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H: Irreducible polynomials with integer coefficients over Q
Suppose p(x) is an irreducible polynomial over Q of degree n, with integer coefficients. If p(x) has two roots r1 and r2 satisfying r1r2 = 5, prove that n is even.
Attempt at solution:
Because the base field is Q, the field extension is separable, hence no r... |
H: Can I only apply the Gauss-Hermite routine with an infinite interval or can I transform the interval?
Short version of my question
Can I only apply the Gauss-Hermite routine with an infinite interval or can I transform the interval?
Long version (reason I'm asking)
I am interested in solving integral equations nume... |
H: Help Solving for Probability in a $2$-Player Multi-Round Game with Unequal Odds
Possible Duplicate:
If a player is 50% as good as I am at a game, how many games will it be before she finally wins one game?
Can anyone help me solve the following problem:
Player A and Player B are playing a game with multiple roun... |
H: Latin squares of even order with all cells only participating in one subsquare.
For even ordered Latin squares, we can create squares in which every cell participates in a $2\times 2$ subsquare by using a simple circulant as for $n=6$ below:
0 1 2 3 4 5
1 2 3 4 5 0
2 3 4 5 0 1
3 4 5 0 1 2
4 5 0 1 2 3
5 0 1 2... |
H: Why does $\sum\limits_{n=2}^{\infty} \frac{ 1}{ n^2 \log (n) }$ converge?
The way I see it you can compare
$$\sum_{n=2}^{\infty} \frac{ 1}{ n^2 \log (n) } < 1/n$$
$1/n$ is a $p$-series in which $p = 1 \leq 1 $
So $1/n$ diverges.
Thus $\sum\limits_{n=2}^{\infty} \frac{ 1}{ n^2 \log (n) }$ diverges.
AI: As $\... |
H: Why does gradient descent work?
On Wikipedia, this is the following description of gradient descent:
Gradient descent is based on the observation that if the multivariable function $F(\mathbf{x})$ is defined and differentiable in a neighborhood of a point $\mathbf{a}$, then $F(\mathbf{x})$ decreases fastest if o... |
H: Evaluate an integral of the Airy function $\operatorname{Bi}(x)$
How do I evaluate this interesting integral with the Airy function:
$$\int_0^x \operatorname{Bi}(u)^2 du$$
More generally, how do I evaluate
$$\int_0^x \operatorname{Bi}(u)^n du$$
AI: The first one is easy. We know that $\operatorname{Bi}^{\prime\prim... |
H: Additive primes
The product of two integers is always an integer. However, the quotient of two integers is not always an integer. This simple fact leads directly to concepts such as "divisibility", "divisors" and "factors", and ultimately to the "prime numbers". Famously, every positive integer can be uniquely writ... |
H: Sorting Grid Right to left
All,
We are having a problem with sorting the results of a database query.
The result of the query will be used to calculate discount to a customer. The table is then read from right to left.
However, the problem we are facing, is that we are unable to find a proper sort method to sort ... |
H: (ZF) Dedekind infinite + Limit Point Compact ⇒ Separable
If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF)
Yesterday, i posted this question and got an answer that 'Limit Point Compact⇒Separable' is unprovable in ZF.
As you can see, the proof in the link is done by the ... |
H: Division is equal to zero
I have this:
$f(x) = 0$
where
$f(x) := \cfrac{3x^2 - 5x + 2}{x + 2}$
How do I solve that?
Do I multiply by $(x + 2)$ and solve $3x^2 - 5x + 2=0$ or solve $3x^3 + x^2 - 8x + 4=0$ with Horner method?
AI: In plain words, if $a/b$ is defined and equal to $0$, then $a=0$. Of course, this does i... |
H: Two combinatorics problems
I have two problems I can't cope with:
Problem 1. How many are ways to divide $n$-convex polygon into triangles using non-intersecting diagonals?
Problem 2. We have $3n$ different balls and $n$ different boxes. How many are ways to put all balls in boxes and in every box there is at leas... |
H: The graph of xy = 1 is connected or not
The graph of $xy = 1$ in $\Bbb C^2$ is connected. True or false?
I know that it is not connected in $\Bbb R^2$, but what is the case of $\Bbb C^2$?
AI: The map $f:\Bbb C\setminus\{0\}\to\Bbb C^2$ given by $f(z)=(z,1/z)$ is continuous and maps $\Bbb C\setminus \{0\}$ onto the ... |
H: Is the limit of uniformly integrable functions integrable?
If $\left\{f_n\right\}$ are uniformly integrable and $f_n\overset{a.e.}{\rightarrow}f$ ($f$ measurable), is $f$ integrable? Can "uniformly integrable" be weakened to "integrable"?
AI: Yes
$\int \left|f\right|\overset{\mathrm{a}}{\leq}\liminf\int |f_n|\overs... |
H: How to prove that the $L^p$ spaces are infinite dimensional
It is well-known that (given a measure space $(S,\mathcal A,\mu)$ and $1\le p\le\infty$) the Banach space $L^p(S,\mathcal A,\mu)$ has infinite dimension.
Is there an easy way to proof this statement (or a suitable reference (preferably a book) where I can ... |
H: When does (Riemann) regularization work?
I've seen the Wikipedia articles on how to sum $1+1+1+1+\cdots=-1/2$ or $1+2+3+4+\cdots=-1/12$.
Is there a theory behind it or is it a random trick? It basically uses analytic continuation of the Riemann zeta?
The article says it may be used in physical applications. So I w... |
H: Integer Partition by Counting Repetition : Conjecture ??
I would like to find informations regarding this way of doing Integer Partitions or
this conjecture,
Suppose you have all the ordered partitions of 5:
5
4 1
3 2
2 2 1
3 1 1
2 1 1 1
1 1 1 1 1
Then extend those partitions to 5 digits by adding 0:
5 0 0 0 0
4 ... |
H: Subtract matrix from scalar
Is this even possible? Since you can subtract on the right-hand side I think there must be a way to do it from left-hand side too.
I would like to calculate this:
3 - [2 1] = ??
AI: You can't actually add a scalar and a matrix. In general, you can't add two matrices unless they are of t... |
H: Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$, but cannot be mapped onto $D(2^\omega)$?
Possible Duplicate:
Two questions on Sorgenfrey line
This is an exercise from Engelking's Book.
Verify that the Sorgenfrey line can be mapped onto $D(\aleph_0)$, but cannot be mapped onto $D(2^\omega)$?
I d... |
H: Show that the Sorgenfrey line and the Niemytzki plane are not homeomorphic
This is an exercise from a topological book.
Show that the Sorgenfrey line and the Niemytzki plane are not homeomorphic.
Thanks for any help.
AI: HINT: The Niemycki plane has a closed, discrete set of cardinality $2^\omega$. Does the Sorge... |
H: A Question about the strong maximum principle in Evans Partial differential equation
Evans stated the strong maximum principle as follows: $U\subset\mathbb{R}^n$ a bounded and open set. If $u\in C^2(U)\cap C(\overline{U})$ is harmonic within $U$.
Then,
$\max_{\overline{U}}u=\max_{\partial U}u$
if $U$ is in additio... |
H: If a number can be expressed as a product of n unique primes............
If a number can be expressed as a product of n unique primes, in how many ways can the number be expressed as a difference of two squares?
AI: Write $m = p_1 \times \ldots \times p_n$ and for now assume each $p_i$ is odd. Then each factorizat... |
H: Frequency of twin primes
Is there a function that gives the frequency with which twin primes less than a particular number, N (assuming N is not a part of a twin prime) occur?
I've tried with a program but I did not notice any pattern........
:
int pc(long double n) \\pc:prime check
{
long double i=2, f=0;
... |
H: Eventually periodic point and homeomorphism.
The point $x$ is a periodic point of period $n$ if $f^n(x)=x$. The least positive $n$ for which $f^n(x)=x$ is called the prime period of $x$.
A point $x$ is eventually periodic of period $n$ if $x$ is not periodic but there exists $m>0$ such that $f^{n+i}(x)=f^i (x)$ for... |
H: Newton's method - error bounds
I just have a very brief question regarding the formula for error bounds in Newton's method. Depending on where you look, this will either be written as:
$$e_{n+1} \approx \frac{f^{\prime \prime}(r)}{2 f^{\prime}(r)}e_{n}^2$$
or:
$$e_{n+1} \approx -\frac{f^{\prime \prime}(r)}{2 f^{\p... |
H: A problem on indefinite integral: $\int(\cos x)^m\sin(nx) \mathrm dx$
If
$$I(m,n)=\int(\cos x)^m\sin(nx) ~\mathrm dx,$$
How do I get $7I(4,3)-4I(3,2)$?
AI: Not the general answer, but for the specific case
$$
7I(4,3)-4I(3,2)=\int(7\cos^4x\sin 3x-4\cos^3x\sin 2x)dx
$$
using the known formulae
$$
\sin2x=2\sin x\cos ... |
H: Converting imperial-based equations to metric
I have the following equation:
$\frac{n\times2.2046}2\times0.0284130625$
to: convert kg to lbs, divide by $2$, then convert the result from oz to L
is there a shortcut to do this without the conversion to imperial?
yeah, I know this is probably a very elementary questio... |
H: Cardinality of cartesian square
Given an infinite set $A$ - does the cardinality of $A$ equal to the cardinality of $A^2$?
AI: Zermelo proved that every well-ordered infinite set has this property, so if we assume the axiom of choice then the answer is yes. In fact the axiom of choice is equivalent to the assertio... |
H: Conditions under which function is continuous
I have the following graph:
$$h(t) = \begin{cases} 2t+1, & \mathrm{if}\ t \le -1, \\
3t, & \mathrm{if}\ -1 < t < 1, \\
2t-1, & \mathrm{if}\ t \ge 1.\end{cases}$$
The question I have to answer is: Give the conditions which would make the function $h$ continuous at the po... |
H: Two problems: When a continuous bijection is a homeomorphism? Possible cardinalities of Hamel bases?
Let $X$ and $Y$ be topological spaces and let $f : X\rightarrow Y$ be a continuous
bijection. Under which of the following conditions will $f$ be a homeomorphism?
(a) $X$ and $Y$ are complete metric spaces.
(b) $X$... |
H: Understanding an Outer Automorphism of $S_6$
In an article (paper), there is a description of an outer automorphism of $S_6$. There are six pentagons, arranged with a rule, with vertices $1,2,3,4,5$. Any permutation of these vertices will permute the six pentagons, hence giving a map (homomorphism) from $S_5$ into ... |
H: Subspaces of separable normed spaces
Let $X$ be a separable normed space. Is it true that every subspace is separable?
If it was Hilbert space I would take the dense set and then their projections.
It sounds trivial but I cannot prove or disprove it...
Thank you.
AI: Let $X$ be a separable metric space and $x_n$ a... |
H: Easy way to find roots of the form $qi$ of a polynomial
Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem).
Is there an easy way to check if $p$ have any roots of the form $qi$ where $q\in\mathbb{Q}$ (or at least $q\in\m... |
H: Prove that a finite set of points $z_1,z_2,......,z_n$ cannot have any accumulation points.
How can I prove a finite set of points $z_1,z_2,......,z_n$ on the complex plane cannot have any accumulation points.please give me some hints.
AI: Hint: Consider $S = \bigl\{z_1,\ldots,z_n\bigr\}$, and take any element $x \... |
H: What is the reflection in linear algebra
as we know projection $A/B = AB^t(AB^t)\cdots B$
how about reflection? do it have orthogonal reflection or oblique reflection?
what is the reflection in linear algebra
Reflection $= 2(A/B) - A$ where $(A/B)$ is above equation ?
AI: A reflection is an involution on a vector ... |
H: What will be the minimum value of $\frac{p^2}{\tan9^\circ} + \frac{q^2}{\tan27^\circ} + \frac{r^2}{\tan63^\circ} + \frac{s^2}{\tan81^\circ}$?
What will be the minimum value of
$$\frac{p^2}{\tan9^\circ} + \frac{q^2}{\tan27^\circ} + \frac{r^2}{\tan63^\circ} + \frac{s^2}{\tan81^\circ}$$ if
$$p+q+r+s=5$$ where $p, q, ... |
H: Drawing mesh figure in MATLAB
I want to draw a figure of the mesh which the interval (0,1) ared divided into 8 equalsize subintervals. Which comment do I need to use ? The points needs to be on the natural straiht line. Thanks
AI: L = 0:1/8:1;
plot(L,zeros(size(L)),'.','MarkerSize',14);
hold on
plot([0 1],[0 0],'k'... |
H: Markov property w.r.t. a countable state space
Background
Let $\left(X_t\right)_{t \in I}$ ($I\subseteq\mathbb R$) be an $E$-valued stochastic process ($E$ being a Polish space with the Borel $\sigma$-algebra $\mathcal{B}\left(E\right)$) equipped with the filtration generated by $X$, $\left(\mathcal F_t\right)_{t\i... |
H: Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$
I'm having trouble computing the integral:
$$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx.$$
I hope that it can be expressed in terms of elementary functions. I've tried simple substitutions such as $u=\sin(x)$ and $u=\cos(x)$, but it was not very effective... |
H: $x^n+y^n=z^n$ where $x,y,z$ are real numbers
Problem:
If $$ x^n+y^n=z^n$$ where $x,y $ and $ z $ are real numbers $\gt 0$ and n is any integer $ \neq 0$ then $$ \frac {x}{n} + y >z$$ assuming $x \lt y $
Background: While studying Fermat's theorem I saw that the following are the characteristics of $x,y$ and $z$ $... |
H: Pick out the true statements complex analysis
Pick out the true statements:
(a) There exists an analytic function $f$ on $\mathbb{C}$ such that $f(2i) = 0$, $f(0) = 2i$
and $|f(z)|\le 2$ for all $z\in\mathbb{C}$
.
(b) There exists an analytic function $f$ in the open unit disc $\{z\in\mathbb{C} : |z| < 1\}$
such th... |
H: Find $f(x)$ from $f(3x + 1)$
The problem that I have to solve is:
If the following function is valid for every value of $x$
$$f(3x + 1) = 9x^2 + 3x$$
find the function $f(x)$ and prove that for every $x\in\mathbb R$ the following is valid:
$$f(2x) - 4f(x) = 2x$$
AI: Here $$f(3x+1)=3x(3x+1)=((3x+1)-1)(3x+1)$$ $$\i... |
H: Implicit function theorem
Suppose I have the curve $\gamma: \mathbb{R} \rightarrow \mathbb{R}^2$ given by $\gamma: t \mapsto (\gamma_1(t),\gamma_2(t)) =(t^2,t)$. If I want to apply the implicit function theorem to this to see if $\gamma_1$ can be expressed in $\gamma_2$ at $t=0$ then I need to show that $d \gamma_... |
H: How to get the aspect ratio of an image?
I have an image that is:
320 original width
407 original height
I want to let users resize the image via a form I am building on a webpage. They can adjust either the width or height. When they do the other dimension should auto adjust to maintain the aspect ratio.
When a u... |
H: How can I prove that two remainders are equivalent?
I have the following mathematical statement that I believe to be true, but I need to prove that it is:
(1) rem(x·y, z) = rem(x·rem(y, z), z)
Using the definition that:
rem(a, b) = a - b·div(a, b)
I can show that this is equivalent to proving:
(2) div(x·y, z) = x... |
H: Find the Jordan Canonical Form from c(x) and m(x)
Given the matrix B:
$$
\begin{pmatrix}
2 & 1 & -2 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
1 & 0 & 0 & 0 \\
\end{pmatrix}
$$
I have found that the characteristic polynomial is:
$
c(x)=(x-1)^{3}(x+1)
$
and then found that the minimal polynomial is:
$
m(x)=c(x)
$
so t... |
H: Absolute continuity with respect to a Rajchman measure
A measure $\sigma$ on $\mathbb{T}$ (the unit circle in the complex plain) is called a Rajchman measure if $ \hat{\sigma}(n)\rightarrow0$ as $|n| \rightarrow \infty$.
I want to prove that if $\sigma\ll\mu$ and $\mu$ is a Rajchman measure on $\mathbb{T}$, the... |
H: In which topological spaces is every singleton set a zero set?
The title question says it all: if $X$ is a topological space, then a subset $Z$ of $X$ is a zero set if there is a continuous function $f: X \rightarrow \mathbb{R}$ with $Z = f^{-1}(0)$.
Now I know the following:
Every zero set is a closed subset.
Eve... |
H: Conjugate of exponential imaginary number
The conjugate of $e^{-iwt}$ is $e^{iwt}$.
Then, what would be the conjugate of $e^{iwt}$? Would it be $e^{-iwt}$?
Also, for $|e^{iwt}|^2$, what would the value look like?
AI: Complex conjugation is an automorphism of order 2, meaning $\,\overline{\overline z}=z\,\,,\,\,\for... |
H: When to read of the degree of a variety from its defining polynomials
The question concerns algebraic varieties.
I just read the question
The degree of an algebraic curve in higher dimensions
and great answer by user M P. One of the thing he says is that if a curve in $\mathbb{P}^n$ is given by $n-1$ equations (whi... |
H: Identifying which "plain text" is in English
I need to identify which text is in English from a list of possible list of plain texts generated from a brute force attack on a cipher text. I am thinking of using frequency distributions of English letters such as this one from Wikipedia. I can easily generate the freq... |
H: Automorphism groups of real clifford algebras
I'm sure someone has already worked-out what all the relevant groups really are; my question is about how signature duality interacts with these groups.
So, by an awful calculation, and choosing a convenient convention,
$$ Cl_{3,1} \simeq M_{4\times 4} (\mathbb{R}) $$
a... |
H: Surface of genus g is not homotopy equivalent to wedge of cell complexes both with non-trival H_1
I came across this question while studying for a qualifying exam:
Prove that a closed orientable surface of genus $g \ge 1$ is not homotopy equivalent to the wedge $X \vee Y$ of two finite cell complexes both of which ... |
H: Calculating a Multivariable derivative.
I'm trying to work through Spivak's Calculus on Manifolds and I've arrived at Differentiation. While I can usually follow his steps, I find myself lost or stuck when I try to do something on my own. So I decided to work through one of his first examples using $Df$ notation ... |
H: Counting Weak Compositions and Approximating Alternating Sum
I have the following problem:
"Suppose you have a universe of $N$ distinct objects, and you observe $k$ of them, possibly with repetition. The order in which the objects are observed does not matter. $k < \frac{1}{10}N \ll 2^{128}$. What is the expected n... |
H: Functions that generate "easy" matrices of full rank
While explaining how to invert matrices I once used this ill-fated example
$A=\begin{pmatrix} 1&2&3\\4&5&6 \\7&8&9 \end{pmatrix}$ which can not be inverted ($\det(A)=0$). That got me thinking, given a matrix of size $N$, what are some good functions that map to ... |
H: Matrices with elements that are a distinct set of prime numbers: always invertible?
Inspired by a previous question, given a square non-symmetric matrix whose elements are all prime but distinct from each other, does this guarantee that the matrix is invertible? It's easy to see $N=2$ this holds, a counter-example ... |
H: Subgroup criterion.
I've been reading some stuff about algebra in my free time, and I think I understand most of the stuff but I'm having trouble with the exercises. Specifically, the following:
Prove that a nonempty subset $H$ of a group $G$ is a subgroup if for
all $x, y \in H$, the element $xy^{-1}$ is also i... |
H: Range of the solutions for $\sqrt{x} + \sqrt{x+16} = 3$
The given equation is
$$ \sqrt{x} + \sqrt{x+16} = 3$$
What is the range of the solution?
AI: Clearly existence of $\sqrt x\implies x\geq 0$ for $x\in \Bbb R$.Thus, $\sqrt {x+16}$ is atleast $4\implies \sqrt x+\sqrt {x+16}\geq 4$ for $x\in \Bbb R\implies$ no r... |
H: Drunk person walking in 1D desert
Given $f(x)$, a strictly positive monotonically decreasing sequence, converging to 0.
How to check whether a one dimensional random walk with stepsize $f(n)$ in random direction at the $n$-th step, will travel arbitrarily far from the origin with probability 1?
Clearly $\sum_n f(n)... |
H: My proof that a harmonic series diverges..
Suppose $\sum_{n=1}^\infty \frac{1}{n} = S$ where $S$ is finite. Then
$$S =\sum_{n=1}^\infty \frac{1}{n}= \sum_{n=1}^\infty \frac{1}{2n-1} + \frac{1}{2n} > \sum_{n=1}^\infty \frac{1}{2n} + \frac{1}{2n} = S$$
which is a contradiction. Is this valid?
AI: One way to express... |
H: The height of a principal prime ideal
A formal consequence of Krull's principal ideal theorem is the following:
If $A$ is a Noetherian ring, and $I$ is an ideal generated by $r$ elements, then any prime ideal which is minimal among those that contain $I$ has height at most $r$.
This statement implies that for Noeth... |
H: Solovay Randomness
Say that an $x\in 2^{\omega}$ is Solovay random if for all computably enumerable collections of intervals $\{I_n\}$ such that $\sum_n\mu(I_n)<\infty$, then $x\in I_n$ for at most finitely many $n$.
Also say an $x\in 2^{\omega}$ is Martin-Lof random if for all computable collections $\{U_n\}$ of c... |
H: Is failing to admit an axiom equivalent to proof when the axiom is false?
Often, mathematicians wish to develop proofs without admitting certain axioms (e.g. the axiom of choice).
If a statement can be proven without admitting that axiom, does that mean the statement is also true when the axiom is considered to be ... |
H: Plane vertices from normals and centroid
I am trying to visualize a best fit plane for a set of points which is defined by the normals and the centroid.
I have to find the boundary vertices of the plane given the extent of the plane. Is there any easy algorithm which can help in doing so?
AI: Find the convex hull ... |
H: Varieties as schemes
Some questions about schemes and varieties, one really basic. I follow the definitions as given in Hartshorne.
Firstly, my main question. I understood that Grothendiecks introduction of schemes revolutionized the subject. Just out of curiosity, could you give me some examples of theorems of tec... |
H: Prove : $\frac{\cos(x_1) +\cos(x_2) +\cdots+\cos(x_{10})}{\sin(x_1) +\sin(x_2) +\cdots+\sin(x_{10})} \ge 3$
If we assume that: $0\le x_1,x_2,\ldots,x_{10}\le\frac{\pi}{2} $ such that:
$$\sin^2 (x_1) +\sin^2 (x_2)+\cdots+\sin^2(x_{10})=1$$
How to prove that:
$$\frac{\cos(x_1) +\cos(x_2) +\cdots+\cos(x_{10})}{\sin(x... |
H: Symmetry, reflexivity and transitivity in set relations
I am really having a difficult time applying the definitions of the above three set relations terms.
For the following problem $R = \{(x,y)|xy \geq 1, x, y \in Z\}$ I have to determine whether the expression is reflexive, symmetric, antisymmetric and/or trans... |
H: Express $x$ in terms of the other variables.
Now, I got the first one by using the like triangles. This is my work, please tell me if I'm right:
$$\frac{t}{h}=\frac{x+t}{r}\Rightarrow x+t=\frac{rt}{h}\Rightarrow x=\frac{rt}{h-t}$$
Now, I figure that $(b)$ must use the same concept (big triangle=little triangle... |
H: Prove $\lim_{x \to +\infty} \frac{f(x)}{(1 + x^2)}\ = 0$ for all $f(x)$ uniformly continuous on $[0, \infty)$.
This question is from a bank of past master's exams. Here is my initial, albeit handwavy, intuition. In essence, I try to show that the uniform continuity condition on $f(x)$ prevents it from growing faste... |
H: Name of the Number series
Is there are name for the number series which has the following pattern
Starts small, rises in the middle and goes down as in
1,2,3,4,3,2,1
To give you a background of why this question came up. I am relearning sorting algorithms and there was one case on quick sort (sorting is based on ... |
H: Interpreting Probability
Given a variable x, with dom(x) = {1,0}
p(x = 1) refers to the probability of variable x to be in state '1'
p(x = 0) refers to the probability of variable x to be in state '0' = 1 - p(x = 1)
What does it mean by p(x)?
Also, given a distribution
p(a,b,c,d,e) = p(d|a,b)p(e|c)p(a)p(b)p(c)
... |
H: Help with understanding definition of Definite Integral
I am having some difficulty in understanding notation/meaning in the definition of definite integral. Can you guys help clarify/correct and fill in the gaps in my understanding.
Following is from wikipedia entry for the Darboux integral which is similar to wha... |
H: Compute: $\lim\limits_{n\to\infty} \frac{x_n}{\ln {n}}$
Let be the sequence $(x_n)_{n\geq0}$, $x_0$ a real number, and defined as follows: $$ x_{n+1} = x_n + e^{-x_n} $$
Compute the limit:
$$\lim_{n\to\infty} \frac{x_n}{\ln {n}}$$
Luckily, I've found another post here with a very similar question. If you know other... |
H: Use the given equalities to derive trigonometric functions.
(A) $\sin(-x)=-\sin x$
(B) $\cos(-x)=\cos x$
(C) $\cos(x+y)=\cos x\cos y-\sin x\sin y$
(D) $\sin(x+y)=\sin x\cos y+\cos x\sin y$
Use these equalities to derive the following important trigonometric functions:
f) $\left|\cos\dfrac{x}{2}\ri... |
H: Direct sum of compact operators
I am having some trouble proving this:
Let $T_1\in H_1$ and $T_2\in H_2$ where $H_1,H_2$ are Hilbert spaces. Let $T=T_1\oplus T_2$ on $H_1\oplus H_2$. I need to show $T$ is compact iff $T_1$ and $T_2$ are compact.
I was able to prove it in the reverse direction, but I am having troub... |
H: What does this notation represent
$${\bf w}_{k}\sim\mathcal{N}(0,{\bf Q})$$
Note that $\bf Q$ is $2\times2$ matrix. So what is ${\bf w}_k$?
This is from Wikipedia- Kalman example application
AI: Multivariate normal distribution |
H: Is inverse mapping theorem true for locally convex spaces
If $X,Y$ are locally convex spaces, and $f:X\rightarrow Y$ is a continuous linear transformation which is bijective, then is the inverse of $f$ continuous as well?
AI: As pointed out by Matt in the comments, the bounded inverse theorem makes use of completen... |
H: Help me to understand the Gaussian blurring
Here is an unknown luminosity function $f(x,y)$ and its integration results:
$$p_{i,j}= \frac{\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy}{\iint\limits_{D_{i,j}} \,dx\,dy}$$
I need to express the result of the following transformation in terms of $p_{i,j}$ and $g(x,y)=\f... |
H: Explanation of why the height of a binary tree $\theta(\lg n)$.
From Heap Sort chapter of Introduction to algorithms :
Since a heap of $n$ elements is based on a complete binary tree , its
height is $\theta(\lg n)$.
I know this is correct but how can this be proved ?
AI: If we write down the series counting the... |
H: Help me to understand the Gaussian blurring (2)
Here is an unknown luminosity function $f(x,y)$ and its integration results:
$$\begin{align*}
p_{i,j} &= \frac{1}{\Delta_{i,j}}\iint\limits_{D_{i,j}} \! f(x,y) \, dx \, dy,\\
\Delta_{i,j} &= \iint\limits_{D_{i,j}} \!dx\,dy\;.
\end{align*}$$
Let's consider the followin... |
H: Balanced tree number of nodes
This should be a simple one but maybe I'm dumb or maybe I'm just tired, but how to prove that
$$n = 1 + 2^1 + 2^2 + \cdots + 2^h$$
is equal to
$$n = 2^{h+1} - 1$$
?
AI: Look here:
http://en.wikipedia.org/wiki/Geometric_series#Sum |
H: Are there n-th roots of differential operators?
In analogy to a Dirac operator, it seems to me that formally, the equation
$$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$
is solved by
$$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$
Is there a theory surronding the $\sqrt[n]{D_y}$-idea?
AI: The short answer is yes, a... |
H: Operator norm of the sum of a finite collection of bounded linear operator
I recently got some difficulty with my homework question. The question is:
Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le 1$.
Suppose that $T_kT_j^\ast = T_k^\ast T_... |
H: What is the meaning of left hand side is divided by 2
$$\begin{align}\sqrt{2} = \frac{a}{b} \\
2a^2 = b^2\end{align}$$
I have the equation above and was told that since the left land side is divided by $2$, $b^2$ is an even number. But to me, the left hand side is times $2$.
AI: I suspect that you were told t... |
H: Minimize collision of bit strings
I'm sorry, if I got the wrong expressions, I'm gonna describe it:
I got bit-strings of n bits with k ones and want to minimize "collision"
The collision count of two strings $a=(a_1,...,a_n), b=(b_1,...,b_n)$ is defined by
$$Coll(a,b):=\sum_{i=1}^n a_i b_i$$
and $a,b$ are said to ... |
H: Extracting a function from a standard
I have a standard and I need create a function to resolve this, for example:
if my $X < 21$ my $Y$ will be $24$ else if my $x < 28$ my $y$ will be $32$ ...
How do I calculate a function for this?
Thanks all.
AI: You can simply define the function in piecewise terms over its dom... |
H: Has any previously unknown result been proven by an automated theorem prover?
The Wikipedia page on automated theorem proving states:
Despite these theoretical limits, in practice, theorem provers can solve many hard problems...
However it is not clear whether these 'hard problems' are new. Do computer-generated ... |
H: O-notation property - sum of the first n powers growth
I read here that in the tenth property:
http://www.cs.auckland.ac.nz/~jmor159/PLDS210/latex/complexity.pdf
The sum of the first $nr^{th}$ powers grows as the $(r+1)^{th}$ power
This is not very intuitive to me, why is that?
Since the base is changing (not the e... |
H: Doubt about series - which series is this?
Possible Duplicate:
Sum of n consecutive numbers
I really can't remember (if I have ever known this): which series is this and how to demonstrate its solution?
$$\sum\limits_{i=1}^n i = \frac{n(n+1)}{2}$$
AI: This is an Arithmetic Series starting from $1$ with differen... |
H: If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)
If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)
I'm struggling with this problem for a week.
I'm talking about this in Metric space.
Here's the part of argument in Rudin PMA p.45;
Let $X$ be lim... |
H: Is there a name for this identity involving series?
I'm working through Spivak's Calculus, and for one of the problems I had to prove the Scharwz inequality. I derived this identity, and was wondering if this had a name, it seemed quite important. It's factoring the first sums into to squared terms:
$$\sum_{i=1}^{n... |
H: Operations on two conditions
Hi this is an extension of my previous question: Combining two equations for two conditions
I was wanted to know if operations which were to be carried out on both conditions could be placed outside the piecewise brackets. i.e. would the following be valid
$$y = \sum\limits_{i=1}^3 A.\b... |
H: How many positive values of $a$ are possible in $2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$
How many positive values of $a$ are possible in the following case?
$$2^{3}\le a\lfloor a\rfloor \le 4^{2} + 1$$
where $a\lfloor a\rfloor$ such that $a[a]$ is an integer.
AI: It is clear that $\lfloor a\rfloor$ must be $\ge 3... |
H: The longest word in Weyl group and positive roots.
How to write down a reduced decomposition of the longest word in a Weyl group? For example, how to write down a reduced decomposition of the longest word in type B3 Weyl group? For a decomposition of the longest word, how can we write down an ordering of positive r... |
H: $4$ way heat distribution multiplier problem
I'm making a simple heat distribution program. It's a $2D$ matrix with cells holding heat value. Every iteration looks for cells near current which have lower heat value and gives them some of its heat. Each cell gets different amount of heat (based on heat difference an... |
H: Factor Rings in Commutative Rings
Let $R$ be a commutative ring, suppose that $I$ and $J$ are ideals of $R$. Suppose that $R/I\cong S_1$ and $R/J\cong S_2$. It is true that if $S_1\subset S_2$ if then $J\subset I$?
AI: Consider the following. Let $k$ be a field. For all $a\in k$, we have
$$k[X]/(X-a)\cong k.$$
How... |
H: Product of all primes less then x
How to prove that $\prod_{p\leqslant{x}}p\leqslant4^{x-1},\ \forall x\geqslant2$,
where product is taken over all prime numbers $p\leqslant{x}$
AI: One proof of this fact is the proof of Claim 2.2 of these notes on Erdos' proof of Bertrand's postulate by David Galvin. |
H: Inequality $a+b+c \geqslant abc +2$
Assuming $a,b,c \in (0, \infty)$, we need to prove that:
$$a+b+c \geqslant a b c+2 \quad \text{if} \quad ab+bc+ca=3$$
Can you give me an idea, please? This inequality seem to be known, but I didn't manage to solve it.
AI: Let's start by applying AM-GM:
$$\frac{ab+bc+ca}{3}=1 \g... |
H: A question about a metric space
I have a question regarding the following problem.
Let $A,B,C$ be the three independently selected, uniformly distributed points on the unit sphere $S^3$ in $\Bbb R^4$. What's the probability that the unique chordal triangle $ABC$ is acute?
I just started analysis, and I have litt... |
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