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H: Prove that the summation of product of co-factor of an element of a matrix with any other element is 0.
Let A = ($a_{ij}$) be an $n×n$ matrix and $A_{ij}$ be the co-factor of $a_{ij}$.
Show that: $\sum$ $a_{ik}$$A_{jk}$ = det($A$) if i=j, else 0
I couldn't think of any way to do this other than trying to open the... |
H: Enlarging the ring of $S$-integers to get a PID
Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing the archimedean ones. Then we can define:
$$\mathcal O_S:=\{a\in K\colon a\in\mathcal O_{\mathfrak p},\;\forall\mathfrak p\notin S\}\supset \mathcal O_K$$
We know that $\mathcal O_S$ is ... |
H: Why are picewise continuous functions on $[a,b]$ bounded?
Consider a picewise continuous function $f:[a,b] \to \mathbb{R}$, i.e there are $a=t_0<\dots <t_n=b$ such that $f$ is continuous on each open interval $(t_i,t_{i+1})$ and the limits $\lim\limits_{x \uparrow t_i}f(x)$ and $\lim\limits_{x \downarrow t_i}f(x)$... |
H: Prove that the image of $D$ is the superior half plane
I have an exercise left without correction, and I'm struggling; if anyone could help, it would be a lot appreciated.
Let $D$ the open disk centered at $0$ with radius $r = 1$.
We define $h(z)=\frac{1+iz}{1-iz}$
How can we prove that
$h(D) = \left\{z \in ... |
H: Almost sure convergence of a sum of independent r.v
Let $S_n:=\sum_{i=1}^nX_i$ where $X_1,X_2,...$ are indepentent r.v.'s
such that:
$P(X_n=n^2-1)=\frac{1}{n^2}$ and $P(X_n=-1)=1-\frac{1}{n^2}$
Show that $\frac{S_n}{n}\rightarrow-1$ almost sure.
It is easy to see that you can't apply strong law of large numbers... |
H: Determine $\arctan{e^i}$
In this answer, the quantity $\arctan(e^i)$ must be determined.
It's obviously $e^i = \cos(1) + i\sin(1)$, but there is no formula for $\arctan(x + y)$ like $\sin(x + y)$, for example, and I'm stuck.
In the comments, the author suggests:
$$\arctan(e^i) = \frac{1}{2} \arctan \left( \frac{\co... |
H: Why does Stolz- Cesaro fail to evaluate the limit of $\dfrac{n + n^2 + n^3 + n^4 + \ldots + n^n}{1^n + 2^n + 3^n + 4^n + \ldots +n^n}$,
I need to find the limit of the sequence
$\dfrac{n + n^2 + n^3 + n^4 + \ldots + n^n}{1^n + 2^n + 3^n + 4^n + \ldots +n^n}$,
My strategy is to use Stolz's Cesaro theorem for this s... |
H: I would like to know what the topics are involved in this question.
I would like to know what the topics are involved in this question that I can use to solve it. I have tried searching for exponents with variables or using functions although I can't find a way to start on this question.
If $3^{m+1} = 5^{n-1} = 15... |
H: Volume by Rotation Using Integration
How to find the volume if the shown area is rotated around the $y$-axis?
The area will be bounded by $=^2+1$, $y=2x$ and $x=0$.
AI: Hint: $V_y=\pi\int_c^d x^2 dy$.
So, $V_{y_1}=\pi\int_0^2 \left(\frac{y}{2}\right)^2dy$ is the volume of cone, and the $V_{y_2}=\pi\int_1^2\left( y-... |
H: Lottery variance
The chance to win in Lottery game is $0.1$. Michael decided to buy a ticket every week until he will win or until he will buy 5 tickets.
if X is the number of weeks Michael bought a lottery ticket,
what is the variance of $X$?
So I Calculated for X=5, which means LLLLL or LLLLLW. I calculated the p... |
H: Showing that map $\phi(z)=z(z-1)$ is not covering projection
Map $\phi : \mathbb{C} - \{0,1/2\} \to \mathbb{C} - \{-1/4\}$ given by $\phi(z)=z(z-1)$ is not covering projection. Can someone give hint how to proceed.
AI: Hint
For $z \in \mathbb C \setminus \{0, -1/4\}$, the inverse image of $z$ under $\phi$ includes ... |
H: Why having even number of edges in cut imply $\mathcal{B}(G) \subset \mathcal{C}^\perp(G)$ and $\mathcal{C}(G)\subset\mathcal{B}^\perp(G)$?
$\mathcal{C}(G)$ is cut space. $\mathcal{B}(G)$ is cut space.
Diestel say at theorem 1.9.4, having even number of edges in cut mply $\mathcal{B}(G) \subset \mathcal{C}^\perp(G... |
H: How does the linear independence or dependence of the set of column vectors of a matrix depend on that of the set of row vectors of the same?
Let A = ($a_{ij}$) be an mxn matrix. If the set of row vectors of A is linearly independent, is the set column vectors too? What happens if the row vectors are linearly dep... |
H: Show that $f$ can be extended to a Cauchy-sequence preserving continuous mapping on $\overline{A}$.
Question:
Let $(X,d)$ be a metric space and $A\subset X.$ If $f: A\to\mathbb R$ be a Cauchy-sequence preserving continuous mapping then show that $f$ can be extended to a Cauchy-sequence preserving continuous mappin... |
H: The closure of a rational polyhedron is a rational polyhedron.
I'm reading the following proof, where the closure of the rational polyhedron $P$ is denoted $P'$.
I don't get the line where $Y$ is defined. This is a set of linear expressions of the form $y^TA$, and since $y \in \mathbb{R}$, I don't suppose that t... |
H: Define a relation R on Z × N by (a, α)R(b, β) if and only if aβ = bα. Prove that R is a reflexive relation.
I'm a bit confused about how to prove that R is reflexive.
By definition, R, a relation in a set S, is reflexive if and only if ∀x∈S, xRx.
Since (a, α)R(b, β), we know that aβ = bα.
Then to prove that this ... |
H: How to convert this integral to a polar integral, $\int_{0}^{1}\int_{0}^{x}ydydx$
I was browsing through some past final exams, and I ran into this integral:
$\int_{0}^{1}\int_{0}^{x}ydydx,$
The question wants us to convert this integral to a polar integral.
I'm wondering how we convert this integral? I started ... |
H: Determine second degree polynomial by least squares method
Determine the polynomial of the second degree that approximates by the method of least squares in the following function, at the interval [0,5, 1,5].
$f(x)=\frac{3}{\sqrt{x}}$
I´ve done least squares before using the point table,
but now all i have is a fu... |
H: ${d\over dx}f(2x+1)=\sin(4x^2-1)$, then is $f$ differentiable?
Question: ${d\over dx}f(2x+1)=\sin(4x^2-1)$, then is $f$ differentiable?
Since $\varphi:x\rightarrow2x+1$ is bijective, I thought that $f$ is differentiable. But, I wanted a precise solution, so I tried to use the chain rule in order to get the form of ... |
H: Help with calculating the integral $\int_{-\pi}^\pi \cos\left(x/2\right) \cdot e^{ix} dx$ by using Eulers formula
I have to determine the following integral
$$
\int_{-\pi}^\pi \cos\left(x/2\right) \cdot e^{ix} dx
$$
by using Eulers formula
$$
\cos\left(x/2\right) = \frac{e^{ix/2}+e^{-ix/2}}{2}
$$
we have that
\begi... |
H: Are there sufficient conditions on $H,K \lneq G$, $H\ne K$, such that $[G:H]=[G:K]$ for infinite $G$?
Let $G$ be a group and $H,K$ distinct proper subgroups of $G$. If $G$ is finite, then $|H|=|K|\Rightarrow [G:H]=[G:K]$. Now, if we move to the infinite $G$ case, and take e.g. $G=\mathbb{Z}$ and $H_n:=n\mathbb{Z}$,... |
H: Need clarification on Abbott's Proof of the Algebraic Limit Theorem for Sequences
Here is the excerpt of his proof that I do not understand
I don't understand why he concludes that $|b_n| > |b|/2$ from this.
AI: Observe that by the triangle inequality $|b|-|b_n|\leq |b_n-b|<|b|/2$. Then rearranging for $|b_n|$ you... |
H: The Hausdorff measure of the unit interval
I am trying to calculate the Hausdorff measure of the unit interval. Here's my attempt:
Fix $\epsilon>0$.
consider the open balls $B(x,\epsilon)$ with $x\in[0,1]$. How many can cover the unit interval? at least $\frac{1}{2\epsilon}$. So if we use $\lfloor{}\frac{1}{2\epsi... |
H: Show that for every m and n value, $\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$
Show that for every m and n value, $$\int_0^1 x^m (1-x)^n \,dx= \int_0^1 x^n (1-x)^m \,dx$$
I have no idea how to solve a question like that. Do I have to solve both parts of the equation and show that they're equal, or is the... |
H: Roots of product of two quadratic.
consider the quadratics,
$ P(x) = ax^2 + bx +c$ ,and, $Q(x)= -ax^2 + dx+c$ , given that ac $\neq$ 0. What can we say about the roots of PQ? More particularly, does PQ have at least two real roots?
my work:
I don't really know where to start even, I suppose it has something t... |
H: First trigonometric differential equation
Show that:
$$\tan(x) \frac {dy}{dx}-y=\sin^2(x)+2\sec(x)$$ where $y=\sin^2(x)-2\cos(x)$
I get:
$\frac {dy}{dx}=2sin(x)cos(x)+2sin(x)$
=$tan(x)(2sin(x)cos(x)+2sin(x))-sin^2(x)-2cos(x)$
From here I go into many directions but not towards the RHS. Guidance is much appr... |
H: Line graph degree sequence
Let $G= (V,E)$ be a simple graph. The line graph $L(G)$ is the graph its vertices correspond to $G$'s edges and two vertices are connected with an edge in $L(G)$ if the respective edges share a vertice in $G$.
Vertices: Obviously $|V(L(G))| = |E(G)|$ by definition.
Edges: An edge appears ... |
H: Amount of money earned only from interest - recursion POV.
I have a bank account with $0€$. At the beginning of every month I put an additional $250€$ into my account and at the end of the month I get a $0.5\%$ interest on my money. Based only on this information, after $10$ years (or $120$ months), how much money ... |
H: If eigenvalue has $n$ eigenvectors, is the matrix the $0$ matrix?
Let $\lambda_1$ be an eigenvalue of $A \in \mathbb{R}^{nxn}$, with $n$ eigenvectors, and $\lambda_2$ an eigenvalue of $B \in \mathbb{R}^{nxn}$ with $k \leq n$ eigenvectors.
Prove that $\lambda_1 + \lambda_2$ is an eigenvalue of $A+B$ with $k$ eigenv... |
H: How is 'Convergence' defined?
I've noticed that some papers, e.g. in theoretical computer science and numerical mathematics, provide pseudo-algorithms for their proposed methods. Often these pseudo-algorithms have instructions like:
While not converged do:
....
end
or
Repeat:
....
Until convergence
This mak... |
H: Why do we have $Q(\alpha, z) = \mathbb{Q}(\alpha, z, \overline{z})$ where $\alpha, z, \overline{z}$ are the roots of $X^3+X+1 \in \mathbb{Q}[x]$?
We are given the polynomial $f = X^3+X+1 \in \mathbb{Q}[x]$. It is easy to show that $f$ has only one real root, call it $\alpha$, and the other two roots are complex con... |
H: Writing explicitly $(s^2-1)^2+(t^2-1)^2$ as a polynomial in $st$ and $s+t$?
Consider the symmetric polynomial
$$ P(s,t)=(s^2-1)^2+(t^2-1)^2.$$
How can we write $P$ as a polynomial in the variables $st,t+s$?
The Fundamental theorem of symmetric polynomials implies this is possible, but I am having trouble doing it i... |
H: Prove that a normal matrix can be completely determined by its eigen values and a unitary matrix.
Prove that a normal matrix can be completely determined by its eigenvalues and a unitary matrix.
I tried using the Spectral decomposition theorem: Is states that a normal matrix N with spectrum set {$k_1, k_2,.., k_n$... |
H: Induced mapping into mapping cylinder
Can anyone please tell what is the induced map from mapping cone of f into Z? Also please explain how H and g induced G.
Thanks in advance
AI: A map $f: X \to Y$ is nullhomotopic iff there is an extension to $\tilde{f}:CX \to Y$ - you can just define $\tilde{f}\left(x, t\right... |
H: Prove that $\sum_{k=1}^{\infty}\frac{k^2}{e^k}< \sum_{k=1}^{\infty}\frac{k}{2^k}$
Prove that :
$$\sum_{k=1}^{\infty}\frac{k^2}{e^k}< \sum_{k=1}^{\infty}\frac{k}{2^k}$$
Without calculating the value of these series .
The partial sum formula are :
$$\sum_{k=1}^n \frac{k}{2^k} = 2^{-n}(-n+2^{n+1}-2)$$
$$\sum_{k=1}... |
H: Limit superior and inferior of Césaro means are almost surely constant
Reading the book of probability of Achim Klenke I came across the assertion that if $(X_n)$ is a sequence of independent real valued random variables then the limit superior and inferior of the sequence of Cesáro means defined by $S_n:=\frac1n \... |
H: Number of ordered Pairs satisfying $4^m-3^n=1$
Find the Number of ordered Pairs $(m,n)$ of positive integers satisfying $4^m-3^n=1$
Mt try:
Trivially $m=n=1$ satisfies
Let $m \gt 1$
$$4^m-3^n=(1+3)^m-3^n=1$$
$\implies$
$$3\binom{m}{1}+3^2\binom{m}{2}+3^3\binom{m}{3}+\cdots+3^m=3^n$$
Now since LHS is not a power of ... |
H: What do the components of the solution to a system of equations to find the parameters of a circle represent?
The homogenous representation of a circle is given by $x^2 + y^2 + 2gxz + 2fyz + cz^2 = 0$ (or, equivalently, if we set $z=1$, $x^2 + y^2 + 2gx + 2fy + c = 0$). Now, given 3 points (in a homogenous form), w... |
H: Rewrite the feasible set of a linear programming problem
Consider a $K\times 1$ vector $x$ and define
$$
\mathcal{X}\equiv\{x\in \mathbb{R}^K: A*x\leq b\}
$$
where $A$ is a $J\times K$ matrix containing known real scalars and $b$ is a $J\times 1$ vector containing known real scalars.
Is it correct to say that
$$
\... |
H: Variance of Linear Combination of Standard Normals
I have a random variable $X_n=\frac{1}{n} \sum^n_{i=1} Z_i$ for $n\in \mathbb{Z}^+$ where $X_0=0$ and the $\{ Z_i \} \sim N(0, 1)$ for all $i$. The $Z_i$ are independent. I need to find the distribution of $X_n$.
I know that a linear combination of normal random va... |
H: How to find maximal element of a set and its minimal elements?
$X \neq \emptyset$ is a set and on $P(X)$ (the power set of $X$), we know that "$\subseteq$" subsuming relation is a partially ordered set.
For $\forall a\in X$, how can I show the maximal element of subset $A=P(X) \setminus \{X, \emptyset\}$ of $X\set... |
H: Quick Question: Variance of a two related Gaussian distributions
I have two random variables, $X$ and $Y$. Both follow a Gaussian distribution, and $$X \sim N(0,1)\;.$$
After some manipulation, I got that $$P(X \leq z) = P(kY \leq z)$$
where $k$ is some constant.
Would this imply that $X=kY$, and therefore $\operat... |
H: Let p be a prime. If a group has more than $p-1$ elements of order $p$, then prove that the group can't be cyclic.
Result: Let G be a group which has more than $p-1$ elements of order $p$. I need to prove that such a group can't be cylic. $p$ is a prime number.
Let's consider the case when G is finite. I want to... |
H: Greatest integer less than or equal to the 50th term of the series defined below
This question I found in one of my practice papers for the JEE
$$a_{1}= \sqrt[3]{28}$$
and
$$a_{n+1}=\sqrt[3]{a_{n} + 60} \space \space \forall \space n \in\Bbb N $$
Question asks for greatest integer less than or equal to the 50th t... |
H: Maths area question
I want to answer the follow question.
I’ve worked out the area of the circle to be 200.96
Where would I go from here?
AI: Hint $$\text{Area of triangle}=\frac{1}{2}(a)(b)\sin(C)$$ where a,b are sides and C is the included angle. So we get $70=\frac{1}{2}(8)(19)\sin(A)$ so $\sin(A)=\frac{35}{... |
H: If $X$ is symmetric at $a$, then $Y= X + a$ is symmetric at $0$
Suppose $X$ has a density function $f$ that is symmetric about $a$. Let $Y = X + a$. Show that the density function $g$ of $Y$ is symmetric about $0$.
Setting $f(x) = g(x-a)$ gives you the result through basic algebra. But I'm having an inordinate amo... |
H: Intuition behind multiplication of decimals that are greater than 0.1
How can I visualize the multiplication of 2 decimals that both are greater than 0.1? For example, 0.2×0.25.
I understand that 0.1×0.1 is to:
step 1. divide a size into 10 parts
2. divide 1 of those parts into another 10 parts
3. retrieve 1 part ... |
H: A simple faithful module for $k\langle x,y\rangle/\langle xy-1\rangle$
Let $S$ be the free algebra $k\langle x,y\rangle$ and $R$ be the quotient $S/\langle xy-1\rangle$. In talking with someone, they mentioned they thought this algebra is left primitive. So, I set about looking how to prove that by finding a fait... |
H: Calculate the number of triplets in a graph
I try to compute the global clustering coefficient of a graph in Matlab using the adjacency matrix. I know how to find the number of closed triangles: trace(A^3), but I do not have any idea how to find open triplets in a graph efficiently. I wrote a code traversing each n... |
H: Show that pushforward measure is inner regular
Let $X, Y$ be compact Hausdorff spaces, $\tau:Y \to X$ continuous and $\mu^\tau:=\mu \circ \tau^{-1}$ the pushforward measure of the measure $\mu$ which is a inner regular measure. Show that $\mu^\tau$ is inner regular as well.
We have to show that $\mu^\tau(A)=\sup\{\... |
H: Möbius Transformation from a unit disk to the upper half plane
Consider the unit-disk $\mathbb{D} = \{ z : |z|\leq 1 \}$.
I need to find a Möbius Transformation $w=Tz$ that maps $\mathbb{D}$ to the upper-half plane $\mathbb{H} = \{ w : Im(w) \geq 0\}$.
I have searched and found that the linear fractional transform... |
H: Clarification on limits arithmetics
Given a limit in the form of: $\lim_{x \to \infty} ((1+\frac{1}{f(x)})^{f(x)})^{g(x)}$
Assuming we know $f(x)\xrightarrow[x \to \infty]{}\infty, g(x)\xrightarrow[x \to \infty]{}c \in \mathbb{R}$
Why is it true that the limit is $e^c$? or isn't it?
This is just a little thing I'm ... |
H: Can $g\cdot A < A$ for some group action?
Can $g\cdot A$ be a strict subset of $A$ for some group action '$\cdot$' from $G\times G$ to $G$. For some subset $A$ of $G$, for some element $g$ other than identity in $G$.
Where $g\cdot A = \{ g\cdot a \mid a \in A \}$.
AI: Let $\mathbb Q^*$ be the multiplicative group o... |
H: Does polynomial generated by repeated application of matrix divide characteristic polynomial?
Given a square matrix $A\in \mathbb C^{n,n}$ and a vector $v\ne0$, the vectors
$$
v, Av, A^2 v, \dots, A^n v
$$
are linearly dependent. Let now $m\le n$ be the smallest number such that
$$
v, Av, A^2 v, \dots, A^m v
$$
ar... |
H: Let $X$ be compact and $f:X\to\mathbb{R}$ s.t. each $x\in X$ has a nbh where $f$ attains its minimum. Show $f$ attains minimum on $X$.
Consider a compact topological space $X$ and a map $f:X\to\mathbb{R}$
such that each $x\in X$ has a neighborhood where $f$ attains its
minimum. Show that $f$ attains its minimu... |
H: I am not able to understand these lines from deductive reasoning
" Conclusion can only be false if atleast one of the premises is also false. If both premises are true, then conclusion is also true. We will say that argument is valid if the premise cannot be all true without the conclusion being true as well. "
I ... |
H: Not Understanding the Key Step in a Linear Algebra Proof
I'm self-studying some linear algebra from the Schaum's Outline of Linear Algebra by Lipschutz and I came across a problem where I don't fully follow the key step. Below I make a note with respect to where I don't follow:
Problem 6.26: (Note that the notation... |
H: factorizing the given polynomial.
factorize $(bc-a^2)(ac-b^2)-(c^2-ab)^2$. The one method I know is to something like let $ a=x$ and if equation becomes zero then $a-x$ is a root but how to exactly think?
AI: I think it's better to do this using determinants,the given equation can be converted to the following mat... |
H: Does the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converge for $x < e^{-e}$?
As per the answer at https://math.stackexchange.com/a/573040/23890, the infinite power tower $x^{x^{x^{x^{x^{x^{x^{.{^{.^{.}}}}}}}}}}$ converges if and only if $ x \in [e^{-e}, e^\frac{1}{e} ] $.
Is $ e^{-e} = 0.06598... |
H: Show that the sequence $a_n=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$ is rising and is unbounded.
We have a sequence:
$$a_n=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{n}$$
We need to show that it is rising and above unbounded.
So I did the following:
$$a_{n+1}=a_n+\frac{1}{n} $$
$$a_{n+1}>a_n $$
$$... |
H: Finding the diagonal of a rectangle.
What fraction of the rectangle is shaded? (You may assume that each line, other than the diagonal of the rectangle, is parallel to some side of the rectangle.)
Is there a way to solve this without doing algebraic manipulations to get the other side length of the rectangle? How ... |
H: There is a $C > 0$ such that $\vert f(z)-f(w)\vert \leq C\vert z-w\vert$ for a holomorphic map into $B(0, R)$.
Let $0<\delta<r$ and $R>0$, and $a\in\mathbb C$. I want to show that there exists $C>0$ such that for all holomorphic functions $f\colon B(a,r)\to B(0,R)$, we have
$$
\vert f(z)-f(w)\vert\leq C\vert z-w\ve... |
H: Stability of these fixed points
Say we have the set of nonlinear equations, where $\alpha>0$:
$$\begin{matrix}
\frac{dx}{dt}=x[1-\alpha x-y]\\
\frac{dy}{dt}=y[1-x-\alpha y]
\end{matrix}$$
I have determined that the fixed points of this system when $\alpha \neq 1$ are $$(0,0), (0,\frac{1}{\alpha}), (\frac{1}{\alpha... |
H: Show that a function f, continuous on (a,b), has an abs minimum value. The limits as x approaches either bounds of the interval is +Infinity
Can you show that the function f must have an absolute minimum value on the interval (a,b), if f is continuous on (a,b) and the right hand limit as x->a along with the left ha... |
H: Randomizing a regular polygon along its "spokes" while maintaining area
I have a 2d graphics question that seems like it'd fit better here than at stackoverflow. Please forgive any breach of ettiquette as I am new to mathematics exchange. I did search for a solution before posting this.
I am taking a regular poly... |
H: Finding a limit given a function is continous
if $ f(x) = \frac{\sin 3x+ A \sin 2x + B \sin x}{x^5} $
for $x\neq 0 $ is continuous at $x=0$ , then $A+B+f(0)$ is..?
Answer is supposedly '$2$'.. According to my friend we have to find '$A$' and '$B$' such that this limit exists but would that be correct interpretation... |
H: Reference for $p$-adic Haar integral
I have stumbled upon the notion of a $K$-valued Haar integral on a locally compact group, where $K$ is a non-Archimedean field, as well as the $K$-valued modular function, in an article of Schikhof. He cites his thesis "Non-archimedean harmonic analysis", but I have not been abl... |
H: How close $f(n)$ is to $\phi(n)^2$ asymptotically?
Consider the number $f(n)=\#\{(a,b) \mid 1\leq a,b\leq n, (a,n)=(b,n)=1,~\text{gcd}(a-b,n)=1\}.$ It is clear that $f(n) \leq \phi(n)^2.$ Is it it known how close $f(n)$ is to $\phi(n)^2$ asymptotically ?
AI: Well (unless $n=1$) it is at most $\phi(n)^2-\phi(n)$, si... |
H: Majorization of $(5,5,0) $ by $(10,0,0)$
Majorization definition: $x $ is majorized by $y$ iff $x$ is in the convex hull of all the points formed by permuting the components of $y$.
Question: $(5,5,0) $ is majorized by $(10,0,0)$ according to the alternate definition of majorization given at wikipedia.
My confusio... |
H: Is this a correct solution to this problem? 2018 AIME II Problem 10
Problem:
Find the number of functions $f(x)$ from $\{1, 2, 3, 4, 5\}$ to $\{1, 2, 3, 4, 5\}$ that satisfy $f(f(x)) = f(f(f(x)))$ for all $x$ in $\{1, 2, 3, 4, 5\}$.
Solution:
Note that there are $5^5$ possible functions $f(x)$.
Now consider the p... |
H: Finding triangle side lengths given altitudes or lengths of sectioned triangles within triangle.
In right triangle $ABC,$ $\angle C = 90^\circ.$ Let $P$ and $Q$ be points on $\overline{AC}$ so that $AP = PQ = QC.$ If $QB = 67$ and $PB = 76,$ find $AB.$
How do I use ratios and given side lengths to create a proport... |
H: Can I infer this about probability of drawing a queen or heart from deck of a cards (twice in a row) no replacements?
So I know the probability of drawing a queen or a heart from a deck of cards is 4/13
Let A be the event a heart or queen is drawn on the first draw
Let B be the event a heart or queen is drawn on th... |
H: A wine taster claims she can distinguish four vintages of a particular Cabernet. What is prob this can be done by guessing? Feedback
A wine taster claims she can distinguish four vintages of a particular Cabernet. What is prob this can be done by guessing? She is confronted with four unlabeled glasses.
I have a sol... |
H: Why does the integral $\int_{-\infty}^ \infty \frac{1}{x^2}dx$ diverge?
The integral $\int_{-\infty}^ \infty \frac{1}{x^2}dx$ according to many websites diverges. But, by manual calculation, the answer is clearly $0$. Why is this the case?
AI: I just realized it's because the value at x = 0 does not exist. It appro... |
H: Nillradical is prime ideal, then the ring is not a product ring.
Let $A$ be a commutative ring and $nill（A）$ is not a prime ideal.
This is just a characterization of $SpecA$ to be irreducible.
Then, according to the argument of general topology, irreducible deduces connected.
So,to translate this into ring theory,
... |
H: Notations from Rudin's book
This is taken from Rudin's book
Notations: If $x=(x_1,...,x_n) \in R^n$ and $y=(y_1,..,y_m)$ in $R^m$. and $(x_1,...,x_n,y_1,..,y_m) \in R^{n+m}$.
Every $A \in L(R^{n+m},R^n)$ can be split into two linear maps $A_x$ and $A_y$, defined by $A_x h = A(h,0), A_y k=A(0,k)$ for any $h \in R^n,... |
H: Counterexample: Two groups $H$ and $G$, with surjective homomorphismus
So I need to find a counter-example, i.e. I need to find two groups $H$ and $G$, with $N$ being normal subgroup of $G$, with a NON-surjective Homomorphism $\phi: G \rightarrow H$, such that $\phi(N)$ is NOT normal subgroup of H.
I know for sure,... |
H: Show that $M_p^p\equiv 1 \mod p^2$
Can it be shown that $M_p^p\equiv 1 \mod p^2$ where $M_p=2^p-1$ is a Mersenne prime.
I tried to develop the left part into into $2^{p^2}-1-pk2^p$ and use $2^{p^2}\equiv 2^p \mod p^2$, but I get nowhere
Thanks
AI: From FLT
$$2^{p-1}\equiv 1\pmod{p} \Rightarrow
2^{p}\equiv 2\pmod{p}... |
H: For every projection $p$ and normal $a$ in a C*-algebra $A$ (with $ap=pa$), there is a $*$-isomorphism $C(\sigma(a))\to C^{*}(a,p)$ such that ...
Let $a$ be a normal element of a (non-unital) C*-algebra $A$. I am trying to prove that for every projection $p\in A$ that commutes with $a$ (i.e. $p=p^{2}=p^{*}$ and $ap... |
H: Predicate logic: Negation
My book says that the negation of "Everyone likes coffee" is "Not everyone likes coffee".
But if i apply that to quantors (c = likes coffee), then...
$\neg(\forall x(c(x))) \Longleftrightarrow \exists x (\neg c(x))$
Which means that "Not everyone doesn't like coffee".
Can someone explain?... |
H: Find $\int_{0}^{\infty} \frac{\log(x) }{\sqrt{x} (x+1)^{2}}\,dx$
Need solve the next integral
$$\int_{0}^{\infty} \frac{\log(x) }{\sqrt{x} (x+1)^{2}}\,dx$$
Tried something with Laurent’s series, but i can’t conclude anything.
Thanks
AI: METHODOLGY $1$: CONTOUR INTEGRATION
Enforce the substitution $x\mapsto x^2$ to... |
H: Line integral $\int_{AB}y\,ds$
I need to calculate this line integral: $$\int_{AB}y\,ds$$ Where $AB$ is an arc defined by $\begin{cases}
x=t \\
y=\sqrt{t}
\end{cases}$ with $t\in [1,2]$
I got $I=\int_1^2 t\sqrt{1+\frac{1}{4t}} \, dt$ and now I'm stuck with this integral. I tried substitution method but it's get... |
H: Brownian motion increments - are they random variables or random processes
If $W_t$ is a Brownian motion process and $0 \le t_1 \le t_2$ then is the increment $W_{t2} - W_{t1}$ a random variable or a random process? My lectures say "random variable" but I believe it makes more sense to call this a random process be... |
H: Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show that $A = X\backslash B$ and $B = X\backslash A$.
I'm trying to prove this
Let $A,B,X$ be sets such that $A\cup B = X$ and $A \cap B = ∅$. Show
that:
(1) $A = X\backslash B$
and
(2) $B = X\backslash A$.
My proof is
Let $x \in A$. We k... |
H: Help to evaluate the integral $\iint_D\frac{y}{\sqrt{x^2+y^2}}dxdy$
I'm solving a problem about integrals in curves, and I got this integral:
$$\int_1^2\int_1^2\frac{y}{\sqrt{x^2+y^2}}dxdy.$$
I have been struggling to solve it. I'm sure i have to do some variable change to polar coordinates (to simplify the denomin... |
H: I don't understand the Rank–nullity theorem..
$$\dim(U) = \dim (\ker\phi) + \dim(\text{im }\phi)$$
Hey all I'm just a little confused about the above mentioned theorem; namely, how does one measure the dimensions of a kernel? A kernel is normally defined as a set, and I don't see how I could measure the dimensions ... |
H: Is it true that if a function is bound by a constant so is it's derivative?
I have an intuition that I haven't been able to prove nor disprove:
Given a parametric function $p$ if $||p(t)|| < C$ then $||p'(t)|| < C$. The intuition behind that claim is that the change in the magnitude of a function cannot be higher t... |
H: Is $\mathbb A^n(\mathbb Z)$ Zariski dense in $\mathbb P^n(\mathbb C)$?
Since $\mathbb A^1(\mathbb Z)$ is infinite, it is obvious that $\mathbb A^1(\mathbb Z)$ is dense in $\mathbb P^1(\mathbb C)$ with respect to the Zariski topology. Does the same property hold for $\mathbb A^n(\mathbb Z)\subset \mathbb P^n(\math... |
H: We have to prove that $H=${$(x_1,...,x_n)\in{\mathbb{R^n}}|a_1x_1+...+a_nx_n=0$} is an hyperplane of $\mathbb{R^n}$.
I've got $\space$ $V$ $K$ - vectorial space, and $H$ which is a subspace of $V$. We say that $H$ is a hyperplane when $dimH=n-1$. If we've got $\space a_1,a_2,...,a_n\in{\mathbb{R}}$ which are not al... |
H: sequence of entire functions that converges to $\sqrt{q}$ uniformly on compact subsets of the line Re $(z)=q$ for each positive rational $q$
I'm scheduled to host a prep course for the complex analysis qualifying exam at my university this summer, so I'm going through old qualifying exams to prepare. I'm stuck on t... |
H: Showing/finding sumfunction
I will try again asking my question: I have $\displaystyle\sum_{n=0}^{\infty}\frac{n}{n+1}x^n$, for x$\in$R. Then I have used wolframalpha finding the sum function:
https://www.wolframalpha.com/input/?i=sum_%7Bn%3D1%7D%5E%7B%5Cinfty%7D+%5Cfrac%7Bn*x%5E%7Bn%7D%7D%7Bn%2B1%7D
And with Mapl... |
H: What do more than or less than sign with a line through them mean?
What do these symbols where the arrow point to mean? Does it mean not less then? If that's the case, why do we write it that way and not > if they are the same?
AI: $0\not<1$ means ‘$0$ is not less than $1$’, just as $a\ne b$ means ‘$a$ is not equa... |
H: How do you convert a number from base $10$ to binary?
I don't understand how you get the binary representation of a number.
Say we have a number in base $10$, how do you change it into binary?
I used the Google math converter as well.
.
All I know is that $a=a_n2^n+a_{n-1}2^{n-1}+\dotsb+a_0$
AI: You keep dividing... |
H: Minimal polynomial of $\varphi:g+(f)\mapsto xg+(f)$ is $f$.
Let $K$ be a field, $a$ an element in a field extension such that $a$ is algebraic over $K$. Denote by $f\in K[x]$ the minimal polynomial of $a$ over $K$. Consider the $K$-vector space $K[x]/(f)$ and the $K$-linear map
$$
\varphi: V\to V,\quad g+(f)\ma... |
H: Hartshorne's proof of Castelnuovo's theorem
For those of us who have forgotten, Castelnuovo's theorem is the following:
Theorem: If $Y$ is a curve on a surface $X$ with $Y \simeq \mathbb{P}^1$ and $Y^2 = -1$, then there is a morphism $f: X \to X_0$ to a smooth projective surface $X_0$ such that $X$ is the blow up o... |
H: Conplex analysis showing normal
Show that the family of functions $f_n(z)=z^n,n\geq 1$ is normal in $D(0;1)$ but not in any region which contains a point on the unit circle.
I tried to show that the spherical derivatives of $f_n$'s are uniformly bounded so that I can use Marty's theorem but I could not do it as $n$... |
H: A question about derivatives between Euclidean spaces: how should we construct it and interpret its definition?
As it is known from the single-variable calculus, given $X\subseteq\textbf{R}$, a function $f:X\to\textbf{R}$ and a adherent point $x_{0}\in X$ which is also a limit point, we define the derivative of the... |
H: Fourier Series Analysis Decomposition
If I understand Fourier series correctly. The objective is to model a periodic function as a linear combination of sinusoidal functions with different amplitudes, frequencies, etc. Since the collections $\{\frac{\cos(nx)}{\pi}\}_{n=0}^{\infty}$ are orthonormal over in the... |
H: Is $a$ assigned to $b$ or is $b$ assigned to $a$?
Being very interested in relations, functions and the best way on how to teach them at a set theoretic level, the following is something that has begun to puzzle me.
Relations
Let $A$ and $B$ be sets such that $a \in A$ and $b \in B$. Now, let $R \subset A \times B$... |
H: Definition of the equality refered by the definition of a partial order
A partial order $≤$ on a set $P$ must satisfy the following properties:
Reflexivity: For all $a \in P$, $a ≤ a$
Anti-symmetry: For all $a,b \in P$, $a ≤ b \land b ≤ a → a = b$
Transitivity: For all $a,b,c \in P$, $a ≤ b \land b ≤ c → a ≤ c$
B... |
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