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H: Proving that the Fourier coefficients of a functional determine it
Proving that the Fourier coefficients of a functional determine it
I have the following exercise, taken from old homework of a functional
analysis course:
Let $\mu\in C(\mathbb{T})^{*}$. Define the Fourier coefficients of
$\mu$ by $$ \hat{\mu_{n... |
H: If the transitive closures of two sets are equal can they be disjoint?
I came across a problem today that I can't get out of my head: if $\mathcal{TC}(A)=\mathcal{TC}(B)$, can $A$ and $B$ be disjoint? My guess is no, but I can't quite show it. A direct proof seems unsuccessful, while a proof by contradiction going ... |
H: How can I find the domain and the range of a function?
I'm working on a task which has the following question:
What is the domain and the range to the function g?
Here is the function g:
Let A - $\{$1, 2, 3, 4$\}$, B - $\{$a, b, c, d$\}$
Let g : B $\rightarrow$ A be defined so that g - $\{$$\lt$a,
1$\gt$,$\l... |
H: Independence of two (discrete) Random Variables
We know that for two discrete random variables $X$ and $Y$ to be independent, the joint density of random vector $Z=(X,Y)$ has to be equal to product of marginal densities of $X$ and $Y$.
Now, this means:
a) To establish independence of $X$ and $Y$, we gotta test the ... |
H: Convergence of series implies convergence of Cesaro Mean.
Proof. Let $\sum_{k = 0}^N c_k \rightarrow s$, let $\sigma_N = (S_0 + \dots + S_{N-1})/N$ be the $Nth$ Cesaro sum where $S_K$ is the $Kth$ partial sum of the series. Then $s - \sigma_N \\= s - c_0 - c_1(N-1)/N + c_2(N-2)N +\dots+c_{N-1}/N \\ =c_1/N + c_2 2... |
H: Probability distribution of $X_N$, $N = min\{n \geq 2: X_n = $ second largest of $ X_1, \ldots, X_n\}$
Given a sequence $X_1, X_2, \ldots$ of independent, continuous random variables with the same distribution function $F$ and density function $f$, let $N = min\{n \geq 2: X_n = $ second largest of $ X_1, \ldots, X_... |
H: How to integrate this function?
Evaluate
$$\int\sqrt{x^3+x^2}\;dx$$
What I have tried
Using substitution (which I believe was applied incorrectly) I get:
$$\frac{(x+2)\sqrt{4x+4}}{4x+4}$$
How can this integral be evaluated?
AI: For the first integral, suppose that $x\gt 0$. Let $u^2=1+x$. Then
$$\sqrt{x^3+x^2}=x\... |
H: Heat Equation - Similarity Solution
Let $ s = xt^{-1/2} $ and look for the solution to the heat equation $ u_{t} = u_{xx} $ which is of the form: $ u(x,t) = t^{-\frac{1}{2}}f(s) $, which satisfies the condition $ \int_{-\infty}^{\infty} u(x,t) \; \mathrm{d}x = 1.$
This is what I've tried so far:
$ u = t^{-\frac{1}... |
H: continuity of a map on $M(\mathbb{R}^n)$
Let $M:=M(\mathbb{R}^n)$ be the space of probability measures on $\mathbb{R}^n$ with respect to the Borel $\sigma$-algebra. Let $K\subset M$ be a compact convex subset. $K$ carries a natural topological structure, i.e. the weak topology induced by the bounded continuous func... |
H: Solutions of triangles, homework
I have 2 questions in which I have doubt :-
Q1. prove that:- a cosBcosC + b cosAcosC + c cosBcosA = ar(ABC)/R
A1. I have used cosine rule and have put the values of all cosines here and after adding them, this is what I get:-
LHS= [1/abc]* $[2b^2c^2+2a^2c^2+2a^2b^2-($a^4+b^4 +c^4$... |
H: I can't solve this limit...
I tried to solve it as difference of two squares. But I guess I can't move any longer from that. Please help...
AI: Rationalize the numerator to get $$\lim_{x \to \infty} \frac{ \sqrt{x + \sqrt{x + \sqrt{x}}}}{\sqrt{ x + \sqrt{x + \sqrt{x + \sqrt{x}}}} + \sqrt{x}}$$ You can eliminate th... |
H: How do you find out what the function $g(f(2))$ and $f(g(2))$ is?
I'm trying to find what the *g*$(f(2))$ and the f $(g(2))$ is.
Here are the functions for f and g:
Let A - $\{$1, 2, 3, 4$\}$ and B - $\{$a, b, c, d$\}$
Let f : A $\rightarrow$ B be defined so that f - $\{$$\lt$1, b$\gt$,$\lt$2, c$\gt$, $\lt$3, d$\... |
H: $S\subset \mathbb{R}^2$ with one and only one limit point in $\mathbb{R}^2$ such that no three points in $S$ are collinear
Question as in the title, but here it is re-typed just in case not all of the title is visible on your screen (you're welcome):
I am interested if there is a set $S\subset \mathbb{R}^2$ with on... |
H: Convergence of sequence (write a proof)
I need to prove the following affirmation: If $ \lim x_{2n} = a $ and $ \lim x_{2n-1} = a $, prove that $\lim x_n = a $ (in $ \mathbb{R} $ )
It is a simple proof but I am having problems how to write it. I'm not sure it is the right way to write, for example, that the limit o... |
H: Sum-to-one constraint
This is a general question, but I am asking it since I am not able to find any good material online. Can someone please explain what's meant by a "sum-to-one constraint"?
Thanks.
AI: There should be some context to suggest the meaning. Evidently, some things are constrained (perhaps in an opti... |
H: Series Convergence
What does this series converge to?
$$ \sqrt{3\sqrt {5\sqrt {3\sqrt {5\sqrt \cdots}}}} $$
and also this?
$$ \sqrt{6+\sqrt {6+\sqrt {6+\sqrt {6+\sqrt \cdots}}}} $$
And, generally speaking, how should one approach these kind of questions?
AI: Here's for
$$
\sqrt{6+\sqrt{6+\sqrt{6+\sqrt{\cdots}}}}.
... |
H: Genus of Edwards curve
Let us work over a field $\Bbbk$ of characteristic not equal to two. Let $d\in\Bbbk\setminus\{0,1\}$. It is said in the wikipedia article about Edwards curves that the plane quartic defined by the equation
$$x^2+y^2 = 1 + d\cdot x^2y^2$$
is birationally equivalent to a curve in Weierstraß for... |
H: Best applications-oriented introductory calculus textbooks?
Note: I've edited this question on October 9th, after establishing a bounty on it.
What are the best introductory calculus textbooks that
explain why calculus is important in a broad intellectual and scientific context, justifying its inclusion in a liber... |
H: $f\in C^1$ and $K$ compact, prove that $f:K \to ℝ^p$ is Lipschitz continuous
I try to prove Corollary 2.5.5:
Corollary 2.5.5. Let $f: {\bf R}^n \to {\bf R}^n$ be a $C^1$ mapping and let $K \subset {\bf R}^n$ be compact. Then the restriction $f|_K$ of $f$ to $K$ is Lipschitz continuous.
Obviously, I somehow need t... |
H: equivalence of matrices and the image
I was working on a problem and I would like to have someone give me an advice. The question was,
Let $A$ and $B$ be two $n \times m$ matrices. Show that the $img(A)$ contains the $img(B)$ if and only if there is an $m \times m$ matrix $X$ so that $AX = B$.
My answer so far is i... |
H: travelling salesman understanding constraints
I am trying to program TSP problem in R. From wikipedia page section "Integer linear programming formulation", I was able to understand all the constraints except the last one.
Need help to understand the last constraint...what are the variable U/artificial varaibles? a... |
H: Finding a dual basis
This is one of my homework questions - I'm pretty sure I understand part of it.
Let $V=\Bbb R^3$, and define $f_1, f_2, f_3 \in V^*$ as follows:
$$f_1(x,y,z) = x - 2y;\quad f_2(x,y,z) = x + y + z; \quad f_3(x,y,z) = y - 3z.$$
Prove that $\{f_1, f_2, f_3\}$ is a basis for $V^*$ (they are linearl... |
H: Probability of secretary making 4 or more errors on a page
I have this problem, and I want to figure out how to do it, or at least figure out the subject that it deals with.
A secretary who only does word processing makes $2$ errors per page when typing. What is the probability that in the next page she makes $4$ ... |
H: Intuition for dense sets. (Real analysis)
I have been having problems with dense sets as my lecturer didn't really develop an intuition for dense sets in my class. So can any of you please help me with that? And can you please tell me (the general case) how I should go about proving that a set is dense in R.
AI: A ... |
H: Galois group of $\mathbb{Q}[\sqrt{3},\sqrt{2}]$
I am trying to compute the Galois group of $\mathbb{Q}[\sqrt{3},\sqrt{2}]/\mathbb{Q}$ in the following way:
First, $\mathbb{Q}[\sqrt{3},\sqrt{2}]/\mathbb{Q}$ is a Galois extension of the separable polynomial $(x^2-2)(x^2-3)$ (separable because $\text{char}(\mathbb{Q})... |
H: How to find a perpendicular line from a point to a line.
I have something i am trying to take a consistent measurement of, but the issue is that the sample is not always consistent with rotation.
However from the sample I have a point, and 2 parallel lines.
Given A as the reference point, | | are lines B & C respec... |
H: Why does the central limit theorem imply that the standard deviation approaches $\frac{\sigma}{\sqrt{n}}$?
According to the central limit theorem, if one takes random samples of size $n$ from a population of mean $\mu$ and standard deviation $\sigma$, then as $X$ gets large, $X$ approaches the normal distribution w... |
H: An extension of a question about no 3 points in $\mathbb{R}^2$ being collinear
I am interested to see if we can use any of the answers here - or other methods - to answer the following question(s):
Does there exist a set $S\subset \mathbb{R}^2$ whose set of limit points in $\mathbb{R}^2$ contains the line $[0,1] =$... |
H: Prove that $x_{n}$ is convergent if $|x_{n+1}-x_n|\leq(\frac45)^n$
The question says; prove the sequence $x_{n}$ given by:
$\left | x_{n+1} - x_{n}\right| \leq (\frac{4}{5})^n \ \forall \ n \in \mathbb{N} $
is convergent.
Here is how I approached the problem:
Since the difference between the terms applies to all $n... |
H: Linear Combination of Vectors simple problem
So I've embarked on teaching myself Linear Algebra with G. Hadley's book which I found in my mom's library. The first chapter teacher vectors, linear dependence, and define subspaces and bases but I can't solve this simple problem in the problems section:
Express x=[4,5]... |
H: Integral $\int_{-1}^{0}\dfrac{1}{x(x^2+1)}$
Suppose I have to compute $\int_{-1}^{0}\dfrac{1}{x(x^2+1)}$. I use partial fractions to get $\int_{-1}^{0}\left(\dfrac1x-\dfrac{x}{x^2+1}\right)$, which integrates to $\log(x)-\log(x^2+1)$. Now, the $\log$ is not defined for $x\in[-1,0)$. What do I do in this case?
AI: F... |
H: spherical geometry
A mobile, on the surface of the earth, is at a point A. Travels 200 km south arriving at a point B. Later moves 200 km west arriving at a point C. Finally moves over 200 kilometers to the north, back to point A. Assuming that the surface of the earth is a perfect sphere, which the geometric place... |
H: T/F: $v_1,\dots,v_k\in\mathbb{R}^n$ linearly independent, $u\in\mathbb{R}^n,u\neq0$, then the matrices $uv_1^T,\dots,uv_k^T$ are linearly independent.
If $v_{1},v_{2},\dots,v_{k}$ are linearly independent vectors in $\mathbb{R}^{n}$ and if $u$ is any nonzero vector in $\mathbb{R}^{n}$, then the matrices $uv_{1}^{T}... |
H: Solving a definite integral using Integration by parts [simple]
The problem is to evaluate this integral:
$\displaystyle\int_{0}^1 ye^{-3y} dy$
I have uploaded an image of my work as well as my attempt to make my writing clearer in Paint.
!
Thanks so much for any help, not sure what I've done wrong.
AI: You did the... |
H: rank and the span of column matrix
Can someone look over my answer for this question?
If $A$ and $B$ are conformable matrices for the product
$AB$ and if the rank of $AB$ equals the rank of $A$, then the span
of the columns of $AB$ equals the span of the columns of $A$.
Here's my answer:
Span of the columns of $AB$... |
H: Tangential Acceleration Proof
Given $$a = a_TT + a_NN = (\frac{d^2s}{dt^2})T + \kappa(\frac{ds}{dt})^2 N$$ prove that $$a_T = a \cdot T = \frac{a \cdot v}{||v||}$$
Where $a_T$ is tangential velocity, $a$ is an acceleration function, and $s$ is an arc length parametrization.
Where do I begin?
AI: Hint: Try computing... |
H: proof of a limit by using epsilon delta definition
I need to prove that $$\lim_{x\to \infty}\frac{x^2}{2^x}=0$$ using only the formal definition of limit. Can anyone help?
AI: First prove that $2^x>x^3$ for all $x$ in some interval $[M,\infty)$, $M$ possibly large. Then use that $$\lim_{x\to\infty}x^{-1}=0$$
A poss... |
H: Lebesgue integral in two dimensions over fraction
Let $X=[-1,1],Y=[0,1]$, and $$f(x,y)=\dfrac{xy}{(x^2+y^2)^2}$$ for $x\in X,y\in Y$. Let $\mu$ be the Lebesgue measure. Does the following integral exist:
$$\int_{X\times Y} f(x,y)d(\mu\times\mu)$$
It seems very hard to follow the definition of Lebesgue integrals u... |
H: Working out the difference in earnings
I'm mathematically impaired/ignorant and trying to figure out the difference in earnings between my partner and I to work out a fair split of the bills.
So; I earn £2060 per month and partner earns £1650. As a percentage, how much more than her do I earn?
Therefore; If we had ... |
H: Calculating the limit $\lim_{x\to\infty}\frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$
I'm having a really tough time trying to evaluate the limit for this expression
$\lim_{x\to\infty}\frac{\sin^2\left( \sqrt{x+1}-\sqrt{x}\right)}{1-\cos^2\frac{1}{x}}$
The only hint I was given is that $\lim_... |
H: Prove that $\lim_{n\to\infty}a_n^2=a^2$ when ${a_n}$ is a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$
Question:
Let ${a_n}$ be a sequence in $\mathbb{R}$ with $\lim_{n\to\infty}a_n=a$
Prove that
$\lim_{n\to\infty}a_n^2=a^2$
Attempt:
Without using any properties of limits (this is a question in a section ... |
H: Clarification about negation in propositional logic
I am a little stumped on the concept of resolution, and want to clarify that something is correct, primarily negation.
if an expression in CNF is ${x = (a \lor b) \land (\lnot a \lor \lnot b)}$
It's negation should be: ${\lnot x = (\lnot a \land \lnot b \land a \... |
H: Question about the Geometric Sequence Theorem
I am reading my professor's notes, and there is something that I'm not understanding about this. The Geometric Sequence Theorem states that if $r$ is a real number such that $|r| < 1$, then $\lim_{n\to\infty} r^n = 0$. I understand that well enough, but then she gives u... |
H: For any positive integer $n$, what is the value of $t^*$ that maximises the following expression?
For any positive integer n, what is the value of t* that maximises the following expression?
$$\displaystyle \sum_{j=1}^{n-t^*}\left(\frac{t^*-j+2}{t^*+j}\right)$$
where $t^*$ is some integer in the set $\{0,1,2...,n-... |
H: Measure $\mu_f$ for function $f(x)=x^2$
Let $X=[0,1]$ and $\mu$ the Lebesgue measure. Describe the measure $\mu_f$ for the function $f(x)=x^2$.
By definition of $\mu_f$: For any Borel subset $A\subseteq R$, we have $$\mu_f(A)=\mu(f^{-1}(A)) = \mu(\{x\in[0,1]\mid x^2\in A\})$$
Is there anything that can/should be ... |
H: schwarz class and $L^2(\mathbb{R})$
Schwarz and $L^2$ both have the property that the Fourier transform is defined and bijective as a self-map of these spaces. Are they related in anyway or is this coincidence? (i.e. dual in some sense, or perhaps both arising from some more general construction.)
AI: $L^2$ is whe... |
H: Goldilocks Packing type problem
This is a resource allocation problem I am attempting to formulate myself, so bear with me this isn't from the 12th edition of some math book.
A miner is selecting 'rocks' from amongst his mine to haul back to the top. There are three components, or attributes, that define a rock; m... |
H: Having trouble understanding this proposition from my textbook.
I'm seeing this perplexing proposition in my optimization textbook:
Suppose an LP
$$\max\{z(x)=\vec{c}^{T}x+\bar{z}:A\vec{x}=\vec{b},\vec{x}\geq\vec{0}\}$$
and a basis $B$ of $A$ are given. Then, the following linear program is an equivalent linear pro... |
H: Finite cover with zero symmetric difference
The original problem is following: Let J be a finite subinterval in $\mathbb{R}$ and A be a measurable subset of J. Then for any $\epsilon > 0$, there exists a finite union of intervals B such that $d(A,B) = \mu^*((A - B) \cup (B - A)) < \epsilon$
I have solved this probl... |
H: Distance from the origin to a plane... where is my logic wrong here?
I'm going through MIT's multivariable calc online course, and came to the following recitation question:
Compute the distance from P = (0,0,0) to the plane with equation $2x+y-2z=4$
The correct answer is $\frac{4}{3}$, and the TA solves this by ... |
H: Matrix math syntax in wolfram alpha
I'm having trouble getting Wolfram Alpha to do my bidding with regard to matrix manipulations. I am trying to take the derivative of the following matrix expression with respect to m, and was hoping WolframAlpha could be used:
-(x-m)^T * E^-1 * (x-m)
...However, I cannot discove... |
H: How do I prove $2^{n+1} + 2n + 1 = 2^{n+2} - 1$
I am attempting to prove using induction:
$\sum_0^n 2i = 2^{n + 1} - 1$
I have gotten to the point where I need to show:
$2^{n+1} + 2n + 1 = 2^{n+2} - 1$
How do I prove this? Or should I be proving the initial question a different way
AI: You missed the power of $2$ i... |
H: Find the derivative of $f(x)=\frac{1}{\ln x}$ and approximate $f'(3.00)$ to 4 decimals.
Find the derivative of $f(x)=\frac{1}{\ln x}$ and approximate $f'(3.00)$ to 4 decimals.
I've been having trouble with this one for a while, I've been using quotient rule but I keep getting 3 when the answer should be a decimal a... |
H: Isometries on the Banch Space M([0,1]) of regular Borel Measures
I'm trying to define an isometric isomorphism $T:M([0,1])\to M([0,1])$ that is not weak-star continuous (by $M([0,1])$ I mean the Banach space of regular Borel measures). How I can build one? One observation is that $T$ is not weak-star continuous if ... |
H: If $\aleph_\lambda>2^{\aleph_0}$ is a limit cardinal, then $\aleph_\lambda^{\aleph_0}=\aleph_\lambda$?
I know that for successor cardinals, the result holds; i.e. if $\aleph_\alpha$ is a successor cardinal, then because this implies that it is a regular cardinal, then we can use this to show that if $\aleph_\alpha\... |
H: Can someone help me please?
$A^+A=I$ where $A^{+}=(A^TA)^{-1}A^T$, $A_{m \times n}$
I have tried with $(A^TA)^{-1}=A^{-1}{A^T}^{-1}$ but the matrix is not squared
AI: Notice
$A^{+}A=((A^{T}A)^{-1}A^{T})A=(A^{T}A)^{-1}(A^{T}A)$ |
H: Prove $\sum\limits_{i=1}^{n}i\binom{n}{i}=n2^{n-1}$ using induction.
I have already derived the formula $\sum_{i=1}^{n}i\left(\begin{array}{c}n\\i
\end{array}\right)=n2^{n-1}$ directly just by doing some algebraic manipulations to the summand, which is indeed proves the validity of the formula. However, for the sak... |
H: Help with a step in Diestel's proof of Tutte's theorem in Graph Theory
The proof is given on page 8 of the pdf here which has page number 42.
We let $G=(V,E)$ be an edge-maximal graph without a 1-factor—that is, if we add any edge to $G$, the resulting graph has a 1-factor. We need to show that a set $S$ exists suc... |
H: Find all primes of the form $n^n + 1$ less than $10^{19}$
Find all primes of the form $n^n + 1$ less than $10^{19}$
The first two primes are obvious: $n = 1, 2$ yields the primes $2, 5$. After that, it is clear that $n$ has to be even to yield an odd number.
So, $n = 2k \implies p = (2k)^{2k} + 1 \implies p-1 = (2k... |
H: Please explain Monte Carlo method
Generally I understand the idea of the Monte Carlo method.
However, when I read articles about it, there is always shown an example of calculating pi using a square, into which we insert 1/4th of a circle. Then we start putting randomly points into the square, and because the area ... |
H: Showing $[a, \infty) = \bigcap (a - \frac{1}{n}, \infty) $
I am trying to show: $[a, \infty) = \bigcap_{n=1}^{\infty} (a - \frac{1}{n}, \infty) $
If we take $x \in [a, \infty)$, then $x > a > a - \frac{1}{n} $ for all $n$. Hence, $x$ is in the intersection.
Similarly, if $x$ is in the intersection, then $x > a - \... |
H: Prove the indication in Integration
Any body can help me to prove this problem in simple function?
Let if $g_n \geq 0$, $g_n \rightarrow g$ and $\int g_n d\mu \leq E \leq \infty$ then $\int g_n d\mu \leq E$
with $(S, \Sigma, \mu)$ as measurable function and $E\in \Sigma$.
Hope that this is will be clear, thank... |
H: Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?
Does every non-trivial subgroup of $S_9$ containing an odd permutation necessarily contain a transposition?
Here $S_9$ denotes the group of all permutations (i.e. bijections with itself) of the set $\{1,2,... |
H: Power of a number in the difference of two factorials
What is the highest power of 3 available in $58! - 38!$ ( ! stands for factorial)
I can take $38!$ out as common to get $38! ( \frac{58!}{38!} - 1).$
I know how to find out the power of 3 in $38!$ But it is the difference term inside the brackets which I am n... |
H: Example of an infinite complete lattice which is distributive but not complemented
Which is an example of an infinite complete lattice which is distributive but not complemented? Is the set of natural numbers under the relation divides an example? Also is the set of natural numbers under the usual $<=$ (less than o... |
H: Arrangements of 1, 2, 2, 4, 6, 6, 6 greater than 3,000,000
How many numbers greater than 3,000,000 can be formed by arrangements of 1, 2, 2, 4, 6, 6, 6?
Any approach on how to solve this?
AI: HINT: You have only seven digits available, and $3,000,000$ is already a seven-digit number, so you’ll have to use all seven... |
H: Let $S$ be a set and $p$ a prime number and $m\in N$ such that $p$ does not divide $m$. Show $p$ does not divide $|S|$.
I'm currently doing an exercise where this should be shown:
Let $S$ be a set and $p$ a prime number and $m\in N$ such that $p$ does not divide $m$.
Suppose $|S| = p^rm(p^rm-1)...(p^rm-p^r+1)/{p^r(... |
H: Solving system of linear eqaution in special cases
I have to solve for $Ax=B$. Here the diagonal elements of $A$ are $-1$ and all other elements are $1$. $A$ is $n \times n$ matrix . In this special case can we solve for $x$ quickly?
EDIT: quick is in terms of asymptotic complexity.
AI: Note that $A= e e^T -2I$, wh... |
H: Why does my calculator show $2^{-329} = 0?$
On my calculator, I usually get a $0$ when I divide something by $2$, a lot of times if that makes sense, but I was just wondering why does $2^{-329} = 0?$
AI: The precise name for the feature is underflow, which means it's less than the smallest number the registers can ... |
H: Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity when $G$ is infinite?
Let $G$ be an infinite transitive permutation group acting on a set $\Omega$. Is $\text{fix}_\Omega(G_\alpha)$ a block of imprimitivity for $G$ in $\Omega$?
$G_\alpha$ is the set of elements of $G$ that fix $\alpha \in \Omega$
$\text... |
H: What are the possible values of $a$ such that $f(x) = (x + a)(x + 1991) + 1$ has two integer roots?
What are the possible values of $a$ such that $f(x) = (x + a)(x + 1991) + 1$ has two integer roots?
$(x + a)(x + 1991) + 1 = x^2 + (1991 + a)x + (1991a + 1)$
This is of the form $ax^2 + bx + c$. Applying the quadrati... |
H: Why is the restriction map $\mathcal{O}_X(U) \to \mathcal{O}_X(V)$ a flat morphism?
I am reading page 255 of Qing Liu and he claims that if $U,V$ are affine open subsets of a scheme $X$, then $\mathcal{O}_X(U) \to \mathcal{O}_X(V)$ is a flat morphism. Why is this necessarily the case? There are no finiteness assump... |
H: Whats the name of this function?
I read this function in an exercise. It looks quit familiar to me, however I do not know its name.
Whats the name of the $\rho_n$ function and who brought it up first?
AI: Here $\rho_{n}=B\left(\frac{1}{2},\frac{1}{2}n-1\right)^{-1}$. I
do not know a special name for it. |
H: How can a unit step function be differentiable??
Recently, I am taking a Signal & System course at my college. In all of the signal & system textbooks I have read, we see that it is written " When we differentiate a Unit Step Function, we get an Impulse function. " But as far as I have read, a unit step function is... |
H: $f: [0,1]\rightarrow \mathbb{R}$ be an injective function, then :
Question is :
$f: [0,1]\rightarrow \mathbb{R}$ be a one one function, then which of the following statements are true?
$(a)$ $f$ must be onto
$(b)$ range of $f$ contains a rational number
$(c)$ range of $f$ contains an irrational number
$(d)$ range o... |
H: Limits of integral: $\iiint_{D} \frac {\mathrm{d}x\mathrm{d}y\mathrm{d}z}{(x + y + z + 1)^3}$ , where $D =\{ x > 0 , y > 0 , z > 0 , x + y + z < 2\}$
What are the limits of the integral:
$$\iiint_{D} \frac {\mathrm{d}x\mathrm{d}y\mathrm{d}z}{(x + y + z + 1)^3}$$ where
$$ D =\{ x > 0 , y > 0 , z > 0 , x + y + z < 2\... |
H: Is this graph edge-connected?
I would like to know the following: is the following graph still connected even if every edges of two specific colors are eliminated, and is this true no matter how these two colors are chosen? If this graph doesn't satisfy the above condition, would you have some advice to make this g... |
H: union of countable many positive sets has a positive signed measure
Let $P_{1},P_{2},....$ be positive sets for $\nu$, then $P=\displaystyle\bigcup_{n=1}
^{\infty}P_{n}$ is a postive set for $\nu$
Proof:
Let $Q_{1}=P_{1},Q_{2}=P_{2}\setminus P_{1}$....$Q_{n}=P_{n}\setminus\displaystyle\bigcup_{j=1}^{n-1}P_{j}$ So a... |
H: how to find the elements of additive Group - $\mathbb{Z_7^+}$
I am given this additive Group G=$\mathbb{Z_7^+}$
I tried to find all its elements and I did:
$$gcd(1,7) = 1 \\ gcd(2,7) = 1 \\ gcd(3,7) = 1 \\ gcd(4,7) = 1 \\
gcd(5,7) = 1 \\ gcd(6,7) = 1 \\ gcd(7,7) = 7 \\ $$
Then I took all which gives $1$. so $1,2,3... |
H: Is $\Pr(X
Given that $X$ and $Y$ are non-negative random variables, and $a$ is a non-negative constant, is $\Pr(X<Y|X<a)\geq \Pr(X<Y)$? I mean, is that $X<a$ gives useful informaiton on the guess of $X<Y$?
Thanks a lot in advance.
AI: Not necessarily. It holds of course if $X$ and $Y$ are independent.
But if they a... |
H: Quadratic Manipulation
Is there some easy way to transform a quadratic equation like $ax^2+bx+c$ into a quadratic equation of the form $d(x+s)^2+e(x+s)$ where $a,b,c,d,e$ are constants and $a,d,e>0$
$-s$ is a root of the function
Thanks in advance
AI: NOTE: This is an answer to the problem OP asked before editing.
... |
H: Constant functions are measurable
Let $f = C$ a constant, and I want to show $f$ is measurable.
In other words, if we take $(a, \infty)$, we show $f^{-1} (( a, \infty) )$ is measurable.
But $ f^{-1}((a, \infty))$ is just the real line $\mathbb{R}$, and therefore measurable? Am I right?
AI: That's one of the two p... |
H: How to solve the limit $\lim_{n \to \infty} \sqrt[8]{n^2+1} - \sqrt[4]{n+1}$?
How do I show that this limit is zero?
$$\lim_{n \to \infty} \sqrt[8]{n^2+1} - \sqrt[4]{n+1} = 0$$
I've done the multiply by conjugates thing, which seems to lead nowhere:
$$\lim_{n \to \infty} (\sqrt[8]{n^2+1} - \sqrt[4]{n+1}) (\frac{\sq... |
H: Sum of closed and compact set in a TVS
I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space $X$. In case $X=\mathbb{R}$ this is easy, using sequences. However, since I ... |
H: Variance and covariance function of stochastic process
Let $X_t = Z_t + \theta Z_{t-1}$ where $ \left\{ Z_t \right\} \approx WN(0, \sigma^2)$. Find variance $ VarX_t$ and covariance function.
Of course we have $EX_t = 0$. Then $VarX_t = EX_t^2 = E(Z_t^2 + 2 \theta Z_t Z_{t-1} + \theta^2Z_{t-1}^2) = \sigma^2 + \the... |
H: Evaluating $\int{\frac{1}{\sqrt{x^2+y^2}}\mathrm dx}$
Attempting to calculate $\displaystyle \int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}$,
$$\int{\dfrac{1}{\sqrt{x^2+y^2}}\mathrm dx}=\int{\frac{1}{\sqrt{(y\tan\theta)^2+y^2}}y\sec^2\theta \mathrm d\theta}=\int{\sec\theta d\theta}=\ln(\sec\theta +\tan\theta)=\ln\left(\... |
H: Infinite Dihedral Group.
Let $D_{\infty}= \langle x,y \mid x^2=y^2=1\rangle$ be the infinite dihedral group. Are the following statements true?
Since $G$ is not torsionfree, $\mathbb{Q}[G]$ is not a domain.
$D_{\infty}$ is an infinite subgroup of $G= \langle x,y \mid x^2=y^2\rangle$.
The infinite dihedral group... |
H: Are coefficients and values for x in F[x] in the same set?
I'm trying to understand the construction of $F[x]$ where $F$ is a field. As far as I understand it now, all coefficients and roots for all polynomials $f(x) \in F[x]$ lies in $F$. But what about the domain for the polynomials? The set $F[x]$ contains polyn... |
H: Proving a Logic Equation
I have two information.
$x+y = 1$ and $xy = 0$.
Now,I need to prove this equation : $xz + x'y + yz = y + z$
What I tried:
$z(x+y) + x'y = z + x'y$
Thats all
What do you think?
AI: xz+x′y+yz = xz+ xy + x′y+yz .... (adding xy=0)
= xz + y(x + x') +yz
= xz + y + yz .....( x+x' a... |
H: Fourier Series: going from $a_n$ and $b_n$ to $c_n$
I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from:
${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi n x \over P})]$
to
$\sum_{n=-N}^N c_n e^{i{2\pi n x \over P}}$
To... |
H: Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$
The problem: Prove that there exist no positive integers $m$ and $n$ for which $m^2+m+1=n^2$.
This is part of an introductory course to proofs, so at this point, the mathematical machinery should not be too involved. This is supposed t... |
H: Difference between parallel and orthogonal projections
i would like to understand what is a difference between parallel and orthogonal projection?let us consider following picture
We have two non othogonal basis and vector A with coordinates($7$,$2$),i would like to find parallel projection of this vector ... |
H: show that function is convex
Let $f:\mathbb{R}\to\overline{\mathbb{R}}$. Show that
$$f\left(x\right)=\begin{cases}
+\infty & \mbox{ if }x\in\left(0,\infty\right)\\
0 & \mbox{ if }x=0\\
-\infty & \mbox{ if }x\in\left(-\infty,0\right)
\end{cases}$$
is convex.
AI: define $+\infty−\infty=0$
In that case the function ... |
H: Finding normal vector of point $(x,y)$ given $f(x, y)$
Ignoring the codes, this is more or less a calculus question. Given an equation of a surface $z = x^2 + y^2$, I need to find normal vector of a point. I am not good at maths ... so ...
For (a), I believe the $\frac{\partial z}{\partial x}$ is equation of tang... |
H: category of linear maps
Let $V,W$ be vector spaces. Let's define a category whose objects are linear map $f:V\to W$ and morphisms from $f$ to $g$ are pair of linear maps $(\alpha,\beta)$ where $\alpha:V\to V,\beta :W\to W$ such that $g \circ\alpha = \beta \circ f$.
Does this category have some special name?
If $f_... |
H: Is this a justified expression for $\int \lfloor x^2 \rfloor \, \mathrm{d}x$?
By observing patterns in Riemann sums for the following integral, I'm convinced that $$\int_0^\sqrt{n} \lfloor x^2 \rfloor \, \mathrm{d}x = (n-1)\sqrt{n}-\sqrt{n-1}-\sqrt{n-2}-\cdots-\sqrt{1},$$ with $n$ a positive integer. (We choose $\s... |
H: Related Rates question from Pure Mathematics 1 by Hugh Neil
There is a question in my Pure Maths book which seems to confuse me. Ive done the rest but somehow this question seems to confuse me I think enough info is not given.
A water tank has a rectangular base 1.5m by 1.2m. The sides are vertical and water is be... |
H: Divisors of zero in $ \mathbb Z_{p^k}$
Let $p$ be a prime numer and let $k$ be a natural numer such that $k\geq 2$. I wish to descripe all zero's divisors in $\mathbb Z_{p^k}$. Obviously elements of the form
$np$, where $n=0,...,p^{k-1}-1$, are zero divisors, because $p^k|np^k$.
Are there others?
AI: Suppose $ab=0... |
H: Let $f,g$ be two distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $.
Let $f,g$ be two continuous, distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $.
Given $n\in \mathbb{N},$ let $y_n = \int_{0}^{1} \frac{f^{(n+1)}}{g^{(n)}} $ ... |
H: What's the fastest way to solve this inequality?
$|x^2-3x+2|>|x|+1$
Thanks in advance.
AI: $x^2-3x+2=(x-1)(x-2)$.
Thus
If $x \in (-\infty, 0]$ the inequality becomes....
If $x \in (0, 1]$ the inequality becomes....
If $x \in (1, 2]$ the inequality becomes....
If $x \in (2, \infty]$ the inequality becomes....
A f... |
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