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Suppose we have two inner product spaces $V$ and $W$ with dimensions $n$ and $m$, respectively. For some set $\{ \mathbf{v}_i\}_{i=1}^k \in V$ and some nonzero $\mathbf{w} \in W$, where $1 \leq k \leq n$, we let the subspace $U \subset L(V,W)$ be the set of linear maps $T$ between these inner products spaces such that ... |
> The proposition $p \rightarrow q \rightarrow r$ can be defined in
> multiple ways that make sense:
>
> * $(p \rightarrow q) \rightarrow r$ (left associativity)
> * $p \rightarrow (q \rightarrow r)$ (right associativity)
> * $(p \rightarrow q) \land (q \rightarrow r)$
>
> Which one of these definitions is use... |
Consider the unit square $I^2 = [0,1]^2$ and suppose we have choose $n$ points at random from $I^2$ where the points are taken from the uniform distribution on $I^2$. Call this space $X_{n}$. Can anyone recommend to me any papers that deal with the probability distribution of the lengths of the Delaunay triangulation o... |
Distribution of the lengths of edges of the Delaunay triangulation? |
Consider the polynomial $f(x)=x^{2m}-p^{2n}=0$ over the $p$-adic field $\mathbb Q_p$. Note that $m,n$ are positive integers and $\gcd(m,n)=1$. I want to study the field extension by the zeros of $f(x)$.
We can rewrite $f(x)=(x^m-p^n)(x^m+p^n)=f_1(x) f_2(x)$, where $f_1(x)$ and $f_2(x)$ are Eisenstein polynomials and... |
To get the answer $\frac{7}{10}\cdot\frac{1}{10}$, the new question can be: "What is the probability that the randomly chosen student is an irregular student, who scores an A?".
In this case you would have these calculations
$$P(\text{Grade A and No 100%})=P(\text{No 100%})\cdot P(\text{Grade A}|\text{No 100%})=\fr... |
I've gone through many math courses thinking that a function $f(x)=x^2$ could mean $f(anything) = anything^2$, however I realized that that isn't always true.
For example, suppose we have a function $f(x) = kx$ where $f$ represents the opposing force. We can't just substitute $t$ for time into that equation, as tha... |
> The proposition $p \rightarrow q \rightarrow r$ can be defined in
> multiple ways that make sense:
>
> * $(p \rightarrow q) \rightarrow r$ (left associativity)
> * $p \rightarrow (q \rightarrow r)$ (right associativity)
> * $(p \rightarrow q) \land (q \rightarrow r)$
>
> Which one of these definitions is use... |
Problem 9(i) of Spivak's *Calculus* (Chpt 22) asks for the following limit:
>$\displaystyle \lim_{n \to \infty}\frac{\sqrt[n]{e^1}+\sqrt[n]{e^2}+\cdots\sqrt[n]{e^n}}{n}$
The terse solution manual's argument reads as follows:
>$\displaystyle \int_0^1 e^x dx = e-1$
**Note: My book has not yet covered series**... |
Let $V:=\operatorname{Spec}A$ be an affine open subscheme of a scheme $Y$ with $x=\mathfrak{q}_x\in V$. Consider next set
$$\{ Y_0 \cap V : Y_0 \operatorname{is an irreducible component of} Y \operatorname{containing} x=\mathfrak{q} \} .$$
**Then it is the set of irreducible components of $V:=\operatorname... |
For an affine open subscheme $\operatorname{Spec}A$, $\{ \operatorname{ker}(A_q \to (A_0)_q )\} = \{ \operatorname{minimal prime ideals of} A_q \}$? |
we define
$ l_p=\{x=(x_n) : \sum_{i=1}^\infty |x_n|^p < \infty\} (p\in [1,\infty) ) $
I tried to prove that
$$\left\|x\right\|=\left(\sum_{i=1}^{n}\left|x_i\right|^p\right)$$ is a p norm.
I tried 1. and 2. conditon but I can't prove triangle inequality for $p$-norm.
I tried to use minkowski inquality but I can... |
For reflexivity, it is easy to see.
To verify whether it has symmetry, take arbitrary $x,y$ s.t. $(x,y) \in R$, then we have
$$x^2−4xy+3y^2=0$$
Can we derive $y^2 - 4xy + 3x^2 = 0$ from it? No, in general it is not true. So it doesn't have symmetry.
To verify whether it has transitivity, take $x,y, z$ s.t. ... |
Problem 9(i) of Spivak's *Calculus* (Chpt 22) asks for the following limit:
>$\displaystyle \lim_{n \to \infty}\frac{\sqrt[n]{e^1}+\sqrt[n]{e^2}+\cdots\sqrt[n]{e^n}}{n}$
The terse solution manual's argument reads as follows:
>$\displaystyle \int_0^1 e^x dx = e-1$
**Note: My book has not yet covered infinite... |
Using Newton's Method, solve the following equation:
$\displaystyle 0 = \frac{1}{ 2}+ \frac{1}{4}x^2- x\sin(x)- \frac{1}{2}\cos(2x) $
iterate until you achieve an accuracy of $\displaystyle 10^{−5}$ with:
$ \displaystyle p_0 = \frac{\pi}{3}
\\
p_0 = 3\pi$
How do I do the calculations to get the precisi... |
In the maximum modulus therem we have two sets
$E_1=\{z \in E:|f(z)|<M\} \quad E_2=\{z \in E:|f(z)|=M\}$ I know that the set $E_1$ is open because pre image of an open set. But should't be also $E_2$ closed because pre image of only a point ? ma por** di quella **** ma he cazzo ve ce vo a risponde a na ca***** de cr*... |
In the maximum modulus therem we have two sets
$E_1=\{z \in E:|f(z)|<M\} \quad E_2=\{z \in E:|f(z)|=M\}$ I know that the set $E_1$ is open because pre image of an open set. But should't be also $E_2$ closed because pre image of only a point ? |
Prove that
$$4(a+b+c)\ge 3(abc+3); \forall a,b,c\ge 0: a^2+b^2+c^2=a^2b^2+b^2c^2+c^2a^2.$$
I've tried $pqr$ method.
Let $p=a+b+c;q=ab+bc+ca;r=abc.$ The condition gives $$r=\frac{q^2+2q-p^2}{2p}$$
Now, we rewrite the inequality as $$4p\ge 9+3\cdot \frac{q^2+2q-p^2}{2p}$$which is reduced to $$11p^2\ge 18p+3q^2+6... |
I am having difficulty understanding a particular solution in a research paper I am reading.
"Analysis of nonlinear duopoly game with
heterogeneous players" I would appreciate your help in clarifying the steps involved.
Problem:
[enter image description here][1]
On page 142 of the paper, there is an inequality... |
What is the probability of 9 people going into 3 rooms with 4,3,2 people respectively if the rooms can and cannot be differentiated?
I would like to verify my thinking logic on this question.
For the case where the room can be differentiated, I take the sample space to be $3^{9}$, which is the number of ways the ... |
For Brownian Motions, I am trying to prove that:
- For large $n$: The expected total variation is infinite and the expected quadratic variation is finite
- For large $n$ : The total variation equals to the expected total variation and quadratic variation is equal to the expected quadratic variation
**We start ... |
One unbiased die is thrown 10 times. For $1 \leq i \leq 6$, let $X_i$ denote the number of times the number $i$ appears.
Find the conditional expectation of $X_1$ given $X_6 = 5$.
My approach to solve this problem is as follows:
I first of all compute the conditional probability mass function.
$P(X_1=i , X_... |
The integral is from WolframAlpha's ad, and I got curious to what the answer actually is.
I tried using the square-square technique, but that leads me to this:
$$\int_{0}^{\infty}\frac{\sqrt{x^2+x\sqrt{x^2+\frac{1}{x^2}}}}{(x^2+x)\sqrt{x^2+\frac{1}{x^2}}}dx$$
I don't even know how to go about this integral in genera... |
Closed form for $\int_{0}^{\infty}\frac{\sqrt{x^2+\sqrt{x^4+1}}}{(x+1)\sqrt{x^4+1}}dx$??? |
I am looking at a sequence where $a_1>0$ and
$$ a_{n+1}=a_n^{a_n}.$$
I want to figure out for what values of $a_1$ does this sequence converge. My guess is that it converges of $a_1\leq 1$ and diverges for $a_1>1$. I was able to show convergence for $a_1\leq 1$ since if $a_1=1$, then it is trivial, otherwise, we can ... |
we define
$ l_p=\{x=(x_n) : \sum_{i=1}^\infty |x_n|^p < \infty\} (p\in [1,\infty) ) $
I tried to prove that
$$\left\|x\right\|=\left(\sum_{i=1}^\infty\left|x_i\right|^p\right)$$ is a p norm.
I tried 1. and 2. conditon but I can't prove triangle inequality for $p$-norm.
I tried to use minkowski inquality but I ... |
How to prove triangle inequality for this p- norm? |
Consider this
Dice A has the numbers 1, 1, 2, 2, 2, 4. We find the expected value to be $E(X)=2$ and the $Var(X)=1$.
Dice B has the numbers 2, 2, 4, 4, 4, 8. As one would expect, $E(2X)=4$ and the $Var(X)=4$. Slightly unintuitively the variance increases by the square of the factor. However, my intuition for t... |
Difference between random variable $2X$ and the sum of two independent observations of $X$? |
***Background:***
-----------------
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$2, 3, 5, 7, 11, 23, 29, 41, 43, 47, 61, 67, 83, 89$$
Here the proportion of $P$s ... |
Let's break this down into an easier problem to see how to approach this. Imagine we have a d3 and roll it 2 times. What is the expected value of the number of times of each number? Well it's just simply $\frac{2}{3}$. Now, conditioned on there being exactly 2 rolls that are 3s, we know that the roll unaccounted for is... |
The question is :
> How many 7 digits number can be formed by using 1,2,3,4,5,6,7,8,9 such that it is divisible by 3? **Repetition is not allowed**.
I have done a similar question which was something like this :
>How many 7 digits number can be formed by using 1,2,3,4,5,6,7,8,9 such that it is divisible by 3? **Re... |
How do we do this question? |
***Background:***
-----------------
*I recently got an idea. My idea was analyzing primes whose digit sum is prime.*
This is a property of some primes ($> 1$ digit):
**The sum of its digits is prime.**
I will call a special prime like this $P$.
For primes in the range $1<p<100$, these are all $P$:
$$... |
The question is:
> How many $7$-digits number can be formed by using $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that it is divisible by $3$? **Repetition is not allowed**.
I have done a similar question which was something like this :
>How many $7$-digits number can be formed by using $1$, $2$, $3$, $4$, $5$... |
How many $7$-digits number can be formed by using $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ such that it is divisible by $3$? Repetition not allowed |
I am having difficulty understanding a particular solution in a research paper I am reading.
Title: "Analysis of nonlinear duopoly game with
heterogeneous players"(2006)
Authors: Jixiang Zhang, Qingli Da, Yanhua Wang
I would appreciate your help in clarifying the steps involved.
Problem:
On page 142 of th... |
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of spaces over $BG$.
Is there some category with the notion of homotopy $\mathscr X(A)$ one may assign to a homotopy commu... |
As per [Wikipedia][1], the discriminant of a polynomial $f(x)=a_nx^n+$(lower degree terms) is defined in terms of roots of $f(x)$ as
$$
D(f)=a_n^{2n-2}\prod_{i<j} (\alpha_i-\alpha_j)^2.
$$
My question is simple, but I am not able to find answer: why the coefficient $a_n$ is taken with power $2n-2$ in the definition... |
I am trying to prove that the category of modules is cocomplete. It suffices to show that it has all coequalizers and coproducts. It's relatively easy to show that all coproducts exist, and I am left to prove that all coequalizers exist in the category of modules. Thus, let $f, g : M \to N$ be $R$-module homomorphisms ... |
As per [Wikipedia][1], the discriminant of a polynomial $f(x)=a_nx^n+$(lower degree terms) is defined in terms of roots of $f(x)$ as
$$
D(f)=a_n^{2n-2}\prod_{i<j} (\alpha_i-\alpha_j)^2.
$$
My question is simple, but I am not able to find answer: why the coefficient $a_n$ is taken with power $2n-2$ in the definition... |
For example, I am currently reading a sentence that writes:
"let $f$ be a continuously differentiable function over a closed and convex set $C$."
Sentence of this type appears everywhere in analyis, optimization, etc.
So let's imagine $f(x) = x^2$ over a closed set $C$ that is symmetric around $x = 0$.
Then... |
Is the statement "$f$ is continuously differentiable over a closed set" mathematically incorrect? |
Suppose $X$ is a smooth quasi-affine surface over $\mathbb{C}$. Then is it true that there is a smooth affine surface $\bar{X}$ along with an open immersion $X \to \bar{X}$. \
By quasi-affine we mean $X$ is an open subscheme of an affine scheme. For example, if $X$ is an open subscheme of an affine scheme so that the... |
Consider [Hardy field][1] $H$, the field of germs of functions at positive infinity.
Can we define $H_I\subset H$, such that it would have the following properties:
* $H_I$ is an [integral domain][2].
* For each germ $u\in H$ there exists and only one infinitesimally close to it germ $w\in H_I$
* All germs ... |
Is it possible to consistently and naturally define this subset of Hardy field? |
Dot product vs Matrix multiplication, is the later a special case of the first? |
I'm pretty sure that we can trace a conic section like a circle exactly using a NURBS curve. The first quadrant of a unit circle, for example, can be traced by:
$C(t)=\frac{(1-t)^{2}P_{0}+2t(1-t)P_{1}+2t^{2}P_{2}}{(1-t)^{2}+2t(1-t)+2t^{2}}$
where
$P_{0}=\left(0,1\right)$,
$P_{1}=\left(1,1\right)$, and
$P_{1}... |
I am currently learning contour integration and I noticed that certain integrals are not mentioned at all when discussing contour integration and some of those are the following :
$$\int_{0}^{\infty} {e^{-x}\ln(x)}\,dx = -\gamma$$
$$\int_{0}^{\infty} {\frac{\sin(x)\ln(x)}{x}}\,dx = -\frac{\gamma\pi}{2}$$
$$\mathcal ... |
Can contour integration produce mascheroni and Catalan's constant? |
> Is $\text{BRANCH}(n)$ finite for $n > 2$? Define $\text{BRANCH}(n)$ as
> the maximum length of a string that is composed of at most $n$ unique
> characters AND meets the following condition:
>
> Define a substring as all strings from positions $i$ to $2i$
> (inclusive) (where $i= 1, 2, 3...$). Then, a certain s... |
Is $\text{BRANCH}(n)$ finite for $n > 2$? |
Suppose $X$ is a smooth quasi-affine surface over $\mathbb{C}$. Then is it true that there is a smooth affine surface $\bar{X}$ along with an open immersion $X \to \bar{X}$. \
By quasi-affine we mean $X$ is an open subscheme of an affine scheme. For example, if $X$ is an open subscheme of an affine scheme so that the... |
For a topological group $G$, assigning to a $G$-space $X$ the (canonical) map $EG\times_GX\to BG$ establishes an equivalence between the homotopy category of $G$-spaces and the homotopy category of spaces over $BG$.
Is there some category with the notion of homotopy $\mathscr X(A)$ one may assign to a homotopy commu... |
How many even numbers of three different digits less than $500$ can be formed from the integers $1, 2, 3, 4, 5, 7$ and $8$? |
I came across this summation while solving harmonic chain.
\begin{align}
\sum_{l=1}^N \sin^2{(\frac{m\pi}{N+1}\cdot l)}
\end{align}
for an integer m that can range from 1 to N.
So the question is if the value of the summation changes with m? I checked for until N=5. And it remains the same for all integer... |
Can contour integration produce mascheroni or Catalan's constant? |
Consider the diagonal operator $A\{x_n\}$ = $\{a_nx_n\}$. The operator A is defined on the sequence space $\ell_2$. It must be proved that this operator has a cyclic vector if and only if all $\{a_n\}$ are different.
I tried to consider a situation where there are indexes $m$ and $k$, that $a_m=a_k, m \neq k$. Then ... |
The definition of the outer measure of a set $A\subseteq\mathbb{R}$ is as follows:
$$
|A| = \inf \{ \Sigma_{k=1}^{\infty}\ \mathscr{l}(I_k): I_1, I_2,\dots\text{ are open intervals such that }A\subseteq\bigcup_{k=1}^{\infty} I_k \}.
$$
My quick observation is that if two open intervals intersect, we can join th... |
Prove that
$$4(a+b+c)\ge 3(abc+3); \forall a,b,c> 0: a^2+b^2+c^2=a^2b^2+b^2c^2+c^2a^2.$$
I've tried $pqr$ method.
Let $p=a+b+c;q=ab+bc+ca;r=abc.$ The condition gives $$r=\frac{q^2+2q-p^2}{2p}$$
Now, we rewrite the inequality as $$4p\ge 9+3\cdot \frac{q^2+2q-p^2}{2p}$$which is reduced to $$11p^2\ge 18p+3q^2+6q ... |
In a 1966 paper by the physicist Kubo (https://iopscience.iop.org/article/10.1088/0034-4885/29/1/306), I found the following problem: The author considers an OU process of the form
$$m\frac{dv}{dt}=-m\gamma v +R(t)$$
Where the second term is a gaussian stochastic process. He then argues that the probability distr... |
I am trying to help my daughter with this question:
A piece of wire 12 cm long is cut into 2 pieces. One piece is used to form a square and the other a rectangular shape in which the length is twice its width.
a. If x cm is the side length of the square, write down the dimensions in terms of x.
b. Form... |
**NOTE**: I do not have any formal training in Logic, at any level apart from intro to proofs. Without that, I had jumped into my mathematical education.
**Question**: How to correctly write statements like **"$P(n)$ is true for utmost finite $n \in \mathbb{N}$ "**
In particular, I am working with Lim Sup and Lim... |
After reading the [post](https://math.stackexchange.com/q/2076791/732917) with the result that
$$\int_0^{\frac\pi2} {\sin^2{x} \ln{(\tan x)} \,dx}=\frac{\pi}{4}$$
I was attracted by its decency and wanted to generalise it as
$$
I_n=\int_0^{\frac{\pi}{2}} \sin ^{2 n} x \ln (\tan x) d x \stackrel{x\mapsto\frac \pi 2-... |
Consider this
Dice A has the numbers 1, 1, 2, 2, 2, 4. We find the expected value to be $E(X)=2$ and the $Var(X)=1$.
Dice B has the numbers 2, 2, 4, 4, 4, 8. As one would expect, $E(2X)=4$ and the $Var(X)=4$. Slightly unintuitively the variance increases by the square of the factor. However, my intuition for t... |
In a 1966 paper by the physicist Kubo (https://iopscience.iop.org/article/10.1088/0034-4885/29/1/306), I found the following problem: The author considers an OU process of the form
$$m\frac{dv}{dt}=-m\gamma v +R(t)$$
Where the second term is a gaussian stochastic process. He then argues that the probability distr... |
Consider this
Dice A has the numbers 1, 1, 2, 2, 2, 4. We find the expected value to be $E(X)=2$ and the $Var(X)=1$.
Dice B has the numbers 2, 2, 4, 4, 4, 8. As one would expect, $E(2X)=4$ and the $Var(X)=4$. Slightly unintuitively the variance increases by the square of the factor. However, my intuition for t... |
Is this method of solving finite nested square roots of 2 via Gray code correct? |
I've been struggling for days now and I cannot come up with an example of a function $f(x)$ that satisfies the following:
1) $|f(x)−f(y)| < |x−y|$ for any $x, y ∈ R$
AND
2) equation $f(x) = x$ does not have any solutions.
Do you have a grasp as to how construct such examples? Thank you!
|
Alternative method of evaluating $\int_0^{\frac{\pi}{2}} \sin ^{2 n} x \ln (\tan x) d x $? |
This question is from the paper 'A new proof of Spitzer's result on the winding of two dimensional Brownian motion' by R Durrett, 1982.
Let $D_t = A_t + iB_t$ be a complex Brownian motion. Then $\int \vert e^{iD_s} \vert ^2 ds = \int e^{2A_s} ds$.
But why? I think by Eulers equation should follow: $\vert e^{iA_s... |
> Is $\text{BRANCH}(n)$ finite for $n > 2$? Define $\text{BRANCH}(n)$ as
> the maximum length of a string that is composed of at most $n$ unique
> characters AND meets the following condition:
>
> Define a substring as all letters from positions $i$ to $2i$
> (inclusive) (where $i= 1, 2, 3...$). Then, a certain s... |
Let $f,g:I \to \mathbb R$ be positive functions on a finite set, then
$$
\left(\sum_i f(i)\right)\left(\sum_i g(i)\right) \geq \sum_i f(i)g(i)
$$
The proof is rather trivial: by multiplying out the sums, you can write the LHS as $\sum_i f(i)g(i) +$ crossterms, which are positive by assumption.
But when $f,g$ are (... |
Does this inequality for sums hold for integrals too? |
I am reading a proof of the irrationality of $e^2$ and I am stuck on the following inequality:<br>
Let $S := -a\underbrace{\left(\frac{1}{n+1} - \frac{1}{(n+1)(n+2)} + \frac{1}{(n+1)(n+2)(n+3)} \mp ...\right)}_{S^*}$ (just because of space issues) with $a \in \mathbb{Z}$, $n \in \mathbb{N}$. The proof I am reading sta... |
https://math.stackexchange.com/questions/3448271/if-generators-of-sigma-algebra-independent-then-sigma-algebras-are-indepe provides a proof that
Let $(\Omega, \mathcal{A}, P)$ be a probability space and $\mathcal{E}_i\subset \mathcal{A},\ \forall i\in I$. If $(\mathcal{E}_i \cup \{\emptyset\})$ is $\cap$-stable, the... |
Does independence of generators of σ -algebras imply the independence of σ -algebras? Without assuming that the generators are ∩-stable? |
I recall what this theorem says:
Let M be a Riemannian manifold upon which a finite group G acts by isometries; let H and K be subgroups of G that act freely. Suppose that
H and K are almost conjugate i.e., there is a bijection f : H → K carrying every
element h of H to an element f (h) of K that is conjugate in G t... |
For me you proof is very confusing and it is wrong. It is wrong because the equation $|f(x)-f(2)| < \frac{\delta}{\sqrt{3}+\sqrt{2}}$ does not hold. $|f(2.1)-f(2)|\approx 0.034$, but $0.1/(\sqrt(2)+\sqrt(3)\approx 0.032$.
But now to the confusion. You state
> Givens: $\varepsilon > 0$ and $\delta > 0$ and $|x-2|... |
Let $(X,\tau)$ be a topological space.
Does anyone know the name of the following somewhat unusual property of the topological space $(X,\tau)$? (It makes me think of the $T_0$ separation axiom.)
For each $x\in X$ and for each finite subset $F$ of $X$ such that $x\not\in F$, there exists a neighborhood $U_{x,F}... |
On page 3 of Arthur Besse’s book “Einstein Manifolds”, he writes:
> An easy argument shows that the Ricci quadratic form $r:UM \rightarrow \mathbb R$ [where $UM$ is the unit tangent bundle of a Riemannian manifold $(M,g)$] is a constant $\lambda$ if and only if $r=\lambda g$.
This function $r$, the “Ricci quadrat... |
> Show that when $|x|<1$, $\int_{0}^{x}\frac{1}{(1+t^4)}dx = x - \frac{x^5}{5} + \frac{x^9}
{9} - \frac{x^{13}}{13}$ .....
The first step of the solution given in my textbook is
$$\frac{1}{1+t^4}=1 - t^4 + t^8 - t^{12}+......+(-1)^{n-1}t^{4n-4}+\frac{(-1)^nt^{4n}}{1+t^4}$$
I don't understand this equation. T... |
What's the expansion of $\frac{1}{1+t^4}$ with a last term? |
The Local Education Authority here in Wales are running a free voluntary course online for maths dunces like me to help us to help our 7-14 year olds with maths.
There's a free optional paper leading to a meaningless qualification, so I thought "why not", and starting to take a look at it... Not going to post the ac... |
Are these test paper questions as unclear and ambiguous as I think they are? |
Do the series $\sum^\infty_{k=0} \frac{e^{3k\mathrm{log}(k)}}{(k!)^2}$ and $\sum^\infty_{k=1} \frac{(k+1)^{k^2}}{k^{k^2}4^k}$ converge/diverge? |
I just came to know that it is possible to define a _generalized base_ as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\in \mathbb N$ can be uniquely expressed as
$$m=\overline{x_n\dots x_1}^\mathbf b := x_1 + x_2\cdot b_1 + x_3\cdot b_1b... |
While expanding log (1+x) using maclaurin series why
The condition (|x| < 1) is applied and domain discussed is restricted
i understand that for logx , x needs to be positive therefore here 1+x>0 ie x>-1 but why x needs to be less than 1?
|
**"Direct" problem** - Assume a non-linear relation between the random variables $\mathbf{Y} = f(\mathbf{X})$, where the RV $\mathbf{Y}\sim p_Y$ takes values $\mathbf{y} \in \mathbb{R}^M$ and the RV $\mathbf{X}\sim p_X$ takes values $\mathbf{x} \in \mathbb{R}^N$, with $M\leq N$.
Formally, if we know the PDF $p_X$, we... |
problem: $\frac{\partial w}{\partial t}=a_1(x,t) \frac{\partial^2 w}{\partial x^2}+\Phi(x, y, z, t)$
with
$w=f(x, y, z) \quad$ at $\quad t=0$.(initial condition)
$\frac{\partial w}{\partial x}+k(\mathbf{x}, t) w=g_3(\mathbf{x}, t) \quad$ for $\quad x=0$ (Third boundary )
I found the following content in t... |
Is there an analytical solution for the three-dimensional time-dependent parabolic equation (using Green's functions)? |
The Local Education Authority here in Wales are running a free voluntary course online for maths dunces like me to help us to help our 7-14 year olds with maths.
There's a free optional paper leading to a meaningless qualification, so I thought "why not", and starting to take a look at it... Not going to post the ac... |
Suppose $A$ and $u$ are given, can we exactly identify the smallest eigenvalue of $A+uu^T$ and its corresponding eigenvector (based on $u$ and given eigenpairs of $A$)? It is okay to add any meaningful conditions on $A$ in this question. |
problem: $\frac{\partial w}{\partial t}=a_1(x,t) \frac{\partial^2 w}{\partial x^2}+\Phi(x, y, z, t)$
with
$w=f(x, y, z) \quad$ at $\quad t=0$.(initial condition)
$\frac{\partial w}{\partial x}+k(\mathbf{x}, t) w=g_3(\mathbf{x}, t) \quad$ for $\quad x=0$ (Third boundary )
I found the following content in t... |
Let bp(G) be the number of complete bipartite graphs needed to partition the edge set of G. This means, $ \forall e \in E(G) $, e is in exactly 1 complete bipartite graph. Now, how to prove bp(G) $ \geq log_{2}(\chi(G)) $ ?
Some things I thought of is that we can consider $ V_{i} $ the independence sets of G that fo... |
This is how I think about this:
> In question 1b, we do not know the intersection of the number of
> people who did not want both vegetables AND sauce, so how can we
> calculate the total number that DID want both? I asked the online
> tutor, who hadn't noticed this before, and he said "Paper can't be
> changed,... |
Suppose, we have N distinct urns each having some number of balls $\ge$ 1. The distribution of balls in the urns follows a truncated Poisson distribution with given $\lambda$. If we take out all the balls from all the urns (Say, there are $N_1$ such balls) and randomly select $n_1$ balls from them - it turns out those ... |
I have been considering the question of probabilities conditioned on inequalities. My intuition says that I should end up with a double integral, but I'm worried I've missed something in my solution.
Notation: $F_X(x)=\mathbb{P}(X\le x)$ denotes the CDF of RV X and p(x) the corresponding pdf.
I know the joint dis... |
I arrived at the answer CP = 12/5 units.
Since ∆APB — ∆DPC. So there ratio is
**BP/CP = AB/DC**
2/CP = 5/6
1/CP = 5/12
**CP = 12/5**
Is this correct way to solve? |
While expanding log (1+x) using maclaurin series why
The condition (|x| < 1) is applied and domain discussed is restricted i.e how to find the radius of convergence for this , i am new to this topic
i understand that for logx , x needs to be positive therefore here 1+x>0 ie x>-1 but why x needs to be less than 1?
... |
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