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H: Find whether the sequence is convergent .
Find whether the sequence $(a_n)$ given by $a_{n+1}= \sqrt{a_n}+\sqrt{a_{n-1}}$, where $a_1=1$ and $a_2=2$, is convergent.
So , $a_{n+1}-a_{n}= \sqrt{a_n} + \sqrt{a_{n-1}} -a_n \implies \sqrt{a_n}(1-\sqrt{a_n})+ \sqrt{a_{n-1}}.$
Now I assumed the sequence is $>1$ an... |
H: Why is the realification of a simple complex Lie algebra a semisimple real Lie algebra?
Why is the realification of a simple complex Lie algebra a semisimple real Lie algebra?
The realification here means to consider the complex Lie algebra as a real Lie algebra of twice the dimension.
The statement was used in the... |
H: Do the sequences converge or diverge?
$$\sum_{n=1}^{\infty} (-1)^{n} \cdot \frac{n}{5n+3} $$
Using Leibniz $\lim_{n\to\infty} a_{n}= \frac{n}{5n+3}=\frac{1}{5}$ so this is not equal to $0$, divergent
$$\sum_{k=1}^{\infty} \frac{n^6}{3^n} $$
I guess that $3^{n}$ is growing faster than $n^{6}$ so $a_{n}$ decreases.... |
H: Probability: Union and Conditional Union
I have $P(A\cup C|B)$
Does it equal to $P(A|B)+P(C|B)-P(A\cap C |B)$
If A,C are mutually exclusive, is it same as $P(A|B)+P (C|B)$?
AI: The short answer is "Yes." We have
$$\begin{align}
\Pr(A\cup C|B)&=\frac{\Pr((A\cup C)\cap B)}{\Pr(B)}\\
&=\frac{\Pr((A\cap B)\cup (C\ca... |
H: Minimum cost of a connected graph
$G$ is a connected graph with cost $p:E(G)\to\mathbb{R}$ defined on its edges. Let $e' \in E(G)$ be such that $p(e')<p(e)$ for every $e\in E(G)-\{e'\} $.
Is it possible to find two spanning trees of minimium weight, $T_1$, $T_2$ ($T_1 \neq T_2$), such that one of them has $e'$ and ... |
H: Speed of decay of $\zeta(x)-1$ as $x \to \infty$
I am trying to find some numerical bound on the Riemann zeta function
$$
\zeta(x) = \sum_{k\ge 1} 1/n^x.
$$
I am only interested in the case which $x > 1$, so the above expression is valid.
More precisely, what is the decay of the function
$$
\zeta(x)-1 \text{ as... |
H: Proving Linearly Independence from System of Equation
I am trying to understand the proof of Linearly Independence of the basis set $\{1, x, x^2, x^3\}$. It is written that -
Substituting $3$ other values of $x$ into the above equation yields a
system of $3$ linear equations in the remaining $3$ unknowns $c_1, c_2... |
H: Holomorphic function bounded
Let $f \in \mathcal{H}(\mathbb{C} \setminus \{0\})$ be and assume that
$$
|f(z)| \leq |\log|z|| + 1, \quad z \in \mathbb{C} \setminus \{0\}
$$
I have to prove that $f$ is constant. My attempt is next proof. Consider $g(z) = zf(z)$. By hypothesis
$$
|z||f(z)| \leq |z||\log|z|| + |z|,
$$
... |
H: Why is the Penrose triangle "impossible"?
I remember seeing this shape as a kid in school and at that time it was pretty obvious to me that it was "impossible". Now I looked at it again and I can't see why it is impossible anymore.. Why can't an object like the one represented in the following picture be a subset o... |
H: Minimum absolute difference after diving number from 1 to n into two groups
I am trying to solve an algorithmic problem mentioned at https://www.geeksforgeeks.org/divide-1-n-two-groups-minimum-sum-difference/.
In the solution it says "We can always divide sum of n integers in two groups such that their absolute dif... |
H: Proof : Given a finite set of equidimensional proper subspaces of a vector space $V$, $\exists$ $x$ in $V$ that belongs to none of them
I am stuck at this statement in a book of linear algebra. Even though the author casually mentions it, I am having a hard time coming up with a proof, why it must be true. Can you ... |
H: Is there an intuitive way of justifying why the square of an infinite cardinal is itself?
By no means I am an expert in this subject, but I do have some knowledge of ZFC. While there are many proofs which are difficult to recollect, I feel like I have enough knowledge that if I am given a statement about ordinals o... |
H: Understanding memoryless property of exponential distribution
For an Exponential distribution X with mean 500, we could say that P(X>1000 | X>500) = P(X>500) and the mean of the conditional distribution X|X>500 would also be 500. Good so far?
Can the memoryless aspect be extended in the other direction, so that you... |
H: Proving a combinatorial identity involving sum and integral
I want to prove
Therefore I use that the derivatives of both sides ($\frac{d}{\text{d}p}$) are equal (and that for a fixed p the values of both sides are equal).
Has anyone got a clue how to find that the derivative of the left side is equal to $-\frac{n... |
H: Square root of 1 modulo N
Given a positive integer N, how do we compute $card(A)$ where $A = \{x\in\mathbb{Z}, 0 < x < N \mid x^{2}\equiv1\pmod N\}$, when the prime factorization of N is known.
In other words, how many square roots of 1 modulo N exist?
We know that when N is prime, there are only two square roots ... |
H: What is the shape in the complex plane generated by all possible points $z_1 + z_2$, where $z_1$ and $z_2$ can be any two points on the unit circle?
What is the shape in the complex plane generated by all possible points $z_1 + z_2$, where $z_1$ and $z_2$ can be any two points on the unit circle centered at $0$
AI:... |
H: Show that $x^{2}-6y^{2}=523$ has infinitely many integral solutions
I want to show that $x^2-6y^2=523$ has infinitely many solutions. For the special case $x^2-dy^2=1$, I know what I need to do. I can get the result by using continued fractions. Also, in the kinds of $x^2-dy^2=m$ for some examples, I can say that ... |
H: vector of random variables and conditional probability problem?
I truly don't know how to approach the following problem
Consider a sequence of events identically distributed and
independent with probability of success $p$. Let $S_i$ be the
success of the i-th event. Denote $X_1$ the time at which the
first succ... |
H: Show that for a rational $a$ there is a finite amount of rationals $\frac{p}{q} \neq a$ such that $|a - \frac{p}{q}| < \frac{1}{q^2}$
I want to shot that there for rational number $a \in \mathbb{Q}$, there is a finite amount of rational numbers $\frac{p}{q} \neq a$ such that $|a - \frac{p}{q}| < \frac{1}{q^2}$.
I k... |
H: Show that there are two total orderings of $\textbf{Q}(\sqrt{2})$ under which it is an ordered field.
Let $\textbf{Q}(\sqrt{2})$ be the set of all real numbers of the form $r + s\sqrt{2}$, with $r,s\in\textbf{Q}$. Show that $\textbf{Q}(\sqrt{2})$ is a subfield of $\textbf{R}$. Show that there are two total ordering... |
H: Number of functions to receive 2 values
I have a set $A$ with $5$ items: $1,2,3,4,5$
I want to know how many functions are there that make $|f[A]|=2$ if $f: A->A$
So what I thought is that it's the same like to put $5$ balls in $2$ boxes out of $5$ boxes.
My direction to the solution would be: choose $2$ boxes out ... |
H: Find $x$ and $y$ where $ax - by = c$ and $x + y$ is minimum.
How can I find out the natural numbers $x$ and $y$, such that
$$ax - by = c$$
and $x + y$ is minimum? $a, b, c$ are given integers.
example: $a = 2, b = 2, c = -2$.
$$2x - 2y = -2$$
$$ans: (x = 0, y = 1)$$
Also, how can I know when it is impossible? I g... |
H: Interpretation of Simple Uniform Marginal Density Example from Mathematical Statistics by Rice
I am stuck trying to visualize Example B from Mathematical Statistics and Data Analysis 3rd ed by Rice. The examples revolve around the concept of independence of Random Variables which is defined in the following way:
I... |
H: Let $a_{n} = \sqrt{n^{2}+n} - n$, for $n\in\textbf{N}$. Is the sequence $(a_{n})_{n=1}^{\infty}$ monotonic?
Let $a_{n} = \sqrt{n^{2}+n} - n$, for $n\in\textbf{N}$. Show that $a_{n}$ converges as $n\to\infty$. What is the limit? Is the sequence $(a_{n})_{n=1}^{\infty}$ monotonic?
MY ATTEMPT
The answer to the first q... |
H: How can we write the function on definite integral form?
How can we write the the following Stieltjes function on definite integral form?
$$\frac{1}{\log(1+x)}$$
for example :
$$\frac{\log(1+x)}{x}=\int_{0}^{\infty} \frac{t^{-1}}{x+t} dt $$
AI: As people have said in the comments,
$${\int\frac{1}{\log(1+x)}dx}$$
do... |
H: What does it mean when polynomials have closed, exact complex solutions, but not exact real solutions?
I was watching this introduction to peturbation theory. His first example is solving
$$x^5 + x = 1$$
for which he claims there is no exact real solution. I asked WolframAlpha what it thought.
It gives an inexact d... |
H: How to express in Legendre's polynomials?
How do I express $cos(3\theta)$ and $sin^{2}(\theta)$ in Legendre's polynomials, knowing that $x=cos\theta$?
I know that $f(x)=\sum a_{n}P_{n}(x)$ and $P_{n}=\frac{(-1)^{n}}{2^{n}n!}\frac{d^{n}}{dx^{n}}(1-x^{2})^{n}$, but I don't know what to do with them
AI: If the functio... |
H: What is a small change of a function with 2 variables?
I am reading a book and am confused about how this equation is founded. I would have thought that delta C would just be equal to the sum of the partial derivatives?
C is a function depending on v1 and v2.
AI: The partial derivatives just tell you how fast the ... |
H: Prove that there exists a bijective function $i:S \to S$ such that $i \circ g = g$ and $g \circ i = g$ for all bijections $g: S \to S$.
Let $S$ be a set.
($a$) Prove that there exists a function $i:S \to S$ such that $i \circ g = g$ and $g \circ i = g$ for all bijections $g: S \to S$.
($b$) Prove that $i$ is a bije... |
H: How many ways to move around parentheses for finite tensor products?
Suppose I have $n$ elements where $n\in\mathbb{N}$ in a place where tensor products make since and are not strict, say $a_1,\cdots, a_n$. Suppose we only know that $a_i\otimes a_{j}$ is defined for all $i,j\in\{1,\cdots,n\}$. How many ways are the... |
H: Given that for two naturals $p$ and $q$ are coprime, How to show that two naturals $u$ $v$ exist such as $pu-vq =1$
I know by Bézout theorem two integers $u$ and $v$ exist and verify $pu+qv=1$ but to show that $u$ and $v$ are naturals I'm stuck.
AI: If $p$ and $q$ are both positive integers and $pu+qv=1$ with integ... |
H: There exists an $\aleph_0$-coloring of a graph on the real numbers.
I have this question:
Let $G = ( \mathbb{R} , E)$ be a graph such that its vertices are the real numbers and its edge set is given by $$E = \big\{ \{u,v\}\,\big |\, u-v \in \mathbb{Q} \setminus \{0 \}\big\}\,.$$
Prove that the graph has a legal c... |
H: Is there an elementary reason $\mathbb{CP}^{n}$ is not homeomorphic to the sphere for $n\ge 2$?
My question is similar to this one, but I am asking this for the complex case.
In the real case, we can use the fact that the fundamental group of $\mathbb{RP}^{n}$ is non-trivial, so the space cannot be homeomorphic to ... |
H: Using Rule of Inference, How to derive following conclusion from given premises?
Question is from the book: Discrete Mathematical Structures with Applications to CS by Tremblay and Manohar. It is an exercise problem. But, unfortunately, there is no help available on answers, or solutions of this book. I have tried ... |
H: Understanding Serge Lang's Definition of Homotopy
I have been following Serge Lang's Complex Analysis text book and today I came across a chapter on homotopy. I have trouble visualising and honestly, understanding the definition that he has given in his book. Here is the definition from his book
Could somebody exp... |
H: Determining the validity of a basis with unnecessary vectors
If we have $W=\{1-x,1-x+x^2,1+x^2,1-x-x^2\}$ find out of this set forms a basis for $P_2$
I put it into an matrix and row reduce it to get:
$$\begin{bmatrix}1&0&0&2 \cr 0&1&0&-1\cr 0&0&1&0\end{bmatrix}$$
I noticed that while the first three columns are in... |
H: Why do three non collinears points define a plane?
I've just started looking at the axioms of 3D Geometry. The first one that I encountered is this one:
"Three non collinear points define a plane" or " Given three non collinear points, only one plane goes through them"
I know that it is an axiom and it is taken to ... |
H: Why are interior products necessary when 1-forms are already dual to vectors?
I am reading a book about differential geometry that introduces an operator $i$, called the "interior product," that takes vectors and produces something that can act on 1-forms. Their rules are,
$$
i\left(\frac{\partial}{\partial x_i}\ri... |
H: How can I find $\theta$ where $\sin \theta=X$ and $\cos \theta=Y$?
I have two variables, $X$ and $Y$. Both are between $-1$ and $1$, inclusive, but I need to find the angle, of which the sine is $X$, and the cosine is $Y$. How can I do that? This is probably a dumb question but it's been troubling me for a while no... |
H: Given two points in $\mathbb{R}^d$, A and B, find the point in $\mathbb{R}^d$ that is most nearly x distance from A and y distance from B
Consider two points in $\mathbb{R}^d$, $A$ and $B$.
Now, consider two scalar quantities, $x$ and $y$.
I want to find the point in $\mathbb{R}^d$ that is closest to being $x$ di... |
H: $C^1$ extension of a harmonic function is harmonic
Let $B(0,r)$ denote the ball of radius $r$ centered at the origin in $\mathbb{R}^d$, $d\geq 2$.
Let $0<r_0<1$. Suppose I have a real-valued function $v$ that is harmonic in the annulus $\{r_0<|x|<1\}$ that vanishes on $\partial B(0,1)$. Now let $\tilde{v}$ be an ex... |
H: "Lazy" Random Walk
I would like to discuss a slightly different kind of random walk on $Z$ in which we include the probability of being "stuck" on the same place.
Let us denote as:
$$p/3 \; \text{the prob. of right step (which is +1 on Z)} $$
$$p/3 \; \text{for left step (which is -1 on Z)} $$
$$ 1-\frac{2}{3}p \; ... |
H: Logic - Identifying domain of discourse, variable, and predicate
I have the following question that I am trying to figure out:
'All swans are white'
Identify a natural domain of discourse for this sentence, the variable and the predicate.
I am trying to study this on my own, and this was my own practice question fr... |
H: Using sum-to-product formula to solve $\sin(2\theta)+\sin(4\theta)=0$
Trying to use the sum-to-product formula to solve $\sin(2\theta)+\sin(4\theta)=0$ over the interval $[0,2\pi)$, but I'm missing solutions.
$$\sin(2\theta)+\sin(4\theta)=0$$
Apply sum-to-product formula:
$$2\sin\left(\frac{2\theta+4\theta}{2}\rig... |
H: Create Log function
I'm a programmer. I'm working with a sensor that has given the following graph.
I would like to make a function where i put the value of Rs/R0 in. The outcome of this function is the value of ppm of alcohol.
I understand that precision is imposible so, for the function i'm going for a straigt li... |
H: Vector Space - how to visualize it for understanding?
I read on Wikipedia about vector spaces, but I don't understand them in a way that I can visualize the vector spaces in my head. During the process of understanding, I had several concepts in my head and I am at a point now, where I am totally confused. Maybe I ... |
H: Convergence in laws
i'm currently stuck in this exercice where i don't know how to start.
Let $\{X_n\}_{n \ge 1}$ be a sequence of independent random variables on $(\Omega, \mathbb{A}, \mathbb{P})$ with:
$$
F_{X_n}(t) = \left(1 - \frac{1}{1+t^2}\right)1_{[0,\infty)}(t)
$$
for which $p \in (0,\infty)$ is $X_1$ in $... |
H: How to compare which interest rate is better compounded annual vs compounded 3 times a year
Having a little trouble with getting the answer to this question.
: How much better is the return on a 4% yearly interest rate investment that is compounded 3 times per year as opposed to compounded yearly?
I tried to set up... |
H: Finding global extrema
Click here for question
I don't know how to find the global extrema of this since taking the partial derivatives with respect to x and y leaves me with no x and y's to find zeros of the function. Please help
AI: In calc 1, how did you find extrema? You took the derivative of the objective fu... |
H: If $E[X^p] < \infty$, then $\lim_{x\to \infty} x P(|X|\gt {x}^{1/p})=0$
In this question, the proof of the following claim was solved:
If $E[X] < \infty$, then $\lim_{x\to \infty} x P(|X|\gt x)=0$ .
Now, I want to ask about the following claim:
If $E[X^p] < \infty$, then $\lim_{x\to \infty} x P(|X|\gt {x}^{1/p})=0$... |
H: About essential range and essential supremum
$\newcommand{\esssup}{\mathrm{ess\,sup}}$$\newcommand{\essrng}{\mathrm{ess\,range}}$
I am trying to prove that $\esssup(f) = \sup(\essrng(f))$, where we define
$$
\esssup(f) = \inf \{b \in \mathbb{R}_+ : \mu(f^{-1}((b, \infty))) = 0\}
$$
and similarly
$$
\essrng(f) = \{... |
H: Suppose $a,b\in \mathbb{Z}$ are relatively prime and $c\in \mathbb{N}$ is a divisor of $a+b$. **Verify my proof** that gcd$(a,c)$=gcd$(b,c)=1$.
I am looking for someone to verify my proof of the problem in the title. Of course, if you believe it to be wrong, you are welcome to shred it into a million pieces and if ... |
H: a linear map on $W$
Define $W = \{(a_1, a_2,\cdots) : a_i \in \mathbb{F}, \exists N\in\mathbb{N}, \forall n \geq N, a_n = 0\},$ where $\mathbb{F} = \mathbb{R} $ or $\mathbb{C}$ and $W$ has the standard inner product, which is given by $\langle(a_1,a_2,\cdots), (b_1,b_2,\cdots)\rangle = \sum_{i=1}^\infty a_i \overl... |
H: Singular Measure with Dense Support
Is there a measure $\mu$ with support $S \subseteq [0,1]$ such that it satisfies:
(i) $S$ has Lebesgue measure zero but is dense on $[0,1]$ with respect to the standard metric;
(ii) $\mu(S)<\infty$; and
(iii) $\forall \epsilon>0$, $\forall a,b \in [0,1]$ such that $b-a\geq \epsil... |
H: Suppose that $(z_{n})_{n=1}^{\infty}$ converges to $z$. Show that $\overline{z}_{n}\to\overline{z}$ and $|z_{n}|\to |z|$ as $n\to\infty$.
Suppose that $(z_{n})_{n=1}^{\infty}$ is a sequence in $\textbf{C}$ which converges to $z$. Show that $\overline{z}_{n}\to\overline{z}$ and $|z_{n}|\to |z|$ as $n\to\infty$.
MY A... |
H: Understanding calculation of jth column in matrix AB
This is a theorem found in Friedberg's Linear Algebra which I have trouble understanding.
A is an $m×n$ matrix, B is an $n×p$ matrix, $u_j$ is the jth column of $AB$ and $v_j$ is the jth column of B.
I am having trouble understanding the proof of $u_j=Av_j$.
$$u... |
H: Solving an equation using 'replacement'
I have this equation: $$ a \cdot 2^a = b$$ and need to find solutions for $a$.
And I know it's complicated and has to do something with the $W$ function.
But I've come up with a way which does not work for some-reason... and I would like you to have a look and help me find ... |
H: Is there a rigorous way to describe $g(x)$ continuously deforming into $h(x)$ and could it be useful?
One thing that bothers me about mappings is that they seem to instantaneously transport points from one space to another. I feel like there should be a space of unique, non-intersecting paths that each of the point... |
H: Why is the Projection (cB) of Vector A on B perpendicular to Vector A - cB?
The following excerpt can be found in Serge Lang's Introduction to Linear Algebra. I am trying to understand mathematically why the vector $\mathbf{A}- c\mathbf{B}$ is perpendicular to the vector $c\mathbf{B}$. I suppose there would be a si... |
H: prove this is a closed nowhere dense subset in $L^1$
A UC qualiyfing exam problem goes like this:
Let $f$ be a positive continuous function on $\mathbb{R}$ such that $\lim_{|t|\rightarrow\infty} f(t)=0$. Show that the set $\{hf|\,h\in L^1(\mathbb{R}),||h||_1\leq K\}$ is a closed nowhere dense set in $L^1(\mathbb{R}... |
H: If $\int\limits_0^{\infty}f^2(x)\ dx$ is convergent, prove $\int\limits_a^{\infty}\frac{f(x)}x\ dx$ is convergent
If $\int\limits_0^{\infty}f^2(x) dx$ is convergent, prove $\int\limits_a^{\infty}\frac{f(x)}x dx$ is convergent for any $a\ge 0$
I use the Cauchy-Schwarz Inequality to get :
$$\left( \int\limits_a^A \... |
H: Optimizing expectation, unknown parameters, normal distribution. How many restaurants should I try before choosing one for the rest of my n-m meals?
Suppose you move to a new city with an infinite number of restaurants and you plan to stay there for a predetermined amount of time. You plan to have n meals at resta... |
H: Optimization on multiplication of function of three variables
I need some suggestions on how to go above solving this problem:
Suppose I have $n$ vectors: $X_1, X_2, ..., X_n$, and a known vector $Y$. Each vector has $T$ rows.
I want to select only $3$ out of $n$ vectors, say $\: X_i, X_j, X_k\: $, along with findi... |
H: Motivation behind definition of complex sympletic group
One definition of complex sympletic group I have encountered is (sourced from Wikipedia):
$$Sp(2n,F)=\{M\in M_{2n\times 2n}(F):M^{\mathrm {T} }\Omega M=\Omega \}$$
What is the motivation for imposing the condition $M^{\mathrm {T} }\Omega M=\Omega$ instead of o... |
H: Extend the direct product representation of a subgroup
$G$ be finite abelian group. $H$ be subgroup of $G$. $H$ be direct product of cyclic subgroups $H_1$, $H_2$, .., $H_m$.
Do there exist cyclic subgroups $H_{m+1}$, .., $H_n$ such that $G$ is direct product of $H_1$, $H_2$, .., $H_n$?
I'm quite confused.
AI: No, ... |
H: Separation properties of a topological space vs. characteristics of the continuum
Suppose that a set $X$ has a topology $\mathcal{T}$. Then $$\mathcal{T}\ \text{is T}_1\Rightarrow|\mathcal{T}|\geq|X|.$$ I'm curious about implications in the opposite direction, possibly assuming the negation of the continuum hypothe... |
H: How to Find the $d^4p $ from the four vector?
Lets assume we are given the four vector of momentum P which can be written as:
$$p = (p_0, p_t cos\theta , p_t sin \theta , P_L ) ........(1)$$
Where $P_L$ is the longitudinal competent. The transverse component can be written as easily : $p_t ^2 = p_t cos\theta^2 + p... |
H: What is the meaning of the derivative of a complex function.
For the derivative of the complex function, when it is analytic, then it satisfies C-R equation. So in this case we have $f'(z_{0})=u_{x}(x_{0},y_{0})+iv_{x}(x_{0},y_{0})$. But if we consider the Vector valued function $g:\mathbb{R^{2}}\rightarrow\mathbb{... |
H: Show that $P=(\bar{x},\bar{z})$ is a prime ideal in the ring $A:=k[x,y,z]/(xy-z^2)$.
Note: $\bar{x},\bar{z}$ denotes the image of $x$ and $z$ in A and $k$ is a field.
I have a possible proof but I do not know if its correct: Suppose $\bar{a}\bar{b}\in P$ where $a,b\in k[x,y,z]$. Then $\bar{a}\bar{b}=\bar{c}\bar{x}+... |
H: Concrete Mathematics: Josephus Problem: Odd induction
I am trying to work through the odd induction case of the closed form solution to the Josephus problem. To start with a quick review of the even case - I'm being quite verbose though to help frame the question and also to potentially highlight any mistakes in my... |
H: Can I put $x = \dfrac{π}{2}$ in $ \tan{2x} = \dfrac{2\tan{x}}{1 - \tan^2{x}} $?
In the textbook it is written about the equation that $ 2x \neq n \pi + \dfrac{\pi}{2} $.
Does that mean x can be equal to $ n \pi + \dfrac{\pi}{2} $ ?
AI: Notice, $\tan x$ is undefined at $ x = n \pi + \frac{\pi}{2} $. Similarly $\tan... |
H: If any finite subcollection of finite sets has non-empty intersection, then the infinite intersection is non-empty as well.
The space is R with usual metric.
Since all the sets are finite, they are compact as well.
Suppose {$A_µ$} is the collection of finite sets and the infinite intersection is empty. Let K belong... |
H: Factorization of $x^b+1$
I came across the result that $(x^a+1)|(x^b+1)$ if and only if $\frac{b}{a}$ is odd. Any intuitive reason why, though? What about $\frac{b}{a}$ being odd makes this true?
AI: This follows from the fact that, if $k$ is odd,$$x^k+1=(x+1)\left(x^{k-1}-x^{k-2}+\cdots-x+1\right).$$ |
H: Infinite product of transcendental numbers approaches 1
I am seeking infinite formulas connect transcendentals and rationals. We know
$$e = \sum_{n = 0}^\infty \frac{1}{n!} = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + \frac{1}{4!} + \cdots$$
as an example of infinite sum of rational numbers which a... |
H: Name and interpretation of the $\stackrel{d}{=}$ symbol.
Context: I have the following statement which uses the symbol $\stackrel{d}{=}$.
Let X be a Random Variable, and let $X'$ be an RV that is independent of $X$ and $X'\stackrel{d}{=}X$.
We call the Random Variable
$$X^s = X −X'$$
symmetrized X.
AI: The symbol ... |
H: Exists $f\in L^{\infty}(\Omega)$ such that $G(x, t)\geq f(x) \vert t\vert^{\theta} -\alpha$?
Let $\Omega$ be an open bounded subset of $\mathbb{R}^n$ and let $g:\Omega\times\mathbb{R}\to\mathbb{R}$ be such that
$$ g(x, \cdot)t\in\mathbb{R}\mapsto g(x, t)\in\mathbb{R} \ \mbox{ is continuous for a.e. } x\in\Omega$$
a... |
H: Showing that $U_{A,B}$ is a subspace
Show that
$$
U_{A,B} = \{X \in \mathbb{K}^{n \times n}, f(X^T) = f(X)^T \}
$$
is a subspace of $V = \mathbb{K}^{n \times n}$ with $f: V \to V, X \mapsto AXB$ and $A,B \in V$.
I'm not sure what to show. Showing $f((v+w)^T) = f(v+w)^T $, with $v,w \in V$ does not get me anywhere a... |
H: Maximising the argument of $\sin and \cos$, given a linear relation between them
Here is a problem I have been struggling with for a while,
If $$4\sin\theta \cos\phi+2\sin\theta+2\cos\phi+1=0$$
where $\theta,\phi\in[0,2\pi]$,
find the largest possible value of $(\theta+\phi)$
I have no idea how to do it, i tried su... |
H: $A\in GL_n(\mathbb{C})$ has a unitary triangularization
Let $A\in GL_n(\mathbb{C})$. Show that $A$ has a unitary triangularization, that is there exists $U\in M(n\times n)$ unitary such that $U^{-1}AU$ is an upper triangular matrix.
Since the characteristic polynomial of $A$ splits over $\mathbb{C}$, I immediately... |
H: Prove that none of the integers $11,111,1111,...$ are squares of an integer.
Please check my proof. Thank you!
Proof: $11,111,1111,...$ can all be written as follows
$\underbrace{111...}_{\text{k times}}=1+10(\sum_{i=0}^{k-2}10^n)$
Let us assume $1+10(\sum_{i=0}^{k-2}10^n)=s^2$ where $s\in\Bbb{Z}$.
Then this means ... |
H: $P = \pi_{1}(P) \times \pi_{2}(P)$?
I’m starting my study of functions, I’m following the book “Proofs and Fundamentals”, by Ethan D. Bloch. This is one of the problems of the book and I’m not sure what would be the solution.
Let $X$ and $Y$ be sets. Let $P \subseteq X \times Y$. Let $\pi_{1}:X\times Y \rightarrow... |
H: Proof there aren't $p(x), q(x)$ polynomials such that $\arctan(x)=\frac{p(x)}{q(x)}, \forall x \in (0,+\infty)$
I've tried this, by contradiction, supposing that $p(x)$ and $q(x)$ exist $$\lim_{x\to+\infty} \arctan x =\lim_{x\to+\infty} \frac{p(x)}{q(x)}= \frac{π}{2}$$.Then the degree of both $p(x)$ and $q(x)$ are ... |
H: Relation $S$ is equivalence reltion in set $A$. Is relation $S^{-1}$ equivalence relation in set $A$ either?
Let $S \subset A^2$ and $S=\{(a,b): aSb\} .\ $Then $\ S^{-1}=\{(b,a):aSb\} \subset A^2.$ That means $S^{-1}$ is also equivalence relation, because every pair is in the same relation as in relation $S$ (so it... |
H: If $A, B, H \leq G$ such that $A \triangleleft B$ and $H \triangleleft G$, then $HA \triangleleft HB$
There is a lemma that I'm trying to understand in my algebra class and I can't get it done. It says:
Given $G$ a group, let $A, B, H \leq G$ be subgroups of $G$ such that $A \triangleleft B$ and $H \triangleleft G... |
H: Divergence of Improper integral with trigonometric functions
I need to test the convergence of the following improper integral:
$$
\int_1^{+\infty} |\sin x|^x dx
$$
I tried to find a minorant, but I did not find it immediately. Furthermore, I tried to split the integral into a series, by dividing the interval x>1 i... |
H: Newton Raphson Method Iteration Scheme
My question here is for the 2nd part. The 1st part is straightforward, taking $x^2 - N = 0$ as $f(x)$. How does one go about the second part? What exactly do they mean by applying the scheme two times?
Edit: By second part, I mean how does one derive that formula "easily"!
A... |
H: Addition of differentiable function on regular surface is still differentiable
Given two differentiable functions $f,g:V\to \mathbb{R}^n$ where $V\subset S$ is open subset of regular surface $S$.
Prove that $f+g$ with $(f+g)(x) = f(x)+ g(x)$ is still differentiable.
My attempt:for a given $x\in V$,we need to show $... |
H: Convergence and calculation of a particular series.
Let $$p_n(x)=\frac{x}{x+1}+\frac{x^2}{(x+1)(x^2+1)}+\frac{x^4}{(x+1)(x^2+1)(x^4+1)}+...…+\frac{x^{2^n}}{(x+1)(x^2+1)(x^4+1)(……)(x^{2^n}+1)}$$
I know this can be simplified to $$p_n(x)=1-\frac{1}{(x+1)(x^2+1)(x^4+1)(……)(x^{2^n}+1)}$$
Since it is only a telescoping ... |
H: If $f:X\rightarrow Y$ is not a constant function and if $X$ is first countable then $f$ is not continuous in any isolated point of $X$.
Conjecture
If $X$ is firt countable and if $f:X\rightarrow Y$ is a function then $f$ is not continuous at $x_0$ if this is an isolated point for $X$.
If $X$ is first countable then... |
H: Is there such polynomials that exist?
Let f be a polynomial of degree 3 with integer coefficients such that f(0) = 3 and f(1) = 11. If f has exactly 2 integer roots, how many such polynomials f such exist?
Approach:
f(0) = 3 so constant term is 3
f(x) = ax^3 + bx^2 + cx + 3
and it has exactly 2 integer roots.
sin... |
H: Fixed Point theorem with Lipschitz continuous mapping.
How can we prove that the below function does not have fixed point?
Define $S:=\{(x_m)\in l^1\mid \sum^\infty |x_i|\leq 1)\}$, and consider the self map $\Phi$ on $S$ defined by $\Phi((x_m)):=(1-\sum^\infty |x_i|,x_1,x_2,,,,)$. Show that $S$ is a nonempty, clo... |
H: The order of a finite field
I'm reading a theorem about the order of a finite field:
Here is the proof:
At the end, the author said
It follows that $\mathbb{F}$ is a vector space over $\mathbb{F}_{p}$, implying that its size $q$ is equal to $p^{m}$ for some $m>0$.
I do not understand how $\mathbb{F}$ is a vect... |
H: Show that $\{(1-t)^{\lambda}(1+t)^{2n-1-\lambda}, \lambda=0,1,...,2n-1\}$ forms a basis in $P_{2n-1}$, polynomial vector space
as stated above, I want to check wether $\{(1-t)^{\lambda}(1+t)^{2n-1-\lambda}, \lambda=0,1,...,2n-1\}$ forms a basis in $P_{2n-1}$, where $P_{2n-1}$ is the vector space of polynomials of d... |
H: What's the general formula for the sum total of first k terms of the following?
I have written a naive code to calculate the sum total of the first k terms of a sequence. It's too complicated for me to write it here in "mathematical" language with nested summations and all.
Code:
int foobar(int n, int k)
{
if (... |
H: A general circle through the intersection points of line $L$ and circle $S_1$ has the form $S_1+\lambda L$. What is the significance of $\lambda$?
We write a general line $L$ passing through intersection of two lines $L_1$ and $L_2$ as $L= L_1 + (\lambda) L_2$ where $\lambda$ is a variable.
Even in family of circle... |
H: How to converted $x^3+y^3=6xy$ to parametric equations?
How to converted $x^3+y^3=6xy$ to parametric equations?
The suggested solution is:
$x=\frac{6t}{1+t^2}$
$y=\frac{6t^2}{1+t^2}$
But what is the process?
AI: Taking $x=r \cos^{\frac{2}{3}} \phi$ and $y = r \sin^{\frac{2}{3}} \phi$ we obtain $r=6 \cos^{\frac... |
H: Is a transitive subgroup $H \leq S_n$ of cardinality $n$ automatically cyclic?
It is very reasonable for me to assume that if $H \leq S_n$ is transitive and $|H|=n$, then $H$ must be generated by a $n$-cycle, but I cannot seem to be able to prove it, so I really can not be sure that it is true. Could you please hel... |
H: Help with $\int_{\Bbb D}(x^2 - y^2)\, dx\, dy$ where $\Bbb D=\{|x|+|y|\le2\}$
Consider $\Bbb D=\{|x|+|y|\le2\}$; I'm trying to solve:
$$\int_{\Bbb D}(x^2 - y^2)\, dx\, dy$$
My goal is to solve that through a change of variables.
I was thinking to something like: $\begin{cases} |x|=u^2 \\ |y|=v^2 \end{cases}$. In s... |
H: Evaluate $\iiint_{D}(x^2+y^2)^2\,dx\,dy\,dz$ over cylinder section
Evaluate $$\iiint_{D}(x^2+y^2)^2\,dx\,dy\,dz\,,$$ where $$D=\{(x,y):x^2+y^2\leq 1, 0\leq z\leq 1,0\leq y\}$$
I used GeoGebra to represent my domain.
It is that right section of the cylinder, enclosed by the planes $z=0$, $z=1$. As I have a cylinder,... |
H: Prove that NAND and NOR are the only Universal Logic Gates.
I was watching this lecture: link
H(x,y) is a boolean function.
He's says that H(x,y) is a Universal logic gate if and only if H(x,x) is 1 - x.
I didn't get this part.
So how to prove that NAND and NOR are the only Universal Logic Gates ?
AI: If $H(x,x)\ne... |
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