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H: Random variables X, G are have functional relationship G=g(X). How does g relate the graphs of their distributions?
More specifically, as an educational tool I want to prepare a slideshow showing (in 2-D) the graph of $F_X$ transforming into the graph of $F_G$. (I think it can be done in 4 steps (e.g., graphs on $... |
H: $s(x)$ is a arc length function, find $s'(x)$
Here is the problem is my textbook:
Suppose $s(x)$ is the arc length function for the curve $y=\sin x$ taking
$(0,1)$ as the starting point. Find $s’(x)$.
According to arc length formula, I have :
$$ L = \int_0^1 \sqrt{1+\left((\sin x)'\right)^2}dx = \int_0^1 \sqrt... |
H: Given pairwise distances of $N$ data points and find the minimal dimenion of space can fit the data
Given a set $D$ consist of all pairwise distance of $N$ unknown dimension points.
e.g. If there is 3 points, ${x_a,x_b,x_c}$
$$D=\{||x_a-x_b||,||x_b-x_c||,||x_a-x_c||\}$$
How can I find the minimum dimension of space... |
H: Calculate alpha from $\alpha + \sin(\alpha)$ = K
Sorry for the dumb question, but I'm not involved in math.
I need to reverse the following formula, to calculate $\alpha$:
$$a = b(\alpha + \sin \alpha)/c$$
So I have:
$$(\alpha + \sin \alpha)=ac/b = K$$
Since $a$, $b$, $c$ are constant, I put equal to $K$.
$\alpha$ ... |
H: Question about a recurrence
In a syllabus of mine, they try to find a closed form of the following recurrence relation
$$\begin{align*}
T(2k) &\leq 3T(k) + ck & k \geq 1\\
T(1) &= 1
\end{align*}$$
The method I usually use to find the closed form of a recurrence is expand it a few times and try to find a patt... |
H: How do I solve this problem?
Exercise: If $a+2b=125$ and $b+c=348$, find out $2a+7b+3c$. Here $a$, $b$, $c$ are natural numbers.
The answer is: $2a+7b+3c = 1294$
I tried but just can't figure out how to get to this answer. I have a lot of exercises similar to this one but don't know how to aproach them. Can anyon... |
H: Winning strategy
Is there a winning strategy for player one or two in the following scenario:
The game begins with the number
2012. In one turn, a player can subtract from the current number any natural number
less than or equal to it that is a power of 2. The player who reaches 0 wins.
AI: It seems a player loses ... |
H: Least value for addition
We know that $$0\leq a \leq b \leq c\leq d\leq e\,\,\text{ and}\,\, a + b + c + d + e = 100$$. What would be the least
possible value of $\,\,a + c + e\,\,$ ?
I apologize for poor syntax.
AI: $$2(a+c+e) =a+a+c+c+e+e \geq a+b+c+d+e =100$$
With equality if and only if $a=0$, $b=c$ and $d=e$. |
H: Zeroing the carrier measure of an exponential family
I'm trying to derive the general process of changing variables so that an exponential family has zero carrier measure. Distributions in the exponential family have cdf
$$dF(\mathbf{x}|\boldsymbol\eta) = \exp\left({\boldsymbol\eta \cdot T(\mathbf{x}) - g(\boldsym... |
H: Name for a discrete ordered set with finite subranges
Is there a term for a set that is:
discrete
totally ordered
has finite subranges (not a technical term), i.e. for any a and b, $\{x|a<x<b\}$ is finite
At first glance, it seems this implies that the set is either finite or isomorphic (in some sense) with the i... |
H: Can monotone classes be finite?
I am new to measure theory and real analysis and am trying to double check my understanding of monotone classes.
My question:
Can monotone classes be finite? (It is not clear to me whether the idea of increasing or decreasing sets refers to STRICTLY increasing or decreasing sets.)
A... |
H: Solving $\frac{dy}{dx} = xy^2$
This problem appears to be pretty simple to me but my book gets a different answer.
$$\frac{dy}{dx} = xy^2$$
For when y is not 0
$$\frac{dy}{y^2} = x \, dx$$
$$\int \frac{dy}{y^2} = \int x \, dx$$
$$\frac{-1}{y^1} = \frac{x^2}{2}$$
$$\frac{-2}{x^2} = y$$
Is there anything wrong with... |
H: $1 +1$ is $0$ ?
Possible Duplicate:
-1 is not 1, so where is the mistake?
$i^2$ why is it $-1$ when you can show it is $1$?
So:
$$
\begin{align}
1+1 &= 1 + \sqrt{1} \\
&= 1 + \sqrt{1 \times 1} \\
&= 1 + \sqrt{-1 \times -1} \\
&= 1 + \sqrt{-1} \times \sqrt{-1} \\
&= 1 + i \times i \\
&= 1... |
H: Is the ring of integers in a relative algebraic number field faithfully flat over a ground ring?
Let $L$ be a finite extension of an algebraic number field $K$.
Let $A$ and $B$ be the rings of integers in $K$ and $L$ respectively.
Is $B$ faithfully flat over $A$?
What if $L$ is an infinite algebraic extension of $K... |
H: Why is $X_1 + X_2 +\ldots + X_n$ a martingale?
If we have $X_k$ random variables with average $0$ and independent, why is the $\sum_{k=1}^n X_k$ a martingale for the sigma algebra $\mathcal F_n$ generated by $\{X_1,\ldots, X_n\}$?
I basically only have to prove that the expected value of $X_{n+1}$ knowing $\mathcal... |
H: Friendship theorem and a group of 9 guests
Our task is to prove that there exists 4 strangers OR 4 friends within this group of 9 guests.
Now what's the best way to go about finding this out?
Using the Friendship Theorem? or using the Pigeonhole Principle ?
AI: Here is a counterexample to the claim that nine guest... |
H: Expressing $\sin(2x)-8\cos(2x)$ as a single sine function
I am asked as a part of a question to express $\sin(2x)-8\cos(2x)$ as a single sine function.
I know it has something to do with the trigonometric identity $$\sin(a-b)=\sin(a) \cos(b)-\cos(a)\sin(b)$$ but I can't get my head around it because of that $8$ in ... |
H: existence of a harmonic function
Let $\Omega\subset\mathbb R^n$ open, not bounded and $n\ge3$. Let $\partial\Omega$ bounded and regular concering the laplace operator. Given a continuous function $\phi:\partial\Omega\rightarrow\mathbb R$ and $\gamma\in\mathbb R$ there exists a harmonic function $u\in C^2(\Omega)\c... |
H: Does the isomorphism $k[x]\otimes_k k[x]\cong k[x,y]$ hold?
I think I am finally beginning to understand tensor products of algebras, and I could use a reality check. If I am understanding correctly, then $k[x]\otimes_k k[x]$ is ring-isomorphic to $k[x,y]$ by the map $x^i\otimes x^j\mapsto x^iy^j$. Is this right?
A... |
H: Finite groups of functions under function composition
Over the years I have done many questions along the lines of the following:
"Given functions $\phi, \theta$ (usually defined on $\mathbb{R}$ or $\mathbb{C}$, or a suitable subset of $\mathbb{R}$ or $\mathbb{C}$) prove that the collection of all functions obtaine... |
H: Proving that the magnitude of the sample correlation coefficient is at most $1$
How can you show that the magnitude of the sample correlation coefficient is at most $1$?
The formula is huge, I'm not even sure how to approach this. Can anyone point me in the right direction?
Note that this is the sample correlation ... |
H: Is the complement of a finite dimensional subspace always closed?
Let $F$ be a finite dimensional subspace of an infinite dimensional Banach space $X$, we know that $F$ is always topologically complemented in $X$, that is, there is always a closed subspace $W$ such that $X=F\oplus W$.
I am thinking about the conver... |
H: Basic probability problem
Problem states:
Consider two events $A$ and $B$, with $P(A) = 0.4$ and $Pr(B) = 0.7$. Determine the maximum and the minimum possible values for $P(A \& B)$ and the conditions under which each of these values is attained.
To solve, I considered the event with the lowest probability $A$ to... |
H: Formula to calculate the number of possible positions for $x$ numbers
What formula do I use to calculate the number of possible positions for $x$ numbers?
Let's say I have $3$ people in a race. What are all the possible combinations of the order they can finish in? Let's assume ties are not possible. I heard I use ... |
H: Easy condition for positive definite endomorphism
This problem is taken from Golan's linear algebra book.
Problem: Let $V$ be an inner product space over $\mathbb{R}$ and let $\alpha$ be an endomorphism of $V$. Show that $\alpha$ is positive definite if and only if $\alpha+\alpha^*$ is positive definite.
Definition... |
H: Divisibility is transitive: $\ a\mid b\mid c\,\Rightarrow\ a\mid c$
As the title says, if a number is divisible by a number, is it always divisible by that number's factors?
An example being that $100$ is divisible by $20$, it is also divisible by $10, 5, 4, 2$ as well?
Does this always apply?
AI: Yes. It is indeed... |
H: Proving that $\{0,1\}$ is a field with $1+1:=0$
EDIT: Hopefully question made clearer. Unfortunately this is a question found in analysis book and I do not actually have background on abstract algebra. Sorry for the confusion arisen.
As the title says, I am trying to show that a set of $\{0,1\}$, equipped with the ... |
H: Functions - Set Theory Proof
Let $f$ be a function from $X$ to $Y$, and let $A$, $B$ be subsets (non-proper) of $X$. For each of the following statements, either prove the statement or else give a counter example:
a.) $f(X\setminus A)=Y\setminus f(A)$
b.) $f(X\setminus A) \subseteq Y\setminus f(A)$
c.) $Y\setminu... |
H: "Fully correlated" definition
Really sorry to be a noob, but I'm a programmer, not a mathematician, and all of my knowledge about statistics come from this book "Schaum's Outline of Theory and Problems of Probability, Random Variables, and Random Processes".
I'm implementing an UKF for target tracking using C++. Ev... |
H: Is there a distributive law for ideals?
I'm curious if there is some sort of distributive law for ideals.
If $I,J,K$ are ideals in an arbitrary ring, does $I(J+K)=IJ+IK$?
The containment "$\subset$" is pretty clear I think. But the opposite ontainment doesn't feel like it should work. I couldn't work out a coun... |
H: The function $f(x) = \int_0^\infty \frac{x^t}{\Gamma(t+1)} \, dt$
Does anyone know if this function has a name? I came up with it by looking at the power series for $e^z$, changing the summation to an integral, and substituting the gamma function for the factorial function.
AI: Almost tautological, but we can also ... |
H: Can such a function exist?
Denote by $\Sigma$ the collection of all $(S, \succeq)$ wher $S \subset \mathbb{R}$ is compact and $\succeq$ is an arbitrary total order on $S$.
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that for all $(S, \succeq) \in \Sigma$ there exists a compact interval $I$ with ... |
H: Why is $a$ and $b$ coprime if $a\equiv 1 \pmod{b}$?
$a$ and $b$ are coprime if their greatest common divisor is 1. How do I conclude that from the fact that $b$ divides $a-1$?
AI: If $b\mid a-1$, then there is an integer $n$ such that $a-1=bn$, and therefore $a-bn=1$. Suppose that $d\mid a$ and $d\mid b$: then $d\m... |
H: which are positive definite matrix
Given that $A,B$ are positive definite matrix, are they also PD?
$A+B$
$AB$
$A^2 +I$
$ABA^{*}$
$x^TAx>0, x^TBx>0$ so $1$, is correct, could you tell me about the 2, 3,4?
AI: Mex
in 4. we can argue as follows:
$\langle x, ABA^* x\rangle = \langle A^*x, BA^* x\rangle =
\langle B... |
H: Finding the transformation when given transformation matrix
Lets say, there is a transformation: $T:\Re ^{n}\rightarrow \Re ^{m}$ transforming a vector in $V$ to $W$.
Now the transformation matrix, $A=\begin{bmatrix}
a_{11} & a_{12} &...&a_{1n} \\
a_{21} & a_{22} &...&a_{2n} \\
. & . & .\\
. & . & .\\
. & . & .\... |
H: Quotient sets and tori
My trek through MacLane and Birkhoff's Algebra has brought me to exercise 1.9.3, which defines the equivalence relation $E$ in $\Bbb R^2$ by $(x, y) E (x', y')$ iff $x-x'$ and $y-y'$ are both integers and asks me to prove that the quotient set $\Bbb R^2/E$ may be described as the set of point... |
H: Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$?
Is it wrong to say $$ \sqrt{x} \times \sqrt{x} =\sqrt{x^2}= \pm x$$
I am quite sure that $\sqrt{(x)^2} = \pm(x)$
But, does $\sqrt{x } \times \sqrt{x} =- (x)$ doesn't holds in $\mathbb{R}$ but if we assume $\mathbb{C}$ it holds right?
A... |
H: Notation in Munkres' $\textit{Analysis on Manifolds}$
I am trying to understand Theorem 9.1 of 1991 copy of Munkres' Analysis on Manifolds. I have stated what I don't understand below; there is a heading in bold. This theorem is a precursor to the implicit function theorem and on my copy of the book is on page 73. ... |
H: Characterising argmax of uniform distributions
I was thinking about comparisons of uniform random variables of the type $U(0,T)$, when I began to wonder about the argmax. Consider a sequence of parameters $T_1\le T_2\le\ldots T_n$ and corresponding independent random variables $U_1 \sim U(0,T_1)$, $U_2 \sim U(0,T_2... |
H: Graded Vector Spaces and the Interchange Law
I'm a little confused about how to correctly interchange factors in tensor
products on graded vector spaces.
In particular let $V:= \bigoplus_{n \in \mathbb{N}} V_n$ be a $\mathbb{N}$-graded
vector space and $\bigotimes^k V$ be the graded k-fold tensor product of $V$. ... |
H: Alternative proof or typo?
In Atiyah-MacDonald, we have the claim that $S^{-1}(I+J) = S^{-1}I + S^{-1}J$ and similarly, $S^{-1}(IJ) = S^{-1}I S^{-1}J$. Here $I,J$ are ideals of a commutative unital ring $R$ and $S$ is a multiplicative subset of $R$.
The proof is omitted with the remark (p.42): "...For sums and prod... |
H: MacNeille completion of a totally ordered set: Dedekind cuts
If $X$ is any partially ordered set with $A\!\subseteq\!X$ and $x\!\in\!X$, define $x\!\leq\!A :\Leftrightarrow \forall a\!\in\!A\!: x\!\leq\!a$ and $A\!\leq\!x :\Leftrightarrow \forall a\!\in\!A\!: a\!\leq\!x$, and denote $A^u\!:=\!\{x\!\in\!X; A\!\leq\!... |
H: What is the trellis diagram for a linear block code?
For the convolutional codes there is so-called trellis diagram,
for which the definition is rather clear for me, however in mathematical sense is not.
I have heard that it can be defined for linear block codes also.
Linear codes have very simple mathematical mean... |
H: About the Wasserstein "metric"
I've just encountered the Wasserstein metric, and it doesn't seem obvious to me why this is in fact a metric on the space of measures of a given metric space $X$. Except for non-negativity and symmetry (which are obvious), I don't know how to proceed.
Do you guys have any advices or l... |
H: Four color theorem, 3-regular planar graph, Hamiltonian path and spiral chains
Studying the four color problem, I was analyzing all possible 3-regular planar graphs of 12 faces, with the additional restriction that graphs that have one or more faces with less than 5 edges, are not to be considered.
Note: It counts ... |
H: Prove that: $2^a+3^b<3a+4b$
Let be $a, b$ in $(0,1)$ such that $a+b>1$. I need to prove that:
$$2^a+3^b<3a+4b$$
I'm looking for an elementary proof that doesn't resort to the calculus tools.
AI: From the graph of the function $f(x)=2^x$ we see that on interval $(0,1)$ it is bellow the line $y=x+1$ joining the point... |
H: A question regarding the algebraic closure of a field
I have slight problems understanding a thing about algebraic closures of fields. It seems to me that any algebraic closure $C$ of a field $K$ is a Galois extension, but I read that this is not true. Following are the definitions I use:
An extension $F/K$ is Gal... |
H: Puzzle: concatenation is three times product
The numbers A and B have three digits, while C is an odd number with 5 digits. Say you were asked to calculate the integer $(A B) /C$. Instead though you put A and B next to each
other to form a 6-digit number D, and then divided by C. Your answer is now three
times t... |
H: Example of an application of a theorem about ideals in rings of fractions in Atiyah-MacDonald
In Atiyah-MacDonald, we have the following theorem (p. 41):
Proposition 3.11.
i) Every ideal in $S^{-1}R$ is an extended ideal.
ii) If $I$ is an ideal in $R$ then $I^{ec} = \bigcup_{s \in S} (I : \langle s \rangle )$. Hen... |
H: A wrong reasoning about conditional probability
Two of three prisoner A, B and C will be executed, A asks the name of one other than A himself who will be executed. Jailer says that it is B. Merely by asking the question, A reduced the probability that he will be executed from 2/3 to 1/2, regardless of which answer... |
H: Summation of a series.
I encountered this problem in Physics before i knew about a thing called Taylor Polynomials My problem was that i had to sum this series :
$$\sum^\infty_{n=1}\frac{(-1)^{n+1}}{n}$$
basically $$1,-\frac{1}{2},\frac{1}{3},-\frac{1}{4},\frac{1}{5},-\frac{1}{6},\frac{1}{7}.....$$
So now i know th... |
H: Every absolutely continuous function with integrable derivative tends to zero at infinity
I am given $f,f' \in L^1(\mathbb{R})$, and f is absolute continuous, I want to show that:
$$\lim_{|x|\rightarrow \infty} f(x)= 0$$
Not sure how to show this, I know that $f(x)=\int_0^x f'(t) \, dt+f(0)$, and I can assume witho... |
H: Proving a language is regular
I know to prove a language is regular, drawing NFA/DFA that satisfies it is a decent way. But what to do in cases like
$$
L=\{ww \mid w \text{ belongs to } \{a,b\}*\}
$$
where we need to find it it is regular or not. Pumping lemma can be used for irregularity but how to justify in a ca... |
H: Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?
Is there a relationship between the Compactness Theorem and the upward Lowenheim-Skolem Theorem in FOL?
I was thinking of another post of mine "Why accept the axiom of infinity?" when I though, "Well, if someone ... |
H: does there exist an analytic function such that
If $f$ is analytic in a nbd $\Delta_{\delta}$ of $0$ and $f(z)=-f(-z)\forall z\in\Delta_{\delta} $ Then there exist an analytic function $g\in \Delta_{\delta}$ such that $f(z)=zg(z^2)\forall z\in \Delta_{\delta}$
AI: Expand $f(x)$ at its taylor series you get
$\sum_{... |
H: Fill in odd combinations in triangle
Can you write a number from 1 to 16 in each of the
triangles, using each number exactly once, such that the
sum of the two numbers in the two cells that share an
edge is always odd?
triangle as such:
http://4.bp.blogspot.com/_PnLYRqe0k9g/SnXtrcvf_nI/AAAAAAAAAKk/RRtqQiVRVqw/s320/... |
H: Why does the Fibonacci Series start with 0, 1?
The Fibonacci Series is based on the principle that the succeeding number is the sum of the previous two numbers. Then how is it logical to start with a 0? Shouldn't it start with 1 directly?
AI: One key number-theoretical reason for starting the sequence $(0,1)$ inste... |
H: The definition of $f(z)$ being analytic at point $\infty$
Consider this function $f(z) = \frac{1}{1+z}$. We can define $f(\infty) = \lim_{z \rightarrow \infty}{f(z)}$, which is zero for this case. Since $f(\frac{1}{t}) \rightarrow \frac{t}{t+1}$ is analytic at point $t=0$, we can deduce that $f(z)$ is analytic at $... |
H: Conditioning series with positive real numbers
I have to prove that if $\sum_{n=1}^{\infty} a_{n}$ is a convergent series with positive real numbers, then $\sum_{n=1}^{\infty} (a_{n})^\frac{n}{n+1}$ converges. I also wonder if the converse is true. Any suggestion, hint will
be very welcome. Thanks.
AI: if $a_n \g... |
H: Creating Unique Values based off Two Sets of Sequential Integers
First off, I apologize if this is the wrong board. I'm a heavy StackOverflow user, and this is technically a programming question (or at least serves programming use), but I find it to be based moreso in math.
I have two sets of sequential integers th... |
H: Can we decide a conjecture is decidable without knowing a conjecture is correct or false?
Can we decide a conjecture is decidable without knowing a conjecture is correct or false?
I asked this question because I assume that the millenium prize problem is already to be decidable, otherwise the mathematician would ne... |
H: A question about Euclidean Domain
This is a problem from Aluffi's book, chapter V 2.17.
"Let $R$ be a Euclidean Domain that is not a field. Prove that there exists a nonzero, nonunit element $c$ in $R$ such that $\forall a \in R$, $\exists q$, $r \in R$ with $a = qc + r$, and either $r = 0$ or $r$ a unit."
Ok, I ... |
H: What is the result of sum $\sum\limits_{i=0}^n 2^i$
Possible Duplicate:
the sum of powers of $2$ between $2^0$ and $2^n$
What is the result of
$$2^0 + 2^1 + 2^2 + \cdots + 2^{n-1} + 2^n\ ?$$
Is there a formula on this? and how to prove the formula?
(It is actually to compute the time complexity of a Fibonacci r... |
H: Find volume of region bound by $y=x, y=x^2$ around x-axis
Here is the problem in my textbook:
Find the volume of the solid obtained by rotating the region bounded by
the curves $y=x, y=x^2$ about x-axis.
Here is my solution :
Because equation $x = x^2$ has two roots : $0$ and $1$. we have:
$$ V= \int_0^1{2\pi... |
H: Determining eigenvalues and eigenvectors from a symmetric matrix
Let $A \in \mathbb{R}^{N \times N}$ be symmetric.
a) The respective Eigenvalue $\lambda$ to an approximately defined
Eigenvector $0 \neq x \mathbb \in {R}^n$ from $A$ has to be
calculated, so that $||Ax-\lambda x||^2_2$ is minimal. Specify a
fo... |
H: Recurrence telescoping $T(n) = T(n-1) + 1/n$ and $T(n) = T(n-1) + \log n$
I am trying to solve the following recurrence relations using telescoping. How would I go about doing it?
$T(n) = T(n-1) + 1/n$
$T(n) = T(n-1) + \log n$
thanks
AI: In the first case you have $T(n) - T(n-1) = \frac{1}{n}$. If you sum with $n... |
H: Simple question regarding continuous functions on $\mathbb{Q}$
I am a self-studying masochist and I came across an interesting example (to me), that I think will help me with further results. I have little experience writing rigorous proofs, so any explicit help is appreciated.
What I am trying to prove is the foll... |
H: How to calculate this Frechet derivative
Suppose $F:C^1(\Omega, [0,T]) \to C^1(\Omega, [0,T])$ with $$F(u) = u_t - f(x, t, u, u_x).$$
How do I calculate the Frechet derivative of $F$ at the point $w = f(x,t, 0, 0)t$?
It should be $$F'(w, v) = v_t - \frac{\partial f}{\partial z}\bigg|_{w}v - \frac{\partial f}{\parti... |
H: Proving combinator identity KMN=M
Have a problem proving K MN=M
By the K combinator definition
$ (\lambda x y.x) M N $
Parenthesized
$ ((\lambda x. (\lambda y.x)) M) N $
By the principal axiom of lambda calculus
$ (\lambda y.M) N $
Second application of the principal axiom
$ M[y:= N] $ ?
This would give correct r... |
H: Pointwise convergence of $1_{(a,b]}$ by a sequence of functions
I am trying to work through an exercise (on my own) out of Resnick's A Probability Path.
One question states the following:
Suppose $-\infty<a\le b<\infty$ and assume we have an indicator function of the form $1_{(a,b]}(x)$. Can this function be approx... |
H: Inverse Laplace transform with partial fraction
I have the transform below:
$$\frac{(7s+2)(2s-5)}{{s^2}(s-2)}$$
I think this is should be partial fraction to be solved. Can you please help me figure how to consider A, B, C and denominators?
AI: Before doing partial function expansion, we need to make sure that the ... |
H: Integral of $\int 2\,\sin^{2}{x}\cos{x}\,dx$
I am asked as a part of a question to integrate $$\int 2\,\sin^{2}{x}\cos{x}\,dx$$
I managed to integrate it using integration by inspection:
$$\begin{align}\text{let } y&=\sin^3 x\\
\frac{dy}{dx}&=3\,\sin^2{x}\cos{x}\\
\text{so }\int 2\,\sin^... |
H: upper bounds for binomial
I'm trying to calculate the upper bound of the binomial coefficient:
\begin{equation}
\sum\limits_{j=0}^{k} {n\choose j}<\left( \frac{ne}{k} \right)^k
\end{equation}
Using binomial theorem and for $x\ge0$:
\begin{equation}
\sum\limits_{j=0}^{k} {n\choose j}{x^j}\le(1+x)^n
\end{equation}
di... |
H: Evaluating: $\lim_{n\to\infty} \int_{0}^{\pi} e^x\cos(nx)\space dx$
Evaluate the limit:
$$\lim_{n\to\infty} \int_{0}^{\pi} e^x\cos(nx)\space dx$$
W|A tells that the limit is $0$, but i'm not sure why is that result or if this is the correct result.
AI: Hint: Integrate by parts, letting $u=e^x$ and $dv=\cos nx \,dx$... |
H: Determining if a language is Recursively Enumerable
Here is a problem from John Hopcroft's "Introduction to Automata Theory" that I'm having a hard time trying to understand.
Exercise 9.2.5:
Let L be recursively enumerable and let Overscript[L, _] be non-RE. Consider the > language:
L' = {0w | w is in L} $\cup... |
H: Find Zariski closure of a set
Let $X=\{(x,\sin(x)): x \in \mathbb{A}^{2}\}$. I want to find the closure (with respect Zariski topology) of $X \subseteq \mathbb{A}^{2}$.
OK I've already shown that $X$ is not a closed set. Now consider $cl(X)$ this is a closed subset of $\mathbb{A}^{2}$ so its dimension is $0,1$ or $... |
H: using PDE existence to show a map is invertible
I have an existence/uniqueness theorem for the PDE $$u_t = a(x,t)u_{xx} + b(x,t)u_x + c(x,t)u - g(x,t).$$
Now if I have a Gateaux derivative of a map $F$ at a point $p$ satisfying $$DF(p)v = v_t - f_1v_{xx} - f_2v_x - f_3v$$
(the $f_i$ are functions of $(x,t)$) then h... |
H: Expected number of steps/probability in a Markov Chain?
Can anyone give an example of a Markov Chain and how to calculate the expected number of steps to reach a particular state? Or the probability of reaching a particular state after T transitions?
I ask because they seem like powerful concepts to know but I am h... |
H: convergence of $\alpha$-Hölder-continuous functions
Let $\Omega\subset\mathbb R^n$ be compact and $C^{0,\alpha}(\Omega)$ the space of all $\alpha$-Hölder-continuous functions. Define $||u||_{C^{0,\alpha}(\Omega)}:=||u||_{\sup}+\sup\limits_{{x,y\in \Omega\space\&\space x\ne y}}\frac{|u(x)-u(y)|}{|x-y|^\alpha}$ and c... |
H: Leslie Matrix characteristic polynomial
I´m having problems to prove the Leslie matrix characteristic polynomial.
I have to prove that the characteristic polynomial is:
$$
\ λ^{n}-a_{1}λ^{n-1}-a_{2}b_{1}λ^{n-2}-a_{3}b_{1} b_{2}λ^{n-3} - ... -
a_{n}b_{1} b_{2}...b_{n-1}\
$$
I would apreciate some light!
AI: I'm ass... |
H: Solutions for $P(D)x(t)=0$, $P(x)=(x-x_0)^n$
I am trying to prove the following statement
Let $P(\lambda)=(\lambda-\lambda_{0})^{r}$where $r$ is a positive
integer. Prove that the equation $P(\frac{d}{dt})x(t)=0$ has solutions
$t^{i}e^{\lambda_{0}t},i=0,1,\ldots,r-1$
I thought of three ideas that I am having ... |
H: How do I show this formula involving several variables?
This is from Woll's "Functions of Several Variables," but there's no proof.
If $g$ is of class $C^k$ ($k \ge 2$) on a convex open set $U$ about $p$ in $\mathbb{R}^d$, then for each $q \in U$,
$
g(q) = g(p) + \sum_{i=1}^d \frac{\partial g}{\partial r_i} \bigg|_... |
H: Does Log-Lipschitz regularity imply Hölder continuity?
A function is Log-Lipschitz if there exists a constant $C$ such that
\begin{equation}
|u(x) - u(y)| \le C|x-y| \log|x-y|
\end{equation}
Is a Log-Lipschitz function $C^{0,\alpha}$ for any $\alpha \in (0,1)
$(Hölder continuous)?
If you need, assume hypothesis. Th... |
H: Is this an abuse of set-theoretic notation?
The expression is this:
$$\bigcup_{n\in\mathbb{N}}\ \bigcup_{a_0\in\mathbb{Z}}\cdots\bigcup_{a_n\in\mathbb{Z}}\big\{z\in\mathbb{C}:a_0z^n+a_1z^{n-1}+\cdots+a_{n-1}z+a_n=0\big\}.$$
I hope it's clear what this is meant to denote (the set of algebraic numbers), but I'm uneas... |
H: Proving that $a + b = b + a$ for all $a,b \in\mathbb{R}$
Being interested in the very foundations of mathematics, I'm trying to build a rigorous proof on my own that $a + b = b + a$ for all $\left[a, b\in\mathbb{R}\right] $. Inspired by interesting properties of the complex plane and some researches, I realized tha... |
H: Expectation Values inside absolute value operator
first: are these equality true ?
$$|E[Y]-E[X]|=|E[Y]|-|E[X]|.$$
$$|E[Y]-E[X]|^2=|E[Y]|^2-|E[X]|^2$$
second: what is result of this relation:
$$\sum_{i=1}^{3}p_i.(X_i-\mu)^2=?$$
where the $\mu =\sum_{i=1}^{3}(p_i.X_i)$
AI: "What is the result?" is a bit vague. In so... |
H: Weird qualification about Cauchy derivative-integral formula
It is known that for a holomorphic function $f$ over $\Omega$, if $C$ is a circle inside $\Omega$, then:
$$f^{(n)}(z) = \frac{n!}{2\pi i}\int_C \frac{f(s)}{(s-z)^{n+1}} \, ds$$
A textbook makes the following claim :
$$f:\mathbb{D}\rightarrow\mathbb{C}$$
w... |
H: the solution set of $\left | \frac{2x - 3}{2x + 3} \right |< 1$
what is the solution set of $\left | \frac{2x - 3}{2x + 3} \right |< 1$ ?
I solved it by first assuming: $-1 < \frac{2x - 3}{2x + 3 } < 1$
ended with: $x > 0 > -3/2$
Is that a correct approach?
And how to derive the solution set from the last inequali... |
H: Norms on inner product space over $\mathbb{R}$
Definition of the problem
Let $\left(E,\left\langle \cdot,\cdot\right\rangle \right)$
be an inner product space over $\mathbb{R}$. Prove that for all $x,y\in E$
we have
$$
\left(\left\Vert x\right\Vert +\left\Vert y\right\Vert \right)\left\langle x,y\right\rangle \leq... |
H: Random variables and sigma field
Given $Z_1,Z_2,\cdots$ i.i.d with $E|Z_i|<\infty$. $\theta$ is an independent r.v. with finite mean and $Y_i=Z_i+\theta$.
If we define $F_n=\sigma(Y_1,\cdots,Y_n), F_\infty=\sigma(\cup_n F_n).$ Do we have $\theta\in F_\infty$?
AI: How about: If $E[Z_i] = m$, then
$$
\lim_{n \to \in... |
H: Always a value with uncountably many preimages? (for a continuous real map on the plane)
Let $f$ be a continuous map ${\mathbb R}^2 \to {\mathbb R}$. For $y\in {\mathbb R}$, denote by $P_y$ the preimage set $\lbrace (x_1,x_2) \in {\mathbb R}^2 | f(x_1,x_2)=y \rbrace$.
Is it true that
(1) At least one $P_y$ is uncou... |
H: Is $A^{q+2}=A^2$ in $M_2(\mathbb{Z}/p\mathbb{Z})$?
I'm wondering, why is it that for $q=(p^2-1)(p^2-p)$, that $A^{q+2}=A^2$ for any $A\in M_2(\mathbb{Z}/p\mathbb{Z})$?
It's not hard to see that $GL_2(\mathbb{Z}/p\mathbb{Z})$ has order $(p^2-1)(p^2-p)$, and so $A^q=1$ if $A\in GL_2(\mathbb{Z}/p\mathbb{Z})$, and so t... |
H: What is the probability of two people meeting?
I am trying to figure out a solution to the following problem:
Let there be two groups of people, Group A and Group B. Group A represents x percent (e.g. 1%) of the world's population, and Group B represents y percent (e.g. 2%) of the world's population. What is the p... |
H: Numerically solving 1D Heat
I'm looking at page 10 of http://www4.ncsu.edu/~zhilin/TEACHING/MA402/notes1.pdf
What happened to the boxed term (it doesn't seem to appear in the matrix equation).
(I'm just implementing this and am not familiar with pde/numerical techniques)
AI: Looks like an error in the notes. You d... |
H: How to prove that the tangent to a circle is perpendicular to the radius drawn to the point of contact?
I've tried drawing a parallel chord to the tangent but then how would you prove that the chord is perpendicular to the radius?
AI: Let $O$ be the centre of the circle, let $\ell$ be a tangent line, and let $P$ be... |
H: Definite Integral (Calculus)
this is a revision problem, not a homework problem. Sincere thanks for help.
Question: Evaluate $\int_0^{\pi/2}\frac{\cos x}{\cos x+\sin x}$.
The answer is $\pi/4$, but I am unable to work out the method.
Sincere thanks for help.
AI: Notice that
$$\frac{d}{dx} (\log(\cos x + \sin x)) = ... |
H: Maximum value of the function
Find the minimum value of $f(x) = \max\{x^2-4, x , - 1\}$
I am able to do this question by using graph. But if there is any other method please tell me
AI: You can reason as follows. The $-1$ in $\max\{x^2-4,x,-1\}$ guarantees that $f(x)$ is always at least $-1$, so the minimum value o... |
H: Invert "Gravitational" Force Function or Solve an Intersection
Recall "gravitational"-type force functions, by which I mean anything of the form:
$f(x,y,z) = \frac{k}{((x-x_0)^2+(y-y_0)^2+(z-z_0)^2)^p}, p\in\Re_{>0}, k\in\Re, (x,y,z) \neq(x_0,y_0,z_0)$
(e.g., for gravity, $p=1,k=G m_1 m_2$)
Define a function $g(x,y... |
H: Infinitely many primes are of the form $an+b$, but how about $a^n+b$?
A famous theorem of Dirichlet says that infinitely many primes are of the form:$\alpha n+\beta$, but are there infinitely many of the form: $\alpha ^n+\beta$, where $\beta$ is even and $\alpha$ is prime to $\beta$? or of the form $\alpha!+\gamma$... |
H: Regular covering implies transitive automorphism group action
We already know the theorem
Theorem
Let $p: (Y,y) \rightarrow (X,x)$ be a covering, with $Y$ connected and $X$ locally path connected, and let $p(y) = x$. If $p_*(\pi_1(Y,y))$ is a normal subgroup of $\pi_1(X,x)$, then $\pi_1(X,x)/p_*(\pi_1(Y,y))$ is i... |
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