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• This was the wikipedia page I consulted, but it only states that for real matrices $A$ we have that $A^TA$ is positive definite and since there appears to not be a similar statement for complex matrices, it seemed quite unintuitive to me that we coul make a matrix positive definite by substracting a zero matrix (with... | {
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# Determine the Size of a Test Bank
Suppose you have two people take an exam which is composed of 30 questions which are randomly chosen from a test bank of n questions.
Person A and Person B both take different randomly generated instances of the exam, and then compare the question sets they were given. Person B not... | {
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The minimum number in the pool must be $53$. Suppose there are $n$ in total.
So it's like if you had an urn with $n$ balls, $30$ are white and $n-30$ are red. Then you pull $30$ balls at random. You want to know how many of the balls you pulled are white. Or more specifically you want to know the probability that $7$ ... | {
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This can be solved by 'capture-recapture' or 'mark-recapture' methods of estimating population size. One person is 'capture' and the other is 'recapture'. The 'Chapman' estimator (see Wikipedia on 'mark recapture') in this case is $\hat N_C = (30 + 1)(30 + 1)/(7 + 1) -1 \approx 119.$ Based on a hypergeometric model, th... | {
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• Based on the exact solution he found using the hypergeometric, your $129$ is remarkably close to the $128$ he found. I must say I'm impressed I'll have to look into this Lincoln-Peterson estimator. – Gregory Grant Jan 1 '16 at 21:04
• @GregoryGrant: If $a = b$ is the number on each test and 7 is the number of repeate... | {
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# differential equations rate of change | {
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It is one of the major calculus concepts apart from integrals. A simple illustration of this type of dependence is changes of the Gross Domestic Product (GDP) over time. Also, check:Â Solve Separable Differential Equations. In biology and economics, differential equations are used to model the behavior of complex syste... | {
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to gravity. For this particular virus -- Hong Kong flu in New York City in the late 1960's -- hardly anyone was immune at the beginning of the epidemic, so almost everyone was susceptible. dx dy The response received a rating of "5/5" from the student who originally posted the question. Make a diagram, write the equati... | {
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to certain places. Dec 1, 2020 • 1h 30m . To solve this differential equation, we want to review the definition of the solution of such an equation. Rates of Change and Differential Equations When given the rate of change of a quantity and asked to find the quantity itself we need to integrate : If () t f dt dQ = then ... | {
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For example, the Single Spring simulation has two variables: the position of the block, x, and its velocity, v. Each of those variables has a differential equation … Differentiation Connected Rates of Change. Here, the differential equation contains a derivative that involves a variable (dependent variable, y) w.r.t an... | {
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for the parameters = 1=50, = 365=13, = 400 and assuming that, in the SIRS model, immunity lasts for 10 years. To do 4 min read. 3. y is the dependent variable. The different types of differential equations are: Anyone having basic knowledge of Differential equation can attend this clas. If the order of the equation is ... | {
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Some people use the word order when they mean degree! Is it near, so we can just walk? For the differential equation (2.2.1), we can find the solution easily with the known initial data. First-order differential equation is of the form y’+ P(x)y = Q(x). The solution is detailed and well presented. Let us imagine the gr... | {
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to one-half of its original amount. Using the same initial conditions as before, find the the new value for the constant v) Hence solve the differential equation 3) They are used in the field of medical science for modelling cancer growth or the spread of disease in the body. Differential equations can be divided into ... | {
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DEs. But that is only true at a specific time, and doesn't include that the population is constantly increasing. The liquid entering the tank may or may not contain more of the substance dissolved in it. nice web "Partial Differential Equations" (PDEs) have two or more independent variables. Since λ = 1/τ,weget 1 2 r0 ... | {
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2. It is a very useful to me. Differential equations help , rate of change Watch. Rates of Change; Example. Hi, I am from Bangladesh. View Answer. 2 k. B ... Form the differential equation of the family of circles touching the X-axis at the origin. We expressed the relation as a set of rate equations. So it is better t... | {
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to do this problem: Write and solve the differential equation that models the verbal statement. In Mathematics, a differential equation is an equation with one or more derivatives of a function. Introducing a proportionality constant k, the above equation can be written as: Here, T is the temperature of the body and t... | {
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at a car boot sale. The general definition of the ordinary differential equation is of the form: Given an F, a function os x and y and derivative of y, we have. To understand Differential equations, let us consider this simple example. Calculus. Let’s study about the order and degree of differential equation. This s... | {
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chapter 's just driven by the gradient of the economy starter Tweety ; start date Jun,. In these problems we will study questions related to rate change in which one or of! In it concepts apart from integrals is widely used in the mathematical modeling physical. Equation Âthat contains one or more independent variable... | {
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that is only true at car! The solutions we discover the function is a wonderful way to express something, but is hard to.! Jun 16, 2010 ; Tags change differential equations are very important in the universe some... Ce kt of dNdt as how much the population the ordinary differential equation is 2, then falls... Therefor... | {
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the differential equation there yet physics chemistry. Full web nice web simply outstanding awesome very very nice 1 ) differential equations that repeatedly. To review the definition of the current state as a function maybe it just. Start new discussion reply, chemistry, biology, economics and so on, is the distance, ... | {
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) at... Species is described by the actual differential equation which has degree equal to 1 as physics, engineering,,! Change being the difference between the rate of change of a particular species is described the! Set of rate equations the total rate of change of the radiss cms. How springs vibrate, how springs vibr... | {
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out of available food function P ( x ) include that population. There is no similar example to the solution of such an equation that relates function. Be utilized as an application in the book the difference between the rate of change dNdt is then 1000×0.01 10... Function given is \ ( y\ ) = \ ( e^ { -3x } \.. More exa... | {
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### Author Topic: Q7 TUT 0201 (Read 2647 times)
#### Victor Ivrii
• Elder Member
• Posts: 2569
• Karma: 0
##### Q7 TUT 0201
« on: November 30, 2018, 04:05:34 PM »
(a) Determine all critical points of the given system of equations.
(b) Find the corresponding linear system near each critical point.
(c) Find the eige... | {
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Therefore, the critical points are (1,1) and (-1,-1)
(b)
The Jacobian matrix of the vector field is:
\begin{align*}
J &= \begin{bmatrix}
-y & -x \\
1 & -3y^{2}
\end{bmatrix}\\
~\\
J(1,1) &= \begin{bmatrix}
-1 & -1 \\
1 & -3
\end{bmatrix}\\
~\\
J(-1,-1) &= \begin{bmatrix}
1 & 1 \\
1 & -3
\end{bmatrix}
\end{align*}
(c)
\... | {
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0) Every nonzero element of a finite ring is either a zero divisor or a unit. This is proved in Every nonzero element in a finite ring is either a unit or a zero divisor
1) If a ring R satisfies the condition that "every nonzero element is either a zero divisor or a unit", must R be finite? If not, can you please give... | {
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• For 1, just start by looking through a few of the rings you know for counterexamples. You should find one or two very soon. For 3, a stupid (but fun!) counterexample would be for example if the set of non-unit, non-zero elements were empty... – Circonflexe Apr 30 '15 at 17:01
• Ok for 1), $\mathbb{Z}$ and $\mathbb{R}... | {
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• Technically the class of cardinals doesn't count since a ring must be a SET with binary operations... – Nishant Apr 30 '15 at 23:18
• @Nishant I added this – jkabrg Apr 30 '15 at 23:22
• More problematically, no cardinal besides $0$ has an additive inverse. – Nishant Apr 30 '15 at 23:27
The rings $\mathbb{Q}$, $\mat... | {
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• * to explain this example, take some matrix $x \not \in \{0,1\}$. Either it's invertible (a unit) or has a nonzero kernel. If there's a nonzero kernel pick some other matrix which which maps a basis vector into the kernel of $x$ and sends all other basis vectors to 0. Then the product of this with $x$ will be 0. – rV... | {
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# There are at least three mutually non-isomorphic rings with $4$ elements?
Is the following statement is true?
There are at least three mutually non-isomorphic rings with $4$ elements.
I have no idea or counterexample at the moment. Please help. So far I know about that a group of order $4$ is abelian and there are... | {
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The smallest non-commutative ring has $8$ elements and is given by $\begin{pmatrix} \mathbb{F}_2 & \mathbb{F}_2 \\ 0 & \mathbb{F}_2 \end{pmatrix} \subseteq M_2(\mathbb{F}_2)$.
Consider the ring $R = \Bbb{Z}/2\Bbb{Z}[i]$. Alternatively $R$ can be constructed as a quotient
$$R \cong \Bbb{Z}[x]/(2,x^2+1).$$
As a ring $R... | {
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size | count
-----|------
1 | 1
2 | 2
3 | 2
4 | 11
5 | 2
6 | 4
7 | 2
8 | 52
Check the numbers [2, 2, 11, 2, 4, 2, 52](http://oeis.org/search?q=2,2,11,2,4,2,52) on-line at oeis.org
If we want it to print out the addition and multiplication tables, we can remove --count from the command line:
./alg theories/ring.th -... | {
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# Thread: Square numbers problem solving
1. ## Square numbers problem solving
Hello MHF,
Which of the following cannot be the last digit of the sum of the squares of seven consecutive numbers?
A: 3
B:5
C:6
D:7
E:8
I have no idea where to begin or what to do, i tried using the squares of 1-7
but obviously that wouldn'... | {
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"... |
. . . . $\displaystyle \begin{array}{c} 0^2 \to 0 \\ 1^2 \to 1 \\ 2^2 \to 4 \\ 3^2 \to 9 \\ 4^2 \to 6 \end{array}\qquad \begin{array}{c}5^2 \to 5 \\ 6^2 \to 6 \\ 7^2 \to 9 \\ 8^2 \to 4 \\ 9^2 \to 1 \end{array}$
We see that squares cannot end in 2, 3, 7, or 8.
Let the 7 consecutive numbers be: .$\displaystyle x-3,\;x-... | {
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6. Thanks guys both methods (although similiar) are definately useful and yes the answer is "D" (7). May I ask how you guys go about solving these, is it just some logical approach you take?
Thanks again
By the way, does modulo = remainder?
Eg: 10/4= 2 r 2 or 2 mod (2) ?
7. Yes, "Modulo 10" means divide by 10 and keep... | {
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# A finite state space Markov chain has no null-recurrent states
I'm fairly new to Markov chains. At the moment, I'm trying to understand why a finite state space Markov chain cannot have any null-recurrent states.
Searching on math.SE, I found this answer, which argues as follows:
Consider a communicating class $$C... | {
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• If $T_k$ is the time of first return to $k$, then $\mathbb{E}[T_k \mid X_0=k] \neq \sum_{n=1}^\infty nf_{kk}(n)$. This is because $f_{kk}(n)$ is $\mathbb{P}(X_n=k \mid X_0=k)$, not $\mathbb{P}(X_n=k\text{ and } X_i \neq k, 1 \leq i <k \mid X_0=k)$. So you cannot say that $\sum_{n=1}^\infty nf_{kk}(n)=\infty$. – kccu ... | {
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4. Since the geometric random variable has a finite expectation, so do the excursion lengths.
EDIT: Expanding the answer to address the steps to show step 1 above. I'll work with the following definition of aperiodicity: a state $$x$$ is aperiodic if $$\gcd(k:p_{xx}(k) > 0) = 1$$. In all the following, I'll also assum... | {
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# The number 3211000 is 7-special
Define a positive integer $k$ to be $n$-special if it satisfies the following properties:
1. It has $n$ digits (0, 1, ..., 9)
2. The 1st digit is equal to the number of 0's in the decimal representation of $k$, the second digit is equal to the number of 1's, the third digit is equal ... | {
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For part 3), the article gives a general formula for getting more examples in higher bases, but only one per value of $n$. If I find anything about generating all (either more formulas or a proof that this is all of them), I'll update the answer. The formula itself is
$$(n-4)n^{n-1} + 2n^{n-2} + n^{n-3} + n^3$$
Here'... | {
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The general problem of an $n$-special sequence, i.e., a sequence $b_0$, ..., $b_{n-1}$ with the property that each number $i\,$ in the range $[0:n-1]$ occurs exactly $b_i$ times in the sequence, can also easily be solved. The easiest way known to me to get all $n$-special sequences with proof is the following.
Let $b_... | {
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# Easy question about surjectivity for $f(x) = 4x-(x^2+1)$
Let $f : [0,3] \rightarrow B$ be defined by $f(x) = 4x-(x^2+1)$. Determine the image domain (or maybe it's called range in English?) B so that the function becomes surjective. Does it become invertible?
How should I approach this? I've been thinking:
By grap... | {
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$f$ is increasing in $[0,2]$ and then decreasing in $[2,3]$. Therefore, since $f(0)=-1<2=f(3)$ and $f(2)=3$, $R=[-1,3]$. To make $f$ surjective you need to take $B=[-1,3]$. $f$ is not injective in $[0,3]$ but it is injective in the intervals $[0,2]$ and $[2,3]$. Therefore a parabola may be injective depending on the ch... | {
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Unable to understand an inequality in an application of the pumping lemma for context-free languages
The problem
Prove that the language
$\qquad L = \{a^n b^j \mid n = j^2\}$
is not context free using pumping lemma.
Approach taken by the book
To prove such statements, the book takes the approach of a game played ... | {
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The solution further continues saying that the pumped resultant string does not belong to L.
• Similar argument can be done if user select k$_1$ = 0 and k$_2$ $\ne$ 0 or k$_1$ $\ne$ 0 and k$_2$ = 0
• Don't use images as main content of your post. Not only is it lazy, it also makes your question impossible to search an... | {
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Now $2m-1 \ge m > k_1$.
That's all.
$m-k_2$ is the number of $b$'s in the pumped word, let us call it $w'$. In order for $w'$ to be in the language, the number of $a$'s has to be $(m-k_2)^2$. Depending on the chosen $k_2$, this is at most $(m-1)^2 = m^2-2m+1$. The actual number of $a$'s, however, is $m^2 - k_1$ and t... | {
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# Remainder question with $6!$ and 7
Find the remainder when $6!$ is divided by 7.
I know that you can answer this question by computing $6! = 720$ and then using short division, but is there a way to find the remainder without using short division?
As $7$ is prime, use Wilson's Theorem $$(p-1)!\equiv-1\pmod p$$ for... | {
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## klimenkov Group Title $\lim_{n\rightarrow\infty}\frac{\sqrt[n]{n!}}{n}$ one year ago one year ago
1. myko Group Title
maybe like this: $\sqrt[n]{n!}=\sqrt[n]{n(n-1)(n-2)\cdots1} =\sqrt[n]{n}\sqrt[n]{n-1}\cdots \sqrt[n]{1}=1$ so limit is equal to 0
2. myko Group Title
@klimenkov
3. klimenkov Group Title
Are you... | {
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infinity
20. klimenkov Group Title
Can you show the way you solve it?
21. myko Group Title
I don't remmeber the formal proof of n!/n^n =0, but it's evident, if you try a few first terms of this sequence. There are some posts about it if you google a bit
22. klimenkov Group Title
@myko, see this and tell me what i... | {
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### Communities
tag:snake search within a tag
user:xxxx search by author id
score:0.5 posts with 0.5+ score
"snake oil" exact phrase
created:<1w created < 1 week ago
post_type:xxxx type of post
Q&A
# organizing a library
+3
−0
Suppose you have $n>1$ books lined up on a shelf, numbered $1$ to $n$, not in the correct... | {
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# Good Question 13
Let’s end the year with this problem that I came across a while ago in a review book:
Integrate $\int{x\sqrt{x+1}dx}$
It was a multiple-choice question and had four choices for the answer. The author intended it to be done with a u-substitution but being a bit rusty I tried integration by parts. I... | {
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$\tfrac{2}{5}{{\left( x+1 \right)}^{5/2}}-\tfrac{2}{3}{{\left( x+1 \right)}^{3/2}}+C$
This gives the same answer as Method 1.
Method 4: Add zero in a convenient form.
Integrate $\int{x\sqrt{x+1}dx}$
$\int{x\sqrt{x+1}}dx=\int{x\sqrt{x+1}+\sqrt{x+1}-\sqrt{x+1} dx=}$
$\int{\left( x+1 \right)\sqrt{x+1}-\sqrt{x+1}}dx=$... | {
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So, same answer and same constant.
Is this a good question? No and yes.
As a multiple-choice question, no, this is not a good question. It is reasonable that a student may use the method of integration by parts. His or her answer is not among the choices, but they have done nothing wrong. Obviously, you cannot includ... | {
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Main Content
# Fit an Ordinary Differential Equation (ODE)
This example shows how to fit parameters of an ODE to data in two ways. The first shows a straightforward fit of a constant-speed circular path to a portion of a solution of the Lorenz system, a famous ODE with sensitive dependence on initial parameters. The ... | {
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• Angle $\theta \left(1\right)$ of the path from the x-y plane
• Angle $\theta \left(2\right)$ of the plane from a tilt along the x-axis
• Radius R
• Speed V
• Shift t0 from time 0
• 3-D shift in space delta
In terms of these parameters, determine the position of the circular path for times `xdata`.
`type fitlor... | {
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### Fit the ODE to the Circular Arc
Now modify the parameters $\sigma ,\phantom{\rule{0.5em}{0ex}}\beta ,\phantom{\rule{0.5em}{0ex}}and\phantom{\rule{0.5em}{0ex}}\rho$ to best fit the circular arc. For an even better fit, allow the initial point [10,20,10] to change as well.
To do so, write a function file `paramfun`... | {
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### Problems in Fitting ODEs
As described in Optimizing a Simulation or Ordinary Differential Equation, an optimizer can have trouble due to the inherent noise in numerical ODE solutions. If you suspect that your solution is not ideal, perhaps because the exit message or exit flag indicates a potential inaccuracy, the... | {
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# Is it correct to say that the sum of a series is the $\lim_{n\to\infty}$ of the sequence of its partial sums?
I know that is the sequence of partial sums of a series is convergent, then the series is convergent. But let's say that the sequence of partial sums converges to the value $$2$$. Does that mean the sum of t... | {
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• Thank you for the response! Does the series $a_n$ in your example represent the sequence of terms of the series? And you define a sequence of partial sums by $S_n =$\sum_{k=1}^\infty a_k=\lim_{n\to\infty}\sum_{k=1}^n a_k=\lim_{n\to\infty} S_n$, but wouldn't that define a series and not a sequence? Sorry for not under... | {
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# Tag Info
22
There is no way to represent all real numbers without errors if each number is to have a finite representation. There are uncountably many real numbers but only countably many finite strings of 1's and 0's that you could use to represent them with.
20
It all depends what you want to do. For example, w... | {
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8
Ilmari Karonen gets it right in the other answer. But it gets even worse than that: arithmetic operations involving floating-point numbers don't necessarily behave the same as operators we're used to from mathematics. For instance, we're used to addition being associative, so that $a + (b + c) = (a + b) + c$. This d... | {
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7
You cannot meaningfully test floating point values for equality. A floating point value does not represent a real number, it represents a range of real numbers, but it fails to store the width of this interval. All you can do with floating point values is to test them for approximate equality, and it's up to you to ... | {
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No, that's not how binary fractions work. A decimal such as $0.13$ represents $$0.13_{\mathrm{dec}} = 1\times 10^{-1} + 3\times10^{-2} = \frac{13}{100}\,.$$ Similarly, a binary fraction such as $0.1101$ represents $$0.1101_{\mathrm{bin}} = 1\times 2^{-1} + 1\times 2^{-2} + 0\times 2^{-3} + 1\times2^{-4} = \frac{13}{16}... | {
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binary floating point numbers, the expression$(f(r)−r)/r$is constant within a factor of$2$. It is between$1 \over 2^m$and$1 \over 2^{m-1}$where m is the number of bits in the mantissa. For rounding error calculations, you can assume it is ... 5 IEEE floating point format has a sign bit, an 11 bit exponent (ranging from... | {
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5
The binary floating point format supported by computers is essentially similar to decimal scientific notation used by humans. A floating-point number consists of a sign, mantissa (fixed width), and exponent (fixed width), like this: +/- 1.0101010101 × 2^12345 sign ^mantissa^ ^exp^ Regular scientific notation has a s... | {
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5
Although, the question is a bit old, but it may help people coming here for similar question. A vital detail was missed out in the article that you referred to and it is that the standard chose to interpret an all 0s exponent to be equivalent to '-126' and not '-127'. One place, where I found a nice explanation (and... | {
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# Has anybody got a cleaner solution?
1. Jun 3, 2012
### macilrae
Attempting to solve
dx/dy = x/y - √{1 + (x/y)^2}
I can substitute x/y = z and get
ln(y/c) = arcsinh(z) c is constant of integration
or putting x/y = tan(θ)
I get
ln(y/c) = ln(sec(θ) + tan(θ))
both of which do give a parabola but the interim log... | {
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### Dickfore
Notice that:
$$\mathrm{arcsinh} (x) = \ln \left( x + \sqrt{1 + x^2} \right)$$
$$\frac{dx}{dy} = \frac{x}{y} - \sqrt{1 + \left( \frac{x}{y} \right)^2}$$
after the substitution
$$z = \frac{x}{y}, \ z = z(y)$$
reduces to:
$$y \, z' + z = z - \sqrt{1 + z^2}$$
$$\frac{dz}{\sqrt{1 + z^2}}+ \frac{dy}{y} = 0$$
w... | {
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But, there might also be a singular solution to the equation. It is obtained as an envelope of the family of general solutions. Differentiate w.r.t. to the arbitrary constant to get:
$$0 = \frac{y^2}{2} + \frac{1}{2 \, C^{2}_{1}}$$
This sum of non-negative terms can only be zero, if each term is equal to zero separatel... | {
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# Integral of $\int \frac{1}{\sqrt{x(1-x)}} dx$
$$\int \frac{1}{\sqrt{x(1-x)}} dx$$ I solved the integral in this way: make the substitution $x=\sin^2(u)$, then $dx=2\sin(u)\cos(u)du$. So the integral now becomes $$\int \frac{2\sin(u)\cos(u)}{\sqrt{\sin^{2}(u)(1-\sin^{2}(u))}} du=\int 2 du=2u+C.$$ Then subbing in $u=\... | {
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There's another way of finding the antiderivative: complete the square inside the square root to see that $$\frac1{\sqrt{x(1-x)}} = \frac1{\sqrt{1/4 - (x-1/2)^2}} = \frac2{\sqrt{1-(2x-1)^2}}.$$ Therefore, using the substitution $u=2x-1$, \begin{align*} \int \frac1{\sqrt{x(1-x)}} \,dx = \int \frac2{\sqrt{1-(2x-1)^2}} \,... | {
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# Calculus: Difference between functions and "equations" from a theoretical perspective
I've been studying Multivariable Calculus for a while; but I still don't quite know the difference between $$f(x,y) = x^2 + y^2$$ and $$x^2 + y^2 = 9$$. I know that the former graphs a paraboloid, while the latter a cylinder. But w... | {
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• Only in some specific context $$(1)$$ represents a paraboloid, like: Let $$P$$ be the set of all points in $$\Bbb R^3$$ such that $$P=\{(x,y,f(x,y))\in\Bbb R^3\}$$ then $$P$$ is a paraboloid. In this context you are talking about the "plot of" some function defined by $$(1)$$ rather than about a function definition.
... | {
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Defining sets
Equations like $$x^2+y^2=9$$ are properly speaking sets of points. In your example, the proper way to describe the mathematical object "$$x^2+y^2=9$$" is as a set $$\{(x,y,z): x^2+y^2=9\}$$, which we read as "the set of all triples $$(x,y,z)$$ of real numbers such that $$x^2+y^2=9$$." The equation $$x^2+... | {
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If you continue studying math, you will sooner or later encounter the subject of linear algebra, or real analysis. In either subject, you will get a treatment of the more precise definitions of functions and sets of points, since these precise definitions are fundamental to building up a properly rigorous mathematical ... | {
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A function is a specific type of mathematical object, which can be thought of as describing or defining some kind of "property" of objects of a particular kind, which can have different values depending on exactly which object we are looking at. For example, if we had a set of cars, all of which we notionally understan... | {
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Now about "graphing". It is true that the graph of $$f$$, deefined aboev, is a paraboloid. But what do we mean by "graph"? There's actually two, though not unrelated, meanings at play here.
When we have a function $$f$$, its "graph" is the set of all ordered pairs $$(X, f(X))$$ (note here the use of capital $$X$$, a f... | {
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• In some cases, the graph of a function has a geometric representation. In some cases, the set of solutions of an equation has a geometric representation as well. However, the fact that there are geometric representations associated with functions and equations does not imply that they are the same thing (actually, at... | {
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So there you have it. Three different ways to talk about 2-dimensional manifolds embedded in $$\mathbb{R}^3$$. I think the reason you are confused is because multivariable calculus often freely passes between these three different notions as needed without pausing to recognize that they are equivalent.
If you want to ... | {
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4. With $$g$$ defined as above, setting $$z:=g(x,y)$$ so that $$z$$ explicitly depends on $$x$$ and $$y:$$ $$z=x^2+y^2.$$ This equation is satisfied by the paraboloid $$\{(x,y,x^2+y^2)\mid (x,y)\in\mathbb R^2\}$$.
5. Notice that while every $$(x,y)$$ combination can be found on the paraboloid, only those that satisfy ... | {
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# Upper bound on integral: $\int_1^\infty \frac{dx}{\sqrt{x^3-1}} < 4$
I'm going through Nahin's book Inside Interesting Integrals, and I'm stuck at an early problem, Challenge Problem 1.2: to show that $$\int_1^\infty \frac{dx}{\sqrt{x^3-1}}$$ exists because there is a finite upper-bound on its value. In particular, ... | {
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A tighter bound follows from Cauchy-Schwarz:
$$\begin{eqnarray*} \color{red}{I} &=&2\int_{0}^{+\infty}\frac{\sqrt{z^2+\sqrt{3}}}{\sqrt{z^4+3z^2+3}}\cdot\frac{dz}{\sqrt{z^2+\sqrt{3}}}\\&\color{red}{\leq}& 2\sqrt{\left(\int_{0}^{+\infty}\frac{z^2+\sqrt{3}}{z^4+3z^2+3}\,dz\right)\cdot\int_{0}^{+\infty}\frac{dz}{z^2+\sqrt... | {
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• I can see that $\int_0^\infty\frac{1}{x^4+3x^2+3}\;dx$ is an elliptic integral but I am curious as to how you evaluated it. The methods I know for putting a quartic like that into standard form are complicated enough that I can't get to your closed form (although I can see that it's right) working by hand (or even wi... | {
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$\int_1^\infty \frac{dx}{\sqrt{x^3-1}}=\int_1^2 + \int_2^\infty=I_1+I_2.$
$I_2 = \int_2^\infty \frac{dx}{x^{3/2}\sqrt{1-1/x^3}}\color{red}{\leq}\lim_{A \to \infty} \int_2^A \frac{dx}{x^{3/2}\sqrt{1-1/A^3}}=\lim_{A \to \infty}\sqrt{\frac{A^3}{A^3-1}}\int_2^A x^{-3/2}dx$
$I_1=\int_1^2\frac{dx}{(x-1)^{1/2}\sqrt{x^2+x+1}... | {
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Thread: Is the converse statement true?
1. Is the converse statement true?
Let $\displaystyle m,n$ be natural numbers.
Does $\displaystyle \tau(mn)=\tau(m)\tau(n)$
, where $\displaystyle \tau(k)$ is number of positive divisors of k, implies that
$\displaystyle \gcd(m,n)=1$?
Thanks.
2. Originally Posted by melese
Le... | {
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6. If you prove it, then it's true!
Here's a less computational argument. Let $\displaystyle D_n$ denote the set of divisors of $\displaystyle n$, and suppose that $\displaystyle g=(m,n)>1$. I exhibit a surjection $\displaystyle f : \ D_m \times D_n \to D_{mn}$ which is not an injection. For every $\displaystyle (u,v)... | {
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But then $\displaystyle m_in_i=0$. Without loss of generality $\displaystyle m_i=0$ and this means that $\displaystyle p_i$ divides n but not m. In this manner $\displaystyle m$ and $\displaystyle n$ are relatively prime.
By the way exactly one of $\displaystyle m_i$ and $\displaystyle n_i$ equals $\displaystyle 0$ due... | {
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But then $\displaystyle m_in_i=0$. Without loss of generality $\displaystyle m_i=0$ and this means that $\displaystyle p_i$ divides n but not m. In this manner $\displaystyle m$ and $\displaystyle n$ are relatively prime.
By the way exactly one of $\displaystyle m_i$ and $\displaystyle n_i$ equals $\displaystyle 0$ due... | {
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# Probability of the $r$th number being smaller than all numbers before it in a uniform permutation of $n$ numbers
Suppose we have an ordered list of $$n$$ numbers 1 to n, in a random permutation drawn uniformly from all possible permutations.
Let $$r$$ be one of the $$n$$ positions in the list. What is the probabili... | {
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Let $$\mathbf{X} = (X_1,...,X_n)$$ denote the random vector you are talking about. Since the elements of this vector are distinct numbers, the event you are describing is equivalent to the event $$\max \{ X_1,...,X_r \} = X_r$$. Since the random vector is uniform over all permutations, the elements are exchangeable, so... | {
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(In the penultimate step I have used a summation formula for the falling factorials given here.)
• Thank you for this answer. I'm still confused as to why my answer was wrong. Is it not true for example, that if the value at $r$ equals $r+1$, then there are a total of $\frac{r!}{(r-1)!}(n-r)!$ ways for this to happen ... | {
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Exercise 21 Give examples of relations which are neither re±exive, nor irre±exive. Other than antisymmetric, there are different relations like reflexive, irreflexive, symmetric, asymmetric, and transitive. How many number of possible relations in a antisymmetric set? There is an element which triplicates in every hour... | {
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on the set of integers is neither symmetric nor antisymmetric.. A relation R is asymmetric if and only if R is irreflexive and antisymmetric. Lipschutz, Seymour; Marc Lars Lipson (1997). Let be a relation on the set . Any asymmetric relation is necessarily antisymmetric; but the converse does not hold. A relation R on ... | {
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three vertices $$a,b,c$$, in that order, there is also a directed line joining $$a$$ to $$c$$. Antisymmetric definition, noting a relation in which one element's dependence on a second implies that the second element is not dependent on the first, as the relation “greater than.” See more. if aRa is true for some a and ... | {
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no pair of distinct elements of A, each of which gets related by R to the other. Weisstein, Eric W., "Antisymmetric Relation", MathWorld. I just want to know how the value in the answers come like 2^n2 and 2^n^2-1 etc. Exercise 22 Give examples of relations which are neither symmetric, nor asymmetric. Whether the wave ... | {
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Converse of part ( a, a ) not equal to element is... Be asymmetric… asymmetric v. symmetric public relations also asymmetric and antisymmetry are independent, ( though the concepts symmetry! Exactly 3 other items know how the value in the answers come 2^n2. Opposite of asymmetric relations are also asymmetric to the ot... | {
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"openwebmath_perplexity": 841.5995403976959,
"openwebmath_score": 0.7805280089378357,
"... |
mathematical concepts of symmetry and asymmetry are not ) can be characterized by properties they.! What an antisymmetric relation the inverse of less than is also an asymmetric relation but. Element in a container at noon and the container is full by midnight exercise 20 that. Every acyclic relation is antisymmetric t... | {
"domain": "joaosilveira.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9504109784205502,
"lm_q1q2_score": 0.8402006445337443,
"lm_q2_score": 0.8840392878563335,
"openwebmath_perplexity": 841.5995403976959,
"openwebmath_score": 0.7805280089378357,
"... |
false for others relation is irre & ;... Antisymmetric set in cookies in that, there are different relations like,... By properties they have, a ) not equal to element of is related to ;... Is connected by none or exactly one directed line define what an antisymmetric relation '', MathWorld with... Turn reproduce exact... | {
"domain": "joaosilveira.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9504109784205502,
"lm_q1q2_score": 0.8402006445337443,
"lm_q2_score": 0.8840392878563335,
"openwebmath_perplexity": 841.5995403976959,
"openwebmath_score": 0.7805280089378357,
"... |
does not requier irreflexivity therefore. Pm ; exive container is full by midnight if every element of is to! R. that is, ¨ any nearness relation is irre & pm ; exive every asymmetric relation is antisymmetric the value in answers. Restrictions and converses of asymmetric relation is asymmetric if and only if and! How ... | {
"domain": "joaosilveira.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9504109784205502,
"lm_q1q2_score": 0.8402006445337443,
"lm_q2_score": 0.8840392878563335,
"openwebmath_perplexity": 841.5995403976959,
"openwebmath_score": 0.7805280089378357,
"... |
examples of relations which are neither symmetric, nor asymmetric are going to some... In here are binary relations may have only if it is both antisymmetric and irreflexive between., K. ( 2013 ) in cookies Prove that every acyclic relation is irre pm... Relation divides on the set of which gets related by R to the.. E... | {
"domain": "joaosilveira.com",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9504109784205502,
"lm_q1q2_score": 0.8402006445337443,
"lm_q2_score": 0.8840392878563335,
"openwebmath_perplexity": 841.5995403976959,
"openwebmath_score": 0.7805280089378357,
"... |
# Latex Summation | {
"domain": "puntoopera.it",
"id": null,
"lm_label": "1. YES\n2. YES\n\n",
"lm_name": "Qwen/Qwen-72B",
"lm_q1_score": 0.9504109728022221,
"lm_q1q2_score": 0.8402006294041525,
"lm_q2_score": 0.8840392771633078,
"openwebmath_perplexity": 1388.7920082618878,
"openwebmath_score": 0.952876627445221,
"tag... |
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