \cap B$ so $x \in A$ and $x \in B$. This implies that $x \notin A \setminus B$. But given that $x \in A$ and $x \notin A \setminus B$ this implies that $x \in A \setminus (A \setminus B)$. | {
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• I would like this answer if you hadn't skipped from "$x \in A$ and $x \notin A\setminus B$, but if $x\in A \setminus B$" to "we realize that $x \in A$ and $x\in B$." I added the words ("with some additional work") in between them. I would like this answer more if you explained that $x \in A$ and $x \notin (A\setminus B)$, then $x \in A$ and, it is not the case that ($x \in A$ and $x \notin B$.) By Demorgans, we get that $(x \in A \land \lnot x\in A),$ or $(x \in A \land x\in B)$. The first disjunct is always false (no element can be in a set, and not be in a set. – Namaste Jan 27 '18 at 19:13
• ...That leaves us to conclude, that "we realize this implies that $x\in A$ and $x\in B$. – Namaste Jan 27 '18 at 19:13
Notice that \begin{eqnarray*} x\in A\setminus(A\setminus B) &\Leftrightarrow& (x\in A)\wedge \neg(x\in A\setminus B)\\ &\Leftrightarrow & (x\in A)\wedge \neg((x\in A)\wedge \neg (x\in B))\\ &\Leftrightarrow & (x\in A)\wedge ((x\notin A)\lor(x\in B))\\ &\Leftrightarrow & ((x\in A)\wedge (x\notin A))\lor ((x\in A)\wedge(x\in B))\\ &\Leftrightarrow & (x\in A)\wedge (x\in B)\\ &\Leftrightarrow & x\in A\cap B. \end{eqnarray*} | {
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• Yes, when in doubt go back to the logic – Tom Collinge Jan 27 '18 at 18:28 | {
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• I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between $x \in A) \land$\lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 18:51 • I like this, because the equivalence makes the proof bi-directional, and doesn't skip important steps between$x \in A \land \lnot (x \in A\setminus B$and its equivalent,$x\in A \land x \in B.$Those are likely the steps on which the asker got tripped up, I suspect. Nice answer. – Namaste Jan 27 '18 at 19:01 • @amWhy: Thanks. Indeed, often mistakes are made in these basic logical steps. I've seen many people making mistakes using the notation$x\notin B$. I think it's safer to write$\neg(x\in B)$, indeed, if$B$is a more complicated expression, we still now how to negate the logical expression appearing inside the brackets. This basic logical way of thinking is user friendly and not very prone to errors. – Mathematician 42 Jan 27 '18 at 20:43 Did you know that $$A \smallsetminus B = A \cap \overline{B} \;\;?$$$\begin{align} A \smallsetminus (A \smallsetminus B) &= A \smallsetminus (A \cap \overline{B}) \\ \\ &= A \cap \overline{ (A \cap \overline{B})}\\ \\ &= A \cap (\overline{A} \cup B)\\ \\ &= (A \cap \overline{A} )\cup (A \cap B) \\ \\ &= A \cap B \end{align}$• cmon, why this get downvoted, it is correct. hmm, maybe you want to precise that$\bar B=B^{\complement}$and not the closure in this context. – zwim Jan 27 '18 at 18:55 • @zwim: Not sure why this got downvoted, however this proof assumes knowledge about standard set operations which you should be able to prove directly as well. You might end up running in circles and not proving anything. – Mathematician 42 Jan 27 '18 at 18:58 Let$x\in A\cap B$. Then$x\in A$and$x\in B$. Then$x\not \in A\setminus B$, so$x\in A\setminus (A\setminus B)$. Conversely, if$x\in A\setminus (A\setminus B)$, then$x\in A$and$x\not | {
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A\setminus (A\setminus B)$. Conversely, if$x\in A\setminus (A\setminus B)$, then$x\in A$and$x\not \in (A\setminus B)$. So$x\in A$and$x\in B$. That is$x\in A\cap B\$... | {
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• ditto of @Skills. – Namaste Jan 27 '18 at 18:36 | {
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# Math Help - sequences problem
1. ## sequences problem
A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.
Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).
Is it 10% from each dimension or from the volume? Hard to figure out from the problem.
If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres
I need some feedback.
2. I expect since it says "the dimension of each container is decreased by 10% with respect to the larger one" I would say it's the dimensions and not the volume that is decreased by 10%.
3. Originally Posted by terminator
A hardware supplier has designed a series of six plastic containers with lids where each container (after the first) can be placed into the next larger one for storage purposes. The containers are rectangular boxes, and the dimensions of the largest one are 120 cm by 60 cm by 50 cm. The dimension of each container is decreased by 10 percent with respect to the next larger one.
Determine the volume of each container, and the volume of all the containers. Write your answers in litres (one litre = 1000 cm3).
Is it 10% from each dimension or from the volume? Hard to figure out from the problem.
Why? The problems says "The dimension of each container is decreased by 10 percent". That tells you precisely which is intended | {
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If from the each dimension then:
V 1= 50x 60x 120 = 360000 cm3 = 360 litres
V2=45 x 54 x 108 = 262440 cm3 = 262.4 liltres
V 3 = 40.5 x 48.6 x 97.2 = 191.3litres
V4 = 36.45 x 43.74 x 87.48 =139.4 litres
V5 = 32.81 x 39.37 x 78.74 = 101.7 litres
V6 = 29.61 x 35.47 x 70.94 =74.5 litres
I need some feedback.
4. Hello, terminator!
A hardware supplier has designed a series of six plastic containers with lids
. . where each container (after the first) can be placed into the next larger one.
The containers are rectangular boxes.
. . The dimensions of the largest one are 120 cm by 60 cm by 50 cm.
The dimensions of each container are decreased by 10 percent
. . with respect to the next larger one.
(a) Determine the volume of each container.
(b) Determine the volume of all the containers.
Write your answers in litres (one litre = 1000 cubic cm).
The original box has dimensions $L,\,W,\,H$
. . Its volume is: . $L\!\cdot\!W\!\cdot\!H\text{ cm}^3$
The next box has dimensions: $0.9L,\,0.9W,\,0.9H$
. . Its volume is: . $(0.9L)(0.9W)(0.9H) \:=\:0.729(LWH)$
That is, each box is 0.729 of the volime of the next larger box.
The first box has volume: . $120\cdot60\cdot50 \:=\:360,\!000\text{ cm}^3$
. . That is: . $V_1 \:=\:360\text{ liters.}$
(a) Therefore, the $\,n^{th}$ box has volume: . $V_n \;=\;360(0.729)^{n-1}$
The sum of the volumes of all the boxes is:
. . $S \;=\;360 + 360(0.729) + 360(0.729^2) + 369(0.729^3) + \hdots$
. . $S \;=\;360\underbrace{\bigg[1 + 0.729 + 0.729^2 + 0.729^3 + \hdots\bigg]}_{\text{a geometric series}}$
The sum of the geometric series is: . $\dfrac{1}{1 - 0.729} \:=\:\dfrac{1}{0.271}$
(b) Therefore: . $S \;=\;360\left(\frac{1}{0.271}\right) \;\approx\;1328.4\text{ liters}$ | {
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The Gambler’s Fallacy
Probability seems simple enough to many people that it can fool them into wrong conclusions. We have had many questions that involve the “Gambler’s Fallacy”, both from people who naively assume it without thinking, and from some who defend it using technical ideas like the Law of Large Numbers.
The Gambler’s Fallacy
Here is a question from 2003 that is a good introduction to the idea:
Gambler's FallacyA current co-worker and I are in a friendly disagreement about the probability of selecting the winning number in any lottery, say Pick 5. He states that he would rather bet the same set of five numbers every time for x period of time, but I insist that the probability is the same if you randomly select any set five numbers for the same period of time. The only assumption we make here is betting one set of numbers on any given day. Who is correct?
I tried explaining to him that the probability of betting on day one is the same for both of us. On day two it is the same. On day three it is the same, etc. Therefore the sum of the cumulative probabilities will be the same for both of us.
Doctor Wallace first examined the appropriate calculation, before moving on to the likely underlying error:
You are correct. If you have the computer randomly select a different set of 5 numbers to bet on every day, and your friend selects the same set of numbers to bet on every day, then you both have exactly the same probability of winning.
Tell your friend to think of the lottery as drawing with tickets instead of balls. If the lottery had a choice of, say, 49 numbers, then imagine a very large hat containing 1 ticket for every possible combination of 5 numbers. 1, 2, 3, 4, 5; 1, 2, 3, 4, 6; etc.
On the drawing day, ONE ticket is pulled from the hat. It is equally likely to be any of the C(49,5) tickets in the hat. (There would be 1,906,884 tickets in the hat in this case.) | {
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Since both you and your friend have only ONE ticket in the hat, you both have the same chance of winning.
On the next drawing day for the lottery, ALL the tickets are replaced. Each lottery draw is an event independent of the others. That is to say, the probability of any combination winning today has absolutely NO effect on the probability of that or any other combination winning tomorrow. Each and every draw is totally independent of the others.
That perspective makes it “obvious”: if two people each have one “ticket”, they have the same probability, whether or not it is the same one that was taken last time, since the “ticket” is chosen randomly each time without regard to the past.
So why is the other opinion tempting?
The reason your friend believes that he has a better chance of winning with the same set of numbers is probably due to something called the "gambler's fallacy." This idea is that the longer the lottery goes without your friend's "special" set of numbers coming up, the more likely it is to come up in the future. The same fallacy is believed by a lot of people about slot machines in gambling casinos. They hunt for which slot hasn't paid in a while, thinking that that slot is more likely to pay out. But, as the name says, this is a fallacy; pure nonsense. A pull of the slot machine's handle, like the lottery draw, is completely independent of previous pulls. The slot machine has no memory of what has come before, and neither has the lottery. You might play a slot machine for 2 weeks without hitting the big jackpot, and someone else can walk in and hit it in the first 5 minutes of play. People wrongly attribute that to "it was ready to pay out." In reality, it's just luck. That's why they call it gambling. :)
The same thing comes up in math classes:
This used to be a "trick" question on old math tests: | {
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The same thing comes up in math classes:
This used to be a "trick" question on old math tests:
"You flip a fair coin 20 times in a row and it comes up heads every single time. You flip the coin one more time. What is the probability of tails on this last flip?"
Most people will respond that the chance of tails is now very high.
(Ask your friend and see what he says.) However, the true answer is that the probability is 1/2. It's 1/2 on EVERY flip, no matter what results came before. Like the slot machine and the lottery, the coin has no memory.
That type of question is still valuable. It tests an important idea that students need to think about.
For more about picking the same number in a lottery, see
Lottery Strategy and Odds of Winning
For a very nice refutation of the Gambler’s Fallacy in coin tossing, see
What Makes Events Independent?
The Law of Large Numbers
That question was based on a naive approach to gambling. The next, from 2000, is based on probability theory. (The questions were asked in imperfect English, which I will restate as I understand it, correcting a misinterpretation in the archived version. Doctor TWE got it right.)
Law of Large Numbers and the Gambler's FallacyIf we throw three dice at a time, three times altogether, is the result the same as if we throw nine dice one time?
Do we have the same probability to get a given number in the two cases?
Where and how different is it?
Doctor TWE answered, covering several possible interpretations of “result”:
The sum, the average, and the probabilities of getting a particular value on one die or more aren't affected by whether the dice are rolled one at a time, in groups of 3, or all 9 at once. These are called "independent events," and the order in which they happen doesn't affect the outcome.
Elvino replied, | {
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Elvino replied,
So, I have 42.12% probability to get at least one six (or other number), hen I throw three dice at once. If I throw the same dice, in the same way, again and again I will always have the same probability!
But, there is a law that says, the more I throw, the more probability I have to obtain a certain number. How does this law of probabilities work?
Checking his calculation, and thereby confirming what he meant, I get the probability of rolling at least one six on three dice to be the complement of rolling no sixes: $$\displaystyle 1 – \frac{5^3}{6^3} = 42.1%$$. Next time I will discuss this further.
Doctor TWE confirmed his calculation, then explained what this law means, and does not mean:
There is something called the Law of Large Numbers (or the Law of Averages) which states that if you repeat a random experiment, such as tossing a coin or rolling a die, a very large number of times, your outcomes should on average be equal to (or very close to) the theoretical average.
Suppose we roll three dice and get no 6's, then roll them again and still get no 6's, then roll them a third time and STILL get no 6's. (This is the equivalent of rolling nine dice at once and getting no 6's, as we discussed in the last e-mail; there's only a 19.38% chance of this happening.) The Law of Large Numbers says that if we roll them 500 more times, we should get at least one 6 (in the 3 dice) about 212 times out of the 503 rolls (.4213 * 503 = 211.9).
This is *not* because the probability increases in later rolls, but rather, over the next 500 rolls, there's a chance that we'll get a "hot streak," where we might roll at least one 6 on three or more consecutive rolls. In the long run (and that's the key - we're talking about a VERY long run), it will average out. | {
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