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H: Solve $\sqrt{x+4}-\sqrt{x+1}=1$ for $x$
Can someone give me some hints on how to start solving $\sqrt{x+4}-\sqrt{x+1}=1$ for x?
Like I tried to factor it expand it, or even multiplying both sides by its conjugate but nothing comes up right.
AI: Start by squaring it to get
$$x+4-2\sqrt{(x+4)(x+1)}+x+1=1\;,$$
which s... |
H: Like term reduction
In finding the derivative of $f(x) = 4x - x^2$ we first find the difference of the numerator $f(x + h) - f(x)$. Therefore we have $f(x + h) = 4(x + h) - (x + h)^2 = 4x + 4h - x^2 - 2xh - h^2$ minus $f(x) = 4x - x^2$. This leads to entire expression in the numerator being:
$$4x + 4h - x^2 -2xh ... |
H: Is there an easy way to see that all derivatives are bounded?
Show that all derivatives of $f:\mathbb{R}\to\mathbb{R}$ given by $$f(x):=\frac{1}{\sqrt{x^2+1}+1}$$ are bounded.
It's easy to see that all derivatives are continuous. So the only potential problem is that a derivative might blow up at $\infty$. Is the... |
H: Is it possible to rationalize a denominator containing two cube roots?
The fraction in question is
$$-\frac{12}{\sqrt[3]{12\sqrt{849} + 108} - \sqrt[3]{12\sqrt{849} - 108}}$$
And was reached in calculating the solution to $x^4 - x - 1 = 0$. I've tried all the standard methods, including $(a+b)(a-b) = a^2 - b^2$, bu... |
H: I need help with relations
Let $S$ be the power set $P({1,2,...,10})$; that is, $S$ is the set of all subsets of $\{1,2,\dots,10\}$. define the relation $\mathcal R$ on $S$ by:
For all subsets $A,B$ of $\{1,2,\dots,10\}$, $A\mathcal RB$ if and only if $A\cup B$ has exactly $3$ elements.
a) Is $\mathcal R$ reflexive... |
H: Finding example of sets that satisfy conditions
give examples of sets such that:
i)$A\in B$ and $A\subseteq B$
My answer : $B=\mathcal{P(A)}=\{\emptyset,\{1\},\{2\},\{1,2\}\}$ and $A=\{1,2\}$ then $A\in B$ and $A\subseteq B$
ii) $|(C\cup D)\setminus(C\cap D)|=1$
My answer is: $C=\{1,2,3\}$, $D=\{2,3\}$ then $C\cup ... |
H: Equivalence classes of the relation that the largest digit of integer a = largest digit of integer b.
Define the relation $\mathcal{R}$ on the set of all positive integers by: for all positive integers $a$ and $b$, $a\,\mathcal{R}\,b$ if and only if the largest digit of $a$ is equal to the largest digit of $b$. for... |
H: Unexplainable Divergent area?
Somebody came up to me recently with the following problem. Consider $y = \dfrac{1}{x^2}$ with $x>0$. Now there is this square $S$ sitting under the curve, connecting $(0,0) , (1,0) , (1,1)$ and $(0,1)$. The area under the curve from $x = 1$ to infinity is equal to $1$ through an impro... |
H: find $\theta_{MLE}$ for a function
For
$$
f(x;\theta)=(\theta+1)x^{-\theta-2}
$$
find the maxmimum likelihood estimators (MLEs) for $\theta$ based on a random sample of size $n$.
My work so far:
$$
\begin{align}
\prod_{i=1}^n \log(f(x_i;\theta)) &= \sum \log(\theta+1)-\log(x_i)(\theta+2) \\
&=n\log(\theta+1)-(\the... |
H: Can the second term of the Schur complement of a symmetric matrix be undefined?
Given the next symmetric matrix conformably partitioned
$$\begin{bmatrix}
A &B \\ B^T &C
\end{bmatrix}$$
I know that $A$ and $C$ are positive definite matrices.
The Schur complement is $S=C-B^TA^{-1}B$
What can I say about $B^TA^{-1}B... |
H: Simplifying a Product of Summations
I have, for a fixed and positive even integer $n$, the following product of summations:
$\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = n-5}^{n-1}i \right )\cdot ... \cdot \left (\sum_{i=5}^{n-1}i \right )\cdot \left (\... |
H: Matrices manipulation
I am having difficulty with the following question
I have to determine if the following claim is true or not.
If it is true I have to proof it else I need to give an example
I believe it is not true but I do not find.
$A$ is a matrix (with scalar in real).
$A$ is symmetric.
if there exists $k>... |
H: Subalgebras of certain C*-algebras
Let $A$ be a C*-subalgebra of $C(X, M_{n}(\mathbb{C}))$ where $X$ is a compact Hausdorff space, does it follow that $A$ is isomorphic to $C(Y, M_{m}(\mathbb{C}))$ for some $Y\subseteqq X$ and $m<=n$ ?
AI: This is not even even true for a singleton $X=\{\ast\}$. In this case $C(X,... |
H: Square of sum of matrices
I'm trying to follow these lecture notes on Linear Discriminant Analysis (LDA) but I can't seem to figure out how the author gets from:
$$ \Sigma_{x\epsilon\omega_{i}} (w^{T}x - w^{T}\mu_{i})^2$$
to
$$ \Sigma_{x\epsilon\omega_{i}} w^{T}(x-\mu_{i})(x-\mu_{i})^Tw$$
AI: Since, $w^{T}x - w^{T}... |
H: On Lévy collapsing the reals
Consider the Lévy forcing notion. Let $M$ be a transitive standard model of $\mathsf{ZFC}$. Let $\aleph_n$ be the cardinality of the real numbers $2^\omega$ in $M$. Now collapse $\aleph_n$ to $\omega$. The resulting model $M[G]$ is again a model of $\mathsf{ZFC}$ but it is provable in $... |
H: Using the Chebyshev Inequality
This is the Q:
A 20 fair coins tosses, (f means the "H" of the coin).
I have to block the probability that I will get n/2+n/100 "H"-s by Chebyshev Inequality. [n=20 in this case...], so:
n/2+n/100 = 20/2+20/100 = 10.2
How I'm doing it?
the result it's much higher then one.
What I get ... |
H: Evaluating $48^{322} \pmod{25}$
How do I find $48^{322} \pmod{25}$?
AI: Recall Euler's theorem: $a^{\phi(n)} \equiv 1 \pmod n$, where $\gcd(a,n)=1$. We hence have
$$48^{20} \equiv 1 \pmod{25}$$
Hence,
$$48^{322} \equiv \left(48^{20}\right)^{16} \cdot 48^2 \pmod{25} \equiv 48^2 \pmod{25} \equiv (-2)^2 \pmod{25} \equ... |
H: Minimal Red-Black tree with depth 3
I'd like to ask what is minimal RBT with black depth 3. Is this following RBT ? Values are not important. And that tree can't have depth 2 or 1.
AI: According to Wikipedia, a red-black tree should be "roughly balanced". The rule is "that the path from the root to the furthest le... |
H: Finding the MLE of $\theta$ where $\theta \leq x$
consider the following PDF
$$
\begin{eqnarray}
f(x;\theta) &=&
\left\{\begin{array}{ll}
2\frac{\theta^2}{x^3} & \theta \leqslant x\\
0 & x< \theta; 0 < \theta
\end{array}\right.\\
\end{eqnarray}
$$
Now the answer stats $X_{1:n}$ so the minimum of $X$, but this canno... |
H: On a question chosen at random, what is the probability that the student answers it correctly?
I'm really confused about this question. I appreciate your help.
A student takes a multiple choice exam where each question has five possible answers. He answers correctly if he knows the answer, otherwise he guesses at r... |
H: Continuity of $d(x,A)$
I am doing a head-check here. I keep seeing this theorem quoted as requiring $A$ to be closed (as in Is the distance function in a metric space (uniformly) continuous?), but I don't think that it is needed.
Theorem. Let $(X,d)$ be a metric space and $\emptyset \neq A\subseteq X$. Then $x \lo... |
H: Unity in the rings of matrices
Suppose we are given an arbitrary ring $R$. Then the set $M_n(R)$ of all square matrices with elements from $R$, together with usual matrix addition and multiplication forms a ring. If R is a unitary ring then $M_n(R)$ too.
My question seems to be very trivial but is it possible th... |
H: Prove that if $n\geq\text{lcm}(a,b)$ and $\gcd(a,b)|n$ then $n=xa+yb$ for some integers $x,y\geq 0$
I thought I had it, but then I realized I didn't. Even just a hint—am I going in the right direction or should I try something completely different?
We know that $\gcd(a,b)=wa+zb$ for some integers $w,z$. Then since ... |
H: Does $P(A\cap B) + P(A\cap B^c) = P(A)$?
Based purely on intuition, it would seem that the following statement is true, when thinking of the events as sets:
$$P(A\cap B) + P(A\cap B^c) = P(A)$$
However, I am not sure if this is true, and cannot find out how to prove it, or describe a straight forward intuition of i... |
H: Statistics Question
This is probably super simple to most of you on here, but I was chatted by a friend earlier with a question. It reads just like this:
A sample of n =7 scores has a mean of M =5. After one new score is added to the sample,
the new mean is found to be M =6. What is the value of the new
score? (H... |
H: Does Euler totient function gives exactly one value(answer) or LEAST calculated value(answer is NOT below this value)?
I was studying RSA when came across Euler totient function. The definition states that- it gives the number of positive values less than $n$ which are relatively prime to $n$.
I thought I had it, u... |
H: Analogy between prime numbers and singleton sets?
While trying -- in vain -- to write an alternative answer for another question (If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.), I discovered the following property for sets: $$A \textrm{ is a singleton set} \;\equiv\; \lan... |
H: A proof of $n*0=0$?
The only proof I've seen for this assumes that $0$ follows all the rules of arithmetic. How can we make that assumption when dividing by $0$ is a problem? I know that some people don't agree that all of the numbers follow the rules for arithmetic; for example, people say that the proof of $.9999... |
H: If for every $x_n$ such that $x_n \rightarrow x$, there exists a $x_{n_k}$ such that $Tx_{n_k} = Tx$, is $T$ continuous?
Let $X$ and $Y$ be Banach spaces and $T$ be the (possibly nonlinear) map $T\colon X \rightarrow Y$. $T$ is continuous if for every $x_n \in X$ such that $x_n \rightarrow x$, then $Tx_n \rightarro... |
H: An exercise about zerodivisors
If $A$ is a commutative ring with unity, $f\in A$ and $x\in SpecA$, with the notation $f(x)$ I mean the coset $x+f\in A/x$. Now look at this exercise:
Prove that a nonzero element $f\in A$ is a zerodivisor if and only if there exists a decomposition $SpecA=X\cup X'$ where $X\subsete... |
H: How can I measure variance efficiently?
I have bunch of values, for example {1,2,3,4}.
I need to measure variance in a very efficient way.
On wikipedia variance is defined as sum of squared differences
between the data examples and the mean, and then you normalize
that sum with 1/n, where n is the number of data ex... |
H: Can't understand homework assignment
Let $p\in\mathbb{N}$ be an odd prime. Prove that if $p \equiv 3 \pmod 4$ then $−1$ is not a square modulo $p$.
$\textbf{Hint}$ : recall that $\mathbb{Z}/p\mathbb{Z}$ is a field, so that its multiplicative group is a cyclic group
of order $p − 1$. Prove that in any such group the ... |
H: $f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
$f_n(z)={z^n\over n}$, $z\in D$ open unit disk then
1.$\sum f_n$ converges uniformly on $D$?
2.$f_n$ and $f'_n$ converges uniformly on $D$?
3.$\sum f'_n$ converges on $D$ pointwise?
4.$f_n''(z)$ does not converge unless $z=0$
clearly $1$ is false as $\sum {1\over ... |
H: A question regarding the Poisson distribution
The number of chocolate chips in a biscuit follows a Poisson distribution with and average of $5$ chocolate chips per biscuit. Assume that the numbers of chocolate chips in different biscuits are independent of each other. What is the probability that at least one biscu... |
H: REVISTED$^1$ - Order: Modular Arithmetic
Relevant Literature:
Question:
Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$.
Thoughts:
Direct answers are OK, but I'd like to know if I'm right that what I'm really looking for is this:
$$\inf\left\{\frac{2^x-2}{25}~:~x\in \mathbb{N}\setminus \{... |
H: Solving System of Differential equations
The general solution to differential equation
$$x'=Ax$$ where A is a square matrix is given by solving for the eigenvalues and then eigen vectors of matrix $A$. However, is there a general method if I have $$x'=Ax+b$$ where $b$ is a vector of length same a $x$?
The straight ... |
H: Minimal polynomial matrix
I want to show that $ x^n-1$ is the minimal polynomial of the permutation matrix $P:=(e_2,e_3,....,e_n,e_1)$ where $e_i$ is the i-th unit vector written as a column vector.
And now I have to show that over an arbitrary field $x_n-1$ is the minimal polynomial. This seems to be tough. I hav... |
H: Are absolute extrema only in continuous functions?
The Extreme Value Theorem says that if $f(x)$ is continuous on the interval $[a,b]$ then there are two numbers, $a≤c$ and $d≤b$, so that $f(c)$ is an absolute maximum for the function and $f(d)$ is an absolute minimum for the function.
So, if we have a continuous ... |
H: How are these two equations equal?
$$\dfrac{1}{1+e^{-x}} = \dfrac{e^x}{1+e^x}$$
I was told to sketch a curve but couldn't figure out the first step. The solution manual rewrote the left hand side of the equation above as the right hand side. I cannot figure out what they did to get this. Could someone explain this ... |
H: How to prove $4\times{_2F_1}(-1/4,3/4;7/4;(2-\sqrt3)/4)-{_2F_1}(3/4,3/4;7/4;(2-\sqrt3)/4)\stackrel?=\frac{3\sqrt[4]{2+\sqrt3}}{\sqrt2}$
I have the following conjecture, which is supported by numerical calculations up to at least $10^5$ decimal digits:
$$4\times{_2F_1}\left(-\frac{1}{4},\frac{3}{4};\frac{7}{4};\frac... |
H: Derive Cauchy-Bunyakovsky by taking expected value
In my notes, it is said that taking expectation on both sides of this inequality $$|\frac{XY}{\sqrt{\mathbb{E}X^2\mathbb{E}Y^2}}|\le\frac{1}{2}\left(\frac{X^2}{\mathbb{E}X^2}+\frac{Y^2}{\mathbb{E}Y^2}\right)$$ can lead to the Cauchy-Bunyakovxky (Schwarz) inequality... |
H: Example of ring such that the nil radical is prime and 0 is not
I was just trying to think about an example of a ring that is not a domain and the nilradical is prime, however I could not find anyone.
Thanks in advance.
AI: $\mathbb{Z} / 4 \mathbb{Z}$ works. |
H: Complex Analysis boundedness and limits
True or false. If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then $\lim_{z\to 0} f(z)$ exists.
R If f is bounded means that there is some M∈R such that ∀z ∈C holds that |f(z)|≤M.
My Answer
Would $\frac{z}{|z|+1}$ be an example of such a function? Bounded above by 1 and below by... |
H: Solutions of $x\frac{\partial \psi}{\partial x} + y \frac{\partial \psi}{\partial y} + \psi = f(x)e^{-2\pi i y}$?
I stumbled across the following PDE for a function $\psi(x, y)$:
$$
x\frac{\partial \psi}{\partial x} + y \frac{\partial \psi}{\partial y} + \psi = f(x)e^{-2\pi i y}
$$
where $f(z)$ is some arbitrary fu... |
H: Equation of a line passing through a point and forming a triangle with the axes
How can I find the equation of a line that;
is passing through the point (8, 6) and
is forming a triangle of area 12 with the axes
?
So I tried to start using $A = |{\frac{mn}{2}}|$ and got that $m\cdot n$ is either -24 or 24.... |
H: Finitely generated flat modules that are not projective
Over left noetherian rings and over semiperfect rings, every finitely generated flat module is projective. What are some examples of finitely generated flat modules that are not projective?
Compare to our question f.g. flat not free where all the answers are f... |
H: How do you determine the points of inflection for $f(x) = \frac{e^x}{1+e^x}$?
$$f(x) =\dfrac{e^x}{1+e^x}$$
I know we can find points of inflection using the second derivative test. The second derivative for the function above is $$f''(x) = \dfrac{e^x(1-e^x)}{(e^x+1)^3}$$ I have found one critical point for the seco... |
H: Differential Equations Reference Request
Currently I'm taking the Differential Equations course at college, however the problem is the book used. I'll try to make my point clear, but sorry if this question is silly or anything like that: the textbook used (William Boyce's book) seems to assume that the reader doesn... |
H: How are these two equivalent?
$$\frac{\ln(e^x+x)}{x}=\frac{e^x+1}{e^x+x}$$
I see that they did something to get rid of the natural log. I couldn't find any properties that would allow me to do this. I also think that they raised both the numerator and denominator by $e$. I have tried it and I did not get the same r... |
H: Example of a continuous function which is bounded and not contained in any $L_p$-space ($p\gt 0$)
I'm struggling to find an example of a continuous function $f:(0,\infty)\to \mathbb R$ which is bounded, not contained in any $L_p$-space ($p\gt 0$) and goes to zero when x goes to infinity.
I need help.
Thanks a lot!
... |
H: Finding the limit, multiplication by the conjugate
I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$
I tried and tried... friends of mine tried as well and we don't know how to get out of:
$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$
(this is what we get after multiplying by the conjugate of $2 + \sqrt... |
H: Representing sums of matrix algebras as group rings
Let $A = M_{n_1}(\mathbb R) \oplus M_{n_2}(\mathbb R) \oplus ... \oplus M_{n_m}(\mathbb R)$ be a direct sum of real matrix algebras. Under what conditions does there exist a group ring $\mathbb R[G]$ which is isomorphic to $A$?
I know every group ring is isomorph... |
H: Solving Simple Mixed Fraction problem?
How do you wrap your head around mixed fraction, does anyone knows how to figure out, can someone give me an example how it can be solved?
AI: $$a +\frac bc = \frac {a\times c}c + \frac bc = \frac{(a\times c) + b}{c} $$
What we do first is like when finding a common denominat... |
H: Basic question on the transformation of Exponential distribution.
Why central moments coincide for random variables
$V\sim E(a,h)$ and $Y\sim E(h)$
where a=location parameter
h= scale parameter.
AI: Because $E(a,h)$ is $E(h)$ translated by $a$. Taking central moments (by subtracting the mean) removes the translat... |
H: Solution for Summation of $\cos^2x$
Can you give me the solution for the summation
$$
\sum_{n=0}^{\infty} \cos^2(\pi n)
$$
Edit: Please give me the explanation of how it is calculated and also final answer in integers.
AI: Have you noticed out the terms behaved? $\cos(\pi n) = (-1)^n$, so $\cos^2(\pi n) = 1$. Thi... |
H: Topics to Study and Books to Read?
I'm an undergraduate studying materials science and engineering with a concentration in polymer science. I would like to go to graduate school and focus on theory and computation of synthetic polymer and biopolymer systems. So I'm planning of studying things such as Hamiltonian me... |
H: Does such $A,B$ exist?
true/false test: there're $n\times n$ matrices $A,\ B$ with real entries such that $(I-(AB-BA))^n=0$
I'm cluesless to begin.
AI: Let $C = AB - BA$ and assume that $(C - I)^n = 0$.
Then the minimal polynomial $m(x)$ of $C$ divides $(x - 1)^{n}$. Thus $m(x) = (x-1)^{k}$ for some $1 \leq k \leq ... |
H: A counterexample of normal subgroup with cyclic Sylow 2-subgroup
We know that when a group $G$ has order $2^k m$, where $m$ is an odd integer, $G$ should have a normal subgroup with order $m$ from here. When $k=1$, this implies the index of the normal subgroup is $2$. However, when $k=2$, $m$ is a prime, can we als... |
H: How do I solve this solution-mixing problem?
A chemist has a 55% acid solution and a 40% acid solution. How many liters of each should be
mixed in order to produce 100 liters of a 46% acid solution?
AI: Let $X_1$ denote the total amount of acid solution 1. Let $X_2$ denote the total amount of acid solution 2. Now, ... |
H: Check if $\lim_{x\to\infty}{\log x\over x^{1/2}}=\infty$, $\lim_{x\to\infty}{\log x\over x}=0$
Could anyone tell me which of the following is/are true?
$\lim_{x\to\infty}{\log x\over x^{1/2}}=0$, $\lim_{x\to\infty}{\log x\over x}=\infty$
$\lim_{x\to\infty}{\log x\over x^{1/2}}=\infty$, $\lim_{x\to\infty}{\log x\o... |
H: Example of sequence converging in $d_{l^\infty}$ but not in $d_{l^1}$.
I'll denote by $X$ the space of real sequences $(a_n)$ such that $\sum |a_n|$ converges. Let $d_{l^1}$ be the metric
$$
d_{l^1}((a_n),(b_n))=\sum|a_n-b_n|
$$
and $d_{l^\infty}$ be the metric
$$
d_{l^\infty}((a_n),(b_n))=\sup\{|a_n-b_n|\}.
$$
It ... |
H: Is there a way to eliminate an extraneous index from this sum?
Fix a natural number $n$. Suppose we have a triple sum of the form
$$ \sum_{i=1}^n \sum_{j=1}^n \sum_{k=j+1}^{i-1} f(i,k) $$
where the summands only depend on $i$ and $k$.
Is there a way to rewrite the sum so that it does not include the index $j$?
A... |
H: How do you define definition symbol :=?
How is "$:=$" defined formally and why? "$\iff$", "$=$", ...?
AI: There is no need of a formal definition of a definition. A definition is just an abbreviation. For example, in formal arithmetic (number theory) we write $x\mid y$ ($x$ divides $y$) as an abbreviation for $\ex... |
H: Is there still any hope that the GCH could be equivalent to some large cardinal axiom?
Is there still any hope that the GCH could be equivalent to some large cardinal axiom?
Even a simple yes or not answer will be fine. Thanks!!
AI: No.
The term "large cardinals" has no explicit and well-defined meaning, but it's g... |
H: Lagrange's Theorem for further elementary consequences
Question: Let $G$ be a finite group, and let $H$ and $K$ be subgroup of $G$. Prove: suppose $H$ and $G$ are not equal, and both have order the same prime number $p$, Then $H\cap K=\{e\}$.
This is my proof steps:
Proof: $G$ is finite, $H, K$ subgroup of $G$. the... |
H: Natural logarithm
Can someone please suggest how one proves:
$(1+2x)\ln(1+\frac{1}{x}) -2 >0$ where $x>0$.
I plotted the function in a program and the inequality should be correct.
AI: Here is one approach. Let $f(x)=(1+2x)\ln(1+\frac{1}{x})$. You can show that $\lim\limits_{x\to+\infty}f(x)=2$, for example usin... |
H: Why is Cauchy's integral formula always written with the function as the subject?
Scouring textbooks, lecture notes, Wikipedia, etc., I notice that the standard presentation of Cauchy's integral formula is $$f(w)=\frac1{2\pi i}\int_L\frac{f(z)}{z-w}\,\mathrm dz\tag1$$ rather than $$\int_L\frac{f(z)}{z-w}\,\mathrm d... |
H: How to write $\pi$ as a set in ZF?
I know that from ZF we can construct some sets in a beautiful form obtaining the desired properties that we expect to have these sets. In ZF all is a set (including numbers, elements, functions, relations, etc...).
For example we can define a copy of $\mathbb{N}$, using the empty... |
H: How to find the amount to added every month or year to get the required amount after certain years?
I want to do a Java application for which after giving the current savings, and the rate of interest and and required amount after specified no of years, it has to show how much a person has to earn a month or year... |
H: Natural Deduction proof for $\forall x \neg A \implies \neg \exists xA$
$\forall x \neg A \implies \neg \exists xA$
I won't ask you to solve this for me, but can you please give some guiding lines on how to approach a proof in NDFOL?
There are many tricks that the TA shows in class, that I could not dream of...
P.S... |
H: A theorem about the Poisson Point process.
In the proof of the Levy-Khintchine theorem, I saw a theorem about the Poisson
point process.
The theorem states that if $\Pi$ is a poission point process on $S$ with
intensity measure $\mu.$ Let $f:S\rightarrow\lbrack0,\infty)$ be a measurable
function. And define
$$
Z=\... |
H: Show that If k is odd, then $\Bbb{Q}_{4k}$ is isomorphic to $\Bbb{Z}_k \rtimes_{\alpha} \Bbb{Z}_4$ for some $\alpha$
Show that if k is odd, then $\Bbb{Q}_{4k}$ is isomorphic to $\Bbb{Z}_k \rtimes_{\alpha} \Bbb{Z}_4$ for some $\alpha: \Bbb{Z}_4 \rightarrow Aut(\Bbb{Z}_k)$. Calculate $\alpha$ explicitly.
We know th... |
H: Self-adjoint and eigenvalues properties
I wondering about something.
Let $V$ be an inner product space
$T\colon V\to V$ is a linear map
$T$ is self-adjoint and all the eigenvalues of $T$ are not negative
I need to prove that for all $v$ in $V$, $(T(v),v)\ge0$.
So I think that if all the eigenspaces of all the eige... |
H: Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?
I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a geo... |
H: One of $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$ for odd $n$
Let $n$ be an odd integer greater than 1. Show that one of the numbers $2^1-1,2^2-1,...,2^n-1$ is divisible by $n$.
I know that pigeonhole principle would be helpful, but how should I apply it? Thanks.
AI: Observe that there are $n$ such number with t... |
H: Show that eigenvalues are negative
I have to consider the eigenvalue problem:
$$ L[u] := \frac{d^2 u}{dx^2}= λu,x \in (0,1)\quad u(0)-\frac{du}{dx}(0)=0, u(1)=0.$$
I need to show that the eigenvalues are negative.
AI: Suppose we have $\lambda > 0$, then $u'' = \lambda u$ gives us $$u(t) = \alpha\exp(-\sqrt\lambda... |
H: Product of variables
Suppose we have a set of variables $\{a_i,b_i| i=1,2,3\}$ which can take values $\pm 1$ according to some probabilities.
If there's a constraint that $a_1b_2b_3=a_2b_1b_3=a_3b_1b_2$ must equal to $1$, why then must the value of $$a_1a_2a_3=1$$?
I thought of saying that $a_1b_2b_3a_2b_1b_3a_3b... |
H: Order of convergence of a sum
Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that
\begin{align*}
\mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty.
\end{align*}
Assume that $\mathbb{E}[X_t]=O(t),\;t\to 0$. Clearly for each fixed $N\in\mathbb{N}$,
\begi... |
H: Help Understanding Fields
I came across this problem in a Linear Algebra text today:
Let $u$ and $v$ be distinct vectors in a vector space $V$ over a field $F$. Prove that $\{u,v\}$ is linearly independent if and only if $\{u+v,u-v\}$ is linearly independent.
Working on ($\Rightarrow$), I must show that
$$c_1(u... |
H: Can someone explain to me what are these 2 statements talking about?
I have to prove that these 2 statements are equivalent, but I can't even understand them.
There exist $\epsilon_0>0$ such that for all $k\in\mathbb N$, there exist $n_k\in\mathbb N$ such that $n_k\geq k$ and $|x_{n_k}-x|\geq\epsilon_0$?
There exi... |
H: Conditioning a martingale increment by earlier increments
I have a $L^1$ - martingale ($E[|X|]<\infty$) defined on $(\Omega,\mathcal F , \mathbb P)$, with constant expectation $EX_t$, and I have to prove that $$E\{(X_v-X_u)|(X_t-X_s)\}=0$$ for $0\le s<t\le u<v$.
$$$$
Can I consider $X_t-X_s$ as a filtration, then a... |
H: Determinant of product of symplectic matrices
In optical ray tracing it's possible to use symplectic matrices. I have a problem with them.
If a matrix $M$ is symplectic, this means that for $M$ the following equation hols:
$$M^T\Omega M=\Omega$$
where
$$\Omega =
\begin{pmatrix}
0 & I_n & \\
-I_n & 0
\end{pmat... |
H: weighted average
I'm not sure the correct term for my problem is weighted average. But let me explain.
I've conducted a survey where participants answer on a scale betweeen $1$ and $7$.
The questions fall into three categories. In category one & two there are $12$ questions, and in category three there are four qu... |
H: Is $e^x$ the only morphism for addition and multiplication?
I find it interesting that the only (more or less) function that is equal to its own derivative also happens to be a morphism from the reals under addition to the positive reals under multiplication. This would be even more interesting if it were the only ... |
H: confused about the limit of a trigonometric function
I am trying to calculate the limit of the following function for general $a$:
$$\lim_{x\to a}[\cos(2 \pi x)-\sin(2 \pi x) \cot(\frac{\pi x}{a})]$$
I was believing this is infinite. But Mathematica calculates for $a=3$ for instance the limit is $5$
And for $a$ in ... |
H: Polynomial differential equation
I came across this problem in an old olympiad paper (Putnam?)
Find all polynomials $p(x)$ with real coefficients satisfying the differential equation
$7\dfrac{d }{dx } [xp(x)]=3p(x)+4p(x+1)$ $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ -\infty<x<\infty$
I didn't find any "official" solutio... |
H: Which one is the correct series expansion?
Is
$$p^{n+1} = p^0+p^1+ \dots + p^n$$
or
$$p^{n+1} = p^0\times p^1\times \dots \times p^n\text{ ?}$$
I am confused.
please explain the correct one.
AI: Hints:
$$\begin{align*}\bullet&\;\;\;\;1+q+q^2+\ldots+q^n=\frac{q^{n+1}-1}{q-1}\\\bullet\bullet&\;\;\;\;1+2+\ldots+n=\fra... |
H: A problem on matrices : Sum of elements of skew-matrix
If $A=[a_{ij}]$ is a skew-symmetric matrix, then write the value of $$ \sum_i \sum_j a_{ij}$$
My doubt is that what is the meaning of $ \sum_i \sum_j ?$ Is it the same as $\sum_{ij}?$
Please offer your assistance.
Thank you
AI: As for the notaion. It is easie... |
H: A problem on matrices : Powers of a matrix
If $ A=
\begin{bmatrix}
i & 0 \\
0 & i \\
\end{bmatrix}
, n \in \mathbb N$, then $A^{4n}$ equals?
I guessed the answer as $ A^{4n}=
\begin{bmatrix}
i^{4n} & 0 \\
0 & i^{4n} \\
\end{bmatrix}
=\begin{bmatrix}
... |
H: Necessary condition for local minima; non-negative Hessian matrix
The problem I have is the following. Any results on Taylor expansions etc. can be assumed:
Let F : R^n -> R be a C2 function. Let x_0 be a local minimum of F. Prove that the Hessian matrix of F is non-negative.
Thanks for your help.
AI: Let $\def\R{... |
H: There are infinitely many choices of $(\alpha_1,\dots,\alpha_n)$ such that $f(\alpha_1,\dots,\alpha_n)\neq 0$
I'm trying to solve this exercise in the page 10 of this book
Maybe I'm forgetting something, but I couldn't solve this exercise, I need a hint or something to begin to solve this question.
Thanks in advan... |
H: complex analysis -examples of complex functions that are bounded and the limits
What are some examples of complex functions that are bounded and the limits does not exists as $z\to 0$?
AI: $f(z)=\frac{z}{|z|}$.
Also, let $g(z)$ be any analytic function with $g(0)\neq 0$ and let
$$h(z)=\frac{z}{|z|}g(z) \,.$$ |
H: Are these functions orthonormal?
Are the following set of functions orthonormal over the interval $0$ to $1$?
$$Y_r(x) = \sin{\beta_r x}-\sinh{\beta_r x}-\frac{\sin\beta_r-\sinh\beta_r}{\cos\beta_r-\cosh\beta}\left(\cos\beta_r x-\cosh\beta_r x\right)$$
where $\beta_r$ are the positive solutions to:
$$1-\cos\beta_r\... |
H: $\mathbb{Z}[i]$ is a Dedekind domain
I know that $\mathbb{Z}[i]$ is a PID, and that every PID is a Dedekind. But I want to show that $\mathbb{Z}[i]$ is a Dedekind, without using PID.
One strategy coul be to show that $\frac{\mathbb{Z}[i]}{\mathfrak{P}}$ is a field for every nonzero prime ideal $\mathfrak{P}$. Coul... |
H: Gateaux and Fréchet differentials in $\ell^1$
I am in trouble trying to solve the following:
Let $X = \ell^1$ with the canonical norm and let $f \colon \ell^1 \ni x\mapsto \Vert x \Vert$. Then $f$ is Gateaux differentiable at a point $x = (x_1, x_2, \ldots )$ if and only if $x_i \ne 0$ for every $i \in \mathbb N$... |
H: Finding convergence of the next function: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$
How can i find whether the next function converges: $f(x)=\sum_{n=1}^\infty\frac{(\ln n)^3}{n}x^n$?
I thought about this question for quite a while, What's the trick?
AI: The ratio test here will work nicely:
Recall that we app... |
H: Trace of a differential operator
Given the differential operator:
$$A=\exp(-\beta H)$$
where $$H=\frac{1}{2}\left( -\frac{d^2}{dx^2}+x^2 \right)$$
and $\beta\gt 0$
How can I get the trace of this operator?
Thanks in advance.
AI: The trace of this operator is easily obtained in the following way:
$$
Z={\rm Tr}\ex... |
H: Runge-Kutta method and step doubling
I am studying Runge-Kutta and step size control and came up with a few doubts. Because they are related with this integration method, I will divide it in two parts. First, allow me to introduce the problem.
$1^{st}$ part - questions about Runge-Kutta Method
Consider a $2^{nd}$ ... |
H: What is the purpose to define central moment?
What is the purpose to define central moment?
I searched the google and all i could find is bunch of properties
AI: Not sure I got your question, one of the 'purposes' of central moment is to set $n=2$ and get the variance of the rv, which is a measure of dispersion aro... |
H: Time has dimension $2$ with respect to the Ricci flow scaling
Terence Tao in his lecture notes on Ricci flow has written:
If we are to find a scale-invariant (and diffeomorphism-invariant) monotone quantity for Ricci flow, it had better be constant on the gradient shrinking soliton. In analogy with $\frac{d}{dt}\ma... |
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