instruction stringlengths 12 30k |
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Why do some limits "explode" to infinity? |
For example:
the limit of $e^t$ as $t$ approaches infinity is simply infinity.
but the limit of $e^{t^2}$ cannot be found because it explodes? Is this due to the fact that there are differing types of infinity and the second limit produces a non-"countable" infinity?
[see referenced question](http://math.stac... |
The limit can be found. What are we sayng when we assert that $lim_{t\to\infty} f(t) = \infty$, with $f:\mathbb{R} \to \mathbb{R}$? What does it mean?
It means, *by definition*, that given $M >0$ there exists $\varepsilon >0$ such that if $t>\varepsilon$ then $f(t) > M$. Simply this. In (not so rigorous) words, for ... |
Suppose we look at $k$-digit strings with digits between $1$ and $n$. How many distinct classes of strings are there when digit identity and order doesn't matter?
More formally, what is the number of equivalence classes over strings $d_1,\ldots,d_k$ with $d_i\in \\{1,\ldots, n\\}$ where strings $d,\hat{d}$ are consi... |
Let $f(x) \in L^p(\mathbb{R})$ and $K \in C^m(\mathbb{R})$. Can I then say that $(f \ast K) (x) = \int_{\mathbb{R}} f(t) K(x-t) dt$ is in $C^m$?
I know that this is true if $K$ has compact support, but I was wondering if it is possible to have a stronger result (perhaps $K$ vanishing at $\infty$?). |
I'm attempting to come up with a first order language $L$ that is able to describe vector spaces over fields. I came up with a few sets of nonlogical symbols.
$Rs_L=\\{Scal,Vec\\}$ where $Scal,Vec$ are unary relation symbols which I interpret to be true or false depending on whether $x$ is a scalar or vector.
Als... |
Injective functions with intermediate value property are continuous. Better proof? |
It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives along. I hope I won't offend anybody by saying that one smooth piece of the solution (for a wheel with sides of length 2) is... |
It's kind of an infamous problem in differential equations to find the correct road surface so that a car with square wheels (and an axle located in the center) keeps its axle level as it drives along. I hope I won't offend anybody by saying that one smooth piece of the solution (for a wheel with sides of length 2) is... |
Trying to remember my high school formulas, and coming up dry.
Say I have two choices: A and B.
P(A) = 0.25; P(B) = 0.75
There are no conditional probabilities or anything. Each choice is independent of the last.
How do I go about calculating the probability that I will choose A exactly twice, having chose... |
> **Possible Duplicate:**
> [Boolean Algebra, Simplification: Don't know the method used](http://math.stackexchange.com/questions/7126/boolean-algebra-simplification-dont-know-the-method-used)
<!-- End of automatically inserted text -->
Based on a [previous question][1], the answer I got for the "sum of produ... |
Is there an expansion for the following summation?
$$ (a_1 + a_2 + \cdots + a_k)^n $$ |
I don't know if the following counts as "intuition", but perhaps it
can put things in some perspective.
It's a theorem that any rotation in $R^n$ can be written as the
composition of an even number of reflections through hyperplanes.
To construct a way of doing rotions by algebra (not matrix algebra,
but rather ... |
What is an alternative formulation to a contour integral? |
I don't know if the following counts as "intuition", but perhaps it
can put things in some perspective.
It's a theorem that any rotation in $R^n$ can be written as the
composition of an even number of reflections through hyperplanes.
To construct a way of doing rotions by algebra (not matrix algebra,
but rather ... |
1. Take any number $x$
2. Add 1 to it $x+1$
3. Find its reciprocal $1/(x+1)$
4. Repeat from 2
So, taking $x = 1$ to start:
- 1
- 2 (the + 1)
- 0.5 (the reciprocal)
- 1.5 (the + 1)
- 0.666... (the reciprocal)
- 1.666... (the + 1)
- 0.6 (the reciprocal)
- 1.6 (the + 1)
- 0.625
- 1.625
- 0.61584...
- 1... |
Why does this process, when iterated, tend towards the golden ratio? |
1. Take any number $x$
2. Add 1 to it $x+1$
3. Find its reciprocal $1/(x+1)$
4. Repeat from 2
So, taking $x = 1$ to start:
- 1
- 2 (the + 1)
- 0.5 (the reciprocal)
- 1.5 (the + 1)
- 0.666... (the reciprocal)
- 1.666... (the + 1)
- 0.6 (the reciprocal)
- 1.6 (the + 1)
- 0.625
- 1.625
- 0.61584...
- 1... |
Why does this process, when iterated, tend towards a certain number? (the golden ratio?) |
1. Take any number $x$ (edit: x should be positive, heh)
2. Add 1 to it $x+1$
3. Find its reciprocal $1/(x+1)$
4. Repeat from 2
So, taking $x = 1$ to start:
- 1
- 2 (the + 1)
- 0.5 (the reciprocal)
- 1.5 (the + 1)
- 0.666... (the reciprocal)
- 1.666... (the + 1)
- 0.6 (the reciprocal)
- 1.6 (the + 1)
-... |
Thats it! Thats what I want to know.
If \theta is the angle subtended by an arc of length L at its center with radius R.
We know, \theta = L/R.
How did we get this?
Please don't say we got it from generalizing the idea of perimeter of circle. |
\theta = length of arc/ angle subtended by it. How? |
1. Take any number $x$ (edit: x should be positive, heh)
2. Add 1 to it $x+1$
3. Find its reciprocal $1/(x+1)$
4. Repeat from 2
So, taking $x = 1$ to start:
- 1
- 2 (the + 1)
- 0.5 (the reciprocal)
- 1.5 (the + 1)
- 0.666... (the reciprocal)
- 1.666... (the + 1)
- 0.6 (the reciprocal)
- 1.6 (the + 1)
-... |
What is your definition of an angle? The formula you quoted follows directly from the definition of an angle as the length of the arc on the unit circle, combined with the intercept theorems (for “scaling the circle to a unit circle,” if you want).
Using angles defined this way is, as J.M. said, much more convenient... |
For a probability distribution, its quantile function is defined in terms of its distribution function as
> $Q(p)=F^{-1}(p) = \inf \left{ x\in R : p \le F(x) \right}$
I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distributio... |
Does Quantile Function Uniquely Determine Distribution Function? |
Thats it! Thats what I want to know.
If θ is the angle subtended by an arc of length L at its center with radius R.
We know, θ = L/R.
How did we get this?
Please don't say we got it from generalizing the idea of perimeter of circle. |
θ = (length of arc)/(angle subtended by it). How? |
For a probability distribution, its quantile function is defined in terms of its distribution function as
> $Q(p)=F^{-1}(p) = \inf \{ x\in R : p \le F(x) \}$
I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distribution just as... |
For a probability distribution, its quantile function is defined in terms of its distribution function as
> $Q(p)=F^{-1}(p) = \inf \\{ x\in R : p \le F(x) \\}$
I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distribution just ... |
For a probability distribution, its quantile function is defined in terms of its distribution function as
> $Q(p)=F^{-1}(p) = \inf \\{ x\in R : p \le F(x) \\}$
I was wondering if, conversely, a quantile function can uniquely determine a distribution and therefore fully describe the probability distribution just ... |
I don't know if the following counts as "intuition", but perhaps it
can put things in some perspective.
It's a theorem that any rotation in $R^n$ can be written as the
composition of an even number of reflections through hyperplanes.
To construct a way of doing rotions by algebra (not matrix algebra,
but rather ... |
I'm trying to show that if $f$ and $1/f$ are in $H^2$ then $f$ is an outer function. I think I somewhat have a proof (if it is correct) but it doesn't feel like it is the easiest one.
We know that if $f$ is in $H^2$ and non-zero and $B$ is the Blanschke product of its zeros, then there exists a constant $K$ of modul... |
I'm trying to show that if $f$ and $1/f$ are in $H^2$ then $f$ is an outer function. I think I somewhat have a proof (if it is correct) but it doesn't feel like it is the easiest one.
We know that if $f$ is in $H^2$ and non-zero and $B$ is the Blanschke product of its zeros, then there exists a constant $K$ of modul... |
In $\mathbb{R}^2$ specify a set of vectors, that has many elements as possible, so that no vector is a constant multiple of another vector in the set.
What do such sets have in common?
In $\mathbb{R}^3$ do the same
---
I asked both of these earlier, but people are acting like it is a HW question (giving me ... |
What do these sets have in common? |
In $\mathbb{R}^2$ specify a set of vectors, that has many elements as possible, so that no vector is a constant multiple of another vector in the set.
What do such sets have in common?
In $\mathbb{R}^3$ do the same
---
I asked both of these earlier, but people are acting like it is a HW question (giving me ... |
It is well known that the differential calculus has a nice algebraization in terms of the [differential rings][1] but what about integral calculus? Of course, one sometimes defines integration in a differential ring $R$ with a derivation $\partial$ as a projection $\pi: R\rightarrow \tilde R$, where $\tilde R$ is a quo... |
I was asked to find a solution to
$$\frac{\sin^2(nx)}{n^2\sin^2(x)}=2^{-1/2}$$
where $n$ is a fixed integer greater than 1.
Numerically, there's a solution just above 1/n so I decided to find this one (rather than one of the other roots). I hoped to find an obvious pattern that I could use to find some sort of cl... |
I have some geometry in 3d (like road surface or terrain surface) and there could be several not connected parts of a road.
So I'd like to know if I may unite this two not connected pieces of road in a term "surface".
And it I couldn't, then what is the best mathematical term for this? |
is surface always locally connected area in euklidian 3d ? |
It is well known that the differential calculus has a nice algebraization in terms of the [differential rings][1] but what about integral calculus? Of course, one sometimes defines an integral in a differential ring $R$ with a derivation $\partial$ as a projection $\pi: R\rightarrow \tilde R$, where $\tilde R$ is a quo... |
Is there some place (on the internet or elsewhere) where I can find the number and preferably a list of all (isomorphism classes of) finite connected $T_0$-spaces with, say, 5 points?
In know that a $T_0$-topology on a finite set is equivalent to a partial ordering, and [wikipedia](http://en.wikipedia.org/wiki/Parti... |
Is there a list of all connected $T_0$-spaces with 5 points? |
I'm trying to show that if $f$ and $1/f$ are in $H^2$ then $f$ is an outer function. I think I somewhat have a proof (if it is correct) but it doesn't feel like it is the easiest one.
We know that if $f$ is in $H^2$ and non-zero and $B$ is the Blanschke product of its zeros, then there exists a constant $K$ of modul... |
I'm trying to show that if $f$ and $1/f$ are in $H^2$ then $f$ is an outer function. I think I somewhat have a proof (if it is correct) but it doesn't feel like it is the easiest one.
We know that if $f$ is in $H^2$ and non-zero and $B$ is the Blanschke product of its zeros, then there exists a constant $K$ of modul... |
Fractional derivatives can be used to establish connections between various special functions. The book [An Atlas of Functions][1] makes heavy use of this, especially derivatives of order 1/2 and -1/2.
Also, the *existence* of fractional derivatives is related to the convergence of Fourier transforms. For example,... |
Fractional derivatives can be used to establish connections between various special functions. The book [An Atlas of Functions][1] makes heavy use of this, especially derivatives of order 1/2 and -1/2.
Also, the *existence* of fractional derivatives is related to the convergence of Fourier transforms. For example,... |
I have some geometry in 3d (like road surface or terrain surface) and there could be several not connected parts of a road.
So I'd like to know if I may unite this two not connected pieces of road in a term "surface".
And if I couldn't, then what is the best mathematical term for this? |
Both [fourier transform][1] and [taylor series][2] are means to represent functions in a different form.
**My question:**<br>
What is the connection between these two? Is there a way to get from one to the other (and back again)? Is there an overall, connceting (geometric?) intuition?
[1]: http://en.wikiped... |
The Wronskian lets us determine if a set of functions (possibly the solutions to a differential equation) are linearly dependent or not. But, for every example in the book, it is very obvious if one of the functions is a linear combination of the others. The examples in the book use 3-5 functions. What would be an exam... |
What are the uses of trigonometry in architecture? |
Perhaps this has a simple answer, but I don't know (I wouldn't be asking if I did). Every set of outer measure zero is Lebesgue measurable with Lebesgue measure zero. Is the converse true? That is, if a set has Lebesgue measure zero, does it necessarily have outer measure zero?
And I suppose I should specify that I'... |
Have you tried to prove by hand (i.e., only using the definition of linear independence) that $\sin \theta$ and $ \cos \theta$ are linearly independent? Of course, this can be done with the help of the Wronskian.
And what about $e^{i\theta}$ and $ e^{i (\theta + \frac{\pi}{2})}$? This is geometrically clear, but: c... |
In my attempts to derive the closed form for the arclength of the hyperbola, I wound up with the following integral:
$$\int\frac{\sqrt{1-m\;\sin^2 u}}{\sin^2 u}\mathrm{d}u$$
I am aware that such integrals are expressible as combinations of elliptic integrals and trigonometric functions, and *Mathematica* does ret... |
We can easily see that if a sequence of inner functions converges then certainly the limit is still inner (using the fact that a countable union of measure zero sets still has measure zero). This implies that the set of inner functions is closed in $H^2$.
Now, is the set of the _singular_ inner functions closed?
... |
Suppose $f_{X}(x) = xe^{-x^2/2}$ for $x>0$ and $Y = \ln X$. Find the density function for $Y$. So we want to find $P(Y \leq y)$. This is the same thing as $P(\ln X \leq y)$ or $P(X \leq e^{y})$. Thus $f_{Y}(y) = f_{X}(e^y)$? Or does $f_{Y}(y) = F_{X}(e^y)$ since $P(X \leq x) = F_{X}(x)$? |
We can easily see that if a sequence of inner functions converges then certainly the limit is still inner (using the fact that a countable union of measure zero sets still has measure zero). This implies that the set of inner functions is closed in $H^2$.
Now, is the set of the _singular_ inner functions closed?
... |
I found [this book][1] quite useful. But it does take the background listed. The basic result that the equation is the same as one dimensional heat flow and the graphs of roughly how the value flows are not too tough. There are also free calculators on the web that let you play with it.
[1]: http://www.amazon... |
Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not finitely generated? |
Is there an Artinian module with infinitely generated proper submodule? |
Two players put a dollar in a pot. They decide to throw a pair of dice alternatively. The first one who throws a total of $ \$ 5$ on both dice wins the pot. How much should the player who starts add to the pot to make this a fair game?
So my interpretation of this problem is that I first throw a dice and then the o... |
Maple tells me that $\lim_{n \to \infty} n^2 \log(1 - \frac{c \log n}{n})^n = 0$ for any constant $c$, but I can't find a way to prove it. Any suggestions? |
Miller and Ross looks very nice indeed.
As far as applications are concerned: <a href="http://docs.google.com/viewer?a=v&q=cache:aWfxS6fj7WgJ:www.m-hikari.com/ams/forth/rahimyAMS49-52-2010.pdf+An+Introduction+to+the+Fractional+Calculus+and+Fractional+Differential+Equation&hl=en&gl=mx&pid=bl&srcid=ADGEESg0ul80vQSr4J... |
Two players put a dollar in a pot. They decide to throw a pair of dice alternatively. The first one who throws a total of $ 5$ on both dice wins the pot. How much should the player who starts add to the pot to make this a fair game?
So my interpretation of this problem is that I first throw a dice and then the othe... |
Have you tried to prove by hand (i.e., only using the definition of linear independence) that $\sin \theta$ and $ \cos \theta$ are linearly independent? Of course, this can be done with the help of the Wronskian.
And what about $e^{i\theta}$ and $ e^{i (\theta + \frac{\pi}{2})}$? This is geometrically clear, but: c... |
I know that in general, $H_{n}(X)$ counts the number of $n$-cycles that are not $n$-boundaries of a simplicial complex $X$. So for the sphere, $H_{0}(X) \cong \mathbb{Z}$ since it is connected. Also $H_{n}(X) = 0$ for all $n>0$ (e.g. all $1$-cycles are $1$-boundaries, etc..). How do you use this geometric interpretatio... |
Let R be a ring with identity. An R-module M is Artinian if it satisfies the descending chain condition on submodules. What is an example of an Artinian module with a proper submodule that is not finitely generated?
|
Maple tells me that $\lim_{n \to \infty} n^2 \log^n(1 - \frac{c \log n}{n}) = 0$ for any constant $c$, but I can't find a way to prove it. Any suggestions? |
A group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef? |
Each person from a group of 3 people can choose his dish from a menu of 5 options. Knowing that each person eats only 1 dish what are the number of different orders the waiter can ask the chef? |
i couldn't do the following question for hours
minimize sum of xi^3 for i from one to n
s.t. sum of xi's is equal to zero
and sum of xi^2 is equal to n.
by Lagrange multiplier rule
? |
Reading "Random number generation and Monte Carlo methods" by James E. Gentle, I encountered an example concerning the Bays-Durham Shuffling algorithm whose result I can't reproduce.
Basically, such an example is written in <a href="http://en.wikibooks.org/wiki/Statistics/Numerical_Methods/Random_Number_Generation#B... |
What is a good introduction in gradient flows in metric spaces? I know the book Gradient flows: in metric spaces and in the space of probability measures by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré, but is too hard for an introduction.
Can someone make this a community wiki? I cannot find how you have to do ... |
Am I doing something wrong in making this inequality or do algebra tools just usually not factor? |
So, I thought of something nerdy...I was making an algebra problem to disguise it so that to see the joke, you'd have to a bit of work, but I think that I might be doing something wrong, because neither [Wolfram|Alpha][1] nor [MyAlgebra][2] can solve it...I fixed a few things that I had wrong initially, but I don't see... |
Your mistake lies in trying to cancel $5u - (i+2u)$ without being careful about whether it is positive, negative, or $0$.
Remember that "cancelling factors" is *really* multiplying by the inverse. But when you multiply an inequality by a number, you have to be careful about whether the number is positive or negativ... |
So, I thought of something nerdy...I was making an algebra problem to disguise it so that to see the joke, you'd have to a bit of work, but I think that I might be doing something wrong, because neither [Wolfram|Alpha][1] nor [MyAlgebra][2] can solve it...I fixed a few things that I had wrong initially, but I don't see... |
So, I thought of something nerdy...I was making an algebra problem to disguise it so that to see the joke, you'd have to a bit of work, but I think that I might be doing something wrong, because neither [Wolfram|Alpha][1] nor [MyAlgebra][2] can solve it...I fixed a few things that I had wrong initially, but I don't see... |
Your mistake lies in trying to cancel $5u - (i+2u)$ without being careful about whether it is positive, negative, or $0$.
Remember that "cancelling factors" is *really* multiplying by the inverse. But when you multiply an inequality by a number, you have to be careful about whether the number is positive or negativ... |
Your mistake lies in trying to cancel $5u +(i+2u)$ without being careful about whether it is positive, negative, or $0$. **Added:** There is also a miscalculation in the third line.
Remember that "cancelling factors" is *really* multiplying by the inverse. But when you multiply an inequality by a number, you have to... |
I believe I'm right in saying that an isometry of a convex figure will be convex. There are other automorphisms that have this property, for example "stretching along one axis" (e.g. $f(x,y) = (2x,y)$) won't transform a convex shape into a non-convex one or vice versa.
What other maps have this property? |
What automorphisms of R^n preserve convexity? |
Can you bound the series from above by one that you know converges? The factorials grow very fast, so you should be able to. |
Given the coordinates of two rectangles on a coordinate plane, what would be the easiest way to find the coordinates of the intersecting rectangle of the two? |
Any transformation that sends lines to lines will do, since convexity is defined by the existence of certain lines in a figure. In particular, any transformation of the form $x \mapsto Ax + b$ where $A$ is a matrix and $b$ is a vector works. (It's not clear to me what you mean by "automorphism" here - in what categor... |
Any transformation that sends lines to lines will do, since convexity is defined by the existence of certain lines in a figure. In particular, any transformation of the form $x \mapsto Ax + b$ where $A$ is a matrix and $b$ is a vector works. (It's not clear to me what you mean by "automorphism" here - in what categor... |
In order to simplify the problem, I will assume that the rectangles are orientated "straight up and down." By this I mean that each of the rectangles has an edge which is parallel to the x (or y) axis. Judging by the context, this should be a fair assumption. Let me know if this is not the case (in other words, what cl... |
Let $I$ be a finitely generated ideal of a commutative ring $R$. Assume every element of $I$ is a zero divisor. Does then exist a $x \neq 0$ in $R$ with $xI=0$?
This is true if $0$ is a decomposable ideal, for example if $R$ is noetherian. I wonder if we actually need this. Doesn't it sound plausible? The problem is... |
Let's suppose, imitating the real case, the we want to integrate the expression
$f(z) dz$, where $f$ is a function of the complex variable $z$. What would this mean?
Well, imitating the real case, we find that we have to form Riemann sums
$\sum_{i = 1}^{n-1} f(z_i) (z_{i+1} - z_i)$, and then let the absolute value ... |
This question comes from a test :
For the equation x^2 + |x| - 6 = 0 analyze the four statements below for correctness.
1. there is only one root
2. sum of the roots is + 1
3. sum of the roots is zero
4. the product of the roots is +4
Answer : (3)
**Answer Explanation :**
If x > 0 |x| = x.
Give... |
This question and the described solution are copied from a test-paper :
For the equation $x^2$ + |x| - 6 = 0 analyze the four statements below for correctness.
1. there is only one root
2. sum of the roots is + 1
3. sum of the roots is zero
4. the product of the roots is +4
Answer : (3)
**Answer Expla... |
Let $C$ be a convex subset of $\mathbb{R}^n$. I've been trying for hours to prove that $\dot{\overline{C}}=\dot{C}$. Somehow my intuition completely fails me. I found a proof in a textbook, but just got stuck on another statement the author considered obvious. Could someone please give a proof that uses little more tha... |
Why does a convex set have the same interior points as its closure? |
Why are very large prime numbers important in cryptography? |
A subset $S$ of a metric space $X$ is totally bounded if for any $r>0$, $S$ can be covered by a finite number of $X$-balls of radius $r$.
A metric space $X$ is totally bounded if it is a totally bounded subset of itself.
For example, bounded subsets of $\mathbb{R}^n$ are totally bounded.
Are there any interest... |
Reading "Random number generation and Monte Carlo methods" by James E. Gentle, I encountered an example concerning the Bays-Durham Shuffling algorithm whose result I can't reproduce.
Basically, such an example is written in <a href="http://en.wikibooks.org/wiki/Statistics/Numerical_Methods/Random_Number_Generation#B... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
There is a whole class of cryptographic/security systems which rely on what are called "trap-door functions". The idea is that they are functions which are generally easy to compute, but for which finding the *inverse* is very hard (here, "easy" and "hard" refer to how quickly we know how to do it), but such that if yo... |
Page 13 of [Tropical Algebraic Geometry][1] by Itenberg, Mikhalkin, and Shustin mentions 1-valent vertices, but I haven't been able to find a source that actually defines this term or managed to guess the definition myself. Either a definition or reference would be greatly appreciated. Thanks!
[1]: http://books... |
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