instruction stringlengths 12 30k |
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I usually set this as an exercise when teaching Number Theory.
My hint is to ask the students: what are the solutions of
$z^8=16$ in the complex numbers? |
What do we mean by 'sporadic' ? I suggest that we may find some extension of the
Chevalley programme (Tohoku J 1950+) that brings in the sporadics. This is about
construction - not the classification as we know it. |
Joseph Gallian has decrypted many of the US state license plate and driver's license codes.
http://books.google.com/books?id=PD0clAlF8O4C&pg=PA27
|
Joseph Gallian has decrypted many of the US state license plate and driver's license codes.
http://books.google.com/books?id=PD0clAlF8O4C&pg=PA27
I think he used Markov chain models. As whuber mentioned your problem is similar to the German tanks for which the subject reference is "extreme value statistics". |
I prefer the direction: Dominated convergence theorem -> Beppo-Levi (monotone convergence) -> Fatou.
This direction requires more machinery (like Egoroff and the absolute continuity of the Lebesgue integral) but it is in my opinion more tidy. A book that uses this approach is [Bogachev - Measure Theory](http://books... |
Let $f: X \rightarrow Y$ be a local homeomorphism with X, Y connected, locally path connected, Hausdorff and with X also compact. Then f is also a covering with finite fibers.
I know how to show that the fibers are finite.
Given that f is a surjection, I know how to show that f is a covering map.
**How do I show... |
What is motivation behind the definition of a complete metric space?
Intuitively,a complete metric is complete if they are no points missing from it.
How does the definition of completeness (in terms of convergence of cauchy sequences) show that? |
> Sum of a set of sphenic numbers can't be equal to the sum of *any other* set of sphenic numbers.
By that I meant, Say S is the set of sphenic numbers. Let, S$\_1$ $\subset$ S.
Then, there is no such S$\_2$ $\subset$ S so that,
S$_1$ $\neq$ S$_2$ AND $\sum S_1$ $=$ $\sum S_2$
Question 1 : Is this statement ... |
Let $T:X \times X \to \mathbb{R}$ be a continuous bilinear operator defined on a normed linear space $X$ s.t.
$T(\alpha x + \beta y,z) = \alpha T(x,z) + \beta T(y,z)$) and $T(x,y) = T(y,x)$.
Does there exist a constant $C$ s.t. $||T(x,y)|| \leq C$ $||x||$ $||y|| \forall x,y$?
I know that the result is true ... |
I need to calculate all the numbers in a certain row of Pascal's triangle. Obviously, this is easy to do with combinatorics. However, what do you do when you need to estimate all the numbers in, say, the 100000th row of Pascal's triangle?
Is there any way to estimate the number so that the costly multiplications an... |
Are you computing the factorials and *then* dividing? Don't do that. You can combine the estimates you get from Stirling's formula into
$${n+m \choose n} \approx \sqrt{ \frac{2\pi (m+n)}{mn} } \frac{(m+n)^{m+n}}{m^m n^n}$$
and even then you can optimize, for example rewriting the second factor as
$$\left( 1... |
Are you computing the factorials and *then* dividing? Don't do that. You can combine the estimates you get from Stirling's formula into
$${n+m \choose n} \approx \sqrt{ \frac{m+n}{2 \pi mn} } \frac{(m+n)^{m+n}}{m^m n^n}$$
and even then you can optimize, for example rewriting the second factor as
$$\left( 1 ... |
In the comments to the question: [$a^n+n|b^n+n \Longrightarrow a=b$][1], there was a claim that $5^n+n$ is never prime (for integer $n>0$).
It does not look obvious to prove, nor have I found a counter example.
Is this really true?
**Update**: $5^{7954} + 7954$ has been found to be prime by a computer: http://... |
The Cantor set is uncountable so I expect countably many of them to be able to cover $\mathbb R$, but the set has measure $0$ so countably many of them also has set of measure $0$ and thus can't cover the real line.
Why/where is my intuition broken? |
Why cant Cantor sets cover $\math R$? |
A [perfect number][1] is a number that is the sum of its proper divisors:
`28 = 1 + 2 + 4 + 7 + 14`
Is there a name for numbers that are divisible by 1, 2, and N/2? For example
6 is divisible by 1, 2, and 3
8 is divisible by 1, 2, and 4
34 is divisible by 1, 2, and 12
Is there a name for these numbers?
... |
Is there a name for this kind of number? |
Why can't Cantor sets cover $\mathbb{ R}$? |
A [perfect number][1] is a number that is the sum of its proper divisors:
`28 = 1 + 2 + 4 + 7 + 14`
Is there a name for numbers whose only proper divisors are 1, 2, and N/2? For example
6 is divisible by 1, 2, and 3
8 is divisible by 1, 2, and 4
34 is divisible by 1, 2, and 17
Is there a name for these ... |
I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the... |
I'm not quite sure what the mathematical term for what I'm asking is, so let me just describe what I'm trying to figure out. Let's say that I have two ordered sets of numbers {1, 2} and {3, 4}. I'm trying to figure out the number of possible ways to combine these two sets into one without breaking the ordering of the... |
How do I determine the possible number of combinations of two ordered sets? |
Do you think the following limits are correct?
$\displaystyle\lim_{d\to\infty}\frac{\sum\limits_{k=1}^{d} {\phi(N) \choose k} {d-1 \choose k-1}}{\phi(N)^d}=0$
$\displaystyle\lim_{N\to\infty}\frac{\sum\limits_{k=1}^{d} {\phi(N) \choose k} {d-1 \choose k-1}}{\phi(N)^d}=c$
I plotted the equations and guessed t... |
Do you think the following limits are correct?
$\displaystyle\lim_{d\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=0$
$\displaystyle\lim_{N\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=c$
I plotted the equations a... |
Let's say I have a huge number starting from 1 going forward like 12345678910111213...
Any number in the series occurs multiple times. The first occurence of 3 is in position 3 of the series, 2nd occurence is in position 17.
How do I find the position of 3 in 100th occurence? Is there a pattern? |
I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the... |
Let's call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) written down in base 10.
Thus, $S = 1234567891011121314151617181920212223242\ldots$
Any number in the $S$ occurs multiple times. The first occurence of 3 is in position 3 of the series, 2nd occuren... |
I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable set of reals. "Measurable" involves third-order concepts but I know that there's often a "coding trick" that gets around ... |
Where in the analytic hierarchy does V=L start having consequences? |
Let's call $S$ the (infinite) string that is made by concatenating the consecutive positive integers (starting from 1) written down in base 10.
Thus, $S = 1234567891011121314151617181920212223242\ldots$
Any number in $S$ occurs multiple times. The first occurence of 3 is in position 3 of the series, 2nd occurence i... |
You can solve this by thinking of some lists:
How many 3's is there in 1-10? -- 1
How many 3's is there in 11-20? -- 1
How many 3's is there in 21-30? -- 2
How many 3's is there in 31-40? -- 9
How many 3's is there in 41-50? -- 1
...
How many 3's is there in 91-100? -- 1
Now, how many 3's is the... |
You can solve this by thinking of some lists:
How many 3's is there in 1-10? -- 1
How many 3's is there in 11-20? -- 1
How many 3's is there in 21-30? -- 2
How many 3's is there in 31-40? -- 10
How many 3's is there in 41-50? -- 1
...
How many 3's is there in 91-100? -- 1
Now, how many 3's is th... |
Brute force:
[mariano@godot ~]$ ghci
GHCi, version 6.12.1: http://www.haskell.org/ghc/ :? for help
Prelude> [i | (d,i) <- concat (map show [1..]) `zip` [1..], d == '3'] !! 100 - 1
889
Prelude>
|
A general rule-of-thumb for "is there a prime of the form f(n)?" questions is, unless there exists a set of small divisors D, called a *covering set,* that divide every number of the form f(n), then there will eventually be a prime. See, e.g. [Sierpinski numbers][1].
Running WinPFGW (it should be available from the... |
Let $X_1,....,X_n$ be some positive random variables.
Is $A_{ij} = \frac{E[X_i*X_j]}{E[X_i]E[X_j]}$ a positive (semi) definite matrix?
(note that $B_{ij} = Cov(X_i,X_j) = E[X_i*X_j] - E[X_i]E[X_j]$ is PSD because it is a covariance matrix.) |
is the following a positive definite matrix? |
Here's an horrible drawing that tries to explain what I'm asking:
![Okay, so if you're reading or hearing this you can't see images, so I will take the effort of explaining it with words. Consider four $n$-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as t... |
What is the number of sides of the hollow area generated by juxtaposition of four $2n$-sided regular polygons? |
Here's an horrible drawing that tries to explain what I'm asking:
![Okay, so if you're reading or hearing this you can't see images, so I will take the effort of explaining it with words. Consider four $n$-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as t... |
Here's an horrible drawing that tries to explain what I'm asking:
![Okay, so if you're reading or hearing this you can't see images, so I will take the effort of explaining it with words. Consider four $n$-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as t... |
Can anyone see a nice way to prove the following for $0\le x \le 1$?
$$\sqrt{1-x^2}\ge \text{erf}(\sqrt{-\log x})$$
|
Here's an all-purpose technique for proving your kind of induction problems.
You are trying to prove using induction that 9 divides $n^3 + (n+1)^3 + (n+2)^3$ whenever n is a non-negative integer.
You have already demonstrated the base case, so here's the inductive step.
Assuming it is true for n, is it also t... |
I approached this the wrong way apperantly, help please?
"Consider the word mathematics. In how many ways can you rearrange all the letters so that the vowels are paired and always apart?" worth 6 marks |
Given there are four vowels, list all the ways to split them into pairs. Now for each set of pairs think of each pair as a single character. How many ways can you permute the resulting collection? How many of those have the vowel pairs next to each other? |
Problably a simple question and possibly not asked very well. What I want to know is..
In binary, a decimal value of 1 is also 1.
It can be expressed as x = 1 X 2º
(probably the wrong use of a character symbol above, I was trying to represent "to the power of zero")
Question:
Why is two to the power of zero... |
Why is two to the power of zero equal to binary one? |
Let $y = \sqrt{-\log x}$. Then the inequality reduces to $\text{erf}(y) \leq \sqrt{(1-e^{-2y^2})}$ or equivalently $\text{erf}^2(y) + e^{-2y^2} \leq 1$. Now $\text{erf}^2(y)$ can be written as a double integral $\text{erf}^2(y) = \frac{4}{\pi} \int_{0}^y \int_{0}^{y} e^{- (a^2+b^2)} da db$ (Hopefully, this is correct a... |
Let $y = \sqrt{-\log x}$. Then the inequality reduces to $\text{erf}(y) \leq \sqrt{(1-e^{-2y^2})}$ or equivalently $\text{erf}^2(y) + e^{-2y^2} \leq 1$. Now $\text{erf}^2(y)$ can be written as a double integral $\text{erf}^2(y) = \frac{4}{\pi} \int_{0}^y \int_{0}^{y} e^{- (a^2+b^2)} da db$ (Hopefully, this is correct a... |
Can anyone see a nice way to prove the following for $0\le x \le 1$?
$$\sqrt{1-x^2}\ge \text{erf}(\sqrt{-\log x})$$
$erf$ is defined as
$$erf(z) = \frac{2}{\sqrt{\pi}}\int_{0}^{z} e^{-t^2} \\ dt$$ |
12, 13, 11, 14, 10, 15, 9, 16... 0, maxNum
Start with x then alternate on each side of x until you reach 0 on one side and maxNum on the other. Need to do this in some programming code but can't nail it down. How would you write out the formula for that sequence? |
what is the current state of the art in methods of summing "exotic" series? |
Consider that I have 3 parametric equations as function of time and describe the motion of a body in space:
x = f(t)
y = g(t)
z = h(t)
These curves are pretty simple and can be modeled within a certain interval by a 2nd order polynomial.
Is it as simple as performing individual fitting on each function o... |
The problem is to find:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$
I rewrite the equation as follows:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$
And multiply by $\dfrac{\cos(x)}{\cos(x)}$, producing:
$\lim_{x \to 0}\ \dfrac{\cos(x)*\sin(\cos(x))}{\dfrac{\cos(x)}{\cos(x)}}$
And r... |
What is n here?
Is it no .of digits or the value of the number?
If it is the value of the no,, then just convert each digit of base-9 no. in to its base-3 equivalent. |
The problem is to find:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$
I rewrite the equation as follows:
$\lim_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$
And multiply by $\dfrac{\cos(x)}{\cos(x)}$, producing:
$\lim_{x \to 0}\ \dfrac{\cos(x)*\sin(\cos(x))}{\dfrac{\cos(x)}{\cos(x)}}$
And r... |
The following is a lemma I read online, but I don't understand part of the proof.
Let $d$ be the maximum possible order among integers $a$ prime to $p$. Then for any integer $a$ not divisible by $p$, the order of $a$ divides $d$.
Proof: Let $b$ be an element of order $d$, let $k$ be the order of $a$, and let $g=... |
Below is the key lemma for this and related results (Herstein's hardest problem). See my [post here][1] for an example of how it is applied.
**LEMMA** $\ $ A finite abelian group $\rm\:G\:$ has an lcm-closed order set
i.e. $\rm\ X,Y \in G\ \Rightarrow\ \exists\ Z \in G:\ o(Z) = lcm(o(X),o(Y))$
**Proof**$... |
I think it's helpful to think about sums for a bit, and then come back to products.
With sums… we’re all happy with the idea that you can add up a list of 2 things, or 3 things, or more (so e.g. the sum of the list $(x,y,z)$ is $x+y+z$). And after moment’s thought, we’re happy with the idea that you can add up a li... |
Below is the key theorem. See my [post here][1] for an example of how it is applied.
**THEOREM** $\rm\ \ maxord(G) = expt(G)\ $ for a finite abelian group $\rm\: G$
i.e. $\rm\ max\ \\{ ord(g) : \: g \in G\\}\ =\ min\ \\{ n>0 : \: g^n = 1\ \:\forall\ g \in G\\}$
**Proof** $\ \:$ By the lemma below, $\rm\ S ... |
I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the... |
This is a weird logic problem that was posed to me today. This question isn't about the actual logic problem (or the trivial paradox) but in its implications, so read to the end!
A man has been sentenced to death. The executioner comes to his cell on May 31st and tells him that he will die before June 30th. He als... |
Given two complex varieties over a common base, I can take their fiber product in the category of varieties, or I can take their fiber product in the category of schemes and then take the reduced subscheme. I have heard, that these two operations yield the same.
Has someone a reference? |
I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled?
Say for instance that you have a accelerometer and a gyro and want to present the "horizon level", like in an airplane, a good demo of something like this [here][1].
How do you actually harvest the two sensors positiv... |
I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled?
Say for instance that you have an accelerometer and a gyro and want to present the "horizon level", like in an airplane, a good demo of something like this [here][1].
How do you actually harvest the two sensors positi... |
I'm not going to add nothing directly related to your question and previous answers, but make some propaganda of a theorem I like since I was student and which, I believe, says something stronger than comparing some intuitive notion of completness with its definition.
A somewhat related notion of completeness is the... |
My friend is taking a course on probabilities and come across a problem that has a solution we both have trouble understanding:
A single printer prints, on average, 22 jobs an hour. Let's use a simplification where no job takes more than 2 minutes to complete. Using these assumptions, calculate the upper limit for t... |
I'm taking a course on machine learning and came across a model answer which I can't wrap my head around.
The whole model answer is the 2nd exercise here: https://noppa.tkk.fi/noppa/kurssi/t-61.3050/viikkoharjoitukset/T-61_3050_proposed_solutions.pdf
The part I'm specifically having trouble understanding is the l... |
Weird differentation of probabilities, please explain? |
Using Poisson distribution to estimate if a printer is available? (I have the solution but don't understand it) [SOLVED] |
I encounter a problem recently stated as below:
How many pieces of cheese we can obtain from a single thick piece by making five straight slices?(we can't move the cheese when slicing)If we wanna maximize the number of pieces which is denoted by P(n), is there any recurrence relation for P(n)?(n is number of slices)... |
I often see notation/wording like "let f(x) be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that sin and $x \rightarrow sin(x)$ are functions, while sin(x) is a real number.
Are there any correct or best practices in this regard?
|
In written mathematics, is f(x) a function or a number? |
I often see notation/wording like "let f(x) be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that sin and $x \rightarrow \sin(x)$ are functions, while sin(x) is a real number.
Are there any correct or best practices in this regard?
|
I often see notation/wording like "let f(x) be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that sin and $x \mapsto \sin(x)$ are functions, while sin(x) is a real number.
Are there any correct or best practices in this regard?
|
If a quantity is successively increased by **20%** for two times and another quantity is successively decreased by **10%** for three times, the first quantity is **80%** of the second. What percentage is the original first quantity as compared to the original second quantity?
|
Elevator pitch for a (sub)field of maths? |
If a quantity is successively increased by **20%** for two times and another quantity is successively decreased by **10%** for three times, the first quantity is **80%** of the second. What percentage is the original first quantity as compared to the original second quantity?
|
Say your quantity is $Q$. To get 80% of that, you multiply by $0.8$, so $0.8Q$. How do you get 80% of *that*? Well, you multiply by $0.8$ again, so you get $0.8(0.8Q) = (0.8)^2Q$.
How do you take 80% successively three times? Four times? Five times? $n$ times?
If you first take $k$% ($k$ between $0$ and $100$), t... |
This was a very old Monthly problem - see below. For an excellent introduction to the general topic see Richard Stanley's paper [An Introduction to Hyperplane Arrangements (2004)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.118.5996)
![alt text][1]
[1]: https://i.stack.imgur.com/U07We.jpg |
This was a very old Monthly problem - see below. For an excellent introduction to the general topic see Richard Stanley's paper [An Introduction to Hyperplane Arrangements (2004)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.118.5996) and also Renteln's lecture slides [It All Depends on How You Slice It: An ... |
I'd like to understand the physical meaning of the Feynman's vector potential definition:
$ A_{m}^{(b)}(x) = e_b \int \delta (xb_{\mu}xb^{\mu})db_m(b), \qquad m=0,1,2,3 $
(component m of the vector potential of the particle b at the point x)
Can you give me any clues?
Thanks! |
I'd like to understand the physical meaning of the Feynman's vector potential definition:
$$ A_{m}^{(b)}(x) = e_b \int \delta (xb_{\mu}xb^{\mu})db_m(b), \qquad m=0,1,2,3 $$
(component m of the vector potential of the particle b at the point x)
Can you give me any clues?
Thanks! |
I'd like to understand the physical meaning of the Feynman's vector potential definition:
$$ A_{m}^{(b)}(x) = e_b \int \delta (xb_{\mu}xb^{\mu})db_m(b), \qquad m=0,1,2,3 $$
(component m of the vector potential of the particle b at the point x)
Here
- $\delta$ is the Dirac delta function
- $ xb_{\mu}xb^{\... |
A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the partition function for that system converges non-uniformly (i.e. the partition function converges point-wise). This is because in order for a phase change to occur, the partition function must converge to two diff... |
A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. converges point-wise). This is because in order for a phase change to occur, the partition function must converge to two diff... |
A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. converges point-wise). This is because in order for a phase change to occur, the partition function must converge to two diff... |
A loan for $8,000$ must be repaid with 6 year end payments at an annual rate of $11 \%$. What is the annual payment? I know that the present value of an annuity with end payments is $\frac{1-v^n}{i}$ where $v = 1/(1+i)$. Likewise, the future value is $(1+i)^{n}\frac{1-v^n}{i}$. How do I use this to solve this problem? |
I wanted to prove that $\varphi$ is multiplicative (that for $(a,b)=1$, $\varphi(ab)=\varphi(a)\varphi(b)$) using the following idea:
* Define $\varphi'$ by $n = \varphi'(n) + \varphi(n)$.
* Multiply out $\varphi(a)\varphi(b) = (a - \varphi'(a))(b - \varphi'(b)) = ab - \varphi'(a)b - a \varphi'(b) + \varphi'(a)\var... |
Can this broken proof (of multiplicative of $\varphi$) attempt be fixed? |
**HINT:** When you look at the answer of Chandru1 you must think twice. Do you know any algebraic properties of the absolute value? How do you solve an inequality?
You might also like to think of different points $(x,y)=(5,5)$, $(x,y)=(5,-5)$, $(x,y)=(-5,5)$, $(x,y)=(-5,-5)$ -- are they in you set? $(x,y)=(1/5,5)$, ... |
The Setting
----
Let $K$ be an [Archimedean field](http://en.wikipedia.org/wiki/Archimedean_field). TFAE:
1. $K$ has the least upper bound property.
2. Every [Cauchy sequence](http://en.wikipedia.org/wiki/Cauchy_sequence#In_topological_groups) in $K$'s additive group converges.
Now proving that 1 implies 2 is ... |
Archimedian field $K$ has LUB property iff it's complete requires DC? |
Can we find all positive integers $a,b$ such that $a^{n}+b^{n}$ is an $(n+1)^{th}$ power? I think this question equivalent to solving the statement $$a^{n} + b^{n} = c^{n+1}$$ for $a,b,c \in \mathbb{N}$. But i don't know as to how i can solve this.
I attempted by subsituting $n$ as $n+1$ so that we get $$a^{n+1} +... |
Below is the key theorem as it applies to arbitrary finite abelian groups. See my [post here][1] for an example of how it is applied to deduce the more general result that a finite multiplicative subgroup of a domain is cyclic. The lemma is famous as "Herstein's hardest problem" - see the note below.
**THEOREM** $... |
By "Sylvester's Law of Inertia," I mean:
http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia
I can guess that Sylvester is the name of the dude that proved this theorem, but I have no idea as to why it's called "Law of Inertia."
Where is this name coming from? I guess it's from physics, but... I just don... |
Why is it called Sylvester's Law of Inertia? |
This is a follow-up to <a href="http://math.stackexchange.com/questions/6828/proving-sqrt1-x2-ge-texterf-sqrt-log-x">this</a> question. I came across a tighter potential bound and checked it numerically for $0\ge x\ge 5$. I think it holds for all positive $x$, can anyone see a proof?
$$1-\exp(-4x^2/ \pi) \ge \text{e... |
This is a follow-up to <a href="http://math.stackexchange.com/questions/6828/proving-sqrt1-x2-ge-texterf-sqrt-log-x">this</a> question. I came across a tighter potential bound and checked it numerically for $0\le x\le 5$. I think it holds for all positive $x$, can anyone see a proof?
$$1-\exp(-4x^2/ \pi) \ge \text{e... |
**How many number of three digit even numbers than can be formed out of the digits 0 to 9 ?**
The question seems confusing since there is no mention of whether repetition is allowed or not ?! since if repetition is allowed then the answer would be **450** (`5*9*10`) where as if not then the answer would be **328** ... |
**How many number of three digit even numbers than can be formed out of the digits 0 to 9 ?**
The question seems confusing since there is no mention of whether repetition is allowed or not ?! since if repetition is allowed then the answer would be **450** (`5*9*10`) where as if not then the answer would be **328** ... |
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