instruction stringlengths 12 30k |
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When you say the data fits one distribution or another, what are you plotting?
One hypothesis would be that the two samples are drawn from the same distribution. Then you would expect A/B to be the same over all proteins. If the distribution is the frequency of A/B for all the different proteins you observe you ca... |
Given a $\sigma$-finite measure $\mu$ on a set $X$ is it possible to formulate a topology on the space of functions $f:X \rightarrow \mathbb{R}$ that gives convergence $\mu$-almost everywhere?
I can't seem to find any way to write this and am suspecting that no such topology exists! Is this true? If so, is there so... |
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry data. By labeling proteins of two different samples A and B, we get the relative abundance of proteins in A/B. Alongside with this ratio, we can estimate its variance best on the signal intensities.
W... |
If we have the standard topology T is the countable union of disjoint open sets and so T is an open set.
The complement of S is $(-\infty, 0) U (1/2, 3/4) U (7/8, 15/16) U \cdots$... etc. if this is open then S is closed. Is it open? Why? |
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry data. By labeling proteins of two different samples A and B, we get the relative abundance of proteins in A/B. Alongside with this ratio, we can estimate its variance based on the signal intensities.
... |
How do I solve the following question?
> You are given the identity $x^2-ax+144\equiv(x-b)^2$
>
> Work out the values of $a$ and $b$. |
$x^2-ax+144\equiv(x-b)^2$
Expand the right-hand side:
$(x-b)^2=x^2-2bx+b^2$
Get the coefficients (bit hazy on this, can anyone explain?):
$144\equiv{b}^2$
$-a\equiv-2b$
Square root the first:
$\pm12\equiv{b}$
Then calculate $a$ from $b$:
$a\equiv2b$
$a\equiv\pm24$
$a=12, b=24$ **or** $a... |
The probability that a fire will occur is $0.001$. If there is a fire, the amount of damage, $X$, will have a Pareto distribution given by $P(X>x) = \left(\frac{2(10)^6}{2(10)^6+x} \right)^2$. An insurance will pay the excess of the loss over a deductible of $100,000$. For this coverage the one-time insurance premium w... |
Posted by [Pi R Squared][1] on [Yahoo Answers][2]:
> $a=24$ and $b=12$
>
> $x^2-24x+144=(x-12)^2$
[1]: http://in.answers.yahoo.com/my/profile;_ylt=AuGKDZFCaIla44dugPaZqxXaHwx.;_ylv=3?show=31f743411a42c3a8de21b1c9c1b15871aa
[2]: http://in.answers.yahoo.com/question/index?qid=20090510094003AA5Xzgh |
Given I understand you correctly (topologize almost everywhere convergence), we can show that this is not possible. If we have a topological space, then we have convergence of a sequences if and only if every subsequence has a further convergent subsequence and so on.
So pick a sequence that converges in measure but... |
If we have the standard topology T is the countable union of disjoint open sets and so T is an open set.
The complement of S is $(-\infty, 0) \cup (1/2, 3/4) \cup (7/8, 15/16) \cup \cdots$, etc. if this is open then S is closed. Is it open? Why? |
To me the [Cayley-Salmon theorem](http://en.wikipedia.org/wiki/Cubic_surface#27_lines_on_a_cubic_surface) is an example of a result that still strikes me as rather surprising.
>**Theorem(Cayley-Salmon)**
>
A smooth cubic surface $\mathcal{S} \subset \mathbb{P}^{3}_{k}$ contains exactly 27 lines, where $k$ is an al... |
Can someone explain how to picture or construct the Monster Group?
|
What is the simplest way to fathom the Monster Group? |
To be able to answer these questions you need to know what "open" and "closed" mean. And that depends on context.
In the context you are looking at, in which you are dealing with subsets of the real line, without any further information we usually assume that we are dealing with the *standard* meanings of "open" and... |
Consider the set
$$S = [0,1/2] \cup [3/4, 7/8] \cup [15/16, 31/32] \cup \cdots$$
and the set
$$T = (0,1/2) \cup (3/4, 7/8) \cup (15/16, 31/32) \cup \cdots$$
Is $S$ open, closed or neither? Is $T$ open, closed, or neither? |
Q: $\int \frac{1}{sinxcosx} dx$
I know that the correct answer can be obtained by doing:
$\frac{1}{sinxcosx} = \frac{sin^2(x)}{sinxcosx}+\frac{cos^2(x)}{sinxcosx} = tan(x) + cot(x)$ and integrating.
However, doing the following gets a completely different answer:
$\int \frac{1}{sinxcosx} dx$
$=\int \frac{s... |
Q: \int \frac{1}{\sin x\cos x} dx
I know that the correct answer can be obtained by doing:
$\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely different answer:
$\int \frac{1}{\sin x\cos x... |
Q: \int \frac{1}{\sin x\cos x} dx
I know that the correct answer can be obtained by doing:
$\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely different answer:
$\int \frac{1}{\sin x\cos x... |
Q: \int \frac{1}{\sin x\cos x} dx
I know that the correct answer can be obtained by doing:
$\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely different answer:
\\begin{eqnarray*}
\int \f... |
Q: $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely different answer:
\\begin{eqn... |
Q: $\displaystyle \int \frac{1}{\sin x\cos x} dx $
I know that the correct answer can be obtained by doing:
$\frac{1}{\sin x\cos x} = \frac{\sin^2(x)}{\sin x\cos x}+\frac{\cos^2(x)}{\sin x\cos x} = \tan(x) + \cot(x)$ and integrating.
However, doing the following gets a completely different answer:
\\begin{eqn... |
I am working on a [signal-smoothing algorithm][1] for personal interest. I understand the basic concept of the Savitzky-Golay algorithm but I would like to understand how the [coefficients][2] were discovered. Can someone please explain the the mathematics behind it, so I could calculate the coefficients on my own?
... |
I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations wh... |
I am wondering about particle trajectories for solutions of the Navier-Stokes equation. Is it possible that there is a Manifold $M$ for which fluid particles move along geodesic's or "straight lines on the manifold"? If so, would it be fair to say that the Navier-Stokes equations are partial differential equations wh... |
To be able to answer these questions you need to know what "open" and "closed" mean. And that depends on context.
In the context you are looking at, in which you are dealing with subsets of the real line, without any further information we usually assume that we are dealing with the *standard* meanings of "open" and... |
If you've got multiple arrays like this:
(24,36,28,28,16,27)
(38,38,45,57,35,50)
every array being 6 integers, each integer in range [0,60]
I would like to find the distance between those 2 arrays. The problem is related to computer science, however, I would like to interpret these arrays as "planes" so as... |
Problem interpretation - Distance Formula? |
How can I compute the following limit?
$\displaystyle \lim_{a \to 0+} (a* \int_{1}^{\infty} e^{-ax}cos \left(\frac{2\pi}{1+x^{2}} \right) dx)$
Any hints you can please give?
Cheers |
How can I compute the following limit?
$\displaystyle \lim_{a \to 0+} \left(a \int_{1}^{\infty} e^{-ax}\cos \left(\frac{2\pi}{1+x^{2}} \right) dx\right)$
Any hints you can please give?
Cheers |
I am having some problem wrapping my head around an assignment.
I have to find out how many additions, subtractions, multiplications and divisions are used while solving an NxM linear equation, while using Gauss Jordan elimination.
Could anyone point me in the right direction? |
It depends on how efficient you want to be, and it also depends on the rank of the matrix. But for a "generic" matrix, you can expect the rank to be $\min(N,M)$, and for "most" entries to be nonzero. (And, do you count multiplying zero by a constant as a "multiplication"? Adding two zeros as an "addition"?)
I interp... |
I have an equation system of the form Aix + Biy + Ciz = Di, where (x,y,z) is a unit vector, and (Ai, Bi, Ci, Di) are sets of measurements from a noisy system (with typically 3-5 independant readings).
My first intuition to solve this problem was to pose this as an overdetermined linear equation system AX = B where ... |
I have the following series of linear equations (n+1 equations n+1 variables):
$k_0q_0+\lambda q_0 + c_0 = 0$
$k_1q_1+\lambda q_1 + c_1 = 0$
$k_nq_n+\lambda q_n + c_n = 0$
$q_1+q_2+....+q_n = 1$
The variables are $q_0,q_1,.....,q_n$ and $/lambda$. k and c are series of constants.
How can ... |
I have the following series of equations (n+1 equations n+1 variables):
$k_0q_0+\lambda q_0 + c_0 = 0$
$k_1q_1+\lambda q_1 + c_1 = 0$
$k_nq_n+\lambda q_n + c_n = 0$
$q_1+q_2+....+q_n = 1$
The variables are $q_0,q_1,.....,q_n$ and $\lambda$. k and c are series of constants.
How can I solve... |
This was supposed to be a comment, but it got too long.
I'm personally not too enamored of the NR routine for Savitzky-Golay myself; it has to use Gaussian elimination (and thus a two-dimensional array) for what I think is a well-structured problem. Peter Gorry's proposal in [this paper][1] is a better choice, IMHO ... |
Is there a simple/efficient way of enumerating the elements of a free distributive lattice? |
Is there a simple/efficient way of enumerating the elements of a free distributive lattice?
(I'm doing some computations with them, and listing their elements in what's probably the least efficient way possible...) |
I am working on a [signal-smoothing algorithm][1] for personal interest. I understand the basic concept of the Savitzky-Golay algorithm but I would like to understand how the [coefficients][2] were discovered. Can someone please explain the the mathematics behind it, so I could calculate the coefficients on my own?
... |
I am working on a [signal-smoothing algorithm][1] for personal interest. I understand the basic concept of the Savitzky-Golay algorithm but I would like to understand how the [coefficients][2] were discovered. Can someone please explain the the mathematics behind it, so I could calculate the coefficients on my own?
... |
I am working on a [signal-smoothing algorithm][1] for personal interest. I understand the basic concept of the Savitzky-Golay algorithm but I would like to understand how the [coefficients][2] were discovered. Can someone please explain the the mathematics behind it, so I could calculate the coefficients on my own?
... |
So, I'm relearning Group Theory. And I got the axioms down, I think. So let's make a concrete example:
- The collection of numbers the positive integers less than 7: 1,2,3,4,5,6
- The • operation will be multiplication mod 7.
- Associativity holds.
- The Identity e is 1.
- Every element has an invers... |
What happens to the 0 element in a Finite Group? |
I am seeking conditions on the distribution of the step sizes that guarantee that a random walk on the 2D lattice will return to the origin (with probability 1). Essentially, under what conditions can Pólya's theorem be proved? Certainly it holds if the steps are of size 1 (and equally probable in all directions), a... |
If we can say that we're able to determine the weather at 10ms intervals, and if we observe that over a whole second (1000 samples) that it's raining at every sample, what is the accuracy rate within which we've calculated the weather for the second? I'm sure this sounds like homework to some, but it's not. I was readi... |
How can you calculate the accuracy rate of the following set of statements? |
> In maths, you can use something as
> simple as statistical analysis to
> intuit the theory, and in comp-science
> you can use simulators.
Is the above statement true for maths ? |
Is the above statement true for maths ? |
Say I have a function
**V(x)=A(x-x1)(x-x2)(x-x3)**
where **x1**, **x2**, **x3** are the three roots in increasing order and **A** is positive. Clearly **V(x)** is positive at large **x>x3**, negative between **x2** and **x3**, and so on.
Now, say I wish to
**Integrate Sqrt[-V(x)]** between ... |
Let $a_n$ be a sequence of real numbers such that $exp(ita_n)$ converges to 1 for all real t.
Show that $a_n$ converges to 0.
One can show this by letting $X_n$ be degenerate random variables which take value $a_n$ with prob 1 and then using results about characteristic functions and convergence.
But this seems li... |
Elementary solution of convergence problem ? |
I am CS major, please be patient if my question is not well-stated.
The dataset is quantitative mass spectrometry (MS) data. By labeling proteins of two different samples A and B, we get the relative abundance of 100 to thousands of proteins in A/B. Alongside with this ratio, we can estimate its variance based on t... |
What do I miss in the following?
Let $R$ be a commutative Noetherian ring with unit. A map $f:M\to N$ of $R$-modules is injective/surjective iff the associated map $f_p:M_p\to N_p$ on the localization is injective/surjective for every prime ideal $p$ of the ring $R$.
There is an isomorphism $M_p=M\otimes_R R_p$. ... |
Is there any known bound on sum of independent but not identically distributed geometric random variables?
I have to show that the tail of the sum drops exponentially (like in the Chernoff bounds for the sum of iid geom. variables).
Formally, if X_i ~ Geom(p_i), and X = sum_{i=1}^n X_i,
Is it possible to show th... |
Find the maximum and minimum value of $ \sin ^4 \theta + cos ^4 \theta $ ? |
I have this table:
<pre>
x | 0 1 2 3 4 5
------------------------------------------
f(x) | .7 .2 .05 .03 .01 .01
F(x) | .7 .9 .95 .98 .99 1
</pre>
Where f(x) is the density function for x and F(x) is the cumulative distribution function for x.
<hr/>
... |
Find the maximum and minimum value of $ \sin ^4 \theta + \cos ^4 \theta $ ? |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis (and integer exponent), then the logarithm of **x**, in base **b**, is an irrational number."
Thank you!
Paulo
|
Consider n loaded dice with differing probability distributions. I have constructed an equation that tells me the probability mass, the number of times a certain number is expected to be thrown.
Now i want to extend the numerical range from a line onto a plane. How do i get the expected number of times a number/num... |
I have the following series of equations (n+2 equations n+2 variables):
$k_0q_0+\lambda q_0 + c_0 = 0$
$k_1q_1+\lambda q_1 + c_1 = 0$
$k_nq_n+\lambda q_n + c_n = 0$
$q_1+q_2+....+q_n = 1$
The variables are $q_0,q_1,.....,q_n$ and $\lambda$. k and c are series of constants.
How can I solve... |
http://stackoverflow.com/questions/4113133/changing-a-positive-value-to-a-negative-one
why? is this just a computer science problem , or is this a mathematical property.
|
For a mapping algorithm I'm working on, I'm trying out the effect of sigmoid weightings.
Right now I'm using $$y = \frac{1}{1+x^n}$$ where n is the steepness of the sigmoid function.
This graph shows values of x between 0 and 256 and the following values of n: 2, 4 and 32. X values have been scaled to a range of ... |
Is there a general (logistic?) function for sigmoids over a given range? |
I am seeking conditions on the distribution of the step sizes that guarantee that a random walk on the 2D lattice will return to the origin (with probability 1). Essentially, under what conditions can Pólya's theorem be proved? Certainly it holds if the steps are of size 1 (and equally probable in all directions), a... |
<a href="http://en.wikipedia.org/wiki/Banach_fixed_point_theorem">Banach's fixed point theorem</a> gives us a sufficient condition for a function in a complete metric space to have a fixed point, namely it needs be a contraction.
I'm interested in how to calculate the limit of the sequence $x_0 = f(x), x_1 = f(x_0)... |
As promised in my response to Byron, here is my very partial progress to
date on this problem. Though long-winded, it doesn't amount to much.
I'm still looking for an answer to the general case, or to other special
cases (like $n=2$ or $c=2$) or even just better estimates. I'm also
interested to know about any ap... |
This problem has me scared to my wits, mostly because there are more like it!
<blockquote>The mean of a random sample of n observations drawn from a N $ ( \mu ,\sigma^{2} ) $ distribution is denoted by $ \overline{\chi} $. Given that P $ (| \overline{\chi} - \mu| > 0.5\sigma ) \lt 0.05 $ </blockquote>
Find the s... |
Reading the Wikipedia page for [Random Walk][1], I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as random walk on $\mathbb{Z}^d$, Gaussian random walk and random walk on a graph can fall into this general definition?
(1) In Ross's book for stocha... |
Reading the Wikipedia page for [Random Walk][1], I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as random walk on $\mathbb{Z}^d$, Gaussian random walk and random walk on a graph can fall into this general definition?
(1) In Ross's book for stocha... |
Quoted from [Wikepedia][1]:
> The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as Donsker's theorem.
I was wondering if the scaling limit of any [random walk][2] is always a Wiener process... |
Reading the Wikipedia page for [Random Walk][1], I was wondering what is the definition for a general random walk as a random process, so that all the concepts such as [random walk on $\mathbb{Z}^d$][2], [Gaussian random walk][3] and [random walk on a graph][4] can fall into this general definition?
(1) In Ross's ... |
Let $a_n$ be a sequence of real numbers such that $\exp(it\cdot a_n)$ converges to 1 for all real t.
Show that $a_n$ converges to 0.
One can show this by letting $X_n$ be degenerate random variables which take value $a_n$ with prob 1 and then using results about characteristic functions and convergence.
But this s... |
Could someone provide, please, a proof of the theorem below?
"Being **x** and **b** integers greater than 1, which can not be represented as powers of the same basis(positive integer) and integer exponent, then the logarithm of **x**, in base **b**, is an irrational number."
Thank you!
Paulo
|
I am trying to evaluate this integral or at least get bounds for its aboslute value.
I have where $\tau \to \infty$
$$\int_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$
Integrating by parts gives me,
$$\int_{1}^{\infty} \frac{f(t)}{t^\sigma} d(-\cos(\tau \log t)) = \left[ -\frac{f(t)}{t^... |
Is this the right way to evaluate this integral? |
I am trying to evaluate this integral or at least get bounds for its aboslute value.
I have where $\tau \to \infty$
$$\int_{1}^{\infty} f(t) \frac{\tau \sin(\tau\log t)}{t^{\sigma+1}} dt$$
$f(t) = (t + 1/2) - \lfloor t + 1/2 \rfloor$ (where $\lfloor.\rfloor$ is the floor function)
Integrating by parts gives... |
In light of your <a href="http://math.stackexchange.com/questions/9161/concept-of-random-walk">other question</a>, one should say there are a lot of things referred to by the name "random walk". In its simplest form, Donsker's theorem is about a process $X_n$ in $\mathbb{R}^d$ whose increments $X_n - X_{n-1}$ are iid ... |
I'm looking to strengthen my understanding of the math that is directly useful to practical computer science, as opposed to unsolved computer science problems. In other words, the kind of math that would be useful in day to day algorithmic analysis, and come up during senior level interviews at software companies. My m... |
What are the prerequisites for combinatorics? |
How do I solve the following question?
> You are given the identity $x^2-ax+144\equiv(x-b)^2$
>
> Work out the values of $a$ and $b$.
*Question appears in AQA 43005/1H.*
|
In my opinion the "prerequisite" knowledge for Combinatorics/Discrete mathematics is primarily basic logic.
Many people would say that this falls under discrete math, but I think the main things that you will need to understand are Truth Tables, De Morgan, Induction, set theory and such. Many discrete books cover t... |
@Andy (in reply to your comment/question "Could you provide some example that has a closed form and explain if (and how) it is possible to find the fixed point without solving x = f(x) but trying to calculate the limit of x_n?":
I believe that you would be hard-pressed to achieve this, since your function $f$ is a ... |
In Hatcher's _Algebraic Topology,_ section 3.2, during the computation of the cohomology ring of a $n$-torus, the following assertion is made. Let $Y$ be a space and $R$ a commutative ring. Then the natural map in cohomology
$$ H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$$
is a split injection. This is l... |
Why is $H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$ a split injection? |
Isn't $I$ just the closed unit interval? Then $I\times Y$ is homotopy
equivalent to
$Y$ and $\partial I\times Y$ is just two copies of $Y$. It's
clear that your map then is just the diagonal map from
$H^n(Y)$ to $H^n(Y)\oplus H^n(Y)$. |
This is a follow-up (refinement?) of [this question][1].
In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of terminology because it provides an embedding of $Rings^{op}$ into a larger category.) One construction we can do, for... |
To what extent do the stories on manifolds carry over to schemes? |
In this [post][1] we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question:
- Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
In know these as $\mathbb{Q}$-Vector spaces, are isomorphic from the linked post. But as fields are they isomorphic. I neither know to prove it or to di... |
I was reading John Lee's Introduction to Smooth manifolds, and I came across this question:
- Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous function. Using a partition of unity, show that there is a smooth function $\tilde{\delta} : M \rightarrow \mathbb{R}$ such... |
I have a velocity vector on X, Y and Z axes. The values are represented as M/S (meters per second).
What algorithm should I use to convert this vector into a general MPH (miles per hour) value that takes into account all axes of the vector?
This may (or may not) be a very simple question. But I am far from being ... |
Converting a velocity vector into MPH? |
Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$? |
In this [post][1] we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question:
- Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
In know these as $\mathbb{Q}$-Vector spaces, are isomorphic from the linked post. But as fields are they isomorphic? I neither know how to prove it nor ... |
Let $f$ be in $L^{1}(\mathbb{R})$, $\mathbb{R}$ the real numbers. Show that for every $\varepsilon > 0$ there exists $A \subseteq R$ , measurable, such that $m(A) < \infty$ , $f$ is bounded on $A$ and $ \int_{\mathbb{R}} |f| < \int_{A} |f| + \varepsilon$.
If we take $A$ as the support of the simple function which a... |
This is motivated by [this question][1] and the fact that I have no access to the Timothy Chow's paper [What Is a Closed-Form Number?][2] indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form solution, what does it normally
mean?
To me this is equivalent to say that one cannot solve it fo... |
What does closed form solution usually mean? |
I was reading John Lee's Introduction to Smooth manifolds, and I came across this question:
- Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous function. Using a partition of unity, show that there is a smooth function $\tilde{\delta} : M \rightarrow \mathbb{R}$ such... |
Because you're familiar with complex exponentials, all you needed to succeed is to observe that by setting $w = m - i$ your equation becomes
$$w \exp(i \alpha) + \bar{w} \exp(-i \alpha) = 2.$$
(**Edit** What might motivate this choice of $w$? Well, if you're thinking of complex numbers you're thinking that $\cos... |
I would say it really depends so much on the context, and what tools are at your disposal. For instance, telling a student who's just mastered the usual tricks of integrating elementary functions that
$$\int\frac{\exp{u}-1}{u}\mathrm{d}u$$
and
$$\int\sqrt{(u+1)(u^2+1)}\mathrm{d}u$$
have no closed form solut... |
You are allowed to use +, -, / and * (plus, minus, division and multiplication) signs and bracketing. These signs you can put between the numbers
1963
to form mathematical expressions. You must put at least one sign between two numbers and – cannot be used as “negative”, thus -1+9+6+3 is not allowed, but 1-9+6+3 ... |
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