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H: $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$, find $ a_{1995}$
The sequence $a_0,a_1,a_2,...$ satisfies
$a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n}), a_1=1$ for all non-negative integers $m,n$ with $m\ge n$. Find $a_{1995}$
This is a quick sketch of my solution.
$n=0$ gives $a_{2m}=4a_{m}$
$m=n=0$ gives... |
H: Converting area of a rectangle
This is probably a very simple question and I'm being stupid.
Let's suppose I have a rectangle that is 0.6m by 0.4m. To calculate the area of this rectangle, you do 0.6 multiplied by 0.4, which is 0.24m^2.
However, if I convert the units to cm first, to calculate the area you do 60 mu... |
H: How does the Fundamental Theorem of Calculus (FTC) tell us that $\frac{d}{dx}\left(\ln (x)\right)= \frac{1}{x}$?
According to Wikipedia, one common definition of the natural logarithm is that:
$$
\ln (x) = \int_{1}^{x} \frac{1}{t} dt
$$
The article then goes on to say that because of the first FTC, we can deduce th... |
H: If $\ U_{r} = \frac{1+\ U_{r-1}}{2}$ and $\ U_{0}=0$, Find $\lim_{n\to\infty} \sum_{r=1}^n \ U_{r}$
I am trying to understand fully how drug half-life works. So I derived this relationship:
$$\ U_{r} = \frac{1+\ U_{r-1}}{2}$$
Where $\ U_{0}=0$ and r is a set of natural numbers.
My issue to how to deduce a relatio... |
H: Show that the sequence converges and what is it limit?$x_{n+1}=\frac{1}{2}(x_{n}+\frac{a}{x_n})$
The sequence is: $x_{n+1}=\frac{1}{2}(x_{n}+\frac{a}{x_n})$ for$ , n \in \mathbb{N}_{0}$, $a>0$ and $x_{0}=a$
Hint: Show at first that $x^{2}_{n+1} - a \ge 0$ and than take $x_{n+1}-x_{n}$
I tried this way: $\frac{a}{x_... |
H: Class equation of normal subgroups
I am puzzled with a thought or a question regarding class equation and normality of subgroups.Consider the following situation.
Let $G$ be a finite group and $N\trianglelefteq G$.Let $G$ act on $N$ by conjugation which is allowed since $N$ is normal and let the representatives of ... |
H: Designing an XNOR function for real numbers
I would like to design a function of $(x,y)$ which gives a large output for large values and $x$ and $y$ and for small values for $x$ and $y$. For values of $x$ and $y$, when one is large and the other one small, it should yield a small output value. What would be a suita... |
H: Show the correctness of this derivation to the Taylor's Theorem?
As to this beautiful derivation of the Taylor's Theorem, wouldn't it break the equality when we add the term $f(0)$ to the right side of $f(x) = \int_{0}^{x} f'(t)dt$ ?
AI: No. The reason is as follows: we have
$$ \int_{0}^{x} f'(t) dt = \left [ f(t) ... |
H: Probability of Exiting A Roundabout
I am piggybacking on this question: probability of leaving
The question was closed, but I found it interesting and I would like feedback on what I have done with it, and I have more questions about it. I'm not sure what proper etiquette is for piggybacking on closed questions.
I ... |
H: convergence or divergence of $\sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}$
Finding whether the series $$\;\; \sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}$$ is converges or Diverges
What i try
We know that $\sin^2(x)\leq 1$ for all real number.
So $$\sum^{\infty}_{k=1}\frac{k}{k^2-\sin^2(k)}\geq \sum^{\infty}_{k=1}\f... |
H: Estimative of the measure of the intersection.
Let $(\Omega, \mathscr{F}, \mathbb{P})$ be a probability space. Let $\Sigma_n\subset \Omega$ such that $\mathbb{P}(\Sigma_n^c)<\epsilon_n,$ where, $\sum \epsilon_n<\delta$, where $\delta>$ is as small as we want (much smaller than 1).
Can I estimate $\mathbb{P}(\bigca... |
H: How to apply ratio test to prove convergence of $\sum_{n=1}^{\infty} \frac{(2^n n!)^2}{(2n)^{2n}}$
Can someone please help me with this
I know that we should use the ratio test but can you show me the steps in detail
$$\sum_{n=1}^{\infty} \frac{(2^n n!)^2}{(2n)^{2n}}$$
And I stuck here
$$\lim_{n \to \infty}\left|\f... |
H: Q: When can we say that the starting value is near the root in Newton Raphson method?
I would like to know when can I use the Newton Raphson methos to find an approximation of a root. We know that it can be used or that it is possible to work when the starting value is near of the root; but what is near? I mean, if... |
H: What can we say if the inner products of two vectors ($u$ and $v$) with another vector ($x$) are equal? ($x$ is an eigenvector)
We know that $x$ is an eigenvector and the inner products of $x$ with $u$ and $v$ are equal, I mean:
$\langle u,x\rangle \;= \; \langle v,x \rangle$
On the other hand, we know that $v = [1... |
H: Distinct ways to put $N$ balls in $M$ boxes such that there is no more than $K$ balls in each box?
The question is: in how many different ways can I put $N$ indistinguishable balls into $M$ distinguishable boxes such that each box contains no more than $K$ balls it it?
A more general problem: if $K$ is different fo... |
H: basic notion about schemes.. what is the difference between $s(x)$ and $s_x$ for $s \in \Gamma(U, O_X)$
Let $X$ be a scheme and $U$ an open subset of $X$. Let $s \in \Gamma(U, O_X)$ and $x \in U$.
I am getting confused with what the difference is between
$s_x$ and $s(x)$... Are they the same thing?
AI: It is stand... |
H: What is the definition of the Entscheidungsproblem (Decision Problem)?
I have been trying to find the most “formal” definition of the Entscheidungsproblem for the past couple of days now.
On Wikipedia it states this:
The problem asks for an algorithm that considers, as input, a statement and answers "Yes" or "No" ... |
H: Proof that the sequence $ \{g \circ f_{n} \} $ converges uniformly in compact subsets of $ \Omega $ to the function $ g \circ f $.
Let $ \Omega $ be an open in $\mathbb{C}$, $\{f_{n} \}$ a succession of continuous functions of $\Omega$ in $ \mathbb{C}$ that converge to a
function $ f $ uniformly in compact subsets ... |
H: (Proof-check) Alternative formula for the total variation
Good morning!
I'm reposting a message I posted a bit earlier because it was quite messy and I wanted to make it clearer.
I have a continuously differentiable function $f$ on $[a,b]$, and I am trying to prove the following equality:
$\sup\limits_{\mathcal{P... |
H: There is another function to calculate $T_n$?
I was searching about Fibonacci numbers, and I found that Tribonacci numbers also exist given the following recurrence: $T_{n+3}= T_{n+2}+T_{n+1}+T{n}$ with $T_0=T_1=0,T_2=1$. Then I thought about what would be a function that would calculate the value $T_n$ and it wou... |
H: Linear algebra: Null space Basis
Suppose I have a matrix $A_{3\times 3}$ then if my rank of column space is 1 then rank of null space is 2. Is there anyway I can get the basis vectors of null space from this matrix A itself?
AI: There is no such thing as the basis vectors of the null space, since there are infinite... |
H: Pointwise convergence of ${\tau \wedge n}(ω)$ to the rv ${\tau}(ω)$, given that the stopping time is finite
Does ${\tau \wedge n}(ω)$ convergence uniformly to the random variable ${\tau}(ω)$ too?
My intuition is that, it does not, as a result of ${\tau \wedge n}(ω)$ converging to ${\tau}(ω)$ being dependent on $ω... |
H: Defining discrete set
$\mathcal{S}$ is a set with discrete elements. Is it a good practice to write $|\mathcal{S}| < \infty$ in order to say $\mathcal{S}$ is discrete? This is not a common notation, but is short and effective in my view.
Note: it is also not of infinitely many elements.
AI: This doesn't actually m... |
H: Multiplying $P(x) = (x-1)(x-2) \dots (x-50)$ and $Q(x)=(x+1)(x+2) \cdots(x+50)$
Let $P(x) = (x-1)(x-2) \dots (x-50)$ and $Q(x)=(x+1)(x+2) \cdots(x+50).$
If $P(x)Q(x) = a_{100}x^{100} + a_{99}x^{99} + \dots + a_{1}x^{1} + a_0$, compute $a_{100} - a_{99} - a_{98} - a_{97}.$
I've been quite stuck with this one. If I... |
H: Prove that every measure space has a completion.
I am following the book $\\$ An Introduction to Measure and Integration written by Prof. Inder K Rana for my measure theory course. While going through this book I came across the aforesaid theorem which roughly states that every measure space has a completion. The ... |
H: Proving integral transform using banach fixed-point theorem
I'm currently working on the following problem:
Let $K: [0,1]^2 \to \mathbb{R}$ be continuous with $|K(x,y)| < 1$ for all $(x,y) \in [0,1]^2$. Prove the existence of a function $f \in C([0,1])$ s.t.
$$f(x) + \int_0^1 f(y) K(x,y) dy = e^{(x^2)}$$
for... |
H: How can we find the shortest paths in the boundaries of a matrix?
This is another version of a previous question.
In this image,
we have all the points within the shape (yellow area), which can be represented as a matrix.
We aim to connect the boundary points (black dots) within the shape.
How can we find the clos... |
H: Rolle's theorem, such that $f'(c)=2c$
We have that the function $f: \mathbb R \xrightarrow{}\mathbb R$ is differentiable and satisfies $f(0)=0$ and $f(1)=1$.
I need to use Rolle's Theorem to show that there exits $c\in(0,1)$ such that $f'(c)=2c$.
I am unsure how to proceed considering we do not have $f(a)=f(b)$.
AI... |
H: Simplification of ${0 \binom{n}{0} + 2 \binom{n}{2} + 4 \binom{n}{4} + 6 \binom{n}{6} + \cdots}$
Simplify
$$0 \binom{n}{0} + 2 \binom{n}{2} + 4 \binom{n}{4} + 6 \binom{n}{6} + \cdots,$$ where $n \ge 2.$
I think we can write this as the summation $\displaystyle\sum_{i=0}^{n} 2i\binom{n}{2i},$ which simplifies to $\b... |
H: If $M$ is a martingale, show that $X_t = M_t − kt^2$ is a supermartingale iff $k$ is non-negative.
If $M$ is a martingale, show that $X_t = M_t − kt^2$ is a supermartingale iff $k$ is non-negative.
A supermartingale is defined when $X_s \ge \mathbb E[X_t|\mathcal F_s]$ when $s\le t$. Clearly $X$ must be integrabl... |
H: Intuition behind area of ellipse
This is not meant to be a formal proof, but I just wanted to know if this is a valid way of thinking about the area of an ellipse. It does assume knowledge of the area of a circle, but this can be proven without knowledge of the area of an ellipse. I also don't know how to include p... |
H: Showing that disjoint intervals implies independent number of arrivals for random point set
If we have a number of i.i.d. random variables, $X_1, X_2, ..., X_z$ (where $z$ is the realisation of a random variable $Z\sim Po(\lambda)$ independent of each $X_i$) with pdf $f$ that form a random point set, then I want to... |
H: I do not understand an inequality used to prove that e is bigger/equal to exp(1)
I have seen the following inequalies:
$$e \geq 1+1+\sum \limits_{k=2}^{n}\frac{1}{k!}=s_{n}
\implies e\geq \lim_{n\to\infty}s_{n}=exp(1)$$
Based on this information, how can one explain the implication? Why would $$e\geq \lim_{n\to\inf... |
H: How to find residues of $\frac{1+z}{1-\sin z}$
I have to compute an integral:
$$\int_C \frac{1+z}{1-\sin z}$$
where $C$ is the circle of radius $8$.
I would do this way:
the function is holomorphic in $\mathbb{C}-\{\frac{\pi}{2}+k\pi\}$,
$C$ is homologous to $0$ in $\mathbb{C}$ and the singularities don't intercep... |
H: Equilateral triangle in a regular pentagon
I don't understand how the line segment of $|BF|$ equals to $|BC| = |FC| = |AB| = |AE|$. How can the line segment maintains the equilateral triangle? The instructor says while drawing that it just maintains a equilateral triangle, but not how.
AI: $|BC|=|AE|$ because it's ... |
H: Proof that $\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$ without L'Hospital
I proved that $$\lim_{n\to\infty}\left(1+\frac{x^2}{n^2}\right)^{\frac{n}{2}}=1$$
using L'Hospital's rule. But is there a way to prove it without L'Hospital's rule? I tried splitting it as
$$\lim_{n\to\infty}n^{-n}(n^2+x... |
H: Continuity is a local property: topology, proof
Consider topological spaces $X, Y$ and $f: X\to Y$, $x\in X$.
Show: if $f$ is continuous in $x\in V\subseteq X$, then $f_{\mid V}: V\to Y$ is continuous in $x$.
Show: if $f_{\mid V}: V\to Y$ is continuous in $x$ and $V\subseteq X$ is a neighborhood of $x$, then $f$... |
H: Calculate shore distributions and correlation coefficient
Task
We have depth of random vector:
$$f(x,y) =\begin{cases}\frac{1}{5} & \text{for } 0<x<1\;,2<y<7\\0 & \text{for other } x,y\end{cases}$$
a) Calculate shore distributions
b) Correlation coefficient for X,Y
==================================================... |
H: The intersection of maximal subgroups of a group lies in a maximal subgroup of that group
I am trying to prove that the intersection of maximal subgroups of a finite group lies in a maximal subgroup of that group.
My question: Can someone please verify my proof below? I am afraid that the two statements in blue are... |
H: Unknown index in a model theoretic considerations
What is $k$ in $${}^kM$$ here on the page $11$ in the definition $2.10$?
AI: It appears to be an instance of the standard notation ${{^A}B}$ for the set of maps from $A$ to $B$. Here $k\in\omega$, so it’s maps from $k$ to $M$, i.e., effectively $k$-tuples from $M$. |
H: problem about Cantor-Schroeder-Bernstein theorem
suppose we need to prove$|(0,1)|=|(0,1]|$
First, $x\in(0,1)$,
$(0,1)→(0,1]$, $f(x)=x$,injective
$x∈(0,1]$
$(0,1]→(0,1)$, $g(x)=\frac{x}{2}$, injective
So according to sb theorem,$(0,1)↔(0,1]$ is bjective,but $0.6$ is not covered by $g(x)$ so it is not subjective and ... |
H: Complex inner product space.
Problem from Schaum’s Outlines of Linear Algebra 6th Ed (2017, McGraw-Hill)
I proved that a and d must be real positive, and b is the conjugate of c.
The solution indicates that a.d-b.c must also be positive, but i can't figure that out.
thanks for your help.
AI: Hint 1: $f(u,v) = u \... |
H: Its correct to assume the Sum Set using "for all" quantifier in place of existential quantifier?
From ZFC Axiom of union we have: $$(\forall x)(\exists y)(\forall u)(u \in y \Leftrightarrow (\exists v)(u \in v \land v \in x))$$
My interpretation of this is that for any set $x$ its guarantee a set $y$ exists contai... |
H: Integrate $\frac{e^{itx}}{1-it}$
I need this integral to find the probability density function from the characteristic function but I don't know how to find it, every method that I try fails. I tried integration by parts but I didn't get the result that I need to.
The integral is
$$\frac{1}{2\pi}\int_{-\infty}^{\... |
H: Is Differentiation Operator bounded for polynomials
$P$ is a vector space of all polynomial functions on $C[0,1]$
$p'$ is derivative of $p$
Check whether $T : P \to P \quad T(p)= p'$ is linear bounded or not. Use $|| \; ||_{sup}$ (supremum norm) for polynomials.
T is linear so there is no problem here. I am struggl... |
H: Looking for a formula to compute $\left\lceil \frac{x+1}{2} \right\rceil$
I'm looking for a formula to easily compute:
$$ \left\lceil \frac{x+1}{2} \right\rceil $$
The formula shouldn't use any floor, ceil or round function. I'm looking for something "simple".
AI: No closed-form expression with $+,-,\times,\div$ ca... |
H: Finding $g(N)$ for $T(N)= \frac{\exp(N^3)}{\lg N}$ such that $T(N) = \Theta(g(N))$
Could a correct answer be $$g(n)=\frac{N\exp(N^3)}{N\lg N}$$ for $T(N)=\Theta(g(N))$ if $T(N)= \frac{\exp(N^3)}{\lg N}$?
AI: Sure this will work, it's the same function. Another one is a different way to write it, note that $\log N =... |
H: Two Options in Lottery - Probability
I was thinking about the lottery and came to this question, which is way above my understanding of probability.
Let's say there are 100,000 tickets and 10 IPhone prizes. You can buy as many tickets as you want but can only win one IPhone. (One prize per person.) Let's assume th... |
H: Is Differentiation Operator continuous?
$P$ is a vector space of all polynomial functions on $C[0,1]$
$p'$ is derivative of $p$.
Show that $T : P \to P \quad T(p)= p'$ is closed operator (the graph of T is closed) but not continuous.
Use $|| \; ||_{sup}$ (supremum norm) for polynomials.
On wikepedia it says that d... |
H: Why does $\frac{|\sin\theta|}{2}<\frac{|\theta|}{2}<\frac{|\tan\theta|}{2}$ not imply that $1>\lim_{\theta\to 0}\frac{\sin\theta}{\theta}>1$?
I was watching this proof of the equality $$\lim_{\theta\to 0} \frac{\sin \theta}{\theta} = 1$$
The author says about the following areas that
red area <= yellow area <= blue... |
H: Group cohomology of finite cyclic group
I am currently reading Cohomology of Groups by Brown and I am stuck on page 58. They do an example of the computation of the cohomology group of a finite cyclic group $G=\langle t\rangle$ of order $n$.
My first question is a about the free resolution (I.6.3):
I see the maps $... |
H: If there were a function mapping a set onto its powerset, would the unrestricted comprehension schema be true?
I have the following exercise in a set:
Cantor proved that there can be no function $\phi$ mapping a set onto the set of all of its subsets. Show directly that if there were such a mapping, then we would... |
H: Find the domain and range of $f(x) = \frac{x+2}{x^2+2x+1}$:
The domain is: $\forall x \in \mathbb{R}\smallsetminus\{-1\}$
The range is: first we find the inverse of $f$:
$$x=\frac{y+2}{y^2+2y+1} $$
$$x\cdot(y+1)^2-1=y+2$$
$$x\cdot(y+1)^2-y=3 $$
$$y\left(\frac{(y+1)^2}{y}-\frac{1}{x}\right)=\frac{3}{x} $$
I can't fi... |
H: Determining $(A \times B) \cup (C \times D) \stackrel{?}{=} (A \cup C) \times (B \cup D)$
I am trying to determine the relationship between $(A \times B) \cup (C \times D)$ and $(A \cup C) \times (B \cup D)$. Usually, I begin these proofs with a venn diagram and then formally prove the result thereafter, but there ... |
H: Proving intersection of empty set is the set of all sets
In an exercise, it asks to prove $\bigcap \varnothing$ is equal to the set of all sets. I understand that there are many proofs online but I have a specific question (mostly regarding mathematical logic) regarding my proof.
From my understanding, given a set... |
H: Sequence convergence without metric equivalence
I only have problem in the metric equivalence part. I think there exists an equivalence. $R^\infty$ is a subspace of $R^\omega$. Let's use the uniform topology for $R^\omega$ and use the relative topology for $R^\infty$. Let the metric for the uniform topology be $d$... |
H: Let $G$ be a multiplicative group and $\varnothing\neq H\subseteq G$. Show that $H\le G$ iff both $H\circ H\subseteq H$ and $H^{-1}\subseteq H$.
Question: Let $G$ be a multiplicative group and $H$ be a nonempty subset of $G$. Show that $H$ is subgroup if, and only if, $H\circ H \subseteq H$ and $H^{-1} \subseteq H$... |
H: Gaussian quadrature method when $n=2$. Function to be approximate is $\arcsin$, how to do it properly when weights are not given?
So, there is a an exercise of numerical integration using Gaussian quadrature method. The givens are:
f(x) = arcsin x, a = −1, b = 1, i = 2
I was just wondering, do I need to use the kn... |
H: Converse to a proposition on subgroups
Let $G$ be a group, and let $H$ and $K$ be subgroups of $G$. I read somewhere that if either $H$ or $K$ is a normal subgroup, then $HK$, which is the set of products from $H$ and $K$, is itself a subgroup. Is the converse true? That is, given that $HK$ is a subgroup of $G$, mu... |
H: Eigenvalues of offset multiplication tables
Consider the $n$ x $n$ 'multiplication table' constructed as (using Mathematica language)
$$
M_n^s = \text{M[n_,s_]:=Table[ k*m , {k,1+s, n+s}, {m, 1+s, n+s} ] }
$$
For example,
$$M_4^0 = \begin{pmatrix} 1 & 2 & 3 & 4 \\ 2 & 4 & 6 & 8 \\3 & 6 & 9 & 12 \\4 & 7 & 12 & 16 ... |
H: f is a distribution, does the right derivative, written here $f'$, always exists on the domain?
I am reading the paper from Tehranchi: https://arxiv.org/abs/1701.03897
Page 20, he mentions that for a density $f$, the right derivative always exists on its domain (the definition of $f$ is given page 18 for anyone who... |
H: The convergence of $(n\sin n)$
I think $\lim_{n\to \infty} n\sin n$ does not exist, for $\sin x$ is a wave function and when $n\to \infty$, then so does factor $n$. But I am confused that how to give a strict proof for the divergence.
I would appreciate it if someone could give some suggestions and comments.
AI: E... |
H: Explanation of Geometric Distribution Graph
I understand what the geometric distribution is and that it is calculating the probability of the number of trials needed up to and including the first success. Expressed as the following:
$$P(X = k) = (1-p)^{k-1}p$$
I asked a similar question years ago and for the life o... |
H: Proving that ${\aleph_1}^{\aleph_0}\leq |[\omega_1]^{\omega}|$
Today I was working with some exercises about topology but in some part I need to prove the next inequality: $${\aleph_1}^{\aleph_0}\leq |[\omega_1]^{\omega}|$$Here $[\omega_1]^{\omega}:=\left\{A\subseteq\omega_1 : |A|=\aleph_0 \right\}$. I don't know h... |
H: Consider a set $G\subseteq \Bbb R$ and a binary operation * defined on $\Bbb R$ as $a*b=a+b+ab$, such that $(G,*)$ is an Abelian Group. Determine $G$.
My question differs from this one(also, the binary operation differs slightly). Here, I'm supposed to determine $G$, as opposed to proving a given $G$ to be an Abeli... |
H: Expressing polynomial equations as a set of linear equations
Suppose the $4$-vector $c$ gives the coefficients of a cubic polynomial $p(x) = c_1 + c_2x+c_3c^2+c_4c^3$.
Express the conditions
$p(1) = p(2), p'(1) = p'(2)$
as a set of linear equations of the form $Ac=b$. Give the sizes of A and b, as well as their ent... |
H: Proving that $h\left(x\right)=x^2-5x+\ln \left(\left|-x+4\right|\right)$ has only one real root in $\left[0,2\right]$
As I need to prove that the h function "$h\left(x\right)=x^2-5x+\ln \left(\left|-x+4\right|\right)$" has one real root in $\left[0,2\right]$ i have tried to do this but i would like to know if is it... |
H: Another social network matrix expression assistance please.
We consider a collection of n people who participate in a social network in which pairs of people can be connected, by 'friending' each other. The n x n matrix F is the friend matrix, defined by $F_{ij} = 1$ if persons i and j are friends, and $F_{ij} = 0$... |
H: A wrong understanding of Permutation and Combination
Suppose We have 10 different things that we want to distribute to P and Q where P gets 3 and Q gets 7, in how many ways can we do this ? ( order doesnt matter when P or Q gets them , ie if P gets {a,b,c} then it is same as {b,c,a} )
Now we know the answer will... |
H: Questions regarding inverse trigonometric integration
Intregrate $\displaystyle \int \frac{x+5}{\sqrt{9-(x-3)^2}} \, dx $
I split the integral
$$\int \frac{x+5}{\sqrt{9-(x-3)^2}} \, dx=\int \frac5{\sqrt{9-(x-3)^2}}\, dx+\int \frac{x}{\sqrt{9-(x-3)^2}} \, dx$$
For the first integral I use the integral table which s... |
H: A problem with the definition of $e^x$
I am very experienced in the world of calculus, but there is a certain problem I need to solve that I can't quite get my head around. The canonical definition of $e$ is $$e=\lim_{n\to\infty}{\left(1+\frac{1}{n}\right)^n}$$
Therefore, $$e^x=\lim_{n\to\infty}{\left(1+\frac{1}{n}... |
H: Bijections Question (from TAACOPS) About Integers to Evens
I was working on a problem about bijections (just learned about this today, so please be harsh/uber-specific when answering/correcting me so that I get a handle on the terminology). So far, my very limited understanding of a bijection is that it is both ont... |
H: Finding $\iint_D \sqrt{\left | x-y \right |} \,dx\, dy$ where $D$ is a rectangular region
I was tasked to compute the following double integral:
$$\iint\limits_D \sqrt{\left | x-y \right |}\, dx\, dy\,,$$
where rectangular region $D$ is bounded by $0 \leq x \leq 1$ and $0 \leq y \leq 2$.
Direct integration is futil... |
H: If Ricci tensor has a null eigenvector in dimension $3$, then it has at most two nonzero eigenvalues and $|\text{Ric}|^2 \geq \dfrac{R^2}{2}$
Here's the necessary context:
I assume a null eigenvector would be a vector $v$ such that $\operatorname{Ric}(v, \cdot) = 0$ (i.e $\text{Ric}_{ij}v^{j} = 0$). But I don't u... |
H: Will the largest possible value of the dot product between a vector A and a mystery vector (x, y) of length 1 always be the magnitude of A?
The question is pretty much all in the title, really.
I have a vector (2, 1) multiplied with (x, y) where the length of this vector is 1, and the largest possible value of the ... |
H: If $ \sum_{n\geq 1}\mu( \{|f_n|\geq n\})<\infty $ then $ f_n-f_n 1_{|f_n|\leq n}\underset {n}{\to} 0 $
Let $(E,\mathcal {A},\mu) $ be a finite measure space and $\{f_n\} $ be sequence of bounded function in $L^1$ such that
$$
\sum_{n\geq 1}\mu( \{|f_n|\geq n\})<\infty
$$
Can we say that
$$
f_n-f_n 1_{|f_n|\leq n}\u... |
H: What is $\nabla\wedge\vec{\mathbf F}(x,y,z)$
What is $\nabla\wedge\vec{\mathbf F}(x,y,z) \quad$ where F is a vector field?
I assume its the dual of the curl but I need an equation. And what is it called? Google is getting me nowhere.
AI: You can start with the formula for the curl and work your way backwards. We h... |
H: Is it good to use for all $\delta$ there exists $\epsilon$?
Logically, the following two definitions are exactly the same:
For all $\epsilon >0$, there exist $\delta >0$ such that if $0<\vert x-a\vert<\delta$, then $\vert f(x)-L\vert<\epsilon$.
For all $\delta >0$, there exist $\epsilon >0$ such that if $0<\vert ... |
H: Finding the optimal $\frac pq$ approximation for a real number given upper limits on $p$ and $q$
When answering a question on Stackoverflow I got curious about how to find the optimal $\frac pq$ approximation for a real number, $r$, where $p$ and $q$ are integers that are limited by the integer type's number of bit... |
H: Integral of a floor function.
Well, i was trying to solve this problem, this told me find the integral
\begin{equation}
\int_{0}^{2}f(x)dx
\end{equation}
with $$ x \in <0, \infty>$$
of this funtion:
\begin{equation}
f(x) =\left \{ \begin{matrix} \frac{1}{\lfloor{\frac{1}{x}}\rfloor} & \mbox{if } 0 < x < 1
\\ 0 & \... |
H: If $p$ is a prime and $p\equiv1\pmod 4$, then $x^2+4=py^2$ has solutions
Prove that if $p$ is a prime and $p\equiv1\pmod 4$, then $x^2+4=py^2$ has integer solutions.
This is related to units in quadratic number field $\mathbb Q(\sqrt p)$. As far as I know, $x^2+4\equiv0\pmod p$ has a solution because $-4$ is a qu... |
H: How to show that (-√2,√2)$\cap$Q is closed and bounded subset of Q ; but not compact?
How to show that (-√2,√2)$\cap$Q is closed and bounded subset of Q ; but not compact.
The part that given set is bounded is clear. How to show that it is closed in Q. Now let G=(-√2,√2)$\cap$Q. Then G is closed in Q if and only if... |
H: Math package to help verify subsitutions in finite extended field
This question originates from Pinter's Abstract Algebra, Chapter 31, Exercise B5:
Find the root field of $x^3+x^2+x+2$ over $\mathbb{Z}_3$.
Let $a(x)=x^3+x^2+x+2$. We can show $a(x)$ is irreducible over $\mathbb{Z}_3$ by direct substitution of 0,1 ... |
H: Is the following Leibniz's notation in the chain rule written correct?
I have a doubt about the Leibniz's notation in chain rule.
Suppose that $f(x) = \tan^n(x)$.
I want to use the Leibniz's notation, so I think that I will have:
Let ${u(x)=\tan(x)}$
$${\frac{d}{dx}f(x)}=\frac{d}{du}u(x)^n\cdot \frac{d}{dx}u(x).$$
... |
H: Finding the domain of convergence
Find the domain of convergence
$ \sum_{n=0}^{\infty} n (z-1)^n$
My attempt : $-1 < (z-1) <1 \tag 1$
adding $1$ on $(1)$, we have
$0 \le z <2$
so the domain convergence will be $z \in [0,2)$
Is its true ?
AI: If this is really a quastion about Complex Analysis, then your answer d... |
H: Euclidean Space is a Lie Group under Addition
I attempt to understand the definition and examples of Lie group supplied by An Introduction to Manifolds by Loring Tu (Second Edition, page no. 66). The definition is given below.
The first example is as follows.
The Euclidean space $\mathbb{R}^n$ is a Lie group unde... |
H: How to check the following two connected open sets coincide?
As subsets of a Hilbert space (we don't present the concret space here), if $A_1,B_1,A_2,B_2$ are open sets satisfying
\begin{equation}
A_1 \cap B_1=A_2\cap B_2 =\emptyset, \quad A_1 \cup B_1=A_2 \cup B_2.
\end{equation}
Moreover, $A_1$ and $A_2$ have the... |
H: Find $\int \frac{x}{\sin^2x-3}dx$
I started like this
\begin{align*}
\int \frac{x}{\sin^2x-3}dx & =\int \frac{x\sec^2x}{\tan^2x-3\sec^2x}dx\\
& =-\int \frac{x\sec^2x}{2\tan^2x+3}dx\\
& = -\left [ \frac {x\tan^{-1}\left (\frac {\sqrt {2}\tan x}{\sqrt {3}}\right)}{\sqrt {6}}-\int \frac {\tan^{-1}\left (\frac {\sqrt ... |
H: Bernoulli Series Are Singular Distribution
I have a question of the following:
Assume $Y_{1}, Y_{2} \ldots$ is a sequence of iid rvs so that $\mathcal{L}\left(Y_{j}\right)=\frac{1}{4} \delta_{0}+\frac{3}{4} \delta_{1},$ for $j \in \mathbb{N}$
Show that the distribution $W$ are singular where
$
W:=2 \sum_{j=1}^{\inf... |
H: How to solve the following system of differential equations?
If $x(t)$ and $y(t)$ are the general solution of the system of the differential equations: $$\frac{dx}{dt}+\frac{dy}{dt}+2y=\sin t $$ $$\frac{dx}{dt}+\frac{dy}{dt}-x-y=0$$ Then which of the following conditions holds for $x(t)$ and $y(t)$? ($a$ is any ar... |
H: Graph theory: equivalence relation properties and edge
I'm new to graph theory, and I am studying a question about graphs generated using 4 vertices, and $V = \{a, b, c, d\}$, and let E be a symmetric relation on V.
So at this moment, I have successfully managed to draw all 11 non-isomorphic graphs using the vertic... |
H: Radius of circle that touches 3 circles, which in turn touch each other
I had $3$ circles of radii $1$, $2$, $3$, all touching each other. A smaller circle was constructed such that it touched all the $3$ circles.
What is the radius of the smaller circle?
This is what I did:
I conveniently positioned the $3$ circle... |
H: is a diffeomorphism regular?
I have learned the inverse function theorem which ensures that a regular mapping (which has its inverse) is a (local) diffeomorphism.
But I wonder whether a diffeomorphism is regular.
I guess the answer would be 'yes', but I have no idea.
AI: If $M$ and $N$ are smooth manifolds and if $... |
H: Why does $x=e^t+2e^{-t},y=e^t-2e^{-t}$ plot to a straight line?
This parametrization satisfies $x^2-y^2=8$, so I was expecting a hyperbola. But what I got was a straight line. Why though?
https://www.wolframalpha.com/input/?i=parametric+plot+%28e%5Et%2B2e%5E%28-t%29%2Ce%5Et-2e%5E%28-t%29%29
EDIT- I tried a differen... |
H: Find the largest diagonal of a parallelogram if the area is known
The original question:
In a parallelogram, the length of one diagonal is twice of the other
diagonal. If its area is $50\text{ sq. metres}$, then the length of
its bigger diagonal is...
A) $5\sqrt 2$ metres
B) $15\sqrt 2$ metres
C) $10\sqrt 2$ m... |
H: Given $\kappa = \sup_{\alpha< \lambda} \kappa_{\alpha}$n can we assume the $\{\kappa_\alpha: \alpha < \lambda\}$ is strictly increasing?
Suppose $$\kappa= \sup_{\alpha < \lambda} \kappa_\alpha$$
where $\kappa$ is an infinite cardinal and $\kappa_\alpha$ are cardinals, $\lambda$ is a non-zero limit ordinal, $\lambda... |
H: Prove $\bigcup_{t \in I}P(A_t) \subseteq P(\bigcup_{t \in I}A_t) $
Obviously $ \{A_t\}_{t\in I} $ is basically a family of desired set.
I believe the proof must have something to do with the definition of unions and also the power sets.
I had previously proved a similar problem with almost the same structure where ... |
H: What is the purpose/meaning of parametrizing a set by an index set
My lecture notes define $B=\{x_i|i\in I\}$ as a basis for a vector space, parametrized by the index set $I$, and uses this notation for the rest of the chapter. But what is the purpose of this notation when we can label the ${x_i}$ by anything we wa... |
H: How many tries to get higher number than uniform random variable?
I came across the following problem, and don't know how to solve it:
Player 1 samples from the Uniform(0,1) distribution. Then Player 2 repeatedly samples from
the same distribution until he obtains a number higher than Player 1’s. How many samples
i... |
H: How is 0 = infinite from infinite series?
The infinite series $1/2 + 1/4 + 1/8 + ... = 1$
made me get $$(1-a)+a(1-a)+a^2(1-a)+...=1$$
by dividing both sides with $(1-a)$
I get $1+a+a^2+a^3+...=\frac{1}{(1-a)}$
where $a\neq1$
However, when $$\lim_{a\to 1}1+a+a^2+a^3+...=\frac{1}{(1-a)}$$ the L.H.S. of the equation w... |
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