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I'd like to get the exact value of the following sum:
$\sum_{i=0}^{\lceil \frac{k}{2} \rceil}{({k \choose i}\cdot i)}$
I'd also like to know the asymptotic limits of the function.
|
While reading Munkres' Topology section about integers and reals (Chapter 1, Section 4), he defines the set $\mathbb{Z}_+$ as:
Definition: A subset $A$ of the real
numbers is said to be inductive if it
contains the number 1, and if for
every $x$ in $A$, the number $x+1$ is
also in $A$. Let $\cal{A}$ be the
c... |
Inspired by
this question.
Let $A$ be a subset of ${\mathbb Z}^d$ that generates the whole additive group ${\mathbb Z}^d$, and let $S$ be the additive semigroup generated by $A$.
Prove that there is a $d$-dimensional convex cone $C\subset {\mathbb R}^d$ so that
all elements of $C \cap {\mathbb Z}^d$ with sufficiently... |
This is a problem I'm stuck on that our professor gave us for additional practice (not homework, but its recommended that we understand how to prove it).
We know X and Y are complete metric spaces, and we need to show that $X \times Y$ is complete. I'm really lost on the proof technique. We were given an outline as fo... |
Given are some positive integers $t_1, ..., t_n$ such that $\mathrm{gcd}(t_1, ..., t_n) = 1$. Now we create a linear combination of these: $k_1 t_1 + ... + k_n t_n$ where $k_1, ..., k_n$ are nonnegative integers that you can choose to generate different integers.
What is the smallest $N$ such that all integers greater ... |
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
My attempt at answering the question:
\begin{align*}
x &\equiv 0 \pmod{a}\\\
&\Longrightarrow x\text{ is divisible by $a$}\\\
&\Longrightarrow x = ma\text{ for some integer $m$}\\\
\ \\\
x &\equiv 0 \pmod{b}\\\
&\Longrightarrow x\text{ is divisible by $b$}\\\
&\Longr... |
Suppose n is an integer, such that the sum of the digits of n is 2, and $10^{10} \lt n \lt 10^{11} $. The number of different values for n is:
Let me try to list them :
(1) 11000000000
(2) 10100000000
(3) 10010000000
...
(10)10000000001
So I am getting 10 possible values but the answer is 11.What am I missing here ?
... |
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the 10001st prime number?
|
Define $f$ on $[-1,1]$ by
$$f(x) = \left\{ \begin{array}{ll}
0 & \mbox{if $x\lt 0$;}\\
0 & \mbox{if $x=0$;}\\
1 & \mbox{if $x\gt 0$.}
\end{array}\right.$$
Let the integrator $a$ be defined by
$$a(x) = \left\{\begin{array}{ll}
0 & \mbox{if $x\lt 0$;}\\
0 & \mbox{if $x=0$;}\\
1 & \mbox{if $x\gt 0$.}
\end{array}\right.$... |
Here is a statement of Farkas Lemma from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true:
There exists an $x \in \mathbb{R}^n$ such that $Ax = b$ and $x \geq 0$.
There exists a $y \in \mathbb{R}^m$ such that $A^T y \geq 0$ an... |
In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two statements here below and explain them regardless of the theorem):
"Consider the complex plane. It's not difficult to show th... |
I was recently asked to help someone with the following question on their first year analysis course.
Recall that $\mathbb{N}$ is the set of all positive integers. Use the principle of induction to show that $n \ge 1$ for all $ n \in \mathbb{N}.$ [ Hint: Let $ S = \lbrace n \in \mathbb{N} | n \ge 1 \rbrace.$ ]
I am co... |
How many even positive integers are there that are divisors of 720 ?
I know how to compute the number of divisors but how to compute the number of even or odd positive divisors of a number ?
If we list the divisors of 720 (using mathematica) : {1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45,48, 60, ... |
Please can you explain me why
$$ \nabla^2 G_{\omega}(R) = f(R) $$
is equivalent to
$$ \frac{1}{R} \frac{d^2(RG_\omega(R))}{d R^2} = f(R) $$
Here:
$G_\omega(R)$ is the Fourier transform of $G$
$G$ is the Green's function
We suppose that $G$ and $f$ are spherically symmetric
$R = \sqrt{x^2 + y^2 + z^2}$
Thanks!
|
1.What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this.
2.For how many two-digit numbers, the sum of the digits is greater than the product of the digits?
|
I'm a student and I'm struggling to understand the basics of discrete maths. If such a noob question wont offend you, please help me understand the equation and how to prepare a truth table for this?
$$p \rightarrow (\neg q \lor \neg r) \land \neg p \land q$$
If you use any substitution or such, explain how you derived... |
I have the following problem:
Show that no uncountable subset of $P(\omega)$ is well-ordered by the inclusion relation.
I think they want me to do by embedding it in a separable complete dense linear order (like $\mathbb{R}$), which I have already proved cannot have any uncountable well-ordered subset.
Any hints or ide... |
I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is formal enough, or correct. Any feedback would be appreciated.
Let $\xi$ be an irrational number with continued fraction e... |
Suppose $S^n$ is the n-sphere with basepoint $x$. If based map $f:S^n\rightarrow S^n$ is such that the pull-back $f^{-1}(S^{n}-x)$ is connected do we necessarily have that $deg(f)$ is either -1,0, or 1?
|
How to prove that if you take any 400 consecutive Octobers then exactly 14 % of those years have five Fridays and Sundays?
|
If 72% percent of new apps are reviewed within the last 7 days, then there is a 72% chance that any app is reviewed in the last week.
$$72\% = \sum\limits_{i = 0}^{6} {x\left( {1 - x} \right)^i }$$
where $x$ represents the probability that an app is reviewed each day.
If $p = \left( {1 - x} \right)$, then th... |
Let $(B_t)$ denote the standard Brownian motion on the interval $[0,1]$. For a given confidence level $\alpha \in (0,1)$ a confidence band on $[0,1]$ is any function $u$ with the property that
$$
P(\omega; |B_t(\omega)| < u(t), \quad \forall t\in [0,1])=\alpha.
$$
In other words, the probability that a path of the B... |
Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable
from each other by a rigid motion) parts?
|
I need to find the number of natural numbers between 1 and 1000 that are divisible by 3, 5 or 7.
I know that given a set of numbers, 1 ... n, the number of numbers divisible by d is given by $\lfloor \frac{n}{d} \rfloor$
So if we let $A$ be the set of numbers divisible by 3, $B$ be the set of numbers divisible by 5 and... |
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the quality or power of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conjecture is t... |
In a problem my teacher did the following:
\begin{align*}
5a - 2b &= 3m \\
5b - 2c &= 3n \\
5a - 2c + 3b &= 3m + 3n
\end{align*}
I tried solving for $b$ in the first equation and then plugging it in to the second but could not get the same result.
When is adding full equations like this valid?
Thanks!
|
I want to solve a simple stochastic problem. Imagine there are n agents who want to access a resource, with a probability p at a given time t. What ist the probability that the resource will be free and not accessed by an agent. How does the result change with 2n and 2 available resources?
How do i solve this problem?
|
Is there a relation between the max of a Gaussian random walk of 10
steps vs the max of 10 Gaussian random walks? Specifics (in
Mathematica notation):
(* a Gaussian random walk with standard deviation 1 *)
a[0] := 0
a[n_] := a[n-1] + RandomReal[NormalDistribution[0, 1]]
(* the max of the walk over 10 steps *) ... |
If we have a uniform continuous function $f: X \to Y$, then
$f$ takes a cauchy sequence in $X$ to a cauchy sequence in $Y$.
Now is this statement true: $f$ is uniformly continuous iff, given $\epsilon > 0$, there is an $N > 0$, such that for every $x,y \in I$,where $I$ is an interval ($x \neq y$) we have $$ \Biggl| ... |
http://en.wikipedia.org/wiki/Van_der_Waerden%27s_theorem
The usual approach is to induct on the length of the arithmetic progression, which is difficult to simplify directly to the case of two colors.
Does anyone know a different approach?
Thank you.
|
I'm trying to understand the intuitions behind working with very large numbers. Specifically, I'm talking about numbers of the form $a^b$ where $a > 10,000$ and $b > 10,000$, and in general $a$ and $b$ are small, but $a^b$ has millions of digits. Obviously, computing the number is infeasible, but I still would like t... |
Let $f\in\mathbb{Z}[x]$ be monic and irreducible, let $K=$ splitting field of $f$ over $\mathbb{Q}$. What can we say about the relationship between $disc(f)$ and $\Delta_K$? I seem to remember that one differs from the other by a multiple of a square, but I don't know which is which. On a more philosophical note: why a... |
I know a few proofs over this theorem (where $\mathfrak{c}$ is the cardinality of $[0,1]$ and $\aleph_0$ is the cardinality of $\mathbb{N}$) where they construct two injections and then use Schröder-Bernstein (via the Cantor set or something like that).
Now I was wondering if I could do something like this:
Define $s: ... |
Consider a function $f_1$ defined by $f_1(x)=1-x+o(x)$ and $f_1(2x)=f_1(x)^2 + 0$. It's simple to find that $f_1(x)=e^{-x}$ (for example by writing series near $x=0$).
Consider a function $f_2$ defined by $f_2(x)=2-x^2+o(x^2)$ and $f_2(2x)=f_2(x)^2-2$. It can be proven that $f_2(x)=2 \cos(x)$.
Is there any formula (pro... |
I'm reading through wikipedia about Linear Feedback Shift Registers (Specifically Fibonacci LFSRs) and the only restrictions it mentions about the taps are:
The LFSR will only be maximum-length if the number of taps is even; just 2 or 4 taps can suffice even for extremely long sequences.
The set of taps must be relati... |
Why does the summation
\begin{equation*}
\frac{1}{\cos x}=\sum_{n=1}^\infty \frac{(-1)^n(2n-1)\pi }{x^2-\left (n-\frac{1}{2}\right )^2\pi^2}
\end{equation*}
hold?
|
Let $\wp$ be the Weierstrass function.
From what I understand, $\wp$ maps the torus to $CP^1 \times CP^1$ in the following way:
$a \mapsto (\wp(a),\wp'(a)) = (z,w)$
Furthermore, the image of this map lies on the zero set of the polynomial $P(z,w) = 4(z-e_1)(z-e_2)(z-e_3) - w^2.$
What I don't get is the description of t... |
Given an infinite dimensional manifold modeled on a Banach space, what does it mean for it to have a Riemannian metric? Does it necessarily mean that it is actually a Hilbert manifold?
My understanding is that Hilbert Spaces have inner products, whereas Banach Spaces just have norms. If a Banach manifold has a Riemanni... |
It is known that the number of digits of a natural number $n > 0$, which represent by $d(n)$ is given by:
$d(n)= 1 + \lfloor\log n\rfloor\qquad (\text{I})$
($\log$ indicates $\log$ base $10$)
Well .. the classical approach to the Stirling factorial natural number $n > 1$ is given by:
$$n! \approx f(n) = [(2n\pi) ^{1/... |
Is there a good quick introduction to knot theory? I am relatively mathematically savvy so any level is appreciated.
|
I've just started learning about schemes, so maybe I'm missing something basic.
This is exercise I-24(a):
Take Z = Spec$\mathbb{C}[x]$, let $X$ be the result of identifying the two closed points (x) and (x-1) of |Z|, and let $\phi: Z \to X$ be the natural projection. Let $\mathcal{O}$ be $\phi_* \mathcal{O}_Z$, a she... |
Can we always break an arbitrary field extension $L/K$ into an extension $F/K$ in which the only roots of unity of $F$ are those in $K$, followed by an extension $L/F$ which is of the form $L=F(\{\omega_i\})$ where the $\omega_i$ are roots of unity? What if we restrict to $L/K$ separable, or finite?
|
I find confusing some examples I have seen. Maybe you can help me to determine what is going on with them.
A Generalized Feedback Shift Register (GFSR) sequence defines a sequence $\{W_{i}\}$ satisfying the equation
$$W_{k+p}=c_{0}W_{k}\bigoplus c_{1}W_{k+1}\bigoplus...\bigoplus c_{p-1}W_{k+p-1} \qquad \qquad (1)$$
w... |
There was a problem in Apostol's book namely, to prove that:
$$\sum\limits_{k=1}^{n} \Biggl\lfloor{\frac{k}{2}\Biggr\rfloor} = \Biggl\lfloor{\frac{n^{2}}{4}\Biggr\rfloor}$$
which i could solve. The following probelm is eluding me:
If $a =1,2, \cdots, 7$, prove that there exists an integer $b$ depending on $a$ such tha... |
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler and he gave other proofs.
I believe many of you know some nice proofs of this, can you please share it with us?
|
Suppose $A$ and $B$ are similar matrices. Show that $A$ and $B$ have the same eigenvalues with the same geometric multiplicities.
Similar matrices: Suppose $A$ and $B$ are $n\times n$ matrices over $\mathbb R$ or $\mathbb C$. We say $A$ and $B$ are similar, or that $A$ is similar to $B$, if there exists a matrix $P$ s... |
What is the name for I can only describe as the circumference of a sphere. I mean like the equator on a planet. What is the line called which goes around the entire sphere, it can be anywhere (so not just in the middle of the top and the bottom, if your talking about a planet or something) and the length of it equals 2... |
What requirements should a CRC polynomial of a given degree satisfy to make the CRC catch a maximum of errors?
edit
I'm talking about GF(2) polynomials.
As an example of the kind of requirements I'm looking for: I can imagine (but don't know for sure) that a prime polynomial catches more errors than a composite polyn... |
I hate min and max questions.
I have a function
$$G(x,y)=\frac{1}{x}+\frac{4}{y}+\frac{9}{4-x-y}.$$
I need to prove that $G(x,y)$ has a minimum value of $9$.
I've partially derived them for $x$ and $y$ giving
$$\frac{1}{x^2}+\frac{9}{(4-x-y)^2}=0\tag 1$$
$$\frac{4}{y^2}+\frac{9}{(4-x-y)^2}=0\tag 2.$$
This is where I ge... |
It is a rather surprising fact (to me, at least) that $\displaystyle \binom{14}{4} = 1001$; $\displaystyle \binom{14}{5} = 2002$; $\displaystyle \binom{14}{6} = 3003$.
Actually, this is the only instance where three consecutive binomial coefficients are in the ratio $\displaystyle 1 : 2 : 3$. I found it quite interest... |
Is there any way to decipher, manually of course, whether a (large enough) number is a prime?
|
If we have two Random Variables X and Y, what is the interpretation for the three cases:
a] p(X=x,Y=y) = $p_1$(X=x) * $p_2$(Y=y)
b] p(X=x,Y=y) < $p_1$(X=x) * $p_2$(Y=y)
c] p(X=x,Y=y) > $p_1$(X=x) * $p_2$(Y=y)
where p -> joint pdf of X and Y,
$p_1$ -> marginal pdf of X
$p_2$ -> marginal pdf of Y
|
Given $a > b$, where $a,b ∈ ℝ$, is it always true that $a^2 > b^2$?
|
Could I calculate the answer to either
$$\sum\left[ x^{2} y \right ] - \sum xy$$
or
$$\sum \left[ \left ( x - \sum xy \right )^{2} y \right ]$$
If I had any of these variables
$ \sum xy $; $ \sum x $; $ \sum y $; $ \sum x^2 $; $ \sum y^2 $;
and n ( the amount of items in the list )
adding more info:
I need to calculat... |
I want to count integral of $f(x)=x^2$ in range for example $0$ through $10$.
What should I read to do it by myself and apply it to other funcs ?
|
I would like a family of functions $f_a(x)$ so that $f_a$ is $a\in\mathbb{R}$ fractionally differentiable but not $a+\epsilon$ fractionally differentiable.
Does anyone know such functions which are easily described?
Thanks in advance.
|
While reading a probability paper titled Gröbner bases and factorisation in discrete probability and Bayes, I came across the explanation:
The set of indicator functions $I_i$ form a Boolean algebra that is in one-to-one correspondence with the quotient of the ring $\mathbb{R}[x_1, \ldots, x_n]$ with respect to the re... |
Is there another way to determine monotonicity besides $U_{n+1} - U_n$?
|
Well, this is an exercise problem from Herstein which sounds difficult:
How does one prove that if $|G|>2$, then $G$ has non-trivial automorphism?
The only thing I know which connects a group with its automorphism is the theorem, $$G/Z(G) \cong \mathcal{I}(G)$$ where $\mathcal{I}(G)$ denotes the Inner- Automorphism g... |
Given the following game:
There is a hider and a seeker, each game consists of the hider randomly selecting one of two places to hide that is $A$ with probability $p$ and $B$ with $1 - p$, and the seeker randomly chooses to search in $A$ with probability $q$ and $B$ with $1 - q$. If seeker finds hider then the game end... |
Using $\text{n}^{\text{th}}$ root of unity
$$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$
Prove that
$$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
|
If $G$ is a finite $p$-group with a nontrivial normal subgroup $H$, then the intersection of $H$ and the center of $G$ is not trivial.
|
Simple question-
Is it possible for a linear programming optimization problem possible to have a degenerate optimal solution whereas the dual has a unique optimal solution? I can't find a scenario where this is possible but intuitively it seems that it is possible.
|
I'm sorry if the English isn't correct, I'm not native.
Let's say for example these two:
$$\frac{3n}{2n+3}(-1)^{n+1}-\frac{(-1)^n}{2n}$$
$$((-1)^n+1/n)^2$$
|
In a $365$-day year, Joe, Greg & Dean visit the park multiple times for a walk.
Joe visits every $3$rd day
Greg visits every $5$th day
Dean visits every $7$th day
All three visit the park on the first day of the year
In the year:
On how many days is Greg alone in the park?
On how many days will Dean meet Joe?
On how... |
I'm self-studying field extensions. I ran over an exercise which I can't completely solve. (I haven't yet started studying Galois theory, and I think this exercise isn't meant to be solved using it, just in case):
The problem is:
a) Prove $\sqrt{2}+\sqrt[3]{5}$ is algebraic over $\mathbb{Q}$ of degree 6.
Done: I kn... |
The origin of this question is the identity
$$\sum_{k=0}^n \binom{n}{k} H_k = 2^n \left(H_n - \sum_{k=1}^n \frac{1}{k 2^k}\right),$$
where $H_n$ is the $n$th harmonic number.
Dividing by $2^n$, we have
$$2^{-n} \sum_{k=0}^n \binom{n}{k} H_k = H_n - \sum_{k=1}^n \frac{1}{k 2^k}.$$
The sum on the left can now be interpre... |
I had recently asked the question:
Logistic function passing through two points?
which received a very helpful answer. However, I'd like to ask your help again as the formulation of the problem has slightly changed and i'm lost...
Problem:
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ and $y_l < y_u... |
Please suggest a suitable approach for this problem.
|
I was browsing through Wikipedia for some reason or another, and it says that there is a way to do Galois theory without worrying about whether or not polynomials are irreducible. That's a bit surprising to me.
Is Wikipedia wrong (as is often the case when I read something surprising)? If not, where can I hear more a... |
I'm attempting to implement an equation (for calculating magnetic forces between coils, eqs (22–24) in the linked paper) that requires the use of elliptic integrals.
Unfortunately these equations require the evaluation of the elliptic integrals far outside their standard parameter range of $0\le m\le 1$ and the numeric... |
I was always thinking about this questions when it comes to lottery. lets say you have a number range from 1 - 45. You need 5-6 numbers to win.
Let says you have 2 options
Draw 3 sets of numbers with 10 numbers on each set.
or
1 set of numbers with 19 numbers for that?
Which chance will have higher of winning?
|
Let G=(V,E) be a connex graph. Color it's edges randomly with red/blue.
-prove that there exists an Eulerian circuit, without any two adjacent edges of the same color.. only if for any v vertex of G the number of incident blue edges is equal to the number of red ones.
-if G is complete and x,y,z are three distinct ve... |
pseudocode:
void recursive('k'){ // 'k' and 'i' vertices
sumA = 0;
sumB = 0;
for each non visited 'i' neighbor do{
recursive('i');
sumA = sumA + b['i'];
sumB = sumB + max(a['i'], b['i']);
}
a['k'] = 1 + sumA;
b['k'] = sumB;
}
void main(){
a = b = 0; //initialize tables with 0 (zeros)
... |
Suppose n is an integer. What kind of bounds do we know for how close the closest prime p > n will be?
I'd especially appreciate an answer that pushes me in the right direction of proving a good bound for this without showing the proof fully.
Context: I ran into this while coming up to a solution for https://stackoverf... |
$$\cos A \cdot \cos 2A \cdot \cos 2^{2}A \cdot \cos 2^{3}A ... \cos 2^{n-1}A = \frac { \sin 2^n A}{ 2^n \sin A } $$
I am very much inquisitive to see how this trigonometrical identity can be proved.
PS:I am not much of interested about an inductive proof.
|
I just read this document - Jacobi's Four Square Theorem. It shows how to count the number of representations of a number as the sum of four squares.
I can follow the proof but currently it just appears like a brilliant coincidence, so I am wondering what else can you count with these?
|
What could be the possible way to find the Nth term of following series where the sign toggles after each triangular number?
1 -2 -3 4 5 6 -7 -8 -9 -10 11 12 13 14 15 -16 -17 ....
The series cannot be in a Geometric Progression because there are 4 distinct triangular numbers in the above series.
|
Consider the set $F_n: = [n]^{[n]}$ of all functions $f:[n] \rightarrow [n]$, $[n] = \lbrace 1,2,...,n\rbrace$. It is well known that $|F_n| = n^n$.
Edit: Let two functions $f, g$ in $F_n$ be of the same isomorphism type ($f\sim g$) iff there exists a permutation $\pi$ such that $f\pi = \pi g $
What is the number of ... |
Part1(Solved)
Is it possible to create a function that maps x to y as follows:
x 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
y 0 1 2 3 4 5 4 3 2 1 0 1 2 3 4 5 ...
So far, this is what I have been able to come up with:
|5 sin ((1/10)pix)|
Which will give the following y's for x = 0 to 10:
0.00000
1.5451
2.9389
4... |
How do I solve for $x$ from this equation?
$$ -\frac{1}{x^2} + \frac{9}{(4-x-y)^2} = 0.$$
I need to get this into $x=$"blah"?
|
Claims arrive in an insurance portfolio, according to a homogenous Poisson process $X(t), t\geq 0$. Assume that each of the $12$ months in a year has exactly $30$ days. Calculate the value of the following conditional probability:
\begin{equation*}
P\{X(2) = 3, X(3) = 5 | X(4) = 5\}
\end{equation*}
where $X(k)$ is the ... |
I have a three pairs of points namely : $p_{1,1}, p_{1,2}, p_{2,1}, p_{2,2}, p_{3,1}, p_{3,2}$, I want to find $p$ such that : $|p-p_{i,1}| = |p-p_{i,2}|$ for all $i$.
I know that these represent three planes, but I am not sure whether these planes do meet in a single point. My simple question is even if the three pla... |
Greetings.
$$\sum_{i=x}^{2x} 5 - \sum_{k=2x+1}^{5x+2} 5 = -25$$
How can I discover the value of X?
Thanks in advance!
EDIT: Turns out it was 5x+2, not 5x+1. My bad, apologies for the trouble.
|
$$\sum_{i=1}^{x} x + \sum_{i=1}^{x} 2 = 8$$
Trying to discover $x$ here, so I made:
$$x(x-1) + 2(x-1) = 8$$
Which should give me the $x$, but instead of an integer I get:
$$x = \frac{-1\pm\sqrt{41}}{2}$$
When the answer should be $x=2$, according to the solution.
What am I doing wrong?
|
I need a proof of this well known fact:
If a group $G$ is given as a quotient of a free group $F/R$ then its pro-finite
completion is given by the quotient $\hat{F}/ \bar{R}$ where $\bar{R}$ is
the closure of $R$ in $\hat{F}$. (since $F$ embeds into $\hat{F}$ densely
we can think of $R$ inside $\hat{F}$)
Thanks in ... |
What I'm attempting to do is to rearrange the formula for the sum of a geometric series so as to find the value of its common ratio $r$. I've tried several different methods, all of which have failed; though I can't understand why.
This was my last attempt using logarithms:
$$ S_n=\frac{a(1-r^n)}{1-r}$$
$$S_n(1-r) = a(... |
Find all positive integers n, such that $$\sum\limits_{k=1}^{n}(k+2)^n=(n+3)^n $$
|
The Hoeffding's inequality is $P(S_n - E[S_n] \geq \epsilon) \leq e^{-2\epsilon^2/k'}$, where $S_n = \sum_{i=1}^{n} X_i$, $X_i$'s are independent bounded random variables, and $k'$ depends on the random variables.
In the proof of Hoeffding's inequality, an optimization problem of the form is solved:
$$\min_{s} \ \ e^{-... |
Please imagine that we have a circular segment with some arc length 's' and chord length 'a' (using notation from http://mathworld.wolfram.com/CircularSegment.html):
Provided only 'a' and 's', and placing the left-hand-side point of the chord at the origin of the Euclidean plane (or a more convenient point), is there ... |
Let ${a_k}$, ${b_k}$ be two sequences in $\mathbb{R}$. Suppose ${a_k}$ converges to $a$ s.t. $\liminf_{k \to \infty} {a_k} = \lim_{k \to \infty} {a_k} = a$.
Is it then true that $\liminf_{k \to \infty} (a_k - b_k) = \lim_{k \to \infty} a_k + \liminf_{k \to \infty} -b_k$?
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The series in question is $\sum\limits_{n=2}^{\infty}\frac{(-1)^n n^2}{( \ln(n))^n}$
I think this diverges. So using the divergence test, I am trying to show that limit of the general term of the series is not 0.
By a few algebraic manipulations we get :
$\lim (-1)^n(n^2)( \ln(n))^{-n}$
$\lim (-1)^n(n)/ \ln(n)$
I am... |
Let G be a graph and δ(G) the minimum degree of a peak. Describe an algorithm in pseudocode that, for a given tree T with k<= δ(G) edges, should be build (in polinomial time) a sub-graph H of G so that H is isomorphic with T.
How do I even start ?
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This may sound silly, but... Suppose an aspiring amateur mathematician wanted to plan to move to another city...
What are some cities that are home to some of the largest number of the brightest mathematicians? I'm sure this may depend on university presence, or possibly industry presence, or possibly something surpris... |
Determine all
$ N, |N| \le 9 $ such that $ x^2-94y^2=N $ is solvable in positive integers.
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Is the subset consisting of all integrable (or square integrable) smooth functions of the set of all integrable (or square integrable) functions, dense under the usual Euclidean or integral of absolute difference metric?
By smooth I mean derivatives of all orders exist.
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I define $X_i$ as a random variable that is uniformly distributed between (0,1). What is the expected number of such variables I require to make the sum go just higher than 1.
Thanks
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Consider the sphere $S^2 = \lbrace (x,y,z) :\ x^2 + y^2 + z^2 = 1 \rbrace$. This is a smooth manifold in $\mathbb{R}^3$, and for a given point $s \in S^2$, one can consider its coordinate neighborhood.
There are many ways to put a smooth structure on $S^2$, but all require at least two coordinate neighborhoods. One sim... |
Given a continuous function $g$ on $\mathbb R$. Is it possible to decompose $\mathbb R$ as the union of a countable collection of intervals $I_n=]a_n, a_{n+1}]$ so that $g$ is monotone on each $I_n$?
The same question may be asked for a continuous function defined on a bounded interval.
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I was wondering what the expansion series of the function
$$ f(x) = -\frac{1}{x^3} \cdot \frac{1}{\Gamma(x) \cdot \Gamma(-(\exp(\frac{2}{3}\pi\cdot i))x) \cdot \Gamma(-(\exp(\frac{4}{3}\pi \cdot i))x)} $$ is, at $x = 0$. I'm also interested in the method behind computing the series expansion.
You might think: "why do... |
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