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H: Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$
Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n \geq \varepsilon$ for some $\varepsilon > 0$ this is obvious.
Any idea?... |
H: Show that a function is continuous
Let K be bounded and continuous and bounded on $\mathbb{R}^{n}$ and let $f$ be Lebesgue integrable on $\mathbb{R}^{n}$.
Show that the function $g$ defined on $\mathbb{R}$ by
$g(t) = \int_{\mathbb{R}^{n}} K(tx)f(x)dx$
is well defined and continuous on $\mathbb{R}$.
Well defined:
... |
H: Distribution for Response Times
I have samples from a response time population for a web transaction. I want to be able to use them to describe a distribution for the population but don't know a proper one to use. I have shied away from a Normal since it would result in some probability of getting a negative time. ... |
H: Why are there problems when interpolating $f(x)=\arctan(10x)$?
Given $f(x)=\arctan(10x)$, there would be a problem when we interpolate it by using Lagrange's method. This would have something to do with the derivatives of $f(x)$. I plotted some derivatives of $f$ but I did not come up with an answer. Can anybody te... |
H: How to calculate $\lim_{n \to \infty}\frac{r^n}{n}$
I'm a bit rusty on calculus and I'm not able to solve this rather simple limit:
$$\lim_{n \to \infty}\frac{r^n}{n}$$
In my case $r = -1$, and "just by looking at it" I'd guess that for $\left|r\right| = 1, n \to \infty, \frac{\pm1^n}{n} \to 0$. But I was wondering... |
H: Smooth Quartics in $\Bbb{P}^3$
Algebraic category. Ground field $\Bbb{C}$.
This is a naive question: are all smooth quartic surfaces in $\Bbb{P}^3$ isomorphic ?
The answer is NO if and only if there is a smooth quartic in $\Bbb{P}^3$ containing some (-1)-curve.
AI: A smooth quartic in $\mathbb P^3$ is a K3 surface... |
H: Is this piecewise-defined function on $\mathbb{R}^2$ continuous at $(0,0)$? What about differentiable?
Is this function is a differentiable function, a continuous function at the point $(0,0)$?
How to show that ?
$$
f(x,y)=\begin{cases}
\frac{x^{3}y+xy^{3}}{x^{2}+y^{4}} &\text{if }(x,y)\neq (0,0),\\
0 & \text{if }... |
H: Question about a proof that $\mathbb{Q}$ is dense in $\mathbb{R}$
This is from Ross's elementary analysis book. The statement is if $a,b \in \mathbb{R}$ such that $a<b$ then there exists a rational $r \in \mathbb{Q}$ such that $a<r<b$.
I don't understand an important part of the proof which I will point out.
Here i... |
H: True or false? About unitary operation.
Let $V$ be a finite inner product space. Let $T:V\to V$ be a linear transformation. Suppose that $v_1,...,v_n$ is an orthonormal basis of $V$ such that $(Tv_i,Tv_i)=1$ for every $1\leq i\leq n$. Then $T$ is unitary operation.
Is that statement true?
AI: Unitary operators are ... |
H: Application Church-Turing thesis
I would like to give examples of problems which are solvable with an algorithm, for example the function $f$ which maps the tuple $(n,m)$ to the greatest common divisor. This map is recursive. I would do it in the following way: 1st step: If $m$ equals $0$, then $\mathrm{gcd}(n,m)=n... |
H: What is the terminology for the non-repeating portion of a rational decimal?
Given a number co-prime with 10, such as thirteen, we can construct a repeating decimal from its reciprocal: $\frac{1}{13}$ = 0.(076923). If we successively divide this number by a factor of 10 (i.e., 2 or 5) we get a sequence of numbers w... |
H: Ideal of finite intersection of algebraic sets
In general if $X_1$ and $X_2$ are two algebraic sets on $k^n$ with $k$ a field of characteristic zero, we have that $I( X_1 \cap X_2 ) = \sqrt{ I(X_1) + I(X_2) }.$
Is posible in general compute $I(X_1 \cap X_2 \cap \dots \cap X_n)$ in terms of $I(X_1), \dots , I(x_n)$?... |
H: Is it possible to make integers a field?
Is it possible to define addition and/or multiplication on the set of
a) natural numbers (including $0$: $0,1,2,3,...$)
b) integers $(..., -2, -1, 0, 1, 2, ...)$
in such way that they will become fields?
Thanks in advance.
AI: Let $X$ be any countably infinite set (such as $... |
H: a group is not the union of two proper subgroups - how to internalize this into other categories?
A well-known fact from group theory is that a group cannot be the union of two proper subgroups. I wonder, does this statement internalize into other categories than the category of sets? That is, is there some corresp... |
H: Need help with algebra portion of calculus finding slope of secant line
The example problem is: Given f(x) = $x^2$,find and simplify the slope of the secant line for a = 1 and h = any non-zero number.
The answer is as follows:
For a = 1 and h any non-zero number, the secant line goes through (1, f(1)) = (1,1) and ... |
H: convergence of series with $k!$
check if the following series converges:
$\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$
where $k!!=k(k-2)(k-4)(k-6)...$
I came across this exercise while going trough some old exams. I'm pretty sure we have to bound the sequence and apply Leibniz-criteria but after a while ... |
H: Image of the Norm on a Finite Dimensional Extension of $\mathbb{Q}_p$
I've been trying to see whether following assertion is true in order to give a quick proof of another problem I was doing: if $K$ is a finite dimensional extension of the $p$-adic numbers $\mathbb{Q}_p$, we have the multiplicative field norm $N_{... |
H: Introductory/Intuitive Functional Analysis Book
Can you recommend a gentle introduction to the abstract thinking and motivation of functional analysis? I'm looking for a book that holds you by the hand and shows the details of exercises, etc.
Thanks.
AI: Of the well written books on Functional analysis that I've se... |
H: Cauchy's Theorem- Trigonometric application
any help would be very much appreciated. The question asks to evaluate the given integral using Cauchy's formula. I plugged in the formulas for $\sin$ and $\cos$ ($\sin= \frac{1}{2i}(z-1/z)$ and $\cos= \frac12(z+1/z)$) but did not know how to proceed from there.
$$\int_0^... |
H: Ordered combinations
If I have 2 sets of elements, say $\{A, B, C, D\}$ and $\{P, Q, R, S\}$, how can I calculate combinations of combined set $$\{A, B, C, D, P, Q, R, S\}$$ such that order $A\to B\to C\to D$ and $P\to Q\to R\to S$ is maintained in each combination.
For example $$\{A, B, C, D, P, Q, R, S\},\\\{A, ... |
H: What does area represent?
Since any two Euclidean shapes have an infinite number of points inside of them, shapes with different area have the same infinite number of points in them (and any object has the same number of points in it as are inside the plane it is inside). So area isn't a measure of the amount of po... |
H: Is second derivative of a convex function convex?
If $f$ is twice differentiable and convex, is it true that $f''$ is a convex function ?
AI: I thought I'd take a personal challenge to find a counterexample on the entire real line. Here's a method for constructing such a function: let $g_1(x)$ be any nonconvex but ... |
H: Continuous Linear Mapping $C[0,1]\rightarrow C[0,1]$
Show that $L(f)(x)= \int_0^x f(t) dt $ is a continuous linear mapping from $C[0,1]$ into itself.
Do I only have to show that the operator is bounded? How to do I explicitly choose my $M$ such that $\|L \ f\|<M\|f\|$?
AI: Yes, continuous is equivalent to bounded,... |
H: Proving Bijections in $\mathbb{R}^n$
I have a question that may seem trivial or silly, however I'll try to make my point clear. I'm a student of Mathematical Physics and unfortunately my college doesn't offer a set theory course, so that we are not taught to prove bijections between arbitrary sets. The single varia... |
H: Any practical difference between the metrics $d_1=\sup\{\left|{x_j-y_j}\right|:j=1,2,...,k\}$ and $d_2=\max\{\left|x_j-y_j\right|:j=1,2,...k\}$?
I've been asked to do a proof showing that $d_1\left({x,y}\right)$ is a metric on $\mathbb{R}^k$, but is there any difference between this and $d_2\left({x,y}\right)$, for... |
H: Show that $4mn-m-n$ can never be a square
Let $m$ and $n$ be positive integers. Show that $$4mn-m-n$$ can never be a square.
In my attempt I started by assuming for the sake of contradiction that $$4mn-m-n=k^2$$ for some $k \in \mathbb{N_0^+}$. Then I considered $k^2 \pmod3$ (I couldn't find a way for $\pmod4$ or ... |
H: Five digit re-write game
In the habit of factoring numbers, a notebook I bought had a five digit item number $77076$, which factors as $2^2 3^2 2141$, which may also be $9 \cdot 8564$, and in this form the count of digits is again five. (Repeated digits count separately). [Note I thank Calvin Lin for pointing out I... |
H: Questions about $\mathrm{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})$
Consider the extension $\mathbb{Q} \subset\mathbb{Q} (\sqrt{2}, \sqrt{3})$. How many elements are there in $\text{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathbb{Q})?$ Describe all elements in $\text{Aut}(\mathbb{Q} (\sqrt{2}, \sqrt{3})/\mathb... |
H: For a given real square matrix $A$ what is meant by $e^{kA}$ where $k$ is real.
For a given real square matrix $A$ what is meant by $e^{kA}$ where $k$ is real?
I've problem involving this notion and I wondered if $e^{kA}=(e^{ka_{ij}})$ where $A=(a_{ij}).$
AI: It's the matrix exponential. By definition,
$$e^A = \sum... |
H: Total area of squares.
We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
AI: Consider a square of side $a$. If the new square rotated by $\theta$ has side of length $b$, we ... |
H: Is there a procedure to solve Diophantine Equations?
How would you go about solving a multivariable, non-linear Diophantine Equation?
AI: By a famous result of Matiyasevich, there is no universal algorithm which, when fed any Diophantine equation, will determine whether or not that equation has a solution in intege... |
H: What would be the expected number of targets which didn't get hit by any of the shooters?
Assume there are 5 shooters and 5 targets. Each shooter would choose a target randomly and when the signal is given all shooters will fire at the same time and hit their target. What would be the expected number of targets wh... |
H: Order of the largest cyclic subgroup of $\mathrm{Aut}(\mathbb{Z}_{720})$
Back to easier problems for a bit...
I have been told that it is possible to find the order of the largest cyclic subgroup of $\mathrm{Aut}(\mathbb{Z}_{720})$ without considering automorphisms of $\mathbb{Z}_{720}$. Here's what I have:
We hav... |
H: Why does $3+ (-1)\left(\left\lfloor\sum_{k=1}^{|\{x\in P,\;x\le n\}|} \frac{P_k}{1-P_k}\right\rfloor\right) = \pi(n),\quad n\ge1223$?
Let $P$ denote $\text{primes}$, and $\pi(x)$ denote $|P| \le x$.
Here's my first question: Why does
$$3+ (-1)\left(\left\lfloor\sum_{k=1}^{|\{x\in P,\;x\le n\}|} \frac{P_k}{1-P_k}\r... |
H: A paradox? Or a wrong definition?
Let $A$ be a commutative ring with $1 \neq 0$. Then writing $V(1) = V((1))$, we have $\bigcap_{\mathfrak{p} \in V(1)}\mathfrak{p} = \sqrt{(1)} = (1)$.
But then $\bigcap_{\mathfrak{p} \in V(1)}\mathfrak{p} = \{x : x \in \mathfrak{p} \text{ for all prime ideals } \mathfrak{p} \supset... |
H: Finding an example of a bounded sequence in a complete metric space such that the sequence has no partial limit
I'm working through an analysis text and I've come across this exercise:
Give an example of a complete metric space $X$ and a bounded sequence $\left(x_{n}\right)$ in $X$ such that the sequence $\left(x_{... |
H: Diffeomorphism preserves dimension
I read from Milnor's book $\textit{Topology from the Differentiable Viewpoint}$ this assertion
"If $f$ is a diffeomorphism between opensets $U\subset R^k$ and $V\subset R^l$, then k must equal l, and the linear mapping
$$df_x:R^k\rightarrow R^l$$ must be nonsingular."
The proof w... |
H: If $F$ is a finite field of size $q=p^n$ and $b\in K$ is algebraic over $F$ then $b^{q^{m}} = b$ for some $m > 0$
I want to apply a similar type of argument to show that $\alpha^{q} = \alpha$ for all $\alpha\in F$. When we know that the characteristic of $F$ is $p$ and $|F| = q = p^{n}$. But I dont know how to in... |
H: Find the flaw in my proof that $z^2 =1$ has more than $2$ distinct solutions.
Let $z \in \mathbb{C}$ be any number that satisfies the equation $z^2=1$. Certainly, $z=\pm1$ are two possible solutions to this equation. I claim that $z^k$ is also a solution to this equation for any $k \in \mathbb{R}$, resulting in (pr... |
H: Continuity of the real and imaginary parts of a continuus complex-valued function
If a complex-valued function is continuous, are the component real and imaginary parts $u(x,y)$ and $u(x,y)$ necessarily continuous? If so, why?
AI: The functions $\operatorname{Re}, \operatorname{Im}: \mathbb{C} \to \mathbb{R}$ are c... |
H: Give an example of the $a,b,c$ which satisfies conditions in the generating set
How to derive the specific case of the generating element of a group given its generating set. For example, when
$$G=\langle a,b,c|a^3=b^3=c^2=1,ab=ba,ca=a^2c,cb=b^2c\rangle$$
we can let $G\subset S_3\times S_3$, and let
$$a=((123),1),... |
H: Square bracket $[X]$ with finite fields and polynomial rings
I understand that parentheses are used for functional notation. I do not have any confusion about this one.
However, in some literature, I find square brackets ($[X]$) after some finite field notation.
Source: Encyclopedia of Cryptography and Security, ... |
H: What are morphisms of functors
I am not been able to understand, what is a morphism between two functors. I have gone through the formal definition involving a commutative diagram. Can someone explain that to me in a bit more details or in pictures?
AI: Given categories $\mathcal{C}$ and $\mathcal{D}$, a functor $F... |
H: simplifying equation with logs
I have the following equation:
I would like to solve this for Ze. I have found the same equation expressed in terms of Ze in another paper:
I can't get my head around how this works. This is my attempt:
However, there is something about the last step that doesn't seem right to me, ... |
H: Calculation Of Integral Related To Sequence
Let's evaluate the following integral. Many trials but no success.
$$\int_{-\pi}^{\pi}\dfrac{\sin nx}{(1+\pi^{x})\sin x}dx$$
AI: Hint:
Fix $x=-x \implies dx=-dx$
$$I=-\int_{\pi}^{-\pi}\dfrac{\sin n(-x)}{(1+(\pi)^{-x})\sin (-x)} dx$$
$$I= \int_{-\pi}^{\pi}\dfrac{\sin n(-x)... |
H: Prove that the tangent space has the same dimension as the manifold
I asked this question a couple of days ago. And I thought that I totally understood the question. However it turned out that I didn't, since the argument I constructed was proved to be wrong just now: Diffeomorphism preserves dimension
The original... |
H: Invariant subspaces of tensor product of SU(2)
Let $\varphi_n$ denote the standart irreducible representation of $SU(2)$ group with highest weight $n$.
I know that irreducible representations of $\varphi_2 \otimes \varphi_3 = \varphi_5 \oplus \varphi_3 \oplus \varphi_1$ (according to Clebsh-Gordan decomposition).
W... |
H: Symmetric, transitive and reflexive properties of a matrix
Say I had a relation
\begin{align}
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\end{align}
where $a,b,c,d \in \mathbb{R}$, where $X$ is related to $Y$ if and only if $\det(X) = \det(Y)$ (where $\det(A) = ad-bc$)
So I say it is reflexive since $xRx$ since... |
H: Does a product measure on a product space constructed from two sub-fields of the same space determine a measure on the underlying space?
Let $\mathcal{A}_1,\mathcal{A}_2$ be $\sigma$-algebras on $\Omega$. Let $P$ be a probability on $\mathcal{A}_1$ and let $Q$ be a Markov kernel from $\mathcal{A}_1$ to $\mathcal{A}... |
H: Question Regarding the Axiom of Extensionality
Jech's text on Set Theory states the following:
If X and Y have the same elements, then X = Y :
∀u(u ∈ X ↔ u ∈ Y ) → X = Y.
The converse, namely, if X = Y then u ∈ X ↔ u ∈ Y, is an axiom of
predicate calculus. Thus we have
X = Y if and only if ∀u(u ∈ X ↔ u ∈ Y).
The... |
H: $\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost.
Let $x_1 , x_2, \dots, x_k$ be complex numbers satisfying:
$$x_1 + x_2+ \dots + x_k = 0$$
$$x_1^2 + x_2^2+ \dots + x_k^2 = 0$$
$$x_1^3 + x_2^3+ \dots + x_k^3 = 0$$
$$\dots$$
Th... |
H: Linear algebra classic, Farkas lemma application
$A \in M_{m \times n}(\mathbb{R})$ and $b \in \mathbb{R}^m$.
Farkas' lemma says exactly one of the following holds:
(a) there exists some $x \in \mathbb{R}^n$, $x \geq 0$, such that $Ax = b$
(b) there exists some vector $p \in \mathbb{R}^m$ such that $p^TA \geq 0$ an... |
H: Show that for all $a,b,c>0$, $\frac 1 {\sqrt[3]{(a+b)(b+c)(c+a)}}\geq\frac 3 {2(a+b+c)}$.
Show that for all $a,b,c>0$, $\displaystyle\frac 1 {\sqrt[3]{(a+b)(b+c)(c+a)}}\geq\frac 3 {2(a+b+c)}$.
I tried to cube the both sides, and expand it, but that'll be too troublesome, is there simpler ways? Thasnk you.
AI: HIN... |
H: Being inside or outside of an ellipse
Let $A$ be a point $A$ not belonging to an ellipse $E$. We say that $A$ lies inside
$E$ if every line passing trough $A$ intersects $E$. We say that $A$ lies otside $E$
if some line passing trough $A$ does not intersect $E$. Let $E$ be the ellipse
with semi-axes $a$ and $b$. Sh... |
H: Number of equivalent rectangular paths between two points
I am trying to determine the number of paths between two points.
I am representing the paths as a list of steps "ruru" = right -> up -> right -> up
For my purposes, we can assume that there will never be "backwards" steps (lrlr), and we will only be working ... |
H: property of complementary cumulative distribution function
$$\Bbb{E}(X) = \int_0^\infty xf(x)dx \ge \int_0^c xf(x)dx + c \int_c^\infty f(x)dx$$
I'm having trouble understanding the above formula.
I do understand that $ \Bbb{E}(X) = \int_0^\infty xf(x)dx \ge \int_0^c xf(x)dx$ but I don't yet get where $c \int_c^\inf... |
H: For which values is $x^3$ less than or equal to $3x$?
The title says it all. The answers say:
$x\le -\sqrt{3}$ and $0\le x\le \sqrt{3}$
(can someone edit this so all the $<$ have an 'or equal to' sign. Edit the roots as well please.
I'm not sure how to attempt this question. When I simplify, I get $x^2\le 3$, so $... |
H: Is it true that a monotic, differentiable function with non-zero derivative has a continuous inverse?
Is it true that a strictly monotic, differentiable function on $\mathbb{R}$ with non-zero derivative has a continuous inverse?
This is a small caveat in a problem I'm working on, if this is true then I'm all good.
... |
H: Continuity of an $\mathbb {R}^2$ function
Let $f$ be an $\mathbb{R}^2$ endomorphism and $N:\mathbb{R}^2\to\mathbb{R}^+$
defined by $$\forall u \in \mathbb {R }^2, N(u) = ||f(u)|| $$
I need to show $N$ is continuous.
The problem is that $N$ is only a seminorm, otherwise it would have verified Lipschitz criterion an... |
H: Proving $f(x) = x^2 \sin(1/x)$, $f(0)=0$ is differentiable at $0$, with derivative $f'(0)= 0$ at zero
I need a solution for this question. I've been trying out this question for days and I haven't been able to find out its solution yet. And some explanation would help too.
Show that the function f defined by:
$$... |
H: Plane geometry tough question
$\triangle ABC$ is right angled at $A$. $AB=20, CA= 80/3, BC=100/3$ units. $D$ is a point between $B$ and $C$ such that the $\triangle ADB$ and $\triangle ADC$ have equal perimeters. Determine the length of $AD$.
AI: Add up the the sides of $\triangle ADB$ and $\triangle ADC$ and you g... |
H: Is there any simple way to find out all divisors of $n+1$ under the given conditions?
Aussuming I have given a really large number $n \in \mathbb{N}$ (let's say, $10^{80} \le n \le 10^{100}$) and I know all the divisors of every number $x=0,1,\ldots,n-1$.
Is there any simple, universal and not too time-consuming wa... |
H: I don't understand equivalence classes with relations
I am not quite understanding equivalence classes. For example I have this problem:
Let $A$ be the set of integers and
$\quad a\;R\;b\quad$ if and only if $\quad |a| = |b|$.
I have proved that this is an equivalence relation, (its reflexive, symmetric and transi... |
H: There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart.
There are 50 rooms in a line. If there are 26 rooms with girls, prove there are two girls exactly 5 rooms apart.
My idea was place 25 girls in into pairs of rooms, and there is no scenario which there ar... |
H: Find the limiting distribution
Find the limiting distribution for $n\rightarrow \infty \text{ of} \prod\limits^n_{i=1}X_i$. Given is that $f(x)=\frac{1}{2x\sqrt{2\pi}}e^{-\frac{1}{8}(\ln x-\theta)^2}, x\geq 0$.
AI: The density of the individual $X_i$ is nearly the density of a normal distribution, but it has $\lo... |
H: a recurrence equation interpolating linear and exponential
$f(n+1) = f(n) + f(n)^{a}$ where $a \in (0,1)$ and $n \ge 1$ with $f(1) = m$.
If $a=0$, we see $f(n) = m + n - 1$ and if $a=1$, we see $f(n) = 2^{n-1}m$. So the recursion seems to interpolate between linear and exponential forms.
Is there a closed form or ... |
H: Curve fitting: $a f_1(x)+b f_2(x)+c f_3(x)+d f_4(x)$
I want to fit my data $y$ with a method such as
$$a f_1(x)+b f_2(x)+c f_3(x)+d f_4(x)$$
However I don't know how to obtain coefficients $(a,b,c,d)$ of the known functions $(f_1(x),f_2(x),f_3(x),f_4(x))$. I don't know least squares very well. Therefore I want a ma... |
H: is this symmetric
A is the set of all functions $\mathbb{R}$ $\to$ $\mathbb{R}$
f is related to g if and only if f(x) $\le$ g(x) for all x $\in$ $\mathbb{R}$
I said its reflexive since it is less than OR equal, so f(x)=f(x)
However would it be symmetric and transitive?
I said it would not be symmetric (counter-exam... |
H: Simple meaning to Center of a group
Recently I was learning Center of groups and on referencing the group table, I observed is that all the rows that are also present as columns are the centers of any group.
So, I made a small program to check it for various groups and found somewhat consistent answers. I wrote a ... |
H: 3 cards be selected from a pack of 52 playing cards if at least one of them is an ace? not more than one is an ace? Distributing 9 books to 3 peoples…
I am currently studying Extension 1 Mathematics. I missed two classes and I figured out that tomorrow I will have a quiz. Can you help me to solve this permutation a... |
H: How to show $x^4 - 1296 = (x^3-6x^2+36x-216)(x+6)$
How to get this result: $x^4-1296 = (x^3-6x^2+36x-216)(x+6)$?
It is part of a question about finding limits at mooculus.
AI: Hints: $1296=(-6)^4$ and $a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+\ldots+ab^{n-2}+b^{n-1})$. |
H: How to evaluate limiting value of sums of a specific type
We know that if $f$ is integrable in (0,1) then
$$
\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}f(k/n) = \int_{0}^{1}f(x)dx.
$$
Recently I found the following sum
$$
\lim_{n \to \infty} \sum_{k=1}^{n}\frac{k}{n^2 + k^2} = \frac{\ln 2}{2}.
$$
This sum cannot... |
H: Series of Vectors
In $\mathbb{R}^n$ we define sequences of it's elements in a very natural manner, we say that a sequence is a function $x : \mathbb{N} \to \mathbb{R}^n$ and we denote it by $(x_k)$ as in the $n=1$ case. Then defining limit of a sequence and all of that is pretty straightforward and acts like one ex... |
H: Simple conceptual Fundamental Thm of Calculus question
When applying the Fundamental Thm of Calculus in complex analysis, what does it mean for an open connected set to contain a loop? For example, does my red-color open annulus contain the black colour loop? I think so, but I'm struggling with understanding this:
... |
H: Question about an almost split sequence.
On page 124 of the book Elements of representation theory of associative algebras, volume 1, Example 3.10, I computed the modules in this example.
$$
S(3)=0\leftarrow 0 \rightarrow K \leftarrow 0, \\
P(2) = K\overset{1}{\leftarrow} K \overset{1}{\rightarrow} K \leftarrow 0... |
H: Infinite prime numbers from a sum of powers
I am not sure if it's possible to get infinite prime numbers from this sum:
$$p=k^j+j^k$$
with $j\in\mathbb{N}, k\in\mathbb{N}$
I tried for $j=1,2,...9,k=1,2,...9$ and I get only eleven prime numbers.
If I consider the matrix:
$$A(k,j)=k^j+j^k$$
in which the components $A... |
H: If three corners of a parallelogram are known solve for the 3 possible 4th corners.
An example would be three corners being the points: (1,1), (4,2) and (1,3). I understand the specific solution for this example: (4,4), (4,0) or (-2,2). Which I reasoned when i drew it out. The example came from a linear algebra tex... |
H: How do you solve for Y?
I know there has got to be a way to solve for $Y$ but I just can't seem to figure it out. Does anyone know how to solve this? Please help :)
$$5(Y(8))=C$$
$$C(Y(4))=B$$
$$B(Y(2))=A$$
$$A(Y(1))=0.50$$
AI: Hint: Substitute for $C$ from the first to the second equation to get:
$160Y^2 = B$
Then... |
H: Simple Linear Regression Question
Let $Y_{i} = \beta_{0} + \beta_{1}X_{i} + \epsilon_{i}$ be a simple linear regression model with independent errors and iid normal distribution. If $X_{i}$ are fixed what is the distribution of $Y_{i}$ given $X_{i} = 10$?
I am preparing for a test with questions like these but I... |
H: Vector space: $\forall a\in K, v \in E ( a \cdot_E v=0_E \to a=0_K \lor v=0_E)$
I need to prove the following:
let $E$ vector space on $K$, then $\forall a\in K, v \in E ( a \cdot_E v=0_E \to a=0_K \lor v=0_E)$
Thanks in advance!
AI: It suffices to prove that if $a v = 0_E$ (I take there's a misprint in you formul... |
H: How do I solve $x^2 + y^2 + xy = z$ for $y$
How do I solve the following equation for $x$ or $y$ (does not matter because you can swap them):
$$
x^2+y^2+xy=z
$$
AI: To solve for $x$, say, just consider $y$ and $z$ as constants, and use the usual formula for the quadratic equation. |
H: Probability of two variables of having the same value
Let $X$ and $Y$ be two random variables, whose PDFs $f_X$ and $f_Y$ are uniform. $f_X$ and $f_Y$ may overlap. For instance, they could represent two score distributions for two tuples $x$ and $y$ in a database.
Which is the probability for $X$ and $Y$ of having ... |
H: Why to obtain the coordinates of vectors in the basis that themselves belong?
Let $ \space T: \mathbb{R^2} \to \mathbb{R^3}$ a linear transformation defined as $ \space T(x,y)=(3x+2y,x+y,-2x-y)$, where $\beta=\{(1,-1),(0,1)\}$ is a basis of $\mathbb{R^2}$. Is not specify a basis for $\mathbb{R^3}$.
By the express... |
H: Prove that expected value of X is greater than Y, if given that $P(X\ge Y)=1$
I have to prove that $E(X)$ (Expected Value of a random variable X), is greater than $E(Y)$, if given that $P(X\ge Y)=1$.
my thoughts so far:
I know from the $P(X\ge Y)=1$ statement, that the values that X "receives" are always greater th... |
H: Embedded Lp spaces
Let $L^\infty(Ω,F,P)$ be the vector space of bounded random variables $(X ∈ L^\infty (Ω,F,P)$ means that there exists a constant C such that $|X(ω)|≤C$, a.s.$)$. Show that $$L^\infty(Ω,F,P)⊂L^2(Ω,F,P)⊂L^1(Ω,F,P)$$
AI: It is a consequence of Holder inequality
$$
E[|XY|]\leq E[|X|^p]^{1/p}E[|Y|^q]^... |
H: Regarding $\lim_{n \to \infty} n^{\frac{1}{n}}$
Suppose $\lim_{n \to \infty} n^{\frac{1}{n}} = l \in \mathbb{R}$. The function $f(x) = x^n$ is continuous, then
$$l^n=\left (\lim_{n \to \infty} n^{\frac{1}{n}} \right)^n=\lim_{n \to \infty} \left ( \left (n^{\frac{1}{n}} \right)^n \right ) =\lim_{n \to \infty} n = \i... |
H: $\sum_{k=1}^n m(k)$, where $m(k)$ is defined by $2^{m(k)} || k$.
I'm looking at the sum:
$$f(n) = \sum_{k=1}^n m(k),$$
where $m(k)$ is defined by $2^{m(k)} || k$, i.e. $2^{m(k)}$ is the largest power of $2$ that divides $k$. For example, we have $f(8) = 0+1+0+2+0+1+0+3 = 7$. Here's a table of $n$, $m(n)$, and $f(n)... |
H: Find an interval of convergence and an explicit formula for $f(x)$
Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$
If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$.
The answers are $I = (-1,1)$ and $f(x) = \frac{1 + 2x}{1 - x^2}$
Can anyone please gi... |
H: Behaviour of $f$ in the neighbourhood of $c$ if $f'(c)= \cdots = f^{(n)}(c)=0$, and $f^{(n+1)}(c) \gt 0$
What can I say about the behaviour of $f$ in the neighbourhood of $c$ if
$f'(c)= \cdots = f^{(n)}(c)=0$, and $f^{(n+1)}(c) \gt 0$? I know the behaviour of $f$ if $n \le 2$, but I do not know how to generalize th... |
H: Show that a vector that is orthogonal to every other vector is the zero vector
I have the following question, and I'd like to get some tips on how to write the proof. I know why it is, but I'm still not so great at writing it mathematically.
If $u$ is a vector in $\mathbb{R}^n$ that is orthogonal to every vector i... |
H: Can a finitely generated $\mathbb{Z}$-algebra contain $\mathbb{Q}$?
Is there a ring between $\mathbb{Q}$ and $\mathbb{R}$ that is finitely generated as an algebra over $\mathbb{Z}$? My guess is there isn't.
I can see that it would have to be finitely generated over $\mathbb{Q}$ as well, and I think I can deal with ... |
H: calculate $ \lim_{k\rightarrow\infty}\sin\left(kx\right) $
if $ x \notin \pi\mathbb{N} $ why the limit
$ \lim_{k\rightarrow\infty}\sin\left(kx\right) $
does not exist?
AI: so when $x\rightarrow \infty $ what you see ? is there any existed value for f(x)
note thate $-1\leq sin(kx)\leq 1$ :$k\in R$ |
H: A probability question regarding combinatorics
From a class of $300$ students, three are selected at random to receive three identical prizes. Of the students, $200$ are from department $A$, $60$ from department $B$ and $40$ from department $C$.
Find the probability that the three winners come from different depar... |
H: Derivatives for a piecewise defined function
$f$ is a function on real line and $f(x)=\begin{cases} x^2& x<0\\ 2x+x^2&x\ge 0\end{cases}$. Could any one tell me which of the following is/are true?
1.$f'(0)$ doesn't exist
2.$f'(x)$ exist other than at $0$
3.$f''(x)=2$
4.$f''(0)$ does not exist
I have myself checked ... |
H: Integral with delta Dirac power
Is it possible to calculate the integral:
$$J=\int_{-\infty}^{+\infty}f(x)\delta(x-x_0)^kdx$$
wih $k\in\mathbb{R}$?
I know that in the Colombeau algebra the distribution $\delta(x)^2$ is defined. What happens if the Delta function is raised to a real number different from $2$? Thanks... |
H: Is there a text book containing a self-contained and complete proof of the Jordan Curve theorem?
I seem to remember (in my undergraduate years) encountering a book on complex analysis which contained a proof of the Jordan Curve Theorem, building up from first principles - so self-contained and complete. I am now lo... |
H: $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$
Let $\sum a_n$ be convergent but not absolutely convergent, $\sum_{n=1}^{\infty} a_n=0$,$s_k$ denotes the partial sum then could anyone tell me which of the following is/are correct?
$1.$ $s_k=0$ for infinitely many $k$
$2$. $s_k>0$... |
H: How to show that every complex matrix with orthonormal columns can be supplemented into an unitary matrix?
Show that every matrix $A \in M_{n,k}(\mathbb{C})$ whose columns are orthonormal vectors in $M_{n1}(\mathbb{C})$ can be supplemented with additional n-k columns to an unitary matrix $U \in M_{n}(\mathbb{C})$
... |
H: Prove that a cut edge is in every spanning tree of a graph
Given a simple and connected graph $G = (V,E)$, and an edge $e \in E$. Prove:
$e$ is a cut edge if and only if $e$ is in every spanning tree of $G$.
I have been thinking about this question for a long time and have made no progress.
AI: Suppose $e$ is not... |
H: Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$
Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to expand?
Ideas?
AI: Divide $$(7x-4)-(7x-5)=(4x-1)-(4x-2)\ \ \ \ \ (... |
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