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$K$ numbers are chosen with replacement from the numbers $1,2,3, \cdots N$.In how many ways we can choose the numbers so that the maximum number chosen is exactly $R \le N $ ?
The explanation given in my module is like this :
Since the maximum number is $R$, each of $K$ numbers can be chosen in $R$ ways.so, the number ... |
If $X$ is a Banach space, and $T:X \to X$ is a bounded linear operator with norm < $1$, then $I-T$ has a bounded inverse defined by $(I-T)^{-1} = \sum_{n=0}^\infty T^n$.
Thinking in terms of a converse, if $T$ is any bounded linear operator defined on $X$, then does the existence of a bounded inverse $S=(I-T)^{-1}$ im... |
i was reading wikipedia article on Naive Bayes classifier
but i didn't understand some of the calculations.
can anybody explain. how they have calculated this
P(height|male) = 1.5789 (A probability distribution over 1 is OK. It is the area under the bell curve that is equal to 1.)
P(weight|male) = 5.9881e-06
P(foot siz... |
How does someone approximate something like
$$\frac{1}{n}\sum_{k=1}^n e^{-k I(k)^2}$$
where I(k) is equidistributed on $[0,1]$. Ideally I would like to show its about $K/\sqrt{n}$ for some constant $K$. This is not homework, but a question I stumbled upon "in nature" and I don't really have any ideas on how to tackle... |
The formula for finding the roots of a quadratic polynomial is as follows
$$x = \frac {-b \pm \sqrt{ b^2 - 4ac }}{2a} $$
what happens if you want to find the roots of a polynomial like this simplified one
$$ 3x^2 + x + 24 = 0 $$
then the square root value becomes
$$ \sqrt{ 1^2 - 4\cdot3\cdot24 } $$
$$ = \sqrt{ -287 } $... |
Is the average of i.i.d. random variables with zero mean and finite variance convergent to 0 in probability or $L^2$? How do I show this? I am just beginning to learn convergence of random variables, so an example would be helpful.
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Suppose $h$ is a function on $Z$ that is periodic with period N, that is $h(n+N)=h(n)$ for all $n$. How to prove: for any $m \in Z$,
$\sum_{n=m}^{m+N-1} h(n) =\sum_{n=0}^{N-1} h(n)$. In other words, any sum over an interval of length $N$ yields the same result.
This is problem from the area of linear algebra and is a ... |
One way to tessellate a 3D sphere is by iterated subdivision of an icosahedron. I am wondering whether this method gives a homogeneous surface density of vertices. To the eye, it seems to do so, and logic indicates that too (each face has the same are in the icosahedron, and faces in each subdivision are created of equ... |
How to prove that $N^{-1/2} W_N$ is an unitary matrix( Suggestion: Use lemma 2.2 and lemma 1.105 ii.) \
Lemma 1.105
Let $A$ be an $n \times n$ matrix over $C$. Then the following statements are equivalent:
i. A is unitary
ii. The columns of A form an orthonormal basis for $C^n$
iii. The rows of A form an orthonormal ... |
If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ and thus the phase function $\phi(x)$ be extracted? (with e.g. $N=1/(2\pi)$)
As Marek already stated, this is even not u... |
I have two sets (a,b) of data and I am restricted to using the
values in the sets once and only once when they are combined.
For example, the 2nd highest value possible from
a data set of a{3,2} and b{2,0} is a1+b2=3.
There are two possible ways of combining these two data sets:
(1) a1+b1 and a2+b2 which results ... |
I am trying to understand (simplify) the following space
$Z = S^2 \times S^2 / E$
Let $(a_1 , a_2 , a_3) \in S^2$ and $(n_1 , n_2 , n_3) \in S^2$. Here ${a_1}^2 + {a_2}^2 + {a_3}^2 = 1$ and ${n_1}^2 + {n_2}^2 + {n_3}^2 = 1$.
The equivalence relations $E$ are the following
1) $[(a_1, a_2, a_3);(n_1, n_2, n_3)] \sim [(-a... |
it is a question Convergence/Divergence of calculus II! Please give me a hand!
Determine convergence or divergence using any method covered so far.
$$\sum_{n = 1}^{\infty} \sin (1/n)$$
|
Suppose two manifolds $X$ and $Y$, both orientable of dimension $n$, and a map $f:X\to Y$.
Is there a relationship between the degree of $f$ calculated with respect to homology (the induced map on the top homology groups) and the degree of $f$ calculated with respect to cohomology (the induced map on the top cohomolog... |
I think it is desirable to have that $M_{m\times n}\left(\mathbb{K}\right)\not=M_{m'\times n'}\left(\mathbb{K}\right)$ if $m\not=m'$ or $n\not=n'$. In other words, the set of all $m\times n$ matrices on $\mathbb{K}$ should be different from the set of all $m'\times n'$ matrices on the same field when $m\not=m'$ or $n\n... |
How to convert the following partial differential equation (pde)
$$\frac{\partial V}{\partial t}=aV-as\frac{\partial V}{\partial s}-b^2s^2\frac{\partial^2 V}{\partial s^2}$$
into a pde of the form
$$\frac{\partial u}{\partial t}=x^2\frac{\partial^2 u}{\partial x^2}+cx\frac{\partial u}{\partial x}$$
by some change o... |
Let $$\cdots \rightarrow A_{n+1}\rightarrow^{f_{n+1}} A_n \rightarrow^{f_{n}} A_{n-1}\rightarrow \cdots $$ be an inverse system of finite dimensional vector spaces with the property that the $A_i$ are 'eventually constant', i.e., there is an $m$ such that the maps $f_i$ are isomorphisms for every $i\ge m$. Does it foll... |
Let $A,B \in \mathbb{R}^{n \times n}$ be two positive semi-definite matrices and let $a > 0$ be a constant.
Consider the following maximization problem
$$
\max_{x \in \mathbb{R}^n, \gamma}\ x^T \left(\frac{\gamma}{a + \gamma} A - \gamma B \right)x
\qquad \text{subject to } \gamma > 0, \|x\| = 1
$$
and suppose that ... |
I understand that for a number $a$ to have a multiplicative inverse in mod $n$, it must be coprime to $n$; therefore, all numbers that do not have a multiplicative inverse mod $n$ must share a factor with $n$.
Now, my prediction was that if $n$ is even, then the sum of all numbers that do not have a multiplicative inve... |
I wanted to use this fact in a proof, it seems obvious, but I should probably prove this as well. Any proof will do. Also for the future is there a way to search for these proofs google does not work very well.
EDIT: Ah, it may also be good to show that the n-th root of any real number a s.t. $0\lt a\lt1$ is also $\l... |
I am learning the basics of category theory, so this question is probably obvious to anyone who knows the subject.
The resources I've seen all take the following approach:
0) A category is a collection of objects and morphisms between those objects that satisfy some rules.
1) A functor is a morphism in the category of ... |
Could someone provide a good reference for learning matrix calculus? I've recently moved to a more engineering-oriented field where it's commonly used and don't have much experience with it.
|
In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves contain itself? (see, Russell)
On the other hand, people have no trouble casually talking about the "category of all categ... |
Given a diffusion $dX_t = \mu(X_t)dt + \sigma(X_t)dW_t$, if one applies the transform
$f(x) = \int_a^x \frac{1}{\sigma(u)}du$, one can use Ito's lemma to show that
$df(X_t) = \left( \frac{\mu(X_t)}{\sigma(X_t)} - \frac{\sigma'(X_t)}{2} \right)dt + dW_t$.
Is there a way to explicitly invert this transform in the general... |
We all know that $\displaystyle \sum_{n=1}^{\infty} \frac{1}{n^s}$ converges for $s>1$ and diverges for $s \leq 1$ (Assume $s \in \mathbb{R}$).
I was curious to see till what extent I can push the denominator so that it will still diverges.
So I took $\displaystyle \sum_{n=2}^{\infty} \frac{1}{n\log n}$ and found that ... |
I know this means that there are two cosets. I also know that one must be H itself.
This means that the remaining coset(s) must be equal for right and left. Also since there is only one possible coset, this means that for all elements, a,b in G, then $aH = bH$ and $Ha = Hb$, since each element in G acting on the subgro... |
Find all continuous functions $f$ on $\mathbb{R}$ satisfying $f(x)=f(\sin x)$ for all $x$.
|
In the derived category $D(C)$ of an abelian category $C$, one formally inverts quasi-isomorphisms. In the context of model categories, one inverts weak equivalences.
What does one gain by doing so?
Is there a big-picture way to think about what is being done?
|
Let $G = \hbox{proj.lim.}_{\alpha} \{ G_{\alpha} , \varphi^{\alpha}_{\beta} \}$ be a projective limit of simple groups (i.e., each $\varphi^{\alpha}_{\beta}\colon G_{\alpha}\to G_{\beta}$ is a surjective group homomorphism between simple groups).
It is clear that if $\varphi^{\alpha}_{\beta}$ is not the trivial map ($x... |
Someone recently asked me why a negative $\times$ a negative is positive, and why a negative $\times$ a positive is negative, etc.
I went ahead and gave them a proof by contradiction like this:
Assume $(-x) \cdot (-y) = -xy$
Then divide both sides by $(-x)$ and you get $(-y) = y$
Since we have a contradiction, then ou... |
Cyclic groups have at most one subgroup of any given finite index. Can we describe the class of all groups having such property?
Thank you!
|
I know a proof that $\frac{(30n)!n!}{(15n)!(10n)!(6n)!}$ is an integer.
The proof goes as:
If a prime $p$ divides $(15n)!(10n)!(6n)!$, then the power of the prime dividing $(15n)!(10n)!(6n)!$ is lesser than the $(30n)!n!$. It is relatively easy to prove this.
$\textbf{My question is there a counting argument to prove t... |
$H$ is a subgroup of $G$ and $a,b$ are in $G$.
I was rather lost for a while but I think I may have
actually proved it, though I am unsure… Does this make sense?
\begin{aligned}
aH &= bH \\
&\implies b \in aH \\
&\implies \exists h \in H : ah = b \\
&\implies a^{-... |
I'm trying to find a general solution for $\displaystyle\iint_{S}[[x+y]] dA$ where $S=[a,b] \times [c,d]$ and $[[x]]$ is the greatest integer function. I know that if $a,b,c,d \in \mathbb{Z}$, then the answer is just $\frac{(d-c)(b-a)(a+b+c+d-1)}{2}$ (proven below), but how do I account for the extra volume when $a,b,c... |
Why is the following inequality true: if $x \geq 0$ then $(1-\frac{x}{n})^{n} \leq e^{-x}$ ? here $n$ is a positive integer. Is there a quick way to see this?
|
Possible Duplicate:
About irrational logarithms
Please help proving that $\log_{10}(2)$ is irrational.
|
I would like to know sources, and examples of good "challenge" problems for students who have studied pre-calculus and some calculus. (differentiation and the very basics of integration.) Topics could be related to things such as:
Taylor Series.
Product Rule, Quotient Rule, Chain Rule.
Simple limits.
Delta Epsilon Pro... |
How does one show for complex numbers u and v, and for p>1 that
\begin{equation*}
||u|^{p-1}u - |v|^{p-1}v|\leq c_p |u-v|(|u|^{p-1}+|v|^{p-1}),
\end{equation*}
where $c_p$ is some constant dependent on p. My intuition is to use some version of the mean value theorem with $F(u) = |u|^{p-1}u$, but I'm not sure how to ma... |
Given a sequence $(x_n)$ in a metric space $M$, call it uniformly separated if all pairwise distances $d(x_n,x_m)$ between distinct terms are uniformly bounded away from zero.
Suppose that a given sequence $(x_n)$ has the property that it has no Cauchy subsequence. Must it have a uniformly separated subsequence?
A c... |
Let we have a competitive survival game in which a player has choice between different resources to earn. The question here is which resource should he prefer to maximize the chance of survival. I tried to find out if there already established measures of such sort, but so far without success.
Some thoughts led me to t... |
I have been sick for my real analysis class and I am a having a tough time getting the book.
Could someone explain the what open, closed and compact sets are and how they are related with uniform continuity? I think the hinging point was a poor understanding of accumulation points.
Thanks
|
I apologize for posting such an untechnical question, but with responses it could surely be posed in a better form.
I'm a math noob, but I've seen (as we all have) a few examples of "connections" between different areas of math by way of an equality or some other relating device. An example is:
e^iπ + 1 = 0
These five... |
A graph G is planar if and only if xxx.
What can xxx be substituted for? Note that this is from a topological POV so a graph is a 1-dim cw complex and I guess the fundamental group should be used somehow.
|
Solving a heat equation with central symmetry i got the following result:
The problem is to find a sphere center temperature vs. time given that the surface of the sphere is kept constant at $$T_1$$ and at initial moment t=0 the temperature inside the sphere is distributed uniformly with value $$T_0$$
A solution is f... |
I am not good with vector spaces so I would be grateful for any help.
As I've been told I need to take $v \in \mathrm{Im}(T)$, $v\neq 0$, and show that if $T(v) = \mathbf{0}$ then $T^2 = 0$. But if $T(v) \neq \mathbf{0}$, then need to show that if $\{e_1,\ldots,e_{n-1}\}$ is a basis of $\mathrm{Ker}(T)$, then $\{e_1,\d... |
I'm trying to prove that $\frac{\prod \mathbb{Z_p}}{\bigoplus \mathbb{Z_p}}$ is a divisible $\mathbb{Z}$-module (p is prime, and the direct sum and direct product are taken over the set of all primes). It is an exercise from Rotman, An Introduction to Homological Algebra. Here's what I've done so far:
$\bigoplus \mathb... |
The standard way (other than generating up to $N$) is to check if $(5N^2 + 4)$ or $(5N^2 - 4)$ is a perfect square. What is the mathematical logic behind this? Also, is there any other way for checking the same?
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This question is a follow-up of this one.
Now I'm trying to prove that the torsion submodule of $\prod \mathbb{Z}_p$ is not a direct summand of $\prod \mathbb{Z}_p$.
I found this interesting because it's an example that over a PID, a non-finitely generated module may fail to be decomposed as torsion $\oplus$ somethi... |
I have a problem that I cannot solve regarding interest payments, and payment holidays.
The scenario is this
If I purchase a product which is paid for over a 12 month period, I need to calculate the monthly interest payments. However, the complexity here, is that I have the option to take one month payment holiday (is ... |
By the mean value theorem it's easy to show that $|a_{n+1}-a_{n}| \leq \frac{5}{6}|a_{n}-a_{n-1}|$ for every n.
Next, I thought of saying $|a_{n+1}-a_{n}| \leq ... \leq (\frac{5}{6})^{n}|a_{1}| \to 0$ and somehow show that ** if $M_{n}$ is the closed interval whose end points are $a_{n}$ and $a_{n-1}$ then $a_{n+1} \i... |
Ok, so this might not be a permutation or combination but I'm curious as to what the answer is.
Question: There are 3 groups and 5 Objects in each group. How many combinations are there if your choosing one from each group?
Yes this is in reference to Call of Duty: Black Ops perks lol.
|
I'm trying to show that the expression
$\omega_i^2 \mid \omega \mid^{-\beta}$
(where $\mid \omega \mid = ( \omega_1^2 + \omega_2^2)^{\frac{1}{2}}$ and $i = [1, 2]$) is defined at the origin under certain conditions. Here's my attempt. I'd like to know if its correct.
Because $\mid \omega \mid^2 = \omega_1^2 + \omega_2... |
We are playing a game with my friend and trying to determine who is the better player.
The game has an luck element to it.
The current score is 9-7.
The following questions have risen:
1) How many games we should play to get statistically significant result that one is better than the other? We can assume p=9/16 or p=0... |
I want to prove that:
If $(x_{k})\to L$ and $\forall x_{i}\in (x_{k})$, $x_{i}$ is a subsequential limit of $a_{n}$ then $L$ is also a subsequential limit of $a_{n}$.
I came up with the following: Let $\epsilon\gt0$; if $(x_{k})\to L$ then we simply pick $x_{i}\in(L-\epsilon, L+\epsilon)$ and because $x_{i}$ is a sub... |
I along with one of my friends were just discussing some basic things in group theory, when this question came up:
What are some fundamental results in group theory?
We happened to list out some:
Fundamental Theorem of Group Homomorphism:
Cayley's Theorem
Sylow's Theorem
There may be many more, but as far as my lit... |
In Stein-Shakarchi's book Fourier Analysis (p. 170), the solution of the Black-Scholes equation
$$\frac{\partial V}{\partial t}+rs\frac{\partial V}{\partial s}+\frac{\sigma^2s^2}{2}\frac{\partial^2 V}{\partial s^2}-rV=0$$ for
$0\lt t\lt T$
with boundary condition $V(s,T)=F(s)$, is given as
$$V(s,t)=\frac{e^{-r(T-t)}}... |
One proof of the halting problem goes by contradiction like this :
Assume there is a Turing machine $H$ that can decide the halting problem, then construct a Turing machine $Q$ that takes as input a Turing machine $M$ and an input $I$ to $M$, such that if $H$ decides that $M(I)$ halts, then $Q$ doesn't halt, and vice v... |
Find all functions $f:\mathbb{R} \to \mathbb{R}$, that are continuous at the point $x=0$ and satisfy:
$$f (x+y)=f (x)+f (y)+xy (x+y) \ \ \forall x,y \in \mathbb{R} $$
|
Take projective real space $\mathbb P_n (\mathbb R)$ of ODD dimension. It is easy to proof that all his Stiefel-Whitney numbers are zero . So according Thom theorem
there must exists manifold $M$ with boundary such that boundary is
$\partial M= \mathbb P_n (\mathbb R)$. I should like to see directly such $M$, without... |
For what value of $\alpha$ is $W_t^{\alpha}$ a martingale where $\alpha \in \mathbb{Z}$? ($W_t$ standard Brownian motion)
I can prove for $\alpha = 0$ and $\alpha = 1$, it is a martingale. (rather it is obvious). But my question is:
How to prove that $W_t^{\alpha}$ is not a martingale for $\alpha \in \mathbb{Z}\cap\{0,... |
In exercise 36 Miscellaneous Taylor Coefficients using Bernoulli
numbers on pages 88-89 of Louis Comtet's Advanced Combinatorics, 1974,
one is asked to obtain the following explicit formula for the Bernoulli numbers:
$$B_{2n}=(-1)^{n-1}\dfrac{1+\left[ \varphi _{n}\right] }{2(2^{2n}-1)},$$
where
$$\varphi _{n}=\dfrac{2(... |
Let $S$ be a sheaf over $X$ and $r$ an element in $S_x$ for some $x$ in $X$. Must there exist a section $s$ in $S(X)$ such that such that $s$ equals $r$ when mapped to $S_x$ by the canonical map?
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I'm sorry for this kind of specific question, I'd love if you could link to resources (prime lists, etc) that can answer similar questions more generically.
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I started by showing that $1\leq a_{n} \leq n$ (by induction) and then $\frac{1}{n}\leq \frac{a_{n}}{n} \leq 1$ which doesn't really get me anywhere.
On a different path I showed that $a_{n} \to \infty$ but can't see how that helps me.
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Let $f(x)$ continuous function on $R$ wich can be in different signs. Prove, that there is exists an arithmetic progression $a, b, c (a<b<c)$, such that $f(a)+f(b)+f(c)=0$.
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Say we have the function $$G(z) = \sum_{n \geq 0} g_n z^n.$$
Is there a name for the transform T defined so that $$(T(G))(z) = \sum_{n \geq 0} g_n z^{n^2}?$$
Is there anything known about this kind of transformation?
|
In the book Weak convergence and Empirical Processes, by Aad W. van der Vaart and Jon A. Wellner, on page 81, the notation $l^\infty(\mathcal{F})$ appears, where $\mathcal{F}$ is a set of measurable functions, $f \in \mathcal{F}$ then $f \colon \mathcal{X} \rightarrow \mathbb{R}$.
I am not sure what $l^\infty(\mathcal{... |
In complex analysis, we care about the functions of a complex variable. Are such functions just special case of the complex valued function defined in Rudin's Principles of Mathematical Analysis, since they have many similar properties such as uniformly convergence in series. Moreover, I wonder the relationship between... |
If I have a distribution mass function p(Y) , and values of $X_0,X_1,...,X_i$, I want to assign values to $X_{i+1},...,X_n$ such that the series of X follows p(Y) distribution where Y is the set of all distinct values of X. Is there any technique to do this?
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Possible Duplicate:
The cow in the field problem (intersecting circular areas)
A farmer ties a cow with a rope to a pole at the edge of a round field of radius r. How long must the rope be compared to r if the farmer wants to ensure that the cow can roam exactly half of the field?
The point is to find a closed-form ... |
I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. So if we have a set $\Omega$ and $|\Omega| = \aleph_i$, then (assuming continuum hypothesis) the cardinality of $2^{\Omeg... |
If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For example, the set of all sets must be excluded.
I recall that I came across a paper in Arxiv some time ago which explained that ... |
Are all dense open sets in R cocountable?
That is, are all dense open sets in R such that their complements are at most countable?
It would seem like they must be since the closed sets that are uncountable are all intervals, so their complements are not dense.
|
Having read on Wikipedia about generating functions, I still don't understand what I can do with a generating function.
For example, if I want to compute the first spread polynomial, how can I do that from a generating function, such as these (taken from the Wikipedia article):
$$
\sum_{n=1}^\infty S_n(s)x^n = {sx(1... |
One of the answers to this question indicates that large cardinals are useful for destructive testing of set theory. That aside, and not assuming GCH, are there any sets known that have a cardinality of $\aleph_{n}$, $n>0$, or that of any of the large cardinals? There are a few examples of sets that have these cardinal... |
Let $f:\mathbb{R} \to \mathbb{R}$ is a function with these special properties. $f$ is continuous everywhere. $f$ is not smooth (not infinitely differentiable). $f$ is differentiable only finitely many times everywhere. $f$ belongs to $L^p$. For any $k$ belongs to $\mathbb{N}$ let $E$ be the set of all points where $f$ ... |
A friend of mine is a differential geometer, and keeps insisting that he doesn't need the axiom of choice for the things he does. I'm fairly certain that's not true, though I haven't dug into the details of Riemannian geometry (or the real analysis it's based on, or the topology, or the theory of vector spaces, etc...)... |
Let any Amicable Pair be represented by $(a, b)$, with $a \lt b$. In a search for Amicable Pairs, let $(n, m)$ with $n < m$, be a candidate to be tested for being an amicable pair. Given $n$, is there way to calculate both the least upper bound and greatest lower bound for the m that would make $(n, m)$ an Amicable Pai... |
When operating with sigma algebras, what does it mean when we talk about a countably infinite set? And, what are then closed countably infinite interesections?
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The thing is, since I do a job, and work on mathematics in my spare time, I do not have any connection with the academia. Secondly, the internet is the only way I can interact and reach out to other mathematicians or math enthusiasts like me.
If I need to discuss my paper with someone, where do I do so over the interne... |
During lesson we were given a starter activity which was to try and factorise polynomials and see what happened.
The polynomials were
$x^3-8$,
$x^3-3x^2+ x -3$
and
$x^4 - 16$.
I could not work out what happened to them, and it's bugging me. If anyone can explain then it'd be helpful! Thank you.
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I've got a solution to a economy class calculation, but I can't figure out how they came to the answer. Could you help me?
-5p + 80 = 8p +-50
.......? = ......?
120 = 12p
p = 10
Extra: do you know which rule they are using to get from the first line to the 3d?
Many thx
Math beginner,
Frank
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This is actually a generalized version I wrote of a homework question that intrigued me:
Let $f$ be continuous in $[0,1]$ and
$\forall x\in [0,1], \ f(x)=f(1-x)$.
If $r\in [0,1]$ and $1-r$ are the only
sub limits of $a_{n}$ then $f(a_{n})$
converges.
(the original question states $r=\frac{1}{3}$)
I solved th... |
Suppose I have a set $\mathcal{S}$ of $N$ distinct items. Now consider the set $\mathcal{P}$ of all possible pairs that I can draw from $S$. Naturally, $|\mathcal{P}| = \binom{N}{2}$. Now when I draw $k$ items (pairs) from $\mathcal{P}$ with a uniform distribution, what is the expected number of distinct items from $S$... |
This is a follow-up to the question Undecidability in Conway’s Game of Life I posted at mathoverflow.
For some cellular automata it can be proven that they can simulate a Turing machine, normally by explicit construction. For some of them it can be proven that they can not simulate a Turing machine.
Are there computab... |
Let $f:[0,1] \rightarrow [0,\infty)$ be a measurable function such that:
$\mu (\{x \in [0,1]: f(x) > t \}) \leq \frac{1}{t(ln(t))^{2}}$
holds for each $t>3$.
Show $f$ is an integrable map.
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How do I show that a monomorphism $F \rightarrow G$ of sheaves induces an injection on stalks?
When showing that monomorphism is an injection on sets one uses the maps $x \mapsto a$ and $x \mapsto b$ where $a,b$ are some fix elements such that they have the same image under the monomorphism. Then by using the monomorp... |
Often I have wondered about this question, but today I had a chance to recollect it and hence I am posting it here. During high-school days one generally learns Integration and I still loving doing problems on Integration. But I have never understood this idea of integration by parts. Why do we adopt the ILATE or LIATE... |
How to prove that finite additivity follows from countable additivity!!?
|
I've written Haskell function, that helps me to compute $x^{-1} \mod m$.
invM x m = invMhelp x 0 m; --modular multiplicative inverse x^-1 (mod m) = inv x m
invMhelp x i m = if (x*i `mod` m == 1) then i else (invMhelp x (i+1) m);
For example:
Main> 13 `invM` 2109
649
That means $13^{-1} = 649 \mod 2109$.
How can I c... |
Please observe the triangle shown in the figure below (In red). I consider the angle between $5$ and $4$. The base should be $5$, hypotenuse $4$ and perpendicular $3$, but according to the solution manual the base is $4$ and the hypotenuse $5$. Please tell me how? Thanks!
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Can somebody please explain continuity and discontinuity in fourier series?
|
I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus.
I have searched for "operator calculus" and have not found any accessible references that introduce the topic. Maybe operator calculus g... |
Let $g(t)$ be a real function of a real variable such that
$$\sup |p(t)g(t)|\lt \infty$$ for any polynomial $p$. Is it true that for any nonngetive
integer $k$, one has $$\sup_{(x,y)\in\mathbb{R}^2}(\frac{|x|}{1+|y|})^k |g(x-y)|\le A_k, $$
for some positive constant $A_k\lt \infty$?
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I was wondering if this question should be posted in this forum or in Stack Exchange, but reading both "What kind of questions can I ask here?", in Math I found
Software that mathematicians use
So I guess it should fit here.
I have a TI-89 (a --powerful, I guess-- Texas Instruments calculator) which I've had since hi... |
Let $A$ be a finitely generated Abelian group. Let $tA$ denote the torsion subgroup. Prove that $A$ has a subgroup isomorphic to $A/tA$.
I know that $A/tA$ is torsion free, so my thinking so far has been to take the non-torsion elements of $A$ and consider the subgroup generated by them. Then show that subgroup and $A... |
If I understand correctly, the Euclidean Dirac operator is given by
$$D=\sum_{i=1}^n e_i \frac{\partial}{\partial x_i},$$
where $e_i$ are bases for $Cl_{0,n}(\mathbb{R})$, i.e., the $n$-dimensional Clifford algebra with negative-definite signature over the reals (so $e_i^2=-1$), and $x_i$ are the corresponding coordina... |
What are some families of graphs with known tree width k?
For instance, for tree k=1, for cycle k=2, for series-parallel graph which is not a tree k=2, what are some other examples?
I'm looking for some easy to construct families (or famous graphs which I can find in Mathematica's GraphData or on the internet) for test... |
All,
Given any number, how can we retrieve each digit of that number?
i.e. For number = 4321, how can I get individual digits 4,3,2 and 1?
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I'm trying to follow a proof about immersing/embedding $\mathbb{RP}^n$ into $\mathbb{R}^{n+1}$, which goes roughly as follows:
Write $\tau=T\mathbb{RP}^n$. The normal bundle $\nu$ has rank 1, so its Steifel-Whitney class is $w(\nu)=1$ or $w(\nu)=1+x$. In every case, we need $w(\nu)\cdot w(\tau) = w(\nu \oplus \tau) =... |
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