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H: Limit $\lim _{x \to 0}\sqrt {x+\sqrt {x+\sqrt{x+\sqrt{x...}}}}=1$
I've been investigating some interesting infinite square roots, and I've arrived at the hypothesis that $$\lim_{x\to 0}\sqrt {x+\sqrt {x+\sqrt{x+\sqrt{x...}}}}=1$$
However, I have tried to prove this but have found myself unable to do so. For example... |
H: How To Think About Measurability in $\mathbb{R}$
How do Platonist-leaning mathematicians think about the measurability/non-measurability of subsets of $X=\mathbb{R}\cap [0,1]$? For clarity, let's use "size" for the informal concept of length/area/volume, and "measure" for usual formalized version of this concept. I... |
H: Show that $\ker\sigma \subset \varphi_1(\ker\rho)$ and $\operatorname{im}\tau \subset \psi_2(\operatorname{im}\sigma)$.
Given the homomorphism of short exact sequences, I must show that $\ker\sigma \subset \varphi_{1}(\ker\rho)$ and $\operatorname{im}\tau \subset \psi_{2}(\operatorname{im}\sigma)$. For the first p... |
H: Inequalities in matrix norm.
My book says, for any $t$
$e^
{tA} = C diag(e^
{tJ_1}
,··· , e^
{tJ_k} ) C^{
−1}$
.
Hence,$
|e^
{tA}
| ≤ |diag(e^
{tJ_1}
,··· , e^
{tJ_k} )
|$
Where $J_i$ are exponential of jordan blocks of A.
I didn't understand why this has to be true.please help
|A| here denote induced p norm of A
A... |
H: Finding distribution function when pdf is $f(x) = |x|$ for $ -1 < x < 1$
In my probability/stats course, they have defined a probability density function as:
$$f(x) = |x|\quad,\quad -1 < x < 1$$
I am having difficulty with how they have integrated this to find the cumulative distribution function:
$F(x) = (1-x^2)/2... |
H: A Systematic way to solve absolute value inequalities?
So, I had to solve this problem: $\left\vert \dfrac{x^2-5x+4}{x^2-4}\right\vert \leq 1$
I factored it in the form: $\left\vert \dfrac{(x-4)(x-1)}{(x-2)(x+2)} \right\vert \leq 1$.
After that I found the intervals in which the expression is positive: $x \in(-\inf... |
H: Showing openess in topology of point-wise convergence
I want to check if the set
$$ P_{x_0}=\{f\in C([a,b])\mid f(x_0)=0\},\quad x_0\in[a,b] $$
is open in the topology of point-wise convergence.
I already have a problem with an intuitive picture of this topology. I know it is has the basis
$$ O(x_1,\ldots,x_n,t_1,\... |
H: Prove $f(x) = \sqrt{x}\ln(x)$ is uniformly continuous for $x = [1, \infty)$
Original question is to show this is true for all $x > 0$ with the hint to split cases on $x \in (0,1]$ and $x \in [1, \infty)$. I can show this is true for $x \in (0,1]$ by extending the interval to include $0$ (because $f(0) = 0 = \lim_{x... |
H: Showing "Right hand continuity" , critique and help on solution.
Suppose that $\lim_{x \to a^{+}}f(x) = f(a)$ and $f(a) > 0$. Prove there is a number $\delta > 0$ such that $f(x) >0$ for all $x$ satisfying $0 \leq x - a < \delta$
The issue I'm having is linking the ideas properly to make the solution sound.
So the ... |
H: Is there a shortcut to find a Taylor series not centered at 0 with a Taylor series centered at 0?
We know that the Taylor series of $\ln(1+x)$ centered at 0 is $x-\frac{x^2}{2} + \frac{x^3}{3} - \dots$.
We can find the Taylor series of $\ln(2+x)$ by writing $\ln(1+(1+x))$, so this is equal to $(x-1)-\frac{(x-1)^2}{... |
H: Converse to a proposition on algebraically closed fields
This is a follow up to a previous question. Let us call a field $F$ root-closed if every element $x$ of $F$ has at least one $n$-th root for every positive integer $n$. It is very easy to show that every algebraically closed field of characteristic $0$ is roo... |
H: Show for some subsets of $G$ we have subgroups of $(G, \ast)$
Let $G$ be an abelian group. Show that for the following subsets $H_n$, we have subgroups of $G$.
$H_1= \lbrace g \in G | g^n=e \rbrace $, with $n$ being a certain fixed natural number.
$H_2 = \lbrace g \in G | g^{-1}=g \rbrace$
$H_3 = \lbrace g \in G | ... |
H: Question regarding the Quotient Rule, How does the textbook reach this intermediate step?
$$\begin{align*}
\left(\frac{f(x)}{g(x)}\right)’ &= \frac{(x-3)^{1/3}\frac{1}{2}(x+2)^{-1/2}}{(x-3)^{2/3}} - \frac{(x+2)^{1/2}\frac{1}{3}(x-3)^{-2/3}}{(x-3)^{2/3}}\\
&= \frac{(x-3)^{-2/3}(x+2)^{-1/2}}{(x-3)^{2/3}}\cdot\left[\f... |
H: Coin Flip Problem
So my friend gave me this question this other day, and I've tried to start it (I'll show my logic below), but I couldn't find any efficient way to do the problem.
You start out with 1 coin. At the end of each minute, all coins are flipped simultaneously. For each heads that is flipped, you get an... |
H: finding the point of tangency for two circles
The two circles $x^2 + y^2−16 x−20 y + 115 =0$ and $x^2 + y^2+8 x−10 y + 5 =0$ are tangent.
How could I find the point of tangency?
AI: $x^2+y^2-16x-20y+115=x^2+y^2+8x-10y+5$
$\Longleftrightarrow -16x-20y+110=8x-10y$
$\Longleftrightarrow -24x-10y+110=0$
$-24x-10y+110=0$... |
H: Volume of a solid of revolution with change of variable
I want to calculate the volume of the solid of revolution around the x-axis of this figure
$x = (1-t^2)/(t^4+4)$
$y = (t+1)*(1-t^2)/(t^4+1)$
for t between -1 and 1. In the figure below the plot is shown
plot
In my opinion to do this I have to calculate the int... |
H: Clarification on mutual singularity of probability measures
Let $P_1$ and $P_2$ be two probability measures on a measurable space, $(\Omega, \mathcal{F})$. Then $P_1$ and $P_2$ are mutually singular (denoted $P_1 \perp P_2$) if there exists $A \in \mathcal{F}$ such that $P_1(A) = 1$ and $P_2(A) = 0$. The book Gauss... |
H: Find the equation of the two tangent planes to the sphere $x^2+y^2+z^2-2y-6z+5=0$ which are parallel to the plane $2x+2y-z=0$
Find the equation of the two tangent planes to the sphere $x^2+y^2+z^2-2y-6z+5=0$ which are parallel to the plane $2x+2y-z=0$
My Attempt
We need to find a point which is shortest distance fr... |
H: Partial Derivatives from Loring Tu
I attempt to understand the definition of partial derivatives from An Introduction to Manifolds by Loring Tu (Second Edition, page no. 67). The definition is given below.
My Confusions & Questions
I am confused about how the following argument works.
The partial derivative $\p... |
H: Is the set $\{\langle \varnothing, a \rangle ,\langle \{ \varnothing \}, b \rangle \}$ considered as function?
The set defined as
$$F=\{\langle\varnothing,a \rangle, \langle \{\varnothing\},b\rangle\}$$
a function?
AI: Assuming you're using the "functions-as-sets-of-ordered-pairs" approach, then yes, it is: its dom... |
H: If $a\frac {dy}{dx} + by = c$ has constant coeffcients, does that means that $a=b=c$?
I am trying to identify if a differential equation has constant coefficients.
Let $A = a\dfrac {dy}{dx} + by = c$
The $A$ has constant coefficients only if $a=b=c$ correct?
AI: The general form of an ODE is
$$F(x,y,y',y'',...,y^{[... |
H: E is a collection of sets. How to prove a class/collection of all sets that can be covered by finite union of sets in E is a ring?
Given E is any collection of sets $F_i$ ,
Let E1 be a collection of all sets that can be covered by a finite union of sets in E.
How do we show that E1 is a ring?
Or how can we show tha... |
H: Show that $E(|S_n-np|) = 2vq b(v; n, p) $.
(Feller Vol.1, P.241, Q.35) Let $S_n$ be the number of successes in $n$ Bernoulli trials. Prove
$$E(|S_n-np|) = 2vq b(v; n, p) $$
where $v$ is the integer such that $np < v \le np+1$ and $b(v; n,p)$ is a binomial distribution with $v$ successes of $n$ trials. Hint: The le... |
H: Can $\{(x,y) \mid x^2 + y^2 < 1\}$ can be written as the cartesian product of two subsets of $\mathbb{R}$?
We consider the set
$$S := \{(x,y) \mid x^2 + y^2 < 1\}.$$
The exercise in Munkres asks whether it is possible to write this set as the cartesian product of two subsets of $\mathbb{R}$. We, naturally, consider... |
H: Maximum and Supremum
Please, I want to know
$\max_{x \in \{0,\frac{3}{4},1\}} (x-1/2)^2$, $\max_{x \in [0,1)} \min\{x,1/2\}$ and their $\arg \max$.
Thanks!
AI: For the first part
$max_{x\in\{0,\frac34, 1\}}(x-\frac12)^2$=$max\{\frac14, \frac1{16}, \frac14\}$=$\frac 14$ and the argmax are $0,1$.
Note you could have ... |
H: An artinian ring is a product of local rings
I am rather confused with the last line of the argument used in 00JA, Stacks Project.
Lemma 00JA. Any ring with finitely many maximal ideals and
locally nilpotent Jacobson radical is the product of its localizations
at its maximal ideals. Also, all primes are maximal.
P... |
H: Which sequence ${a_n}$ does $\sum_{n=1}^\infty a_n$ is conditionally convergent and $\sum_{n=1}^\infty (-1)^n a_n$ converges
Which sequence ${a_n}$ does $\sum_{n=1}^\infty a_n$ is conditionally convergent and $\sum_{n=1}^\infty (-1)^n a_n$ converges?
I tried with ${a_n}= \frac{\sin(n)}{n}$ and it seems that it doe... |
H: Do Biconditionals Have to be Logically Related?
I'm studying real analysis from Terence Tao's book, Analysis 1, and was familiarizing myself with mathematical logic that Tao explains in the appendix. In it, he covers the biconditional, or "if and only if" statements. From what I understand, a biconditional is only ... |
H: Does $\frac{\sum_{k=1}^{n-1} k! }{n!}$ converge?
I want to know if $$\frac{\sum_{k=1}^{n-1} k! }{n!}$$ converge as $n \to \infty$. I know that the sequence is bounded by one since $\sum_{k=1}^{n-1} k! \leq (n-1)(n-1)!$.
Any help is appreciated.
AI: If you take your inequality one step further,
\begin{align*}
\sum_{... |
H: Norm of sesquilinear form bounded by norm of associated quadratic form
I have the following question from Teschl's "Mathematical Methods in Quantum Mechanics":
A sesquilinear form is called bounded if $$\|s\|=\sup_{\|f\|=\|g\|=1}|s(f,g)|$$ is finite. Similarly, the associated quadratic form $q$ is bounded if $$\|q... |
H: Is it true that minimizing the square of the expectation is the same as minimizing the expectation of the square?
Is it true that minimizing the square of the expectation is the same and minimizing the expectation of the square?
Consider a random variable $X_c$ depending on some parameter $c$. Do we have that $$\... |
H: Birthday Problem Proof?
I was looking at the Birthday Problem (the probability that at least 2 people in a group of n people will share a birthday) and I came up with a different solution and was wondering if it was valid as well. Could the probability be calculated with this formula:
$$1-(364/365)^{n(n+1)/2}$$
The... |
H: Proof of convergence of series (progression)
First, I'm a beginner in this site. In addition, my mother tongue is not English. Thus, I'm sorry if the sentences I write are difficult to understand.
I'll move on to the main topic.
I can't solve this problem.
Precondition(Given);
There exists a progression ${x_n}$ suc... |
H: Explicit bijection between $\mathbb{N}$ and $\mathbb{N}^2$
I am having some trouble writing a bijection between $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ and $\mathbb{N}^2$, particularly using the definition that $\mathbb{N}$ includes $0$. (Otherwise, it is straightforward.) Here is what I have so far.
Consider an arb... |
H: Why is $\frac{a-ar^n}{1-r}$ always an integer when $a$, $r$ (except $1$), and (positive) $n$ are integers?
If $a_1$ and $r$ are integers, explain why the value of $\dfrac{a_1-a_1r^n}{1-r}$ must also be an integer.
Does anyone have any ideas to rigorously explain/prove it? I can't really think of anything. (Also t... |
H: Find the flaw in this proof that $1$ is the greatest natural number
I think the flaw is in assuming that $N^{2} \in \mathbb{N}$, but I don't know.
AI: The proof is logically correct, and leads to a contradiction (we know that $1<2$). Thus it is a valid proof that there is no greatest natural number. |
H: Lambda Calculus: What is the difference between a $\lambda$ term with and w/o parenthesis?
Eg. what is the difference between $(\lambda y.M)[x:=N]$ and $\lambda y.M[x:=N]$?
AI: The substitution applies to the whole expression in the first term.
The substitution only applies to M in the second term.
In this case the... |
H: How can I evaluate this complex integral equation on Wolfram?
I need to evaluate the complex line integrals in the following equation:
$$g(z)=\frac{\int_0^z\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}{\int_0^1\zeta^{-5/6}(\zeta-1)^{-1/2}d\zeta}.$$
Can someone advise me on how to evaluate this expression on Wolfram? For thos... |
H: How to compute $P(X>Y\mid Y<1)$ given pdf of $(X,Y)$?
I have the following function
$$f(x,y)=\begin{cases} e^{-(x+y)} &, x,y > 0 \\ 0 &, \text{otherwise} \end{cases}$$
I want to compute the following conditional:
$$P(X>Y\mid Y<1)$$
I'm trying to solve this using the following :
$$f(x\mid y) = \frac{f(x,y)}{f_2(y)}$... |
H: In how many ways can you form a committee of three from a set of $10$ men and $8$ women, such that there is at least one woman in the committee?
My textbook employs a brute force method: add the number of committees that could be formed with one woman, two women, and three women in them. Then, the total number of s... |
H: Is it reasonable to say that each random variable has one and only one variance?
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.
Is it reasonable to say that each random variable has one and only one variance?
AI: No. Variance is defined... |
H: Gaussian distribution with mean zero and variance $\sigma^2$
For a Gaussian distribution with mean zero and variance $\sigma^2$, let $X \sim N(0, \sigma^2)$. Is it integral
$$P(X>0)=\frac{1}{2}?$$
AI: For any random variable with a continuous symmetric distribution this is true. $N(0,\sigma^{2})$ has these propert... |
H: Prove that distance between foot of perpendiculars from an arbitrary point on circle to two given diameters is constant.
https://www.geogebra.org/geometry/xaj5mjuz
ORIGINAL QUESTION : You have given a circle with two diameters drawn. Select any point $P$ on the circle and drop perpendiculars to given diameters. Th... |
H: Questions about parametric equations
Consider the parametric equations: $$x=t^3-3t, \; \; y=t^2+t+1.$$
What is the lowest point on this parametric curve?
For what values of $t$ does the curve move left, move right, move up and move down?
When is the curve concave up?
Find the area contained inside the loop of this... |
H: $T:\Bbb{R}^2\rightarrow\Bbb{R}^2$ has 2 distinct eigenvalues. Showing that $v$ or $T(v)− \lambda_1v$ are eigenvectors of $T$
$T:\Bbb{R}^2\rightarrow\Bbb{R}^2$ which is diagonalizable with 2 distinct eigenvalues. Showing that either $v$ is an eigenvector for $\lambda_1$ or else $T(v)− \lambda_1v$ is an eigenvector f... |
H: n vertex graph without isolated vertices - maximum vertex degree
Provided there is a $n$-vertex graph without isolated vertices which is disconnected, prove that the maximum vertex degree does not exceed $n-3$
AI: The graph has at least 2 connected components, which have (since there is no isolated vertex) at least... |
H: Majorisation inequality/upper bound
I saw the following relation and now I'm trying to prove it
$$\sum_{i=1}^l a^{\downarrow}_i \geq \sum_i \{a_i | a_i \geq 1/l \} \, ,$$
but I'm stuck. Here $a^{\downarrow}_i$ is an element, in non-increasing order, of a probability vector $\textbf{a}$. For example, considering th... |
H: Combinatorics counting problem- double counting?
How many different numbers can be formed by the product of two or more of the numbers 3,4,4,5,5,6,7,7,7?
My answer is 138.However the book says 134.
Which answer is correct? My working is -
(2)(3)(3)(2)(4)-6=138
However , the book minuses 10 rather than 6 .
Pls help.... |
H: For $C$ open, bounded, and convex, is it true that $x+r C\subset x + 3rC$, $x\in \mathbb R^d, r>0$?
I have a question regarding transformations of convex sets. Given an open, bounded and convex set $C$, is it true that
$$
x+r C\subset x + 3rC \qquad x\in \mathbb R^d, \ r>0 \quad?
$$
The reason I am asking is tha... |
H: $|z+2|=|z|-2$; Represent on an Argand Diagram
Represent on an Argand Diagram the set given by the equation $|z+2|=|z|-2.$
My attempt:
Apparently the answer is $x\leq 0$ $(z = x + yi)$ and $y = 0$, based on the idea that $-x = \sqrt{(x^2 + y^2)}$, but I am struggling to derive this.
I originally assumed the answer w... |
H: Show if $f_x \rightarrow \eta \,\,\,$ and $g_x \rightarrow \zeta$ so $f_x+g_x \rightarrow \eta + \zeta$
Let $(f_x)$ and $(g_x)$ be two nets on a directed set $X$.
Show if $f_x \rightarrow \eta \,\,\,$ and $g_x \rightarrow \zeta$ so $f_x+g_x \rightarrow \eta + \zeta$
For $(f_x)$ holds:
$$\forall \epsilon > 0 \,\,\,... |
H: Rouché's theorem example verification
I have to use Rouché's Theorem to check how many zeros in $D(0,2)$(disk with center 0 and radius 2) do the following functions have
$z^3+6z-1$
$z^3+6z+1$
Now, the first one: $f(z) = 6z, g(z) = z^3+6z-1$ so for $|z| = 2$
$$
|f(z)-g(z)| = |-z^3+1| \leq |-z^3| + |1| \leq 9 < 12 ... |
H: Statistics and Probability- Cumulative Distribution
There are $10,000$ people in front of you in line at the airport. Each person takes $\text{Exp}(1/3)$ minutes to be served once they get to the front of the line. Approximate the probability that you get to the front of the line in less than $29,000$ minutes.
I am... |
H: Give an example of a non-zero linear operator $T$ on a vector space $V$ such that $T^{2}=O$ but $ \operatorname{Ker} T \neq \operatorname{Im} T$.
Give an example of a non-zero linear operator $T$ on a vector space $V$ such that $T^{2}=O$ but $ \operatorname{Ker} T \neq \operatorname{Im} T$.
AI: $T(e_1)=T(e_2)=0, T... |
H: If $\cos 3x=\cos 2x$, then $3x=\pm 2x + 2\pi k$. Why the "$\pm$"?
There is an equation I'm solving at the moment which involves $\arccos$. In the correction my teacher gave me, it seems after taking the $\arccos$ of an angle, you must take the positive and negative value of the angle plus a multiple of 2π:
Hence, ... |
H: Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$.
Let $G$ be the group of all the maps from closed interval $[0,1]$ to $\mathbb{Z}$. The subgroup $H= \left \{ f \in G :f(0)=0 \right \}$
Then
$1)$ $H$ is countable
$2)$ $H$ is uncountable
$3)$ $H$ has countable index
$4)$ $H$ has u... |
H: Where to find English translations of Euler's collected works?
I will highly appreciate if someone can provide me the link to the book or link to the paper where I can find the following papers $(90-94)$ with English translation.
Any help would be appreciated. Thanks in advance.
AI: The Euler Archive provides link... |
H: Does $\lim \limits_{n\to\infty} \int_0^1 \sin(\frac{1}{x}) \sin(nx)dx$ exist?
Is $\lim \limits_{n\to\infty} \int_0^1 \sin(\frac{1}{x}) \sin(nx)dx$ convergent and if so, what is the limit?
Neither Riemann-Lebesgue lemma nor Dirichlet lemma can be applied directly.
The limit seems to be 0, but I'm not completely cert... |
H: Values of c for which the given quotient ring is a field.
I am stuck with the problem :
Find all values of 'c' in $F_{5}=\frac{\mathbb{Z}}{5\mathbb{Z}}$ such
that the quotient ring $\frac{F_{5}}{⟨X^3 + 3X^2 + cX + 3⟩}$ is a
field. Justify your answer.
My approach was, we've got a theorem for commutative ring R t... |
H: Given a set with 6 vertices, can you prove or disprove G is planar
$p,q$ prime numbers
On set $V=\{1,p,q,pq,p^2q,pq^2\}$ of verticies
Given $G(V,E)$ a undirected simple graph
We define :
$\forall x,y \ \in \ V $
$\exists \ xy \ \in E \ \text{ iff }\ x|y$
Prove or disprove that the graph is planar .
I tried to prov... |
H: Given $f\in Hol(|z|<1)$ and $image(f(z))\subset K\subset D_0(1)$ where K is compact subset Prove that $f$ has one fixed point.
Given $f\in Hol(|z|<1)$ and $image(f(z))\subset K\subset D_0(1)$ where K is compact subset
Prove that $f$ has one fixed point.
My try is:
$g(z)=z-f(z)$ and then we know that $|f(z)|<1$ usin... |
H: Propositional logic - wrong proof
Given the propositions A, B and C, we assume that
\begin{equation}
(A\wedge B)\rightarrow C \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)
\end{equation}
I want to demonstrate the other way around implication
\begin{equation}
C \rightarrow (A\wedge B) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \... |
H: convergence of $ \sum_{n=1}^{\infty} \frac{n !}{n !+3} $?
determine the convergence of
$$
\sum_{n=1}^{\infty} \frac{n !}{n !+3}
$$
I tried using the ratio test and also for n! , I use Stirling approximation.Still I got stuck.
AI: Since $\lim_{n\to\infty}\frac{n!}{n!+3}=1\ne0$, your series diverges. |
H: Finding maximal rings and Laurent Series - solution verification
I am to find regions where $f(z) = \frac{1}{z-2}$ around $i$ has Laurent series. As Jose pointed out, the natural regions are rings $A(i, 0, \sqrt{5})$ and $A(i, \sqrt{5}, \infty)$ because $\sqrt5$ is the distance between the center and singularity.
S... |
H: how do I differentiate this function implicitly
How do I differentiate $$\frac{(x^2 - 4y^2)} {(x^2 + xy^2)} = 2$$ implicitly? I did it by bringing the denominator over to the other side, and I got $$\frac{-(2x + 2y^2)}{(4xy + 8y)}$$
Here are the images of the question and the suggested answer.
AI: Ok, here's how y... |
H: Primitive elements in fields and finite fields
I have the following two definitions:
If $K$ is an extension field of $F$ and $K = F(a)$ for some $a \in K$, then $a$ is a primitive element of $K$.
If $K$ is a finite field and $a$ is a generator for its multiplicative group $K^*$, then $a$ is a primitive element of ... |
H: Show that the map is a linear functional and determine the dimension of the quotient space $V/ker(y)$
During an exam I was given the following task:
Consider the vector space
$$V:=\{p \in \mathbb{Q}[x]\ |\ p \text { has degree at most } 3\}$$
Show that the map
$$y(p)=p(0),\quad V\rightarrow\mathbb{Q}$$
is a linear... |
H: How do you prove that $\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t}$?
I got the following result using the technique "Integral Milking":
$$\ln(x) = \int_0^\infty \frac{e^{-t}-e^{-xt}}{t} dt= \lim_{n\to0}\left(\operatorname{Ei}(-xn)-\operatorname{Ei}(-n)\right)$$
for $x > 0$. So, I have a proof of it the result, ... |
H: Universal completions of *algebras
I am dealing with two "universal completions" but I am not sure if they are the same thing and would appreciate some guidance.
Let $\mathcal{A}$ be a unital *-algebra. A $\mathrm{C}^*$-seminorm on a $\mathcal{A}$ is a seminorm $p:\mathcal{A}\rightarrow \mathbb{R}^+$ that satisfie... |
H: How to get information from events per week to events per day?
I was given the following homework question:
If a company's computer has around 28.14 errors per week. What is the probability of less than 3 errors at one day ?
Can someone give me a little start up aid to solve the question? (no demand here, for solvi... |
H: How can this implication be proved?
What I have given is:
(i) $\operatorname{f}(a_{n})\leq0$ and $\operatorname{f}(b_{n})\geq0.$
(ii)$\forall n\in\Bbb N_{0}: a\leq a_{n}\leq a_{n+1}\leq b_{n+1}\leq b_{n}\leq b.$
(iii)$\forall n \in \Bbb N_{0}:b_{n}-a_{n}=\frac{b-a}{2^{n}}.$
Now it is stated $a_{n}$ and $b_{n}$ are ... |
H: System $\,x+y+z=1\,$ and $\,\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$
I'm trying to solve the following system of equations:
I. $\,x+y+z=1$
II. $\,\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$
And an elegant solution just eludes me.
It should be a rather easy problem, but I'm having slight problems solving it.
I tried just b... |
H: Probability that construction fails
I have the following problem:
The resistance D has a normal distribution with expectation 11 and
variance = 2. The force B has a normal distribution with expectation 9
and variance 1. Assume independence. What is the probability that the
construction doesn't fail.
The answer is... |
H: Induce Metric From Topological Space
Given any topological space $(X, T)$,
can we induce a metric $d$ on $X$, such that the set of all open sets in the metric space $(X, d)$ is equal to the set $T$?
Just trying to grab the intuition behind topological spaces.
AI: Below I'll focus on metrizable topologies - plenty o... |
H: Function that produce a spike in 3D
I am looking for a function that would produce a spike, of any size, in the z-direction at a given x and y coordinate. A little bit like the https://en.wikipedia.org/wiki/Dirac_delta_function but that could be graphed in 3D.
I am not sure if those specifications are clear, if not... |
H: How can I scale a sigmoid curve to fit the criteria I would like
Is there a way I can make a scaled sigmoid function $f(x)$ such that $f(0) \to -1$ (or as close to it as possible) and $f(n) \to 1$ (or as close to it as possible), for whatever $n$ I choose?
AI: While I agree with 5xum, I will give another solution, ... |
H: Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k= k^2+k+1$. Prove that $a_1a_2\cdots a_{p-1}$ is congruent to $3$ modulo $p$.
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k= k^2+k+1$ for $k=1,2,\ldots,p-1$. Prove that product $a_1a_2\cdots a_{p-1}$ is congruent to $3$ modulo $p$.
I tried to ... |
H: Regarding ln|f| being upper semicontinuous
Let $f$ be a holomorphic function on the open unit disc $\mathbb{D}$ In $\mathbb{C}$. Can anyone tell why $\ln|f|$ is upper semicontinuous but not continuous? In particular $\ln|z|$, $z\in \mathbb{D}$.
AI: $\{z: \ln |f(z)| < a\}=\{z: |f(z)| <e^{a}\}$ is open by continuit... |
H: Isomorphisms Between Multiplicative and Additive Modulo Groups.
While working with the fundamental theorem of Abelian groups, I noticed something and would like to confirm my assumption.
$$(\mathbb{Z}/n\mathbb{Z})^* \cong \mathbb{Z}/φ(n)\mathbb{Z} $$
Is the above true? If so why exactly?
AI: Your statement is true... |
H: solve $x' = t^\alpha +x^\beta$
I need to solve the following ode for some non zero $\alpha \beta$ :
$x' = t^\alpha + x^\beta$. I don't have any initial conditions.
I am not sure how to proceed with this ,I tried doing$ x=y^m$ to get $x'=my^(m-1) y'$ but I don know if it helps.
Any help will be appreciated
AI: I do... |
H: Exact value of limit
Find the exact value of
$ \lim_{x\to \infty } (1+ \frac {1}{2x} - \frac {15}{16x^2})^{6x}$.
Feel tempted to reduce $(1+ \frac {1}{2x} - \frac {15}{16x^2})$ to 1 as $x$ appoaches infinity, but a quick check on the GC yields the final answer as $e^3$.
I'm not sure how to arrive there.
AI: When ... |
H: Absolute convergence and continuity of $\sum_{n=1}^\infty \sin(\frac{x}{n^4})\cos(nx)$
Question:
Prove that the series $$\sum_{n=1}^\infty \sin\left(\frac{x}{n^4}\right)\cos(nx)$$ converges absolutely, and it is continuous on $\mathbb{R}$.
Attempt: I can readily see that the series converges absolutely on some cl... |
H: Show that $f_{n}(x):=nx(1-x)^{n}$ is uniformly bounded on $[0,1]$ for all $n\geq 1$.
Consider $f_{n}(x):=nx(1-x)^{n}$ defined for $n=1,2,3,\cdots$ and $x\in [0,1]$. The exercises have two parts:
(a) Show that for each $n$, $f_{n}(x)$ has a unique maximum $M_{n}$ at $x=x_{n}$. Compute the limit of $M_{n}$ and $x_{n... |
H: Solution of the ODE $x'=t+x$
How can i find the general solution of
$$x'=t+x$$
I tried a few things, but I couldn't get to the solution, I don't think I'm remembering the right method to solve this problem.
AI: You can to use "variation of constants technique". You start by solving the homogeneous equation $x'-x = ... |
H: Show that $U:=\left\{x_{1} \otimes v | v \in V\right\}$ is a subspace
Let $V$ be vector space with basis $\{x_1,...,x_m\}$. I want to show that
$$U:=\left\{x_{1} \otimes v | v \in V\right\}$$
Is a subspace of $V\otimes V$.
My attempt:
To show that $0\in U$, I pick $v=0$:
$$x_{1} \otimes 0=0, \quad \Rightarrow 0 \in... |
H: Semicircle law theorem (Math notation)
I will put part of the sentence here because I am interested in something very specific. It follows that
Let $I\subset \mathbb{R}$ be an interval. Define the random variables
$$
E_n(I)=\frac{\#\left( \{\lambda_1(\mathbf{X}_n),...,\lambda_n(\mathbf{X}_n)\}\cap I\right)}{n}.
$$
... |
H: Question about a proof of a theorem regarding montoness and countable additivity of a measure
I have a question regarding the proof of the following theorem:
If $\mu$ is a measure on a ring $\mathbb{R},$ if $E \in \mathbb{R},$ and if $\left\{E_{i}\right\}$ is a finite or infinite sequence of sets in $\mathbf{R}$ su... |
H: Three-Handed Gambler’s Ruin - end of the game
Three players start with $a,b$, and $c$ chips, respectively, and play the following game. At each stage, two players are picked at random, and one of those two is picked at random to give the other a chip. This continues until one of the three is out of chips, and quits... |
H: Twice differentiable function f(x) satisfying $f(x)+f''(x)=2f'(x)$
Consider a twice differentiable function f(x) satisfying $f(x)+f''(x)=2f'(x)$ where $f(0)=0,f(1)=e$.
Find the value of $f'(-1)$ and $f''(2)$
I used the following concept $f(x)=\alpha e^{ax}-\alpha $ as it satisfies $f(0)=0$ and $\alpha =\frac{e}{e^a... |
H: A conditional probability question: Three identical jewelry boxes with two draws each. Each draw contains a watch
The following problem is from the book "Probability and Statistics" which is part of the Schaum's outline series. It can be found on page 30 and is problem number 1.57.
Problem:
Each of three identical ... |
H: Weak convergence implies norm inequality.
When I was reading Mathematical Methods in Quantum Mechanics With Applications to Schrodinger Operators by Gerald Teschl. Link here:
http://www.ams.org/bookstore-getitem?item=gsm-157.
I found in page 56 that if $\psi_n \rightharpoonup \psi$ on a Hilbert space $H$ then
$$\li... |
H: Determine values of $\theta$ for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions
So there is this question that's asking for a "range of values for theta from $-\pi$ to $\pi$, for which $\arg(z-4+2i)=\theta$ and $|z+6+6i|=4$ have no common solutions."
I'm not really sure how to do it as my tea... |
H: How to find values of variables for which $f(x)= x^3+3x^2+4x+b \sin x + c \cos x$ is one-one?
I have a question that tells me to find the range of $b^2+c^2$ for which $$f(x)= x^3+3x^2+4x+b \sin x + c \cos x,$$ $\forall x \in R$ is one-one function.
[The answer says that $b^2+c^2 \leq 1$]
My book does this question ... |
H: Is limit of a an increasing function is a bound?
I have a function $f(x)$ which is defined for positive $x$. I know that $f'\geq 0 $ and $lim_{x \to \infty} f(x) = m$. Can I say that $f(x)$ is bounded by $m$?
AI: No. "$f(x)$ is bounded by $m$" (for $m \geq 0$) means that all values of $f$ have magnitude no greater... |
H: Examples of functions satisfying two particular properties
Give some examples of a general function $f(a,b)$ with $b > 0$ satisfying the following two properties:
$f(a,b) > a$;
$f(a_1, b) - f(a_2, b) \leq a_1 - a_2$.
Obviously $f(a, b) = a + g(b)$ with $g(b) > 0$ satisfies those two properties. Are there any othe... |
H: Prove that for any positive integer $a,$ $a^{561} \equiv a \pmod{561}.$
Prove that for any positive integer $a$, $a^{561} \equiv a \pmod{561}$.
(Hence, $561$ is a pseudoprime with respect to any base. Such a number is called a Carmichael number.)
This obviously works for $1$ but how do I find $2^{561}$ or any other... |
H: How to prove that f(w)=0
Let I=[a,b] , let f:$\rightarrow$R is continuous on I,and assume f(a) < 0 and f(b)>0 .let W={x$\in$I:f(x)<0},and let w:=supW .prove that f(w)=0.
My work: since w=supW then either w$\in$W or w is a limit point of W. If w is in W then this question does not make any sense so w is a limit poin... |
H: Probability of $(0,...,9)$ balls never being drawn on $10$ draws from $10$ balls with putbacks.
I want to calculate the expected value and variance for the random variable
$$X = \text{number of balls which were never drawn}$$
when drawing from $10$t times from $10$ different balls with putbacks.
To calculate the ex... |
H: What is a simple example of a reduced, noetherian, local ring of dimension $0$ which is not Gorenstein?
As the title says, I am looking for a noetherian local ring $R$ of dimension 0 which is reduced (and thus Cohen-Macaulay) but not Gorenstein.
Due to Bruns, Herzog $-$ Cohen-Macaulay Rings Theorem 3.2.10 every noe... |
H: Annihilator, vector space
Let $\dim_K(V)\geq 1$ and $M \subset V$ with $\emptyset \neq M \subsetneq V$.
Can $M$ exist with $M^0=V$? $M^0$ refers to the annihilator of $M$ and $V$ is a finite dimensional vector space.
AI: Edit: the spoiler was wrong! The set $M$ is assumed to be non empty but it could very well be z... |
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