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H: Proving the inverse of a matrix equals $I_n-\frac{1}{n-1}A$
Question:
Let $A$ be a matrix whose elements are all $1$. Prove that
$$(I_n-A)^{-1}=I_n - \frac 1{n-1}A.$$
Thought:
I tried using this identity, but couldn't get any further (computing the adjoint looks pretty nasty):
$$(I_n-A)^{-1}=\frac 1{\det(I_n-A)}\op... |
H: Rule for divergence of a vector field
I need a proof for this theorem and have absolutely no idea how to do it.
Let $ U \subseteq \mathbb{R}^n $ be an open set, $ F : U \to \mathbb{R}^n $ a $C^1$ vector field. Let $ A_k \subseteq U, k \in \mathbb{N}$ be a series of compact, non-empty subsets with smooth border, whi... |
H: Differentiating $y=x^{2}$
I am reading in a book about differentiating, but I am confused with one of the steps he takes. We start with:
$$\begin{align}
y &= x^{2} \\
y + \mathrm{d}y &= (x + \mathrm{d}x)^2 \\
y + \mathrm{d}y &= x^2 + x\mathrm{d}x + x\mathrm{d}x + (\mathrm{d}x^2)
\end{align}$$
Now the author simpli... |
H: Answer-verification: Show that $f(x,y)=1+2x+3y$ for all $(x,y)\in \Bbb R^2$
Define the function $f: \Bbb R^2\to \Bbb R$ has first order partial derivatives and that $f(0,0)=1$
While $\frac{\partial f}{\partial x}(x,y)=2$ and $\frac{\partial f}{\partial y}(x,y)=3$ for all $(x,y)\in \Bbb R^2$
Prove that $f(x,y)=1+2... |
H: Why does a local minimum or maximum need to be between an open interval?
Why does it need to be an open interval if we want to define a local maximum or minimum?
Does it have to do with limits and that you cannot find the derivative of one point (let's say x) without knowing some values close to x?
AI: It doesn’t: ... |
H: Is tensor product of Sobolev spaces dense?
My question is: is $W_2^k(\mathbb{R})\otimes W_2^k(\mathbb{R})$ dense in $W_2^k(\mathbb{R}^2)$, and more generally is this true in $\mathbb{R}^d$?
I found this post:
Tensor products of functions generate dense subspace?
which shows the above type of result for $C_c^\infty$... |
H: Is $2+5x$ a primitive root in $\mathbb{F}_7[x]/(x^2+1)$?
The question I'm inquiring about is all in the title, but I would be more interested in a few things related to the question which I don't know. I know what a primitive root of $\mathbb{F}_p$ is for any prime - an element $g \in \mathbb{F}_p$ such that $\{g^1... |
H: Calculate Binomial Probability of number 75 out of total 100 rows and each row containing 3 random numbers
Im trying to calculate the chance or probability of number 75 appearing. I have total 100 rows in microsoft excel and have frequency of number 75. Each row contains 3 cells and each cell contains a random numb... |
H: If $V \subset H$ compact, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too?
As the question states, if we have the compact embedding of Hilbert spaces $V \subset H$, is $L^2(0,T;V) \subset L^2(0,T;H)$ compact too?
If not true in general, is it true for $V=H^1(\Omega)$ and $H=L^2(\Omega)$?
AI: You want to know whethe... |
H: Characterization properties of number sets $\mathbb{N},\mathbb{ Z},\mathbb{Q},\mathbb{R},\mathbb{C}$
When people say that a structure is defined up to isomorphism means, accordingly, that they assume certain properties that make it completely determined under certain operations and relations.
So, I'd like to know... |
H: Mean value theorem application for multivariable functions
Define the function $f\colon \Bbb R^3\to \Bbb R$ by $$f(x,y,z)=xyz+x^2+y^2$$
The Mean Value Theorem implies that there is a number $\theta$ with $0<\theta <1$ for which
$$f(1,1,1)-f(0,0,0)=\frac{\partial f}{\partial x}(\theta, \theta, \theta)+\frac{\pa... |
H: Convergent sequences coming back from infinity
This may seem like a dumb question, but is it possibly for a sequence to jump to infinity and after some term later we can find an $N$ such that $n > N \implies |a_n - \ell| < \epsilon$? Because if it is, doesn't that mean convergent sequences does not necessary have t... |
H: Calculate the probability of one ball out of 5 being number 80
Im trying to write a lottery game program and want to have probabilities calculated for the players before they choose numbers.
Lottery rules:
5 Total Balls
Each ball has to be a number between 1 and 80
No ball can repeat on one drawing (5,10,20,55,55) ... |
H: Lights Out Variant: Flipping the whole row and column.
So I found this puzzle similar to Lights Out, if any of you have ever played that. Basically the puzzle works in a grid of lights like so:
1 0 0 00 0 0 00 1 0 0 0 0 1 0
When you selected a light (the X), it toggled itself and all the lights in its row and co... |
H: Interpretation of proposition in textbook
I'm reading Algebra by Michael Artin and I have a question about a section of text.
Proposition 1.2.13: Let $M'=[A'\mid|B']$ be a block row echelon matrix, where $B'$ is a column vector. The system of equations $A'X=B'$ has a solution iff there is no pivot in the last colum... |
H: How to Define Product Orientations for Topological Manifolds
When working with smooth manifolds, $M^m$ and $N^n$, it is straightforward to see how orientations at points $p\in M$ and $q\in N$ (i.e. ordered bases for the tangent spaces) give rise to an orientation at the point $(p,q) \in M \times N$, given a conven... |
H: Term for changing properties in higher dimensions
Somewhat simple question, but it's the following. Consider D-volumes (that is, the equivalent volume measurement in D dimensions) of spheres of ever-higher dimensions. The percent of D-volume concentrated in a $\epsilon$ crust around the sphere is
$1-(1-\epsilon)^D$... |
H: $f$ is either regular or $df_x = 0$.
In the text, it says:
Consider smooth functions on a manifold $X$: $f: X \to \mathbb{R}$, at a particular $x \in X$, $f$ is either regular or $df_x = 0$.
So I am not certain here: if $df_x \neq 0$, then all we know here is $df_x$ is injective. Hence how can we conclude that $d... |
H: How to prove gaussian-like integral equation true.
Integrals is definitely not my strong point, and I'm having trouble proving that:
$${\int_{-\infty}^\infty (e^{\large{\pi}n})^{-\large{x}^2} dx = {1\over\sqrt{n}}}$$
It has similarities to the gaussian integral, but I have no idea on how to prove it correct.
What c... |
H: Closure of a connected set is connected
Let $(X,d)$ be a metric space and let $E \subseteq X$ connected. I want to show that $\overline E$ is connected.
How can I prove this in a nice way ?
AI: PROP Let $E\subseteq X$ be connected. Then $\overline E$ is connected.
P Let $f:\overline E\to\{0,1\}$ be continuous. Si... |
H: What is an example of a bounded, discontinuous linear operator between topological vector spaces?
I am thinking there might be an example between the space of compactly supported smooth functions on the real line (chosen because it is non-metrizable under the standard topology for this space of test functions) and ... |
H: Characterization of $C_0^\infty(\mathbb{R}^N)$ in terms of Fourier transform.
Let $C_0^\infty(\mathbb{R}^N)$ denote the space of infinite differentiable functions with compact support. My question is: Is there any characterization of $C_0^\infty(\mathbb{R}^N)$ in terms of Fourier tranform, i.e. is there a statemant... |
H: Are there Hausdorff spaces which are not locally compact and in which all infinite compact sets have nonempty interior?
Here is the background material from which I am working:
The Cantor set is an uncountable compact Hausdorff space with empty
interior.
In a locally compact Hausdorff space, each countable set has... |
H: Simplifying trig expression for Laplace transform
I'm working on the following Laplace transform problem at the moment, and I'm a little stuck.
$$\mathcal{L} \{\sin(2x)\cos(5x) \}$$
I don't recall any trig identity that would apply here. I know that
$$\sin(2x) = 2\sin(x)\cos(x)$$
But I'm not sure if that applies in... |
H: Number of rays on finite grid?
Let's have a set $M = \{ (i,j) : i,j \in \{0,\dots,m\}\}$. Define equivalence on $M$, $(i,j) \sim (k,l)$ iff there is $r \in \mathbb R$ that $(ri,rj) = (k,l)$.
Question is what is the number of elements of $M/_\sim$?
Motivation:
This is math/programming question. I have a grid of p... |
H: Why are nonsquare matrices not invertible?
I have a theoretical question. Why are non-square matrices not invertible?
I am running into a lot of doubts like this in my introductory study of linear algebra.
AI: I think the simplest way to look at it is considering the dimensions of the Matrices $A$ and $A^{-1 }$ an... |
H: Proving the existence of a bridge in a tree
Let $G$ be a connected graph, and let $e \in E(G)$. Prove that $e$ is a bridge if and only if every spanning tree of $G$ contains $e$.
Can someone help me with this please? Thank you!
AI: Suppose $e$ is a bridge in $G$ and $T$ a spanning tree on $G$ not containing $e$. T... |
H: Can the triangle inequlity extened to show the distance inequlity of a trapezium
$AB // CD$. What are the angle conditions (acute, obtuse or right angle) of $a,b,c,d$ to be satisfied the inequality $ |AB+BC| > |CD|$?
$AB,BC,CD$ are distances.
AI: BI think the inequality holds iff $a+\frac{b}{2}<180$. Let $P$ be a... |
H: The product of integers relatively prime to $n$ congruent to $\pm 1 \pmod n$
Problem: Let $1 \leq b_1 < b_2 <...< b_{\phi(n)} < n$ be integers relatively prime with n.Prove that
$$B_n = b_1 b_2 ... b_{\phi(n)} \equiv \pm 1 \bmod n $$
I was thinking of Fermat's Little Theorem or Euler's Theorem.
AI: The cases $n=1$... |
H: Recursion Question - Trying to understand the concept
Just trying to grasp this concept and was hoping someone could help me a bit. I am taking a discrete math class. Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 ... |
H: homeomorphic $\sin \frac{1}{x}$ and $\mathbb{R}$
Is it really true that graph of the function
$$f(x)=\begin{cases}\sin \frac{1}{x}& x\neq0\\0&x=0\end{cases}$$
homeomorphic to $\mathbb{R}$?
This function is not continuous, else it is true.
What's the main ways of prooving non-homeomorphic we have?
AI: This is false.... |
H: Given a fourier series in $L^2$ and using it to determine a particular integral
Suppose $g \in L^2 (-\pi,\pi)$ has Fourier series is $b_0 + \sum_{n=1}^\infty (b_n\cos(nx)+c_n\sin(nx))$. From this we want to determine what $\frac{1}{2\pi}\int_{-\pi}^{\pi}|g(x)|^2dx$ equals.
I was trying to use the Parseval's formul... |
H: Finding the value of Pr(X
Consider the joint density $f_{X,Y}(x,y)=1/40$ inside the rectangle 0< x <5 and 0< y< 8. How do I calculate $Pr[X<Y]$ ?
AI: The probability that $X$ is $\lt Y$ is $\frac{1}{40}$ times the area of the part of the rectangle that is above the line $y=x$. Finding this area is a simple geometri... |
H: want to study permutation groups, only background is linear algebra and calculus
My background is in computer science, specifically software engineering, and not really math heavy. I know the basics of calculus (the Thomas book) linear Algebra (Strang), and some Discrete Math, Graph Theory, Complexity and Algorithm... |
H: Design data structure
3-13. Let A[1..n] be an array of real numbers. Design an algorithm to perform any sequence of the following operations:
Add(i,y) -- Add the value y to the ith number.
Partial-sum(i) -- Return the sum of the first i numbers, i.e. $\sum_{j=1}^j A[j]$
There are no insertions or deletions; the onl... |
H: What does "isomorphic" mean in linear algebra?
My professor keeps mentioning the word "isomorphic" in class, but has yet to define it... I've asked him and his response is that something that is isomorphic to something else means that they have the same vector structure. I'm not sure what that means, so I was hopin... |
H: Writing and expression to find the probability
Suppose (x) and (y) are independent lives. $T(x)$ and $T(y)$ denotes the future lifetimes. How do I write an expression to find out the probability that exactly one of the dies within the next 10 years? the formula i thought of is $Pr[T(x)<10$ or $T(y)<10]$-$Pr[T(x)$ ... |
H: Finding $\,\%\,$ of salary relative to another salary
Sam makes $\$65$ per week and Don makes $\$138$ per week.
How do I express, as a percentage, how much more Don makes per week than Sam does?
AI: $$\text{Don earns}\;\;\left(\frac{\text{Don's earnings}-\text{Sam's earnings}}{\text{Sam's earning's}}\; \times 100 ... |
H: Some questions about $\gcd(n,m)$ and $\phi(n)$
I was messing around in Excel at the end of work today and made a table where the $(i,j)$ entry $a_{i,j}$, for $j \geq i$, is 1 exactly when $i$ and $j$ are coprime (see snapshot of a portion of the table below):
My questions are:
Is there an explanation for the high... |
H: Problem related to Chinese Remainder Theorem
I'm not sure if there is a typo in the question or if I am incorrect (will point out as I get to it), but I am given that $a,b,m,n$ are integers with $\gcd(m,n) = 1$ and that
\begin{equation}
c \equiv (b - a) \cdot m^{-1} \;(\!\!\!\!\!\!\mod n)
\end{equation}
and am ask... |
H: Is there a structure theorem for nonempty, compact, nowhere dense subsets of the real line?
Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line.
Is there a theorem that describes the form of the elements of $X$?
Context
For open subsets of the line, such a result is well-known: every o... |
H: Show that weighted average of a function involving expected values is not equal to the function of the weighted average
How can I show that, given $w_j$ sum to $1$
$$\sum_{j=1}^n w_j \frac{a}{E(x_j)+b}\ne \frac{a}{E(\sum_{j=1}^n w_j x_j)+b}$$
unless $x_j=x_k \forall j,k \\$ where a and b are constants, $x_j$ are ar... |
H: What are zero divisors used for?
This is the first time I hear this term. Specifically the assertion is that $\mathbb{Z}$ has no zero divisors. So, from my understanding this is because there are not two non-zero numbers $a,b\in \mathbb{Z}$ such that $ab=0$.
Also I can see that this definition is related to the on... |
H: permutation of the elements of a matrix with respect to sign?
Let matrix $\mathcal A$ with a $(m\times n)$. Every element $a_{i,j}$ of the matrix is either a positive or negative integer, or zero.
Question: How many distinct matrices could be generated with respect to the above distinction for matrix $\mathcal A$ -... |
H: Inversion of a power series without a linear term
Could someone explain me how to invert
$$
z = y e^{-y} = e^{-1} - \frac{1}{2e}(y - 1)^2 + \frac{1}{3e}(y - 1)^3 - \frac{1}{8e}(y - 1)^4 + \cdots
$$
around $y=1, z=e^{-1}$, so that $y$ is expressed as a series of $(1 - ez)$ ?
This is a part of example VI.8 in "Analyt... |
H: How do we know that $\sum_{k=0}^{\infty}\frac{x^k}{k!}=e^x$?
I've been taught that the definition of the exponential function is the following power series: $$\sum_{k=0}^{\infty}\frac{x^k}{k!}$$ Here's my question: how do we know that this series is equal to $e^x$? That is to say, how do we know that the function d... |
H: Is it true that $\mathbb R \setminus \bigcup A\not= \emptyset$?
Let $X$ be the set of all nonempty compact nowhere dense subsets of the real line. Given any countable subset $A$ of $X$. Is it true that $\mathbb R \setminus \bigcup A\not= \emptyset$?
Thanks for your help.
AI: Yes: that’s just (a special case of) the... |
H: Finding the slope at a point $P(x_1,y_1)$ on a parabola
Given a point $P(x_1,y_1)$ on the graph of a parabola $y^2=4px$, prove that the slope at point P is $$\frac{y_1}{2x_1}$$
AI: How 'bout this: from $y^2 = 4px$, we have, by implicit differentiation, $2yy' = 4p$; dividing the latter equation by the former, we g... |
H: Will the ball hit the wall?
There is a ball starting at point $A$ going forward in the direction towards point $B$ (so it moves along the $(AB)$ line).
A wall is represented by its two ends $W_1$ and $W_2$.
I have to solve in a general way the question "Will the ball hit the wall ?". The question seemed quite simpl... |
H: Why does this polynomial related to Hamming Codes have integer coefficients?
Question: Why does
$$
\frac{(1 + x)^{2^k - 1} - (1 - x^2)^{2^{k-1} - 1}(1-x)}{2^k}
$$
have integer coefficients?
Details: For a question I'm thinking about, I needed to know all the real numbers $c$ such that the generating function
$$
p(x... |
H: Analytic Function vs Exponential Order function
We say that a function $f$ is of exponential order $\alpha$ if there exist constants: $M$, $\alpha$, $T$ such that for $x>T$
$$f(x)<M\cdot e^{\alpha x}$$
Polynomials are of exponential order. Then, is it true that analytic functions in a neighborhood of the origin are... |
H: Proving that $\|A\|_{\infty}$ the largest row sum of absolute value of matrix $A$
I am studying matrix norms. I have read that $\|A\|_{\infty}$ is the largest row sum of absolute value and $\|A\|_{1}$ is the highest column sum of absolute values of the matrix $A$. However, I am not able to prove this. Are there an... |
H: Prove that every integer is either prime or composite
In the book I'm reading, the following proof is given for the stated theorem:
Let n be any integer that is greater than 1. Consider all pairs of positive integers
$r$ and $s$ such that $n = rs$. There exist at least two such pairs, namely $r = n$ and $s = 1$ and... |
H: What is the domain of the following function?
Please tell me the domain of $y = \sin^{-1}(\sin(x))$
P.S. I think domain is $(-\infty, \infty)$ But my teacher says it is $(-\frac{\pi}{2}, \frac{\pi}{2})$. He says since $sine$ is a many one function, so its domain has to be confined in the interval of $(-\frac{\pi}{2... |
H: Connectedness of disjoint union
Consider the topological space $Z$ defined as the disjoint union $Z=X\cup Y$. Please tell me if these statements are true:
1) If both $X$ and $Y$ are open in $Z$ or both are closed in $Z$ then $Z$ must be disconnected
2) If one of them is open and the other is closed then $Z$ may or ... |
H: Prove that every irreducible cubic monic polynomial over $\mathbb F_{5}$ has the form $P_{t}(x)=(x-t_{1})(x-t_{2})(x-t_{3})+t_{0}(x-t_{4})(x-t_{5})$?
For a parameter $t=(t_{0},t_{1},t_{2},t_{3},t_{4},t_{5},)\in\mathbb F_{5}^{6}$ with $t_{0}\ne 0$ and {$t_{i},i>0$} are ordering of elements in $\mathbb F_{5}$ (t1~t5 ... |
H: Definition of complex number
In many situations (problems as well as solutions) I encounter the complex number $i$ which many times is defined as $i^2=-1$ instead of $i=\sqrt{-1}$, since it is "more preferred". Does anyone know why? Is it because there are two solutions to the equation $x^2+a=0, \ \ a>0$?
AI: Basic... |
H: self adjoint properties
I looking for a proof for the theorem but I have not find yet.
A link or even sketch for How it goes will be very appreciate.
A linear map is self adjoint
iff
the matrix representation according to orthonormal basis is self adjoint.
by the way is not that true for all self adjoint matrix a... |
H: Prove that $T$ is a subgraph of $G$
Let $T$ be a tree with $k$ edges, and let $G$ be a graph where every vertex has degree at least $k$. Prove that $T$ is a subgraph of $G$.
Can someone give me tips/help on how to solve this problem?
AI: Hint:
Start with any vertex $t_1 \in T$ and map it to any vertex $g_1 \in G$... |
H: Reference a single element within a set
Is there a notation to reference a single element within a set? Let's say I have a set n = {1, 2, 4, 8, 16}. If I wanted to use a single element from this set, is there a certain notation to do so? In computer programming, if I have an array int x = {1, 2, 4, 8, 16} I could r... |
H: Proof of the statement "The product of 4 consecutive integers can be expressed in the form 8k for some integer k"
I am slowly diving into simple number theory and learning how to craft direct proofs. I needed to proof the statement "The product of 4 consecutive integers can be expressed in the form 8k for some inte... |
H: Prove that there exists walks that each edge is in $G$
For some $k \in\mathbb{N}$, let $G$ be a connected graph with $2k$ odd-degree vertices, and any number of even-degree vertices. Prove that there exists $k$ walks such that each edge in $G$ is used in exactly one walk exactly once, assuming that the main theorem... |
H: log transformation for dummies
I have a question which is probaly very simple to answer for most people here:
We have a formula:
y = -log(x)
Then this happens to x:
= -log(x^1.5) or ( = -log(x^(15/10)) )
How do I now write up y?
Many thanks!
AI: We can make use of the following property of $\log$:
$$\log(a^... |
H: Translation from Spanish - Simple differentiation exercise
Could anybody please translate this exercise into English?
A friend of mine sent me the translation via Facebook but I still don't understand why I have to first do the logaritmization of both sides of the equation and then the differentiation...
Another th... |
H: Distributions on manifolds
Wikipedia entry on distributions contains a seemingly innocent sentence
With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any (paracompact) smooth manifold.
without any reference cited. I went through Vladimirov, Demidov, G... |
H: equivalent condition for moment generating function
Consider a random variable $x$ with pdf $f(x)$, and have $x \ge 0$. The moment generating function is defined as $M(t)=\int^{\infty}_{-\infty}e^{-tx}f(x)dx$ (noted that we change the sign of $t$ compared with common case ). For convenient we consider only when $t>... |
H: For which rationals $x$ is $3x^2-7x$ an integer?
The following exercise is from [Birkhoff and MacLane, A Survey of Modern Algebra]:
For which rational numbers $x$ is $3x^2-7x$ an integer? Find necessary and sufficient conditions.
I think I was able to obtain the set of rationals $x$, but I am not sure what the n... |
H: Dual space of $K[X]$
Let $k[X]$ be the space of polynomials over a field $k$ (regarded as a vector space over $k$). What is the dual space of this vector space? My guess is that it is somehow generated by the derivations $d/dX$?
AI: It is possible to identify $k[X]^*$ with $k[[X]]$ (power series).
Consider the map ... |
H: equivalence between axiom of choice and Zorn's lemma in a particular case.
Define $A(x)$ and $Z(x)$ as follows.
$A(x)\Leftrightarrow $for every indexed family $(S_i)_{i\in I}$ of nonempty sets s.t. $\# I =x$, there exists an indexed family $(s_i)_{i\in I}$ s.t. $s_i \in S_i$ for every $i\in I$.
$Z(x)\Leftrightarrow... |
H: Examples of homeomorphisms between the real numbers and the positive real numbers?
I'm interested in homeomorphisms between the real numbers, $\mathbb{R}$, and the positive real numbers, $(0,\infty)$--where both spaces have the topology induced by the metric $d(x,y)=|x-y|$.
Here is a couple of easy examples, $f:(0... |
H: measurability w.r.t. Borel on extended real line
Following Schilling I have shown for measurable functions $$u, v \in m \mathcal{A}/ \mathcal{\hat{B}}$$ that sums, differences, products and maxima/minima are again measurable whenever they are defined.
( here $\mathcal{\hat{B}}$ is the Borel sigma algebra on the ext... |
H: Find open sets.
Consider the set $X = \{1,2\} \times \mathbb{Z}_+$ in the dictionary order. Then this will be an ordered set with smallest element.
Denote $a_n = (1,n)$ and $b_n = (2,n)$.
Then elements of $X$ will be $a_1, a_2, \dots, b_1, b_2, \dots$.
Singleton sets are open except $\{b_1\}$.
This is not clear to ... |
H: Solution to linear equations as parameterized matrices.
I want to solve the following matrix equation:
$$
\begin{pmatrix} a & b \\ c & d \end{pmatrix}\begin{pmatrix} 3 & 2 \\ 4 & 2 \end{pmatrix}=\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}
$$
and I thus get for equations:... |
H: Question on integral closure in $\mathbb{Q}[\alpha]$
Let $\alpha$ be a root of $f(x) = x^{3} -2x +6$, $ \ \mathbb{K} = \mathbb{Q}[\alpha]$.
Prove that $ O _{\mathbb{K}} = \mathbb{Z[\alpha]}$.
What I've done: $f$ is irreducible, so $disc\{1,\alpha, \alpha^{2}\} = disc(f(x)) = - Res(f(x), f'(x)) = -940$.
But $-... |
H: Divisibility of the difference of powers
Consider the following theorem:
For any $a, b \in \mathbb{Z}^+$, there exist $m, n \in \mathbb{Z}$ such that $m > n$ and $a\ |\ b^m - b^n$.
What's the best way to prove it? I have an idea (and I know it's true because of that idea), but I don't know how rigorous it is to c... |
H: If P=NP, then NP = coNP. Why is this so?
I read that if we assume that P = NP, then NP = coNP. I am unable to understand why this is so.
AI: It is easy to show that $P=coP$ (think about it). Now, if $P=NP$ then $coP=coNP$, so that $NP=P=coP=coNP$. |
H: Expressing pushforward of a flow in integral form
Let $\phi(t,x)$ be a flow of a vectorfield $V$ on some compact domain $\tilde{U} = U\times I \in R^n \times R$. Let X be a vector field. If one wants to write
$(\phi(t,x))_{*}X)(\phi(t,x)(q)) = \phi(t,x))_*(X(q))$
in integral form, is there anything wrong in writing... |
H: Question about Independent events
If we know that $A,B$ are independent events, how can we proof that $\bar{A},\bar{B}$ are independent events to?
We should using the definition: $\Pr(A\cap B)=\Pr(A)\cdot \Pr(B)$
Thank you!
AI: $$\begin{align}\operatorname{Pr}(\bar A\cap\bar B)&=1-\operatorname{Pr}(A)-\operatorname... |
H: Confused between multiple representations of Fourier Series' formula
I have never used the formula for Fourier Series with the representation that the instructor of the above video is using that involves $k$ and $\omega$. Instead, I use $n$ and $\pi$.
Now, suppose that I want to write the formula for complex for... |
H: Lebesgue density for other probability measures on $[0,1]$
Does the Lebesgue density theorem hold for arbitrary (Borel) probability measures on $[0,1]$?
Following Downey & Hirschfeldt's proof leads me to believe that the answer is "yes". (Recall every probability measure on $[0,1]$ is regular, which is a key step i... |
H: Is the unit disk in $\Bbb R^2$ a subspace?
This is the original Spanish version of the exercise:
My understanding is that if we make a circle which has a center in [0,0] and a radius of 1 and we take all the points of that circle.... is this set of points a vector space?
How do they define addition of 2 members of... |
H: Prove that $(n+1)(n+2)(n+3)$ is $O(n^3)$. (Big-o notation)
Will someone help me prove that
$(n+1)(n+2)(n+3)$ is $O(n^3)$?
Thank you.
AI: Hint: For all positive integers $n$, your expression is $\le (n+n)(n+2n)(n+3n)$.
Remark: A fancier way of doing the same thing is to show that
$$\frac{(n+1)(n+2)(n+3)}{n^3}$$ is... |
H: geometric interpretation of a height inequality of prime ideals
Theorem 15.1 in Matsumura's Commutative Ring Theory states that if $f:A\rightarrow B$ is a homomorphism of Noetherian rings, $P \in \operatorname{Spec}B, P \cap A=p$, then $ht (P) \le ht(p) + \dim B_P/pB_P$. He proceeds to say that we can replace $A$ b... |
H: Isomorphisms of $\mathbb P^1$
Prove that every isomorphism of $\mathbb P^1$ (over an algebrically closed field $\mathbb K$) is of the form
$$
\phi(x_0: x_1) = (ax_0+bx_1 : cx_0 + dx_1)
$$
where $\begin{pmatrix} a & b \\c & d\end{pmatrix} \in GL(2, \mathbb K).$
There are some hints; I show you what I've done.... |
H: Some discrete math questions, Big-O, Big-Omega, Asymptotics, etc.
I don't understand how to prove:
$(n-1)(n-2)(n-3)$ is $\Omega(n^3)$.
Also, what am I supposed to do here?
For each of the following predicates, determine, if possible, the smallest
positive integer b where $n \ge b$ implies $P(n)$ appears to be true... |
H: what does $|x-2| < 1$ mean?
I am studying some inequality properties of absolute values and I bumped into some expressions like $|x-2| < 1$ that I just can't get the meaning of them.
Lets say I have this expression
$$ |x|<1.$$
This means that $x$ must be somewhere less than $1$ or greater than $-1$ which means tha... |
H: Why is that interior points exist only inside intervals on $\mathbb{R}$?
I'm reading a book on real analysis that has a chapter on open sets, closed sets, limit points and compact sets (for the sake of generality, according to the author).
If a set $X$ has some interior point, it must have at least one open inter... |
H: Gauss Kronrod quadrature rule
Given the abscissae and weights for 7-point Gauss rule with a 15-point Kronrod rule (Wikipedia); Can anyone provide me a working example how to numerically integrate a function given below:
$$\int_0^1 x^{-1/2}\log(x) \textrm{d}x = -4 $$
Provided abscissae and weights for 6-point Guass ... |
H: Find the adjoint of this operator.
Given $\mathbb C^2$ with the standard inner product, an operator $T(x,y) = (3x+4y, -4x+3y)$. Find $T^{*}$ and prove that $T$ is normal.
So, I took the standard basis $B = \{(1,0),(0,1)\}$ and we know it's orthogonal in respect to the standard inner product. I displayed $T$ in th... |
H: A simple question regarding $df(E)$ for $f(A)=A^{-1}$
I am trying to follow the document on http://web.mit.edu/people/raj/Acta05rmt.pdf, and I got a simple question that
why for $f(A)=A^{-1}$ we have $df(E)=-A^{-1}EA^{-1}$ on page 5? (where $E$ is a small perturbation, and on page 4 the author says the $df$ is j... |
H: Prove that $\sin 10^\circ \sin 20^\circ \sin 30^\circ=\sin 10^\circ \sin 10^\circ \sin 100^\circ$?
$\sin 10^\circ \sin 20^\circ \sin 30^\circ=\sin 10^\circ \sin 10^\circ \sin 100^\circ$
This is a competition problem which I got from the book "Art of Problem Solving Volume 2". I'm not sure how to solve it because t... |
H: Is the function $f:\mathbb{Z}_{8}\rightarrow\mathbb{Z}_2$ where $f([x]_8)=[x]_2$ a group morphism?
Let $[x]_n$ denote the equivalence class of $\mathbb{Z}_n$ that contains $x$.
I am not certain where to begin with this question:
$f:\mathbb{Z}_8\rightarrow \mathbb{Z}_2$, where, $f([x]_8)=[x]_2$.
I am not sureif the... |
H: Contour integration of complex number confuses me, still.
Given $f(z) = (x^2+y)+i(xy)$ and
we integrate it using the Parabola Contour.
For a parabola, $\gamma(t) = t + it^2$.
So, $f(\gamma(t)) = 2t^2 + it^3$. What was done here is all the $y$ were replaced with $t^2$. Now, my understanding is this:
t +... |
H: Find $\int_{-5}^5 \sqrt{25-x^2}~dx$
I need to evaluate $$\int_{-5}^5 \sqrt{25-x^2}~dx$$
How would I do it?
AI: If you are at the earliest stages of integration, you are probably supposed to do this problem without using any integration techniques.
The curve $y=\sqrt{25-x^2}$ is the top half of a circle with centre... |
H: Proof that $ \lim_{x \to \infty} x \cdot \log(\frac{x+1}{x+10})$ is $-9$
Given this limit:
$$ \lim_{x \to \infty} x \cdot \log\left(\frac{x+1}{x+10}\right) $$
I may use this trick:
$$ \frac{x+1}{x+1} = \frac{x+1}{x} \cdot \frac{x}{x+10} $$
So I will have:
$$ \lim_{x \to \infty} x \cdot \left(\log\left(\frac{x... |
H: How to construct a non-diagonalizable matrix with a particular set of eigenvalues
Given a set of eigenvalues, how would you go about constructing a matrix with those particular eigenvalues?
I know that you can construct a diagonalizable matrix with those eigenvalues using a linearly independent basis vector $B$:
$$... |
H: Is $\sqrt1+\sqrt2+\dots+\sqrt n$ ever an integer?
Related: Can a sum of square roots be an integer?
Except for the obvious cases $n=0,1$, are there any values of $n$ such that $\sum_{k=1}^n\sqrt k$ is an integer? How does one even approach such a problem? (This is not homework - just a problem I thought up.)
AI: ... |
H: normal subgroup generated by a subgroup
Let $G= \langle g_1,g_2 \rangle$, and let $H\leq G$ given by $H=\langle g_1 g_2 g_{1}^{-1}g_{2}^{-1}\rangle$.
What is the normal subgroup of $G$ generated by $H$?
AI: Let $N$ be the normal closure of $H$. Since $[g_1,g_2] \in N$, $[g_1N,g_2N]=1N$ and $G/N$ is generated by tw... |
H: question involving analycity of $f=u+iv$
Let $f=u+iv:\mathbb C\to\mathbb C$ be analytic. Then is it true that $\dfrac{\delta^2 v}{\delta x^2}+\dfrac{\delta^2 v}{\delta y^2}=0?$
AI: Let us consider the Cauchy Riemann conditions
$\frac {\partial u} {\partial x}$ = $\frac {\partial v} {\partial y}$
and
$\frac {\parti... |
H: Proving a triangle is a right triangle given vertices, using vector dot product
I want to to show that this triangle is a right triangle.
I know that the dot of the vectors need to be $0.$ I tried to dot between them but I don't get zero.
Claim: Triangle $\bigtriangleup MNP,\;\;\,M(1,-2,3),\;\;N(0,0,4),\;\; P(4,2,... |
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