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# Two definitions of Saddle point
Let $f\colon[a,b]\to\mathbb{R}$ be a differentiable function and $x\in[a,b]$ with $f'(x)=0$. Is there a counterexample with respect to the equivalence
$x$ is not a local extremum of $f$
$\Leftrightarrow$
There is an $\epsilon>0$ such that $f'$ is monotonically increasing on $[x-\ep... | {
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# Energy Calculator Physics | {
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Efficiency is the ratio of the amount of useful energy produced (energy output- Eout) to the amount of energy used (energy input- Ein), expressed as a percentage. KE = 1/2*m*v^2. 5mv^2 and also compare E to the energy at the bottom of the projectile, E3 = 0. The principle of hydro electricity generation is quite simple... | {
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ENERGY CALCULATOR. (a) Calculate the change in the internal energy of the block-belt system as the block comes to a stop on the belt. The unit of energy is the joule, the unit of power is the watt, and the unit of time is the second. Assume that the earth is a uniform sphere and that its path around the sun is circular... | {
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teachers and students, The Physics Classroom provides a wealth of resources that meets the varied needs of both students and teachers. Search for: 7. Quantity of heat This calculator can find missing values in the relationship between heat and temperature: heat added or removed, specific heat, mass, initial temperature... | {
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flows, compressible aerodynamics, thermodynamics of air, boundary layer analysis, heat transfer). (Yes, even Passive House projects. If you heat a balloon (carefully), the molecules of air in the balloon gain energy and strike the inner walls of the balloon with greater force. 2 Chapter4 ChemicalEnergy amoleofoxygenato... | {
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course invariably spends a lot of time on two big ideas: The momentum principle and the work energy principle. Please spread the word about this completely free resource by linking to us. 626x10-34 J s). We provide step by step Solutions for ICSE Physics Class 10 Solutions Pdf. At its peak, a severe storm may have a to... | {
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so far all we have is a kind of elegant messiness. Calculate Energy Density Collection of important formulas of math and physics, with calculators: energy density. You need to make sure the units of work and energy match. LE PHYSICS 111 CONSERVATION OF MECHANICAL ENERGY Calculate the potential energy, kinetic energy, m... | {
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you to input any unit. 00 V? (See Figure 2. At NASA's Zero Gravity Research Facility in Cleveland, Ohio, experimental payloads fall freely from rest in an evacuated vertical shaft through a distance of 132 m. Horsepower is a unit of measurement of power developed by engineer James Watt in the late 18 th century. This s... | {
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amount of energy that an 11. The term is a little ambiguous because it can refer to the amount of energy you can get from breaking atomic bonds, as in the energy you get out of propane when you burn it, or the amount of energy stored as an ionic potential between molecules, such as in a neuron. E = energy measured in j... | {
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heat energy, mechanical energy, electrical energy, sound energy and many more. Efficiency is the ratio of the amount of useful energy produced (energy output- Eout) to the amount of energy used (energy input- Ein), expressed as a percentage. 1099 as of January 2011 (found at U. The next part of the question is asking m... | {
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energy photon undergo Thomson or Compton scattering, and high energy photons pair produce. Energy, in physics, the capacity for doing work. Efficiency The ratio of energy which was transferred to a useful form compared to the total energy initially supplied is called the efficiency of the device. 6: carbon (to CO) 22. ... | {
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1-2. University Physics Volume 1. The energy stored between the plates of a charged capacitor is electrical potential energy. Is this correct? What is another way to calculate loss of energy?. The more usual formula is given for an ideal gas. Every building has them. h = m, then the velocity just before impact is v = m... | {
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- Online conversion of area, currency, density, energy, force, length, mass, power, pressure, speed, temperature, volume and bytes. Home » Nuclear-physics » Compton-scattering Science Calculators. Spring physics calculator solving for potential energy given spring force constant and spring stretch length. 4 calculate t... | {
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light (CFL) bulb, Wlight , uses when left on for 10. Calculate the kinetic energy again when the speed is doubled. Energy Range: Energy Range 0. 01L Physics I: Classical Mechanics, Fall 2005 Dr. At time 2, the gravitational potential energy equals the kinetic energy. The Kinetic energy calculator allows you to calculat... | {
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Dynamics (motion on curved paths, projectile motion), and Fluid Dynamics (ideal flow, shock tube flows, airfoil flows, compressible aerodynamics, thermodynamics of air, boundary layer analysis, heat transfer). Do not use calculations for anything where loss of life, money, property, etc could result from inaccurate. (b... | {
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heat released, you use the specific heat formula. Although its original purpose was to compare the output of steam engines with the power of horses (hence its name), it has since been adopted as a unit of measurement for all sorts of engines used to power things such as vehicles, lawn mowers, boats, chainsaw, and airpl... | {
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bar) trigger is released can be determined from the principle of conservation of energy. Energy Units and Conversions by Dennis Silverman U. The term is most commonly used in relation to atoms undergoing radioactive decay, but can be used to describe other types of decay, whether exponential or not. (a) Calculate the c... | {
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is called Conservation of Energy: energy just gets transformed and the total stays constant. We have also taken a look at how to calculate energy and mass balances with COMSOL Multiphysics in order to check the accuracy of simulation results. KE = 1/2*m*v^2. kilowatt-hour to watt-hour (kW·h—W·h) measurement units conve... | {
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Implantation into Transition-Metal Dichalcogenide Monolayers to Form Janus Structures. 2 Chapter4 ChemicalEnergy amoleofoxygenatomswouldhaveamassof16grams. Moment of Inertia & Rotational Energy Physics Lab IX. At time 2, the gravitational potential energy equals the kinetic energy. to calculate the kinetic friction coe... | {
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For JEE Main other Engineering Entrance Exam Preparation, JEE Main Physics Work, Energy And Power Previous Year Questions with Solutions is given below. Bond dissociation energy (BDE) is a measure of the bond strength in a chemical bond. In thermodynamics, the Gibbs free energy is a thermodynamic potential that can be ... | {
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energy is the energy of an object in motion. What is the potential energy of the barbell when it is lifted to this height?. Energy which an object processes due to its motion is called Kinetic Energy and to explain further it is work done to accelerate a body of given mass from rest to its current or desired velocity, ... | {
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how physics is done. Tweets by @HuntOnly. Our appliance and electronic energy use calculator allows you to estimate your annual energy use and cost to operate specific products. The SI unit of energy is the joule , which is the energy transferred to an object by the work of moving it a. Where KE is kinetic energy; m is... | {
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using conservation of energy. Three things that affect the extent of heat transfer to an object are mass, type of substance and the amount of heat applied. The Digital Dutch Unit Converter - Online conversion of area, currency, density, energy, force, length, mass, power, pressure, speed, temperature, volume and bytes.... | {
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calculator. Yes you use the specific heat capacity of water to calculate, the formula for specific heat is Q = cmθ, Q - is the energy require to raise 1 Celsius of 1 kg o substance c - is the specific heat o a substance m - is the mass of the substance θ - is. zip: 1k: 09-11-03: Velocity after Falling enter initial hei... | {
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data. average cost per KWH \$0. Important: Jump-Start Your Practice Order the Official SAT Subject Test Study Guide in Physics and get two full-length practice tests, detailed answer explanations, tips, and more. Kinetic energy is one half the objects mass times its linear velocity squared, but when an object like a wa... | {
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in Joules. Energy Units and Conversions by Dennis Silverman U. 8 m//s^(2)). Thus, as shown in the Manuscript, “Trebuchet Mechanics,” the greatest range possible is Rm = 2*(m1/m2) h, where h is the distance the counterweight of. Einstein's famous equation relates energy and mass: E = mc2. Calculate values of different e... | {
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Energy Matters. org is the ultimate resource for unit conversion. Calculator that calculate the photon energy using Plancks constant. Solve for the unknown variable. The electron is ejected from its orbital position and the x-ray photon loses energy because of the interaction but continues to travel through the materia... | {
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with calculation examples. (g=10ms-2). You can think of a trebuchet as a see saw! Yes, a see saw is really all that a trebuchet is. kilowatt-hour to watt-hour (kW·h—W·h) measurement units conversion. Year 7 lesson on energy in food Investigating the energy content in food practical instructions not included as it was n... | {
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Output in Joules (J, kJ, MJ), Watt-hours (Wh, kWh), calories (Cal, kCal) and foot-pounds (ft-lbs). Engineering Physics Resources. 8 m//s^(2)). Can you calculate the energy needed to increase the temperature of 100kg of iron by 40°C? Extension. In physics , energy is the quantitative property that must be transferred to... | {
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a trebuchet as a see saw! Yes, a see saw is really all that a trebuchet is. Example of Few questions where you can use this Mechanical Energy Formula calculate the Mechanical energy of the object have mass 10 kg and velocity 3m/s and height above the ground is 10 m calculate the Kinetic Energy,Potential energy and Mech... | {
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# Why can't I get an accurate polynomial equation for these data points?
I have 30 data points that I have digitised from the red dashed line in the graph below. My goal is to find an approximate equation to represent the line.
I have tried to get a 6 degree polynomial trendline through Excel for these points, but if... | {
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• why degree 6? have you tried higher degrees? – Glougloubarbaki Jun 18 '18 at 9:56
• The function doesn't look like a polynomial would be the first choice to approximate it. Have you tried fitting a sum of inverse powers? Or perhaps just subtract out the singularity at the origin and approximate the rest with a polyno... | {
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In order to better represent the left part of the plot, we could consider $$y=\frac b {(x-d)^c}$$ which would lead to $R^2=0.999914$ and $$\begin{array}{clclclclc} \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\ b & 1166.87 & 21.3244 & \{1122.96,1210.79\} \\ c & 0.09707 & 0.00132 & \{0... | {
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# Why is $\sum_{p \in S_n} 2^{c(p)}$ equal to $(n+1)!$?
It is obvious that $$\sum_{p \in S_n} 1=n!$$ because it is just counting how many permutations there are of $$n$$ symbols.
But I have also observed that $$\sum_{p \in S_n} 2^{c(p)}=(n+1)!$$, where $$c(p)$$ is the number of cycles of $$p$$.
What is the combinato... | {
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First let's verify that the sum does actually count what I say that it does. Suppose that we have already chosen the permutation $$p$$. A function $$f$$ satisfies the conditions above if and only if for each cycle of $$p$$, we have that $$f$$ maps every element of that cycle to the same element of $$\{1, 2\}$$. We can ... | {
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• I hadn't heard of Polya's Enumeration Theorem before, but now that I have, I think that this may be essentially the same solution as @Milten's. – Dylan Jan 31 '20 at 22:03
• For your last paragraph, suppose that $f^{-1}(1)=\{x_1,x_2,\ldots,x_k\}$ and $f^{-1}(2)=\{y_1,y_2,\ldots,y_l\}$ with $x_1<x_2<\ldots<x_k$ and $y... | {
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"openwebmath_score": 0.8838981986045837,
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EDIT: Let me just include the immediate generalisation. If we have $$m$$ colours, then there are $$\binom{n+m-1}{n}$$ different colourings (by stars and bars). So: $$\sum_{\sigma\in S_n}m^{c(\sigma)} = n!\binom{n+m-1}{n} = \frac{(n+m-1)!}{(m-1)!} = m(m+1)\cdots(n+m-1)$$ This only works for $$m\in\mathbb N$$ of course. ... | {
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# Permuting $n$ points in a $2$-manifold
Given $$n$$ points on a connected $$2$$-manifold $$M$$, I'd like to consider the homotopy classes of paths that "permute" these points.
Edit (Clarifying what I mean by this):
Given a set of $$n$$ distinct points $$T=\{x_{1},\ldots,x_{n}\}\subset M$$, to each point we assign a... | {
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A survey of surface braid groups and the lower algebraic K-theory of their group rings
I think my notion of $$\text{Mot}_{n}(M)$$ coincides with the Definition in Section 2.2 of the above paper. If so, then the answer to 2(i) is yes (according to the paper). Building on this, I believe Theorems 12 and 13 of Bellingeri... | {
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The comments seem to have answered questions 1 and 2i, to show that the group $$\operatorname{Mot}_n(M)$$ is indeed the surface braid group $$B_n(M)$$.
To answer 2ii, consider a disk $$D \subset M$$ such that $$T \subset D$$. Then the inclusion map $$D \hookrightarrow M$$ induces a homomorphism $$B_n \to B_n(M)$$, so t... | {
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## anonymous one year ago Half-Life reaction help?
1. anonymous
The purpose of this hands-on lab is to model the concept of half-life using a sample to represent radioactive atoms. Materials 200 M&M® candies, pennies, or other small candy/item with two distinct sides shoe box or other small box with a lid Procedure P... | {
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3. anonymous
After how many time intervals (shakes) did one-half of your atoms (candies) decay? Trial 1: It took 8 Trial 2: 6 shakes What is the half-life of your substance? <-------------------(This I need help with)
4. anonymous
The answer is B
5. cuanchi
@alante in the first shake you went from 200 to 107-113 r... | {
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12. anonymous
I'll just post these xD If the half-life model decayed perfectly, how many atoms would be remaining (not decayed) after 12 seconds? If you increased the initial amount of atoms (candies) to 300, would the overall shape of the graph be altered? Explain your answer. Go back to your data table and for each ... | {
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15. anonymous
I'm still really confused ._.
16. abb0t
that the whole decay process is not a constant rate half-life is just a description whence the amount is halved (or close) of the original number of species from when the decay process started at a given time.
17. anonymous
but it ask what the half life is to t... | {
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# Thread: Differentiation and Need HELP with LATEX!
1. ## Differentiation and Need HELP with LATEX!
g(x) = {-x, x<=0
3x^2,x>0
(a) Evaluate the limit of {g(x+delta x) - g(x)}/{delta x) for x<=0 and x>0 as delta x tends to 0
(b) evaluate the limit of {g(delta x) - g(0)}/{delta x) as delta x tends to 0 from the right ... | {
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Then take x> 0 and assume that delta x is small enough that x+ delta x is also positive- then you can use " $3x^2$ as the formula for both g(x) and g(x+ delta x). If $g(x)= 3x^2$, what is g(x+ deltax)?
For (b), where x=0, taking the limit "from the left" means using g(0+ delta x)= -delta x and taking the limit "from t... | {
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Thank you!
6. Originally Posted by cyt91
For (a) :
x<0,
g(x+delta x)= -(x+delta x)
Is this correct?
Then, the limit for [g(x+delta x)-g(x)]/[delta x] as delta x tends to 0 is
{-(x+delta x)-[-x]}/{delta x}=-1
Correct?
Yes, if x< 0 and $|\delta x|< |x|$, which we can assume since we are taking the limit as $\delta x$ go... | {
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# Integrating a function using u-substitution
I'm trying to figure out how to integrate this function. I tried several tricks from my toolkit, but I can't seem to figure it out.
$$\int\ \frac {e^{2x}-6e^x}{e^x+2}\ dx$$
So let's say that I factor out some terms and split the equation to:
$\displaystyle\int\ \frac {e... | {
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The trick is to see that $e^{2x}$ can be easily changed and factored. We start with the integral. $$\int \frac {e^{2x}-6e^x}{e^x+2}~ {\rm d}x$$ We can use the fact that $a^{mn} = (a^m)^n$ to change this to $$\int \frac {(e^x)^2-6e^x}{e^x+2}~ {\rm d}x = \int \frac {e^x(e^x - 6)}{e^x+2}~ {\rm d}x$$ We can then use the su... | {
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• Thanks Pure, that does make sense but what about the left side? $u$ at $e^x+2$ would result in a derivative that is not $e^{2x}$ – user472288 May 11 '18 at 23:13
• It does not result in that derivative, however it does not need to, you still get a cancellation – pureundergrad May 11 '18 at 23:16
• Your problem is tha... | {
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# Math Help - A VERY VERY HARD probability problem.!!!
1. ## A VERY VERY HARD probability problem.!!!
Inside a box, there are 16 marbles in which 7 are red and the rest are blue. 16 people that consists of 11 males and 5 females lined up to randomly pick one marble at a time. The females occupied the 3rd, 4th, 8th,9t... | {
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... |
4. ## Re: A VERY VERY HARD probability problem.!!!
Originally Posted by HallsofIvy
The only thing 'hard' about this problem is recognizing that most of the information given is irrelevant. The "a-priori" probability that a given person has a red marble is independent of their place in line. It is sufficient to imagine... | {
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# $S \not= \varnothing \implies \varnothing \subset S$?
If $S$ is a nonempty set, is the following statement correct: $\varnothing \subset S$? It's confusing me becuase $\varnothing$ does not contain any elements so I'm struggling with the logic behind this statement.
Yes, it's correct. To say "$X \subseteq Y$" means... | {
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$\varnothing\subset S$ for any set $S$, since you cannot find a counter-example, even if $S=\varnothing$: an element in $\varnothing$ which would not be in $S$.
Note: The correct mathematical notation is $\varnothing$ (code \varnothing), not $\emptyset$.
• -1 for saying that $\varnothing$ is correct, but $\emptyset$ ... | {
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# How to prove the Squeeze Theorem for sequences
The formulation I’m looking at goes: If $\lbrace x_n\rbrace$, $\lbrace y_n\rbrace$ and $\lbrace z_n \rbrace$ are sequences such that $x_n \le y_n \le z_n$ for all $n \in \mathbb N$, and $x_n \to l$ and $z_n \to l$ for some $l \in \mathbb R$, then $y_n \to l$ also.
So w... | {
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Then we have:
\begin{align} &\forall \varepsilon_1 > 0,\, \exists N_{\varepsilon_1} \in \mathbb N,\, \forall n \ge N_{\varepsilon_1}\space |x_n – l| < \varepsilon_1 \\ &\forall \varepsilon_2 > 0,\, \exists N_{\varepsilon_2} \in \mathbb N,\, \forall n \ge N_{\varepsilon_2}\space |z_n -l| < \varepsilon_2 \end{align}
Let ... | {
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As $n$ grows, you can get the distance from $x_n$ to $l$ to be less than any $\epsilon > 0$, and the distance from $z_n$ to $l$ to be less than $\epsilon$ as well. Just take the max $N$ of the two indices for $x_n$ and $z_n$ that guarantee this, and you will get that both $x_n$ and $z_n$ are within $\epsilon$ of $l$ fo... | {
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Dataplot Vol 2 Vol 1
# EMPIRICAL QUANTILE FUNCTION
Name:
EMPIRICAL QUANTILE FUNCTION (LET)
Type:
Let Subcommand
Purpose:
Compute the empirical quantile function.
Description:
The quantile function is the inverse of the cumulative distribution function, F,
$$Q(u) = F^{-1}(u) \hspace{0.2in} 0 < u < 1$$
Given a set of... | {
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Applications:
Distributional Analysis
Implementation Date:
2017/02
Program:
. Step 1: Define some default plot control features
.
title offset 2
title case asis
case asis
label case asis
line color blue red
multiplot scale factor 2
multiplot corner coordinates 5 5 95 95
.
. Step 2: Create 50, 100, 200, and 1000 no... | {
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# Create Symbolic Matrix Variables
Since R2021a
Symbolic matrix variables represent matrices, vectors, and scalars in compact matrix notation. When mathematical formulas involve matrices and vectors, writing them using symbolic matrix variables is more concise and clear than writing them componentwise. When you do th... | {
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C = A*B'
C =
$\left(\begin{array}{cc}{A}_{1,1} \stackrel{‾}{{B}_{1,1}}+{A}_{1,2} \stackrel{‾}{{B}_{1,2}}+{A}_{1,3} \stackrel{‾}{{B}_{1,3}}& {A}_{1,1} \stackrel{‾}{{B}_{2,1}}+{A}_{1,2} \stackrel{‾}{{B}_{2,2}}+{A}_{1,3} \stackrel{‾}{{B}_{2,3}}\\ {A}_{2,1} \stackrel{‾}{{B}_{1,1}}+{A}_{2,2} \stackrel{‾}{{B}_{1,2}}+{A}_{2,... | {
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Multiply three matrices that are represented by symbolic matrix variables. The result X is a symmatrix object.
syms V [2 1] matrix
X = V.'*A*V
X = ${V}^{\mathrm{T}} A V$
class(X)
ans =
'symmatrix'
You can pass symmatrix objects as arguments to math functions. For example, perform a mathematical operation to X by taki... | {
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syms A [2 3] matrix
a = A(2,3)
a = ${A}_{2,3}$
class(a)
ans =
'symmatrix'
Alternatively, convert the symbolic matrix variable A to a matrix of symbolic scalar variables. Then, index into that matrix.
Asym = symmatrix2sym(A)
Asym =
$\left(\begin{array}{ccc}{A}_{1,1}& {A}_{1,2}& {A}_{1,3}\\ {A}_{2,1}& {A}_{2,2}& {A}_{... | {
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Show that the two norms in $E=\left\{f\in \mathcal{C}^1[0,1]:f(0)=0\right\}$ are equivalent?
I can't show that the two norms defined as $$||f||_{\infty}=\sup_{x\in[0,1]}|f(x)+ f'(x)|$$ and $$N(f)=\sup_{x\in[0,1]}|f(x)| + \sup_{x\in[0,1]}|f'(x)|$$ are equivalent in $$E=\{f\in\mathcal{C}^1([0,1]) \text{ s.t. } f(0)=0\}$... | {
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Since $$N(f_n-f)\geq \|f_n-f\|_{\infty},$$ therefore $$\lim_{n\to \infty}\|f_n-f\|_{\infty}=0.$$
• Equivalence of norms is stronger than this. One needs to find $m, M > 0$ such that $m N(f) \le \|f\|_\infty \le MN(f)$ for all $f$. – Theo Bendit Jan 5 '19 at 11:21
• @TheoBendit. If two norms on a normed linear space ar... | {
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# Could somebody please explain to me how this binary relation is Anti-symmetric?
$$A = \{1, 2, 3, 4\}$$ where $$xRy$$ if $$x \mid y$$
$$R = \{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)\}$$.
I am watching this video about Partial Orders and the guy just said the relation was Anti-symmetric; howev... | {
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The fact that for non-negative integers $x \mid y$ and $y \mid x$ imply that $x=y$ is not surprising, but if you want a full proof, consider this: $x$ evenly divides $y$ if there exists a non-negative integer $n$ such that $y = nx$. Likewise, if $y$ evenly divides $x$, then there exists a non-negative integer $m$ such ... | {
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• But what about (1,1), (2,2) ... wouldn't that violate that clause? Dec 30, 2016 at 21:48
• No, because $1 \neq 1$ is false (as is $2 \neq 2$). Remember: The contrapositive lets us think about pairs $(a, b)$ and $(b, a)$ where $a \neq b$. Dec 30, 2016 at 21:50
• But if the left hand side is false doesn't that means th... | {
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1. Introduction
For good reasons of approximation theory, Chebfun relies on polynomial interpolation in Chebyshev points, which are unequally spaced, to represent nonperiodic functions. However, many people want to work with equispaced data, and we are often asked, how can we do this in Chebfun?
Chebfun can do a pret... | {
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To construct this function, Chebfun has first constructed a rational function $g$ known as a Floater-Hormann interpolant [1] that has good properties with equispaced data, and it has picked the order of this approximation in an adaptive fashion. Then, since Chebfun works with polynomials rather than rational functions,... | {
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4. Discussion
What's nice about these 'equi' approximations is that, as usual with chebfuns, they are globally smooth functions, and can be differentiated, for example, without any anomalies arising. In some applications this is very appealing.
Another globally smooth way to deal with equispaced data, besides the Flo... | {
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How to find limits of integration of polar curves?
Right now I am working on a problem that involves finding the area enclosed by a single loop given the equation $r=4\cos(3\theta)$. I know that the cosine is bounded from zero to $\pi$, but when using a lower limit of $0$, and a upper limit of $\pi/3$, I get the wrong... | {
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It helps if you have an idea of the graph, but even if you don't: it should be clear that at $\theta = 0$, you have $r(0) = 4$ so you're not at the beginning of a loop. A loop starts when $r=0$ and a single loop closes at the next root of $r(\theta)$.
The function $\cos x$ has a root at $-\tfrac{\pi}{2}$ and the next o... | {
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# integral
Numerical integration
## Description
example
q = integral(fun,xmin,xmax) numerically integrates function fun from xmin to xmax using global adaptive quadrature and default error tolerances.
example
q = integral(fun,xmin,xmax,Name,Value) specifies additional options with one or more Name,Value pair argu... | {
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q = integral(fun,0,0,'Waypoints',[1+1i,1-1i])
q = 0.0000 - 3.1416i
Create the vector-valued function $f\left(x\right)=\left[\mathrm{sin}x,\phantom{\rule{0.2222222222222222em}{0ex}}\mathrm{sin}2x,\phantom{\rule{0.2222222222222222em}{0ex}}\mathrm{sin}3x,\phantom{\rule{0.2222222222222222em}{0ex}}\mathrm{sin}4x,\phantom{\... | {
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Data Types: double | single
Complex Number Support: Yes
### Name-Value Arguments
Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matte... | {
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Example: integral(fun,a,b,'RelTol',1e-9) sets the relative error tolerance to approximately 9 significant digits.
Data Types: single | double
Array-valued function flag, specified as the comma-separated pair consisting of 'ArrayValued' and a numeric or logical 1 (true) or 0 (false). Set this flag to true or 1 to indi... | {
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## Tips
• The integral function attempts to satisfy:
abs(q - Q) <= max(AbsTol,RelTol*abs(q))
where q is the computed value of the integral and Q is the (unknown) exact value. The absolute and relative tolerances provide a way of trading off accuracy and computation time. Usually, the relative tolerance determines the... | {
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+0
# Algebra I question
+5
673
8
+3450
Hey guys.
I'm just totally drawing a blank here...
$${\frac{{\mathtt{2}}}{{\mathtt{3}}}}{\mathtt{\,\times\,}}{\mathtt{X}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{4}}}{{\mathtt{5}}}} = {\mathtt{\,-\,}}{\frac{{\mathtt{17}}}{{\mathtt{30}}}}$$
When I solve these equations I ... | {
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"Thumbs up and points!"
#3
+576
+10
It is always best to leave things in terms of fractions for a most exact answer. The first thing that is tough about this is that you dont have a common denominator, so let's fix that!
We want out common denominator to be 30 here because it is easy to turn 3 and 5 into thirty wit... | {
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126/990 = .127(2727....)
Using your example, ND, of .454545
Combining the repeating part with the non-repeating part = 45 (The non-repeating part = 0, in this case)
Subtracting the non-repeating part from the repeating part gives 45 - 0 = 45
Write this over 99 (the number of 9s = the number of repeated digits, w... | {
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Investigating whether a given relation is reflexive, symmetric, and transitive
Let $$X = \{0, 1, 2, ... , 10\}$$, Define the relation $$R$$ on $$X$$ by, for all $$a, b \in X$$, $$aRb$$ if and only if $$a + b = 10$$.
Is $$R$$ reflexive? symmetric, transitive? Give reasons.
Here are my answers, please see if I made an... | {
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• For reflexivity, your work is fine. If $$R$$ were reflexive, then we would have $$1R1$$, but $$1+1 \ne 10$$, so that doesn't hold.
• For symmetry, you have the right idea, but you should bear in mind that a simple example does not mean that it holds all of the time. (For instance, that $$5R5$$ holds doesn't mean the ... | {
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# Proof of the formula for the number of subsets of an n-element set
Given a set $A = \{1,2,...,n\}$, the number of subsets of this set can be given by the cardinality of the powerset of A: $$|\mathscr P(A)| = 2^n$$ This is fairly standard and I'm happy with the concept. I am curious, however, as to how one would go a... | {
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Go by induction, if you like. There is a bijection between $\mathcal{P}\{ 1, 2, \dots, n \}$ and $\mathcal{P}(\{1, 2, \dots, n-1 \}) \times \{0, 1\}$, given by $$\phi: A \mapsto \langle A \setminus \{ n \}, 1[n \in A] \rangle$$ where $1[n \in A]$ is the indicator function which takes the value $1$ if $n \in A$, and $0$... | {
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Trying to explain this in English obscures what's going on. Logically, the proof is this:
FIRST, consider the set of all ordered $n$-tuples of 0 and 1. For example, for $n$ = 3, these are 000, 001, 010, 011, .... This set is $\{0,1\} \times \cdots \times\{0,1\}$ (Cartesian product), which shows the cardinality is $2^n... | {
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So, we got : $\sum_{i = 0}^{n}{\binom{n}{i}} = 2^{n}$, that's easy to prove by Pascal triangle!
• @Joanpemo yes, thanks a lot! – openspace Apr 16 '16 at 11:53
• Last equality: also by the binomial theorem (consider $(1+1)^n$). Also, why start with 1. subsets with 1 element, and not with 0. subsets with 0 elements? – m... | {
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# Proving that a sequence is periodic
Given the following sequence,
$$1,1,2,3,5,8,3,1,4,5,9,4,3,7....$$ $$a_{n+2}=(a_n+a_{n+1}) \mod{10}\;\;\;\; \forall\;\;{n\geq{1}}$$ Prove that it is periodic?
My Attempt:
There can be atmost $10 \times 10=100$ unique pairs of integers $(a,b)$ with $0\leq{a,b}\leq9$.
So pairs wi... | {
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However, there is no need to worry about the predecessors: once you have a repeat of $x,y$ in your sequence, then, given the rule, from then on everything will be the same again, and that is all you need to show it is cyclic.
In fact, there are many sequences that are cyclic, but where the cycle is preceded by a littl... | {
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# Integral of $\csc(x)$
I'm getting a couple of different answers from different sources, so I'd like to verify something.
I misplaced my original notes from my prof, but working from memory I have the following:
\begin{align} \int\csc(x)\ dx&=\int\csc(x)\left(\frac{\csc(x)-\cot(x)}{\csc(x)-\cot(x)}\right)\ dx\\ &=\... | {
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int
Definite and indefinite integrals
Description
example
F = int(expr) computes the indefinite integral of expr. int uses the default integration variable determined by symvar(expr,1). If expr is a constant, then the default integration variable is x.
example
F = int(expr,var) computes the indefinite integral of... | {
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$\frac{1}{4}$
Integrate another expression from sin(t) to 1.
syms t
F = int(2*x,[sin(t) 1])
F = ${\mathrm{cos}\left(t\right)}^{2}$
When int cannot compute the value of a definite integral, numerically approximate the integral by using vpa.
syms x
f = cos(x)/sqrt(1 + x^2);
Fint = int(f,x,[0 10])
Fint =
${\int }_{0}... | {
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To ignore special cases of parameter values, set 'IgnoreSpecialCases' to true. With this option, int ignores the special case $t=-1$ and returns the solution for $t\ne -1$.
F = int(x^t,x,'IgnoreSpecialCases',true)
F =
$\frac{{x}^{t+1}}{t+1}$
Define a symbolic function $f\left(x\right)=1/\left(x-1\right)$ that has a ... | {
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$\int \mathrm{sin}\left(\mathrm{sinh}\left(x\right)\right)\mathrm{d}x$
You can approximate the integrand function $f\left(x\right)$ as polynomials by using the Taylor expansion. Apply taylor to expand the integrand function $f\left(x\right)$ as polynomials around $x=0$. Compute the integral of the approximated polynom... | {
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Using this option can lead to results not generally valid. This option applies mathematical identities that are convenient, but the results do not always hold for all values of variables.
Indicator for ignoring special cases, specified as true or false. This ignores cases that require one or more parameters to be elem... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9814534322330059,
"lm_q1q2_score": 0.8473786503452273,
"lm_q2_score": 0.8633916011860785,
"openwebmath_perplexity": 1760.2719548899947,
"openwebmath_score": 0.8929407000541687,
"tags":... |
• For indefinite integrals, int implicitly assumes that the integration variable var is real. For definite integrals, int restricts the integration variable var to the specified integration interval. If one or both integration bounds a and b are not numeric, int assumes that a <= b unless you explicitly specify otherwi... | {
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"id": null,
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"lm_q1q2_score": 0.8473786503452273,
"lm_q2_score": 0.8633916011860785,
"openwebmath_perplexity": 1760.2719548899947,
"openwebmath_score": 0.8929407000541687,
"tags":... |
# No. of ways to seat round a table (numbered seats)
#### Punch
##### New member
Two families are at a party. The first family consists of a man and both his parents while the second familly consists of a woman and both her parents. The two families sit at a round table with two other men and two other women. Find th... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540734789343,
"lm_q1q2_score": 0.8473607365509791,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 1518.8162034971838,
"openwebmath_score": 0.6196755170822144,
"t... |
Therefore: .$(3!)(3!)(10)(5)(4!) \:=\:43,200$ arrangements.
#### grgrsanjay
##### New member
First,i am ignoring the numbers on the seat,
this is a round combination
So, formula is (n-1)!
no.of.ways is 5!(3!)(3!)= 4320
Now the seat are numbered,
then i can more these combinations 1 seats,2seata,......9 seats apart... | {
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"lm_q1q2_score": 0.8473607365509791,
"lm_q2_score": 0.8652240895276223,
"openwebmath_perplexity": 1518.8162034971838,
"openwebmath_score": 0.6196755170822144,
"t... |
# Finding Eigenvalues of a 3x3 Matrix (7.12-17)
Please check my work in finding eigenvalues for the following problem. I am working out of the textbook Advanced Engineering Mathematics by Erwin Kreyszig, 1988, John Wiley & Sons.
For reference the following identity is given because some textbooks reverse the formula ... | {
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"id": null,
"lm_label": "1. YES\n2. YES",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9793540668504083,
"lm_q1q2_score": 0.8473607223069712,
"lm_q2_score": 0.8652240808393984,
"openwebmath_perplexity": 203.1782854595166,
"openwebmath_score": 0.9871860146522522,
"tag... |
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