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H: Is $\mathbb{Q} [X,Y]/[x^{20},y^{20}]$ a local ring?
Is $\mathbb{Q}[X,Y]/[x^{20},y^{20}]$ is a local ring?
My approach is to look for a maximal ideal, but got stuck how to find a single maximal ideal. Any feedback would be appreciated.
AI: Well, $\Bbb Q[X,Y]/\langle X^{20},Y^{20}\rangle$ is a local ring.
The idea... |
H: Series $\sum_0^{\infty} a_n$ and $\sum_0^{\infty} b_n$ are convergent series then so is $\sum_0^{\infty} c_k$ where $c_k= \sum_{l=0}^{k}a_lb_{k-l}$
Series $\sum\limits_0^{\infty} a_n$ and $\sum\limits_0^{\infty} b_n$ are convergent series then so is $\sum\limits_0^{\infty} c_k$ where $c_k= \sum\limits_{l=0}^{k}a_l... |
H: best approximation of $\sqrt[3]{8.1}$ using cubic function
Approximate value of $\sqrt[3]{8.1}$ using linear approximation of $g(x)=(x+1)^{\frac{1}{3}}$
What i try:
Linear approximation of $f(x)$ at $=a$ is given by
$$L(x)=g(a)+g'(a)(x-a)$$
$$L(x)=\sqrt[3]{a+1}+\sqrt[3]{(a+1)^{-2}}(x-a)$$
I did not understand w... |
H: Probability problem socks
A drawer has 7 different pairs of socks. Each sock is taken randomly and without replacement until the first pair is found. What is the probability of having the first pair until the 8th extraction?
My answer is $\frac{2^7}{\binom{14}{7}}=0.037$, is my answer right?
AI: The probability is... |
H: Is it true that for any natural number n, that nth order Taylor polynomial of any polynomial at $\vec0$ is just the polynomial itself?
Is it true that for any natural number $n$, that $n$-th order Taylor polynomial of any polynomial at $\vec0$ is just itself ? (If $n\ge$degree of the polynomial)
For example if $f(x... |
H: Number of $h$ hop paths between two vertices with shortest path $s$ on an $n$ sided polygon.
We have a regular polygon with $n$ vertices. We start at one vertex and want to get to another. The least number of hops we need to get from the starting to target vertex is $s$. How many ways are there to go from the start... |
H: Unable to explain flow of steps in this basic modular expression?
Consider the expresion $(13 + 11)· 18 (\mod 7)$:
$(13+11)· 18 ≡ (6+4)· 4 (\mod 7)$ Note the transition from $(13+11)·
> 18$$\implies$ $(6+4)· 4$
$≡ 10 · 4 (\mod 7)$
$≡ 3 · 4 (\mod 7)$ Note the transition from $10 · 4$$\implies$ $3 · 4$
$≡ 12 (\mod ... |
H: find $\sum_{n=0}^{\infty}n^2 x^n$
I'm trying to find $\sum_{n=0}^{\infty}n^2 x^n$
What I tried doing is:
$\sum_{n=0}^{\infty}n^2 x^n$ = $\sum_{n=0}^{\infty}n(n-1) x^n + \sum_{n=0}^{\infty}n x^n$
= $x^2\sum_{n=0}^{\infty}n(n-1) x^{n-2}+x \sum_{n=0}^{\infty}n x^{n-1} =x^2(\frac{1}{1-x})'' + x(\frac{1}{1-x})'$
But an... |
H: Calculate $\iiiint_{x^2+y^2+u^2+v^2\leq 1}e^{x^2+y^2-u^2-v^2}\,dx\,dy\,du\,dv$
$$\iiiint_{x^2+y^2+u^2+v^2\leq 1}e^{x^2+y^2-u^2-v^2}\,dx\,dy\,du\,dv$$
So we just learned substitution and i thought maybe for this integral doing 2 polar subs for x,y and for u,v but i'm not sure this is the right way for this.
Any hin... |
H: how many ways to arrange $3$ people to a circular table which has $6$ seats?
As the title, how can I solve this problem?
Should I use combination with repitition or something else different?
Any help would be appreciated
3 people are distinguish. Below here is an example of symmetry situation which only consider a... |
H: Find $\int(\frac{\sin{x}+\cos{x}}{x^2}+\frac{\cos{x}-\sin{x}}{x})dx$
Problem: Find
$$\int\left(\frac{\sin{x}+\cos{x}}{x^2}+\frac{\cos{x}-\sin{x}}{x}\right)dx$$
The possible methods I thought were:
Substitute $t=\tan{\left({x\over2}\right)}$: failed because of polynomials in the denominators
Put $t={\pi\over2}-x$: ... |
H: Why $\lim\limits_{n\to\infty}\Big\{\frac{|X_n-a|}{1+|X_n-a|}\Big\}=0$?
Consider the following theorem regarding convergence in distribution
THEOREM 1: Let $(X_n)_{n\geq1}$, $X$ be $\mathbb{R}^d$-valued random variables. Then $X_n$ converges to $X$ in distribution if and only if $\lim\limits_{n\to\infty}E\{f(X_n)\}... |
H: Ideal $I=\langle x^2+1 \rangle$ in $R=C[0,1]$
Let $R$ be the ring all continous functions from $[0,1]$ to $\mathbb{R}$. Let $I$ be the ideal generated by $f(x)$ where $f(x)=x^2+1$ .Then
$(a) I$ is maximal ideal of $R$
$(b) I$ is prime ideal of $R$.
$(c) \frac RI $ is finite
$(d)char\frac RI$ is a prime number.
My... |
H: Dieudonné: Treatise on Analysis, chapter 16, differentiable manifolds: Common principle used for proofs
In studying Dieudonné's Treatise on Analysis, vol. 3, chapter 16, differentialbe manifolds, I stumbled over a principle which he uses frequently for his proofs. I encounterd it for the first time in the proof of ... |
H: Calculating matrix exponential for $n \times n$ Jordan block
I want to calculate exponential of the matrix which on diagonal has some $a \in \mathbb{R}$ and ones above. The $n\times n$ matrix looks like following
$$ A = \left( \begin{matrix} a & 1 & 0 & 0 & \cdots & 0 \\ 0 & a & 1 & 0 & \cdots & 0 \\ 0 & 0 & a & 1... |
H: Integral of $\int^{\infty}_0 \frac{x^n}{x^s+1}dx$
$$R(s;n)= \int^{\infty}_0 \frac{x^n}{x^s+1}dx$$
From a previously asked question, I know:
$$R(s;0)=\frac{1}{s} \varGamma\left(\frac{1}{s}\right) \varGamma\left(1-\frac{1}{s}\right)$$
The obvious approach is to do integration by parts but I did not manage to find it ... |
H: Calculate $\iint\frac{dxdy}{(1+x^2+y^2)^2}$ over a triangle
Calculate
$$\iint\frac{dxdy}{(1+x^2+y^2)^2}$$ over the triangle $(0,0)$, $(2,0)$, $(1,\sqrt{3})$.
So I tried changing to polar coordinates and I know that the angle is between $0$ and $\frac{\pi}{3}$ but I couldn't figure how to set the radius becaus... |
H: Integral $\int \frac{x\,dx}{(a-bx^2)^2} $
How can I integrate
$$
\int \frac{x\,dx}{(a-bx^2)^2}
$$ I've tried to use partial fraction decomposition, but I'm getting six equations for four variables, and they don't give uniform answers.
AI: Substitute $u=a-bx^2$ so your integral is $-\frac{1}{2b}\int\frac{du}{u^2}=\f... |
H: Find $E(X_1X_2 \mid X_{(1)})$ where $X_i$'s are i.i.d Exponential. Is my solution correct?
Let $X_1, ... , X_n$, ($n\ge 2$) be a sample from exponential distribution with parameter $\lambda=1$ and $X_{(1)}={\rm{min}}(X_1,...,X_n)$. Then we need to find the conditional expectation $E(X_1X_2\mid X_{(1)})$.
Is this c... |
H: Find the convergence radius for $\sum_{n=1}^{\infty}\frac{2^{n-1}x^{2n-1}}{(4n-3)^2}$
Find the convergence radius for $\sum_{n=1}^{\infty}\frac{2^{n-1}x^{2n-1}}{(4n-3)^2}$
I'm trying to understand why this solution is wrong:
We can look at this as a power series with $a_n = \begin{cases}
& \frac{2^{n-1}}{(4n-3)^... |
H: How do I obtain the following upper bound for the number of heads for coin tosses?
I have $n$ coins, each with probability
$p$ of being head and coin flips are independent of each other, the probability of at least $n/2$ heads is at most
${n\choose n/2} p^{n/2}$. Why is this so?
In other words, I am asking the proo... |
H: Atiyah-Macdonald Q1.27 Ring of Polynomial Equations on X.
I do not understand his last statement where he mentions that $P(X)$ is the ring of polynomial functions on $X$. What does he mean by that? Is it the ring of polynomials with coefficients in $X$? If so how is $X$ a ring in the first place.
AI: To avoid con... |
H: Compositeness test using $S_i=6S_{i-1}-11S_{i-2}+6S_{i-3}$ recurrence relation
Can you prove or disprove the following claim:
Let $S_i=6S_{i-1}-11S_{i-2}+6S_{i-3}$ with $S_0=0$ , $S_1=1$ , $S_2=1$ . Let $n$ be a natural number greater than $3$, then:
$$\text{If } n \text{ is a prime number then } S_{n-1} \equiv ... |
H: Assign 25 identical items to 8 different people
In how many different ways $24$ identical gifts can be shared between $8$ different people such that
One of them doesn't accept more than $2$ gifts , and
The other people accept maximum of $5$ gifts.
(Similar problems are in Combinatorics a guided tour pdf book in ... |
H: Inequality question involving maximum value and three variables
If $x, y, z > 0$ and $x + y + z = 1$, prove that
(a) $x^2+y^2+z^2 \geq \frac{1}{3}$
(b) $x^2yz \leq \frac{1}{64}$
For the first part, since
$\begin{align} x+y+z=1 &\geq 3\cdot \sqrt[3]{xyz} \\
\Rightarrow \frac{1}{9} &\geq \sqrt[3]{x^2y^2z^2} \end{alig... |
H: Weak-* convergence in $L^{\infty}$ implies weak convergence in $L^p$ on bounded set
In a lecture I found the following result: "We remark that when $\Omega$ is bounded the weak-* convergence of $u_{n}$ in $L^{\infty}(\Omega)$ to some $u \in L^{\infty}(\Omega)$ implies weak convergence of $u_{n}$ to $u$ in any $L^p$... |
H: Question about Zariski density and polynomials with full Galois group
Let $A_3\subseteq {\mathbb Q}^4$ be the sets of all $q=(q_3,q_2,q_1,q_0)$ such that $P_q=X^4+q_3X^3+q_2X^2+q_1X+q_0$ has no rational root. Let $A_2 \subseteq A_3$ be the subset of all $q$'s such that $P$ is irreducible over $\mathbb Q$. Let $A_1 ... |
H: Triangle with altitudes and projections.
Let $ABC$ be triangle with $AA_1$ , $BB_1$ and $CC_1$ as its altitude. $M$ and $N$ are the projection of $C_1$ to $AC$ , $BC$ respectively. Let $MN$ intersects $B_1C_1$ at $P$. Show that $P$ is midpoint of $B_1C_1$.
I have tried using Menelaus’ Theorem and similar triangle... |
H: A positive Ricci curvature problem from Peter Petersen's book
The book "Riemannian Geoemetry, the third edition" by Peter Petersen says the following on page 304: $S^k \times S^1$ does not admit any Ricci flat metrics when $k=2, 3.$ My question is whether $S^k \times S^1$ admits metrics of positive Ricci curvature ... |
H: Eigenvalues and reducible polynomials
Let $n \geq 3$. Is it true that an $n \times n$ matrix $A$ with entries from a field has an eigenvalue if and only if every monic polynomial of degree $n$ is reducible? I believe it is true, but I couldn't come up with a proof. Is this known?
AI: Apologies if I'm misunderstandi... |
H: If $f:A \to B$ then prove that $|A| \geqslant |f(A)| $
So, $A$ and $B$ are non empty finite sets and there is a function $f:A \to B$ and I need to prove that $|A| \geqslant |f(A)| $. So, $|A|$ is the cardinality of set $A$ and $ |f(A)| $ is the cardinality of the co-domain of the function. So, here is my thinking ... |
H: $|X|=|\mathbb{N}|$ for $X$ finite?
Consider a countable set $X$, which may or may not be finite (note: we define a countable set to be either finite or infinite). Is it necessarily true that a bijection exists from $X$ to $\mathbb{N}$ (i.e. is it true that $|X|=|\mathbb{N}|$) for finite $X$?
I know from definition... |
H: Why can the average/midpoint of two numbers be described as the sum of the numbers divided by two?
Say I have two numbers, A and B. The "average" or the midpoint of the two numbers is given by $$\frac{A+B}{2}$$
My question is, why does this formula work?
Intuitively, I can derive the equation as follows. The "midp... |
H: Integral $\int^{\infty}_{0} \int^{x}_{0}x.e^{-x^2/y} \, dy\,dx$
This question was asked a couple of times in my Engineering Maths exam in last 5 years but strangely I couldn't find any solution of it on the Internet by Googling, as far as I have tried.
Solve by changing the order of integration:
$$\int^{\infty}_{0}... |
H: volume of solid generated by revolving the triangle about $y$ axis
Volume of solid generated by revolving the region formed by triangle $(5,0),(5,2),(7,2)$ about $y$ axis is
What i Try: Let $3$ points be $A(5,0)$ and $B(5,2)$ and $C(7,2)$
When we rotate a $\triangle ABC$ about $y$ axis. We get a cone
Whose out... |
H: How to prove it is a continuous function?
Assume that $f\in L^1(\mathbb{R}^n)$. Let $B(x,r)$ be a ball centered at $x$ with radius $r$. By Lebesgue Differentiation Theorem,
$$ \lim_{r\to 0} \dfrac{1}{m(B(x,r))}\int_{B(x,r)}f(y)dy=f(x)$$
for $x\in\mathbb{R}^n$ almost everywhere. Assume that $x_0\in\mathbb{R}^n$ ... |
H: getting a number correct on 3 successive dice rolls
I'm working through some practice problems. If we guess a number between $1..6$ and then roll 3 regular six sided die, what is the probability that our guess will appear on any of the dice?
I'm new to probability and I'm struggling a bit to understand this....
I k... |
H: Does convergence to zero follow from $x\cdot a_n \to 0,\ x\in H$?
Let $H$ be an arbitrary Hilbert space. Suppose that for some sequence $\{a_n\} \subseteq H$ we have
$$\forall x\in H,\quad x\cdot a_n \to 0\in\mathbb K.$$
Does it follow that $a_n\to 0$?
I have tried various manipulations of the expression $a_n\cdo... |
H: How to find $a, b$ and $c$?
I was given the following two equations and told to find $a, b$ and $c$:
$$\left\{\begin{array}{c}
a+b-c=1\\
a^2+b^2-c^2=-1\end{array}\right.$$
I tried to form a matrix to solve it and I made this:
$$\left[\begin{array}{ccc|c}
1 & 1 & -1 & 1\\
a & b & -c & -1
\end{array}
\right]$$
Then I... |
H: Built a sequence
They ask me to create a sequence $\{A_n\}$ that fulfill:
$P( \limsup A_n) = \alpha$ with $\alpha\in[0,1]$
$\sum_{n \geq 1}{P(A_n)} = \infty $ and $P( \limsup A_n) = 0$
For the first case I don't know how to start and do it, for the second my idea is something that the probabilities are equiprobab... |
H: find the value of $\sec^4(\pi/9) + \sec^4( 2\pi/9) + \sec^4 (4\pi/9)$
find the value of $\sec^4(\pi/9) + \sec^4( 2\pi/9) + \sec^4 (4\pi/9)$
I tried converting $\sec^2(\theta)$ to $\tan^2(\theta)$ but for no vain.
AI: Like Evaluate $\tan^{2}(20^{\circ}) + \tan^{2}(40^{\circ}) + \tan^{2}(80^{\circ})$
$$\cos3t=4\cos^... |
H: Area of parametric curve
I have a parametric curve (shapes like a heart):
$r(t) = (16sin^3(t), 13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t))$
$0\le t \le 2\pi$
And I'm going to find the area limited by the curve.
I have to find
$\int_0^{2\pi} y*x' dt$
$\int_0^{2\pi} (13cos(t) - 5cos(2t) - 2cos(3t) - cos(4t))48sin^2(t)c... |
H: The number of letters between $z$ and $y$ is less than that between $y$ and $x$
QUESTION: Consider all the permutations of the $26$ English alphabets that start with $z$. In how many of these permutations the number of letters between $z$ and $y$ is less than that between $y$ and $x$?
Source: ISI BMath 2017 UGA
... |
H: Cone over $X$, equivalence relation
I have a question about the definition of the cone over $X$.
If $X$ is a space, define an equivalence relation $X\times [0,1]$ by $(x,t)\sim (x',t')$ if $t=t'=1$. Denote the equivalence class of $(x,t)$ by $[x,t]$. The cone over $X$, denoted by $CX$, is the quotient space $X\tim... |
H: Integration of some two-variables differential (quadratic polynomial) - apparent absurdum
Let us consider some function $f(x,y):\,\mathbb R^2\to\mathbb R$, such that
$$\frac{\partial f(x,y)}{\partial x}=a\,x+b\,y\qquad\qquad\frac{\partial f(x,y)}{\partial y}=b\,x+c\,y\qquad\qquad f(0,0)=0$$
where $a,b,c\in\mathbb R... |
H: Do these two commutative diagrams represent the same homomorphism?
Consider the two many-sorted algebras $\mathtt{A}$, $\mathtt{B}$ and the homomorphism $h:\mid \mathtt{A}\mid \to \mid \mathtt{B}\mid$ below. Do the commutative diagrams in Figure 1 and Figure 2 both adequately represent $h$? Can $h_2$ be dropped fr... |
H: Prove sum of two convex funtions is convex using first and second derivatives
Consider if I have two functions f(x) and g(x) which are both convex. How do I prove that their sum is also convex using the first and second derivatives?. I know if the second derivative of a function is positive then it is convex but I'... |
H: Probability chain rule given some event.
I'm dealing with a probability of the form $\mathbb{P}(\cap_{l \in S} E_l | X)$. Using the conditional probability formula and the probability chain rule would give me a product over the set $\{E_l: l \in S\} \cup \{X\}$. What if I just want to write the product over the set... |
H: Proving that if $H$ and $K$ are subgroups of a finite group G, then $|HK|=\frac{|H||K|}{|H \cap K|}\le |G|$.
Notations: $|C|$= no. of distinct elements in $C$= order of group $C$, if $C$ denotes a group.
$HK=\{hk : \forall \;\;h\in H, g \in G\}.$
The problem lies here: $HK$ may not be a subgroup of $G$. If $HK$ i... |
H: Binomial distribution as a function of $n$ and $k$
As we know, the probability that we get exactly $k$ successes after $n$ experiments is: $\mathbb{P}(S_n = k) = C_n^k\cdot p^k(1-p)^{n-k}$, where $p$ is the probability to get a success in $i^{th}$ experiment.
Let us look at $\mathbb{P}(S_n = k)$ as a function of tw... |
H: $P$ and $Q$ are the two vertices of a regular polygon having $12$ sides such that $PQ$ is a diameter of the circle circumscribing the polygon. Then...
QUESTION: Let $P=(\frac{1}{\sqrt{2}},\frac{1}{\sqrt{2}})$ and $Q=(−\frac{1}{\sqrt{2}},−\frac{1}{\sqrt{2}})$ be two vertices of a regular polygon having $12$ sides s... |
H: Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$
For $a,b,c \in (0,1)$ such that $ab+bc+ca=1$ Prove that $\frac{a^2+b^2}{(1-a^2)(1-b^2)} + \frac{b^2+c^2}{(1-b^2)(1-c^2)}+\frac{c^2+a^2}{(1-c^2)(1-a^2)} \geq \frac{9}{2}$
I tried repleace $1$ ... |
H: How this integral is derived?
There is an interesting definite integral
$$\int_0^1{x^m(1 - x)^n} dx = \frac{n!m!}{(m + n + 1)!}.$$
How to derive this integral?
AI: Let
$$I(m,n) = \int_0^1 x^m (1-x)^n \, dx.$$
It's straightforward to prove that $I(n,0)=I(0,n)=\frac{1}{n+1}$, which satisfies the given formula.
Now no... |
H: How to prove elementary set theory problem: $(A \cap B) \cup (A \cap B^c)=A$
I am trying to prove that $A \cap B$ and $A \cap B^c$ are disjoint, with $(A \cap B) \cup (A \cap B^c)=A$, but I do not know how to approach it. Intuitively expressed in Venn Diagrams, the above statement is correct.
AI: Disjoint:
$$(A\cap... |
H: Linear Factorization of $X^n+1$ in $\mathbb{F}p$
I was just playing around with SageMath and noticed that I can find for each $n < 200$ (at least) a prime $p$ such that the polynomial $X^n+1$ factorizes into polynomials of degree one in $\mathbb{F}_p[X]$. I have a feeling that this statement is in general true, but... |
H: How is ($\neg P \lor R$ ) $\land$ ($\neg Q \lor R$) equivalent to ($\neg P \land \neg Q) \lor R $
How is ($\neg P \lor R$ ) $\land$ ($\neg Q \lor R$) equivalent to ($\neg P \land \neg Q) \lor R $ ?
I know this by making truth table but how to solve this using rules of logic ? Thank you
AI: Just use distributivity o... |
H: If $X,Y,Z \sim $ Poisson($\lambda$) are independent, how do I calculate $P(X+Y=2,X+Z=3)$?
I tried only with some algebric manipulation like
$$P(X+Y=2,X+Z=3)=P(X=2-Y,Z=1+Y)=P(X=2-Y)P(Z=1+Y)$$
But it seems to me that this is not going anywhere. Maybe can be useful the fact that $X+Y,X+Z \sim $ Poisson($2\lambda$), bu... |
H: Size of quotient groups
This question will be really easy to answer I think, I know that there are other questions on roughly the same topic but honestly I could not find what i was looking for in an answer.
I am a physics student trying to learn group theory, I am studying on Pierre Ramond's book "Group theory - a... |
H: Is there a real-analytic monotone function $f:(0,\infty) \to \mathbb{R}$ which vanishes at infinity, but whose derivative admits no limit?
A function $f:\mathbb{R} \to \mathbb{R}$ is called real-analytic if for each $x_0 \in \mathbb{R}$ there exists a neighbourhood of $x_0$ where $f$ is given by a convergent power ... |
H: Set of distinct combinations of combinations
I recently solved a task on Stackoverflow. Please see here
Out of an array with 9 values, triplets consisting of 3 distinct values shall be generated. Calculating them is simple. Result is n choose k, 9 choose 3 = 84 triplets.
Next I needed to select 3 triplets, a triple... |
H: Separated morphisms are stable under base change
Suppose that the map $f$ in the following diagram is a separated morphism (i.e. $\Delta_{X/S}:X\rightarrow X\times_{S}X$ is a closed immersion). I want to prove that $p_{2}$ is also a separated morphism.
$$\require{AMScd}$$
\begin{CD}
X\times_{S}Y @>{p_{1}}>> X\\
@V... |
H: How to find the number of elements of order $p$.
This is a follow up to this question.
Let $G = \langle x,y,z\mid{x^{{p^2}}} = {y^p} = {z^p} = 1,{x^y} = {x^{1+p}},[x,z] = [y,z] = 1\rangle$.
By euler phi function, the number of elements of order $p$ is $p-1$.
For each generator, the number of elements of order $p$ ... |
H: The $y(\frac{∂z}{∂x})-x(\frac{∂z}{∂y})=0$ equation
The $y(\frac{∂z}{∂x})-x(\frac{∂z}{∂y})=0$ equation,
$u=x$
$v=x^2+y^2$
Write according to the new u, v variables given.
I don't know the question at all. Can you please help for solution?
AI: Taking $z=z(x, y)=z(x(u,v),y(u,v))$ can be obtained system
$$z_{u}^{'}... |
H: Explicit presentation of a group
$$\langle G\mid \{abc\mid a,b,c \in G, abc = 1\}\rangle$$ is a presentation of $G$.
Why did we specifically choose the relations to be of length 3?
(Why not something like $\{ab\mid a,b \in G, ab = 1\}$ or $\{abcd\mid a,b,c,d \in G,abcd = 1 \}$?)
(This example appears in https://w... |
H: How big is $\{n\in\Bbb N\mid 1\leq n\leq 2000\text{ and the digital sum of }n^2=21\}$?
well,the title of the question makes it clear,
The question is :find the number of natural numbers between 1 to 2000 such that the sum of digits of their squares is equal to 21.
my approach:
just to make the question more clear I... |
H: Assign the number of ways to share 16 identical objects
In how many different ways can be shared 16 identical objects to 7 different persons such that 3 of them can accept maximum of 2 objects, 3 of them at least 2 objects and for the other person don't have restriction.
I don'n know if I'm on the right way but I s... |
H: Let $a$ and $b$ be elements of odd order in a finite group. Show that $a^2$ and $b^2$ commute if and only if $a$ and $b$ commute.
I really don't know how to solve this problem. I just know that if $|a|=2k_1+1$ and $|b|=2k_2+1$, then,
$a^{2k_1+1}=e=a^0$ and $b^{2k_2+1}=e=b^0$. Also, if $|G|=n$, then, $2k_1+1,2k_2+1... |
H: Let $A$ be a normal matrix. Prove that if $|\lambda| = 1$ for all eigenvalues $\lambda$ of $A$ then $A$ is unitary.
Let $A$ be a normal matrix. Prove that if $|\lambda| = 1$ for all
eigenvalues $\lambda$ of $A$ then $A$ is unitary.
Normal is defined as : $A^*A = AA^*$.
I am unable to find a theorem that makes $... |
H: Graph complexity
Can someone please explain briefly or direct me to some relevant tutorials/material what we maen by complexity of a graph('graph' of graph theory) .It seems there is no one definition and that is confusing me .I would be highly obliged for any help in this regard
AI: Posting as an answer since to... |
H: Combinations series: $\frac{{n \choose 1}(n-1)^3+{n \choose 3}(n-3)^3+\ldots}{n^2(n+3)\cdot 2^n}$
Evaluate $\frac{{n \choose 1}(n-1)^3+{n \choose 3}(n-3)^3+\ldots}{n^2(n+3)\cdot 2^n}$ for $n=10$.
Attempt: I'll deal with the case n being even, as we need to evaluate for n=10.
the numerator is
$${n \choose 1}(n-1)... |
H: Trigonometric equation with two variables
I want to find solutions of the following equation
\begin{equation}
4\cos(x) + 5\cos (x+y)=5
\end{equation}
and the best I’ve managed so far is to express $y$ in terms of $x$
$y= \pm\arccos\biggl(1-\frac{4\cos(x)}{5}\biggr) - x + 2\pi n$ with $n\in \mathbb{Z}$
Is there a wa... |
H: Singleton in a complete metric space is a complete metric space, but has no interior point. Baire?
I am confused by the above statement with the follwing version of Baire's category theorem:
If a non-empty complete metric space $(M,d)$ is the countable union of closed sets, then one of these closed sets has non-em... |
H: How does this show that this homomorphism is onto?
How does $\varphi (x,e') = x$ for all $x\in G$ show that this homomorphism is surjective? I'm not sure how this fits the definition of surjectivity.
AI: Surjective means every point in the target of a function lies in the image of the function. Hence picking any $... |
H: Solution of systems of equations with increasing funcitons
For functions $f,g: I \to \mathbb{R}$ increasing on interval $I$, prove that for the solution of the system of equations
$$
\begin{cases}
\begin{align}
f(x_{1}) &= g(x_{2}) \\
f(x_{2}) &= g(x_{3}) \\
&\;\;\vdots \notag \\
f(x_{n}) &= g(x_{1})
\e... |
H: what is the probability for a green marble to be drawn?
Bag A contains 3 green marbles and 2 red marbles.
Bag B contains 4 green marbles and 5 red marbles.
choose one bag randomly and pick one marble from it.
what is the probability for a green marble to be drawn?
AI: Assuming that you are equally likely to pick Ba... |
H: How can I prove that $\left (\dfrac 1f \right)''(x) = -\dfrac {f''(x)}{f(x)^2} + 2\dfrac {f'(x)^2}{f(x)^3}$
I think its need some theorem what idk. Can someone help me?
$$\left (\dfrac 1f \right)''(x) = -\dfrac {f''(x)}{f(x)^2} + 2\dfrac {f'(x)^2}{f(x)^3}$$
AI: Note that:
$$(f^n)'=nf^{n-1}f'$$
And the derivative of... |
H: When is $f(x)$ reducible over $F$?
Suppose $f(x)$ has a factor $(x-a)$ in the field $F[X]$. Then, is $f(x)$ reducible over $F$? I know that we can write
$f(x) = (x-a)q(x)$ for some $q(x) \in F[x]$. Now, of course, deg$[(x-a)] = 1$ and deg$[f(x)] \geq 1$. This means that it is possible that deg$[f(x)] = 1$ (which w... |
H: Shorter notation for sine function
TL;DR: Are there shorter, established notations for the sine function and other trigonometric functions than "$ \sin(x) $" etc.?
In a some homeworks and tests, there are exercises (e.g. differential equations or matrices) which contains sine and cosine, which I have to write very ... |
H: How do I find a slice of the area of a semi-circle?
A semi-circle around the origin, with radius $r$, is given by $$f(x) = \sqrt{r^2 - x^2}$$ The area of this semi-circle can be written as $$\int_{-r}^r \sqrt{r^2 - x^2} dx = \frac{\pi r^2}{2}$$
How do I find only a slice of this area, i.e. $$\int_a^b \sqrt{r^2 - x^... |
H: Tricking the CLT
I am looking for a sequence of i.i.d. RVs $X_{n}$ with mean $0$ and variance $1$, a standard normal RV $X$ and a set $A \subset \mathbb{R}$ such that for all $n$ we have $P(\frac{1}{\sqrt n}\sum_{k=1}^{n}X_{k} \in A) = 1$ but
$P(X \in A)=0$.
By Portmanteau, obviously we need $P(X \in \partial{A})... |
H: Giving a counterexample to $ 2^{n-1}- 1 = n \cdot a \iff n \text{ is prime}$
Fermat's little theorem asserts: $ n \text{ is prime} \implies 2^{n-1}- 1 = n \cdot a $.
However , the converse , $ 2^{n-1}- 1 = n \cdot a \implies n$ is prime, is not true . How can we prove it , taking odd $n$ (without using a compute... |
H: What's a good symbol to represent many summations? Can a tensor product glyph be used?
What's a good symbol for many repeated summations? I vaguely remember seeing something like
$$
\otimes_{j=1}^N \sum_{n_j=-\infty}^\infty f(\vec{n})= \sum_{n_1=-\infty}^\infty\sum_{n_2=-\infty}^\infty\cdots\sum_{n_N=-\infty}^\inf... |
H: Finding Coefficients via Generating Functions
I need to find the coefficient of $x^5$ in $\frac{2x}{1 - 3x}.$ I have found two different solutions, but I am not sure which is correct. Is the coefficient of $x^5$ given by $3^5 \cdot 2x$ under the interpretation $2x \cdot \frac{1}{1 - 3x},$ or is it given by $2 \cdot... |
H: Bayes Formula Problem Solution Check
The occurrence of a disease is $\frac{1}{100} = P(D)$
The false negative probability is $\frac{6}{100} = P(- | D)$, and the false positive is $\frac{3}{100} = P(+ | \neg D)$
Compute $P(D | +)$
By bayes formula, $P(+) = P(+ | D) P(D) + P(+ | \neg D) P(\neg D) = \frac{97}{10000}... |
H: Calculus proof: $0=1$ What is my mistake?
The quotient rule states that:
$$\frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f'(x)g(x)-f(x)g'(x)}{{g(x)}^2}=\frac{f'(x)}{g(x)}-\frac{f(x)g'(x)}{{g(x)}^2}$$
Integrating tells us that
$$\frac{f(x)}{g(x)}=\int\frac{f'(x)}{g(x)}dx-\int\frac{f(x)g'(x)}{{g(x)}^2}dx$$
If I let $f(x)=g(x)$... |
H: Conjecture: All but 21 non-square integers are the sum of a square and a prime
Update on 6/19/2020. This discussion led to deeper and deeper results on the topic. The last findings are described in my new post (including my two answers), here.
I came up with the following conjecture. All non-square integers $z$ can... |
H: Is the rank of a matrix equal to the number of non-zero eigenvalues?
I have studied before that the rank of a matrix = number of non zero Eigen values. But recently i came across a problem and i dont think it is valid there. I know i am going wrong somewhere.
$$A=
\begin{bmatrix}
0 & 4 & 0 \\
0 & 0 & 4\\
0 & 0 & ... |
H: Question about the ratio of two random variables
We have a couple $(X,Y)$ of random variables and its joint density function is:
$$p_{X,Y}(x,y)=\begin{cases} \dfrac{e^{-1}}{3(x-1)!} & \mbox{if }x \in \mathbb{N}\setminus \{0\},\ y \in \{-x,0,x\} \\ 0 & \mbox{otherwise}\end{cases}$$
Find the distribution of $W=Y/X$. ... |
H: Flat modules are torsion-free
I need to prove the following assertion:
Let $A$ be an integral domain. If $M$ is a flat $A$-module, then $M$ is torsion-free.
My definition of a flat module is: an $R$-module $F$ is flat if the functor $F \otimes_R \star \colon M \mapsto F \otimes_R M$ transforms exact sequences in ... |
H: Show for any monic polynomial $p(x)$ and for any $k$ that there are $k$ primes $q_i$ and $k$ integers that $n_i$ such that $q_i|p(n_i)$
Problem: Let $p(x)$ be a monic polynomial with integral coefficients, I want to show using induction for any integer $k$ that there exists $k$ distinct primes $q_1,\ldots,q_k$ and ... |
H: Proving that CDFs of maximum and $\frac{1}{k}$ sum are equal for $X_i \sim \text{Exp}(1)$.
Let $(X_i)_{i \in \mathbb{N}}$ i.i.d random variables such that $X_i \sim \text{Exp}(1)$.
Prove that CDF of $\max_{1 \leq k \leq n} X_k$ is identical to CDF of $\sum_{k=1}^{n} \frac{1}{k} X_k$, i.e $$\mathbb{P}\Big(X_1 + \dot... |
H: Proving $Ext_{\mathbb{Z}}^1(A,B)$ is a torsion-free abelian group, given that $A$ is divisible and $B$ is torsion-free
I have been trying to prove the following:
Let $A,B$ be abelian groups. If $A$ is divisible and $B$ is torsion-free, then the group $Ext_{\mathbb{Z}}^1(A,B)$ is torsion-free.
So, I study homologica... |
H: MLE and log MLE
I'm getting confused on likelihoods and the use of the log function.
negloglik = negative log likelihood
This statement on negloglik seems to be wrong to me:
"
negloglik is an exponential scale, therefore small changes in negloglik represent a very large change in likelihood.
"
Can anyone explain w... |
H: How to calculate the volume between two cylinders
I am asked to calculate the volume between the two cylinder : $x^2+y^2=1$ and $x^2+z^2=1$
So my assumption here that the limit of $y$ and $z$ must be equal, and we are looking on the unit circle so the integral needs to be:
$$\int_{0}^{1} \int_{0}^{1} \int_{0}^{1} 1... |
H: Showing triangle inequality in this metric.
I'm struggling to show that the following function on $\mathbb{Z}$ is a metric: specifically, showing the triangle inequality.
Fix an odd prime $p,$ and define $d:\mathbb{Z}\times\mathbb{Z}\rightarrow\mathbb{R}$ by $$d(m,n)=0 {\text{ if }} m-n=0,$$ or as $$d(m,n)=\frac{1}... |
H: Whats the joint PDF of Z=XY given the a joint pdf f(x,y)?
Let $X$ and $Y$ be random variables with joint pdf:
$$f(x,y)=x + y \quad \text{if } x \ge 0, y \le 1$$
Let $Z=XY$. Calculate the pdf of $Z$.
I'm a bit confused about solving this problem, I'm trying to get to the pdf by calculating the cdf to derive it aft... |
H: Finding the probability transition matrix
Find $P_t$ when Q =
\begin{bmatrix}
-\alpha & \beta \\
\alpha & -\beta
\end{bmatrix}
and $\alpha, \beta > 0$ satisfying the backwards equation
AI: You have to find exp(tQ) to solve this problem. |
H: Evaluating $\int_{0}^{\infty} (\frac{\sin x}{x})^2 dx$ using complex analysis
I need to calculate $\displaystyle\int_{0}^{\infty} \left(\frac{\sin x}{x}\right)^2dx$.
I have started with defining:
$$f(z) = \frac{1-e^{2iz}}{z^2},\quad z\in\mathbb{C}\;.
$$
Then divided it into four contour integrals, just standard stu... |
H: Reasoning with congruences: does a positive integer $x$ exist with the following properties?
Question:
Find $x$ such that the following properties are true or prove that no such $x$ exists.
Let:
$x>0$ be an integer
$p_1, p_2, p_3$ be distinct odd primes
$1 \le a < p_3$ is an integer
Find $x$ with the following p... |
H: Prove that $-X$ is measurable with respect to some sigma field.
I am reading A Second Course in Probability by Ross and Peköz. I came across the following question:
1.10.4. Show that if $X$ and $Y$ are real-valued random variables measurable with respect to some given sigma field, then so is $XY$ with respect to t... |
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