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H: Selfadjoint compact operator with finite trace
I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$.
Can we conclude that $T$ is trace class?
AI: No, we cannot conclude that the operator is trace class.
For exa... |
H: Closure of image of diagonal morphism of S-scheme
Let $X$ be an $S$-scheme with structural morphism given by $f : X \to S$. The image of the diagonal morphism $\Delta : X \to X \times_S X$ is contained in the subset $Z := \{ z \in X \times_S X : p(z) = q(z) \} \subset X \times_S X$ where $p, q$ are the projection ... |
H: Continuous variables and proportionality
I have another problem in a statistics past paper that goes as follows:
Let $X$ be a continuous random variable, taking values in the range $[0,1]$ with pdf given up to proportionality by $f(x) \sim x^4$, $0 \le x \le 1$. What is the value of $E(X)$?
Normally $E(X)$ should... |
H: Zeros of a complex polynomial
The question is:
Show that $$ P(z) = z^4 + 2z^3 + 3z^2 + z +2$$ has exactly one root in each quadrant of the complex plane.
My initial thought was to use Rouche's Theorem (since that's generally what I use to find how many roots a complex polynomial has), but the more I think about it... |
H: Pointwise order of the Cartesian product of two preordered chains
Definitions: (From Categories for Types by Roy L. Crole.)
A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive.
A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot result, ... |
H: Triangulate rectangular parallelepiped in $\mathbb{R}^{n}$
I need to triangulate the n-dimensioned rectangular parallelepiped in $\mathbb{R}^{n}$ into a set of $n$-simplices.
Could you suggest me any known algorithm for that or maybe an extension of Delaunay triangulation for $n > 2$ cases?
AI: The construction out... |
H: Use a Jacobian matrix to differentiate between linear and non-linear transormations
When determining whether or not a map/transformation is linear or non-linear, how can the Jacobian matrix be used? A linear equation in two variables is one that may be written in the form y = ax + b, but how do know if it is non-li... |
H: Does multiplying by $dt$ have any meaning?
Consider, for example, the equation $x'=x$, then it is usually solved
by writing $\frac{dx}{dt}=x\implies\frac{dx}{x}=dt\implies\int\frac{dx}{x}=\int dt$
...
I know that there is a theorem in ODE that justify $x'=x\implies\int\frac{dx}{x}=\int dt$
,but my question is about... |
H: What is the Psi(x) variable binding operator?
In the Free Variable article on Wikipedia, it lists these:
as variable-binding operators. I have seen all of them during my math studies, except for the psi operator. What does $\psi x$ mean in this context?
AI: I don't know what was intended by $\psi$ here, but some W... |
H: What's the equivalent of the adjacency relation for a directed graph?
I've found several sources describing a relation notated $\sim$ signifying adjacency in an undirected graph, but nothing explicitly describing an equivalent for a directed graph. I've been using $\overset{\mathit{member}}{\longrightarrow}$, mainl... |
H: Am I allowed to move around an operator like this?
Can I take this product:
$$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$
And factor out one of the $L$'s to get:
$$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$
Where the operator $\frac{d}{dt}$ now operates on $\frac{d L}{d \dot{x}}$?
Is this allowed?
Thanks
AI: This... |
H: Finding a pair of elements to satisfy an inequation
Let $F$ be a field of characteristic 2 with more than 2 elements. Show that there are elements $a$ and $b$ in $F$ such that $(a+b)^3 \not= a^3 + b^3$.
$F$ couldn't possibly have less than 2 elements, and if it had exactly 2 — that is, $F = \mathbb Z_2$ —, $(a+b)^... |
H: Evaluating $\int_{-1}^1\sqrt{1-x}\,dx$
Evaluate using the Fundamental Theorem of Calculus:
$$\int_{-1}^1 \sqrt{1-x} \, dx$$
I'm not sure what to do with the $\sqrt{1-x}$. Most of the problems I'm doing for this assignment use substitution but I just end up getting $u = 1-x$ and of course $-du = dx$. Where do ... |
H: evaluating $\int_{7}^{10} \frac{xdx}{\sqrt{x-6}}$
Evaluate using the Fundamental Theorem of Calculus:
$$\int_{7}^{10} \frac{xdx}{\sqrt{x-6}}$$
I am stuck with the x on top. I substitute $u = x-6$ and then $du = dx$.
soo...
$$\int_{1}^{4} (u)^{-1/2}xdu$$
is it okay to still have that x there or do I need to sub... |
H: "Base for a neighborhood system at a point" vs. "base at a point"
For a topological space X do the terms "base for a neighbourhood system at a point" and "base at a point" have the same meaning?
AI: Yes. Usually these two things mean the same. |
H: Does $z ^k+ z^{-k}$ belong to $\Bbb Z[z + z^{-1}]$?
Let $z$ be a non-zero element of $\mathbb{C}$.
Does $z^k + z^{-k}$ belong to $\mathbb{Z}[z + z^{-1}]$ for every positive integer $k$?
Motivation:
I came up with this problem from the following question.
Maximal real subfield of $\mathbb{Q}(\zeta )$
AI: Let's go by... |
H: Does this question involve series?
While solving the problem
Half the people on a bus get off at each stop after the first and no one gets on after the first stop.If only one person gets off at stop number 7 .How many people got on at the first stop ?
Here is how I am solving it but I guess I am wrong
Let no o... |
H: Localization and Noetherian property
From page 101 in Atiyah-MacDonald:
"Two of the important properties of localization are that it preserves exactness and the Noetherian property...."
I remember proving that it preserves exactness, it's proposition 3.3. on page 39. But what is meant by "Noetherian property"? Than... |
H: Solving absolute value inequalities.
Prove: If $z,\alpha \in \mathbb{C}$ with $|z|<1, |\alpha|<1$ then $\dfrac{|z|^2+|\alpha|^2}{1+|\alpha z|^2}<1$
AI: Here, $|\alpha|^2+|z|^2-|\alpha z|^2=|\alpha|^2+|z|^2-|\alpha|^2|z|^2-1+1=1-(1-|\alpha|^2)(1-|z|^2)$. Now, Since $|\alpha|\lt 1$ and $|z|\lt 1\implies (1-|\alpha|^2... |
H: Why are $\sin$ and $\cos$ (and perhaps $\tan$) "more important" than their reciprocals?
(My personal "feel" is that $\sin$ and $\cos$ are first-class citizens, $\tan$ is "1.5th-class," and the rest are second-class; I'm sure there are others who feel the same.)
Main question(s): From a purely high-school-geometric/... |
H: Expected value of a continuous random variable: interchanging the order of integration
I have come across a proof of the following in Ross's book on Probability -
For a non-negative continuous random variable Y with a probability density function $f_Y$
$$
\mathrm{E} [Y] = \int_0^\infty P[Y \geq y]dy
$$
The author p... |
H: Intersection of all neighborhoods of zero is a subgroup
Let $G$ be a topological abelian group. Let $H$ be the intersection of all neighborhoods of zero.
How is $H = \mathrm{cl}(\{0\})$? Isn't the closure of a set $A$ the smallest closed set containing $A$ which is the same as the intersection of all closed sets co... |
H: Double Integral Claim
How can I prove the following inequality: (where $f $ is nice enough) -
Given a function $ f(x,y) : \Omega_1 \times \Omega_2 \to \mathbb{R} $ , and $\alpha,C_1,C_2 $ are some constants, ( $\Omega_i$ is equipped with a probability measure $ \mu_i $ respectively) , then there exists a constant $... |
H: Height of triangle inside a parallelogram
I am stumped on the following question
PQRS is a parallelogram and ST=TR. What is the ratio of area of triangle QST to the area of parallelogram (Ans 1:4)
I need the height of the triangle, how would I get that? Any suggestions would be appreciated.
AI: $$A_T=\frac{1}{... |
H: complex torus has topological genus one
could any one give me a hint how to show a complex torus has topological genus one by constructing an explicit homeomorphism to $S^1\times S^1$?
Complex Torus: $\mathbb{C}/L$, where $L=\{\mathbb{Z}\omega_1+\mathbb{Z}\omega_2\}$.
Thank you.
AI: For convenience, I will identify... |
H: $\mathbb{CP}^1$ is compact?
$\mathbb{CP}^1$ is the set of all one dimensional subspaces of $\mathbb{C}^2$, if $(z,w)\in \mathbb{C}^2$ be non zero , then its span is a point in $\mathbb{CP}^1$.let $U_0=\{[z:w]:z\neq 0\}$ and $U_1=\{[z:w]:w\neq 0\}$, $(z,w)\in \mathbb{C}^2$,and $[z:w]=[\lambda z:\lambda w],\lambda\in... |
H: Determining and (dis)proving if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges
I am trying to determine if $ \sum_{n = 1}^{\infty} (-1)^{n + 1} \left( 1 - n \log \left( \frac{n + 1}{n} \right) \right) $ converges using an alternating series test. The test i... |
H: When is the integral of a periodic function periodic?
I'm attempting some questions from Zwiebach - A First Course in String Theory, and have got stuck. I've proved that a function $h'(u)$ is periodic. The question then asks me to show that $h(u)=au+f(u)$ where $a$ is a constant and $f(u)$ a periodic function. I ca... |
H: Cauchy sequences in metric spaces
Let $(X,d)$ be a metric space and let $(x_n)_{n\in\mathbb{N}}$ be a Cauchy sequence in $X$, i.e. $d(x_n,x_m)$ goes to $0$ when $n,m\rightarrow\infty$. The sequence does not necessarily have a limit in $X$, however.
I'm wondering if for fixed $k$, the sequence $d(x_k,x_l)$ has a lim... |
H: What is the unit of the FFT output?
Consider a signal, f(t), with impulse samples taken N times, i.e f[0],f[1],f[2],...f[N-1] Let us perform FFT on it. Now, we have the amplitude on the y-axis and the frequency on the x-axis. I want to know if the unit of the quantity on the y-axis remains the same. If yes, why? If... |
H: Distance between discrete random variables
Let $X_1, \ldots,X_k$ represent k integers between $1$ and $n$ drawn randomly and without replacement (i.e., there are never repeated numbers). What is the PMF of $Y$, the random variable representing the nearest neighbor distance between draws?
This kind of problem falls... |
H: A compact operator in $L^2(\mathbb R)$
Let $g \in L^{\infty}(\mathbb R)$. Consider the operator
$$
\begin{split}
T_g\colon & L^2(\mathbb R)\to L^2(\mathbb R) \\
& f \mapsto gf
\end{split}
$$
Prove that $T_g$ is compact (i.e., the image under $T_g$ of bounded closed sets is compact) if and only if $g=0$ a.e.
... |
H: Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$
$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k) $
$\int\limits_0^{\infty}\int\limits_x^{\infty}\, f(\gamma)\, d\gamma dx=\sum\limits_{k = 2 }^ \infty \frac{1}{k^2} \ln(k) $
$\int\limits_0^{\infty}\int\limits_x^{\infty} f(\g... |
H: Question about $p$-adic numbers and $p$-adic integers
I've been trying to understand what $p$-adic numbers and $p$-adic integers are today. Can you tell me if I have it right? Thanks.
Let $p$ be a prime. Then we define the ring of $p$-adic integers to be
$$ \mathbb Z_p = \{ \sum_{k=m}^\infty a_k p^k \mid m \in \ma... |
H: Should we consider multiplicity while solving this problem?
I am trying to solve the problem :
A single fence is to be constructed from posts 6 inches wide and separated by lengths of chain 5 feet . If a certain fence begins and ends with a post.Which of following could be length of fence in feet ? a)17 b)28 c)35... |
H: Summing Lerch Transcendents
The Lerch transcendent
is given by
$$
\Phi(z, s, \alpha) = \sum_{n=0}^\infty \frac { z^n} {(n+\alpha)^s}.
$$
While computing $\sum_{m=1}^{\infty} \sum_{n=1}^{\infty}
\sum_{p=1}^{\infty}\frac{(-1)^{m+n+p}}{m+n+p}$,
the expression
$$
-\sum_{k=1}^{\infty} \Phi(-1, 1, 1+k)
$$
came up. I... |
H: Generating functions of discrete random variable
I am trying to understand the solution of a problem.
$X_1,X_2,....$ a sequence of independents randoms variables and same probability distribution.
$N$ rv. taking its values in $\mathbf{N}$
Considering $Z=\sum_{i=i}^N X_i$
We have :
$$G_Z(s)=E(s^Z)=E(s^{\sum_{i=0}^N... |
H: Irreducibility of $x^5 -x -1$ by reduction mod 5
Is there a quick way of deducing that $x^5-x-1 \in \mathbb{Z}[x]$ is irreducible by reducing it mod 5, other than verifying that it has no roots in $\mathbb{Z}_5$ and no factorization as the product of a factor of order 2 and a factor of order 3?
AI: Hint $ $ If it h... |
H: A curve that intersects every plane in finitely but arbitrarily many points
Does there exist a piecewise smooth curve in $\mathbb{R}^3$ such that every plane intersects the curve at finitely many points and the number of intersection points can be arbitrary large?
If the number of intersection points for each plane... |
H: Increasing by doubling a number when it is negative
The question is:
Let $x=y-\frac{50}{y}$, where $x$ and $y$ are both greater than $0$. If the value of $y$ is doubled in the equation above, the value of x will a)decrease b)remain same c)increase four fold d)double e)increase to more than double - Ans)e
Here is... |
H: Can a meromorphic function be written as ratio of holomorphic function?
Well, I want to know whether a meromorphic function can be written as ratio of two holomorphic function on $\mathbb{C}$ or on a Riemann surface.
Thank you for help.
AI: a) On a compact Riemann surface $X$ holomorphic functions are constant so ... |
H: $\lim_{n\to\infty}\frac{a_{n-1}}{a_{n}}$
If $\{a_{n}\}_{n\geq 1}$ is a decreasing sequence of real numbers, $a_{n}\in (0,1)$ and $\lim_{n\to \infty} a_{n}=0$. What we can say about $$\lim_{n\to\infty}\frac{a_{n-1}}{a_{n}}$$
AI: Probably not much.
Let $a_n = b^{-n}$ for any constant $b > 1$, so then $a_n \in (0, 1)... |
H: Polynomial Approximation of an Integral
I require a polynomial $p(x)$ such that
$$\left|p(x) - \int_0^x \cos{(t^2)} dt\right| < \frac{1}{10!}$$
for all $x \in [-1, 1]$. I know that I should probably use the fact that if $$m\leq f^{n+1} {(t)} \leq M$$ for $t$ in an interval containing the point $a$, then $$m \frac{... |
H: Calibration of an eye tracking device: transformation from known gaze points
I am creating a calibration system for an eye tracking device. This calibration involves having the user look at five points on a screen. The eye tracker then reports where it believes the user was looking. The result is a map of five co-o... |
H: Probability of Getting a pair of cards
I was wondering what could be the probability of getting at least a pair of cards , when every time you draw 6 cards at random from a fresh deck of cards.
I calculated it as: 3/51 * 48/50 * 44/49 * 40/48 * 36/47 * 15(6C2) = 0.48
Let me know if it is correct?
AI: What's the pro... |
H: Direct products in the category Rel
Please describe direct products in the category Rel.
AI: As Dylan has already mentioned, the category-theoretic product in $\textbf{Rel}$ is the disjoint union of sets. We can verify this by hand:
\begin{align}
\textbf{Rel}(X, Y \amalg Z)
& = \mathscr{P}(X \times (Y \amalg Z)) \... |
H: Conjugate of matrix
Over an arbitrary ring $R$ with unit, is the matrix\begin{pmatrix}
a & 0 & b & 0\\
0 & 0 & 0 &0\\c & 0 & d &0\\ 0 &0&0&0
\end{pmatrix}
conjugate over $GL(R)$ to
\begin{pmatrix}
a & b & 0&0\\
c & d & 0&0\\0 & 0 &0 &0\\0&0&0&0
\end{pmatrix}
?
AI: View the matrices as linear maps to see ... |
H: Having trouble understanding proof of a theorem involving limits of functions and sequences
I found this theorem in a book:
Theorem: Let $F: A \subset \mathbb{R}^n \to \mathbb{R}^m.$ Let $P \in \bar{A}$ and $L \in \mathbb{R}^m$. Then the following assertions
are equivalent:
$ \lim\limits_{X\to P} F(X) = L $
Fo... |
H: Support of the divisor of zeros of a global section of a invertible sheaf
This is related to Hartshorne's book Algebraic Geometry, Lemma II.7.8, page 158.
Let $X$ be a nonsingular projective variety over an algebraically closed field $k$ and let $\mathcal{L}$ be an invertible sheaf on $X$. For any $s \in \Gamma(X, ... |
H: Sizes of Sumsets and Dilates
Let $A$ be a finite subset of an abelian group $G$. For an integer $n$, define the dilate $n \cdot A = \{na \mid a \in A\}$ and, given another $B \subseteq G$, the sumset $A + B = \{a + b \mid a \in A, b \in B\}$.
I would like to show that
$$
\frac{|n \cdot A + (nm) \cdot A|}{|n \cdot ... |
H: Find the integer solution of $ a^b = 2^{2 c + 1} + 2^c + 1 $
Find the possible number of integer solution for this equation, such that $ b>1$
$$ a^b = 2^{2 c + 1} + 2^c + 1 $$
From $1$ to $1000$, $ {a = 2, b = 2, c=0} $ and $ {a = 23, b = 2, c=4} $ computationally. Are there any other possible solutions? How to sho... |
H: How to prove that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer
As the title says, I'm trying to show that $\frac{10^{\frac{2}{3}}-1}{\sqrt{-3}}$ is an algebraic integer.
I suppose there's probably some heavy duty classification theorems that give one line proofs to this but I don't have any of t... |
H: About Perimeter - Circumference of Quarter Circle
ASB is quarter circle. PQRS is a rectangle with side PQ=8 and PS=6 . What is length of ARC AQB ? Ans $5\pi$
Here is how I am solving it:
Radius of Quarter circle = diagonal of rectangle = $\sqrt {100} = 10$
For a Full Circle : $Length_{Arc}=\frac{Arc_{Angle}}{... |
H: What's the analogue of Sierpinski triangle to disk?
What's the (closest) analogue of Sierpinski triangle to disk?
AI: You could replace each disk with seven disks of $1/3$ the radius packed inside it, like this:
After four iterations, it looks like this: |
H: Inverse limit by example
I'm trying to understand inverse limits. For this I am looking at the example (mentioned in Atiyah-Macdonald, page 102): We start with the topological abelian group $G = \mathbb Z$ (endowed with the topology induced by $|\cdot|_p$) and observes that we have an inverse system, that is, a seq... |
H: An inequality in a proof of Kunen's Inconsistency
This may be a silly question, but I keep coming back to it.
Let $j:V\prec M$ be a non-trivial elementary embedding with $M$ a transitive class and $\kappa$ the critical point of $j$. Define the critical sequence for $j$ as usual, setting $\kappa_0=\kappa$ and $\kap... |
H: Which infinite cardinals can be defined using partition relations?
Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then
$\kappa$ is weakly compact if $\kappa$ satisfies $\kappa\to(\kappa)^2_2$
$\kappa$ is Ramsey if $\kappa$ satisfies $\kap... |
H: Finding the closest number to the power of 2 for x.
What is the fastest way to calculate x given y as a large integer?
$y = 100$
$z = 2$
$x = 64$ (Power of z and smaller than or equal to y).
$x = f(100, 2) = 64$
$x = f(128, 2) = 128$
$x = f(90, 3) = 81$
AI: If you want to avoid a loop you may use :
$$\displaystyle ... |
H: How to show that this set is compact in $\ell^2$
Let $(a_n)_{n}\in\ell^2:=\ell^2(\mathbb{R})$ be a fixed sequence. Consider the subspace $$C=\{(x_n)_{n}\in\ell^2 : |x_n|\le a_n\text{ for all }n\in\mathbb{N}\}.$$
According to the book [Dunford and Schwartz, Linear operators part I, page 453] $C$ is compact in the $\... |
H: Computing a complex integral potentially using residues
The question is:
Compute:
$$\mbox{p.v.}\int_{-\infty}^{\infty}\frac{x\sin4x}{{x^2}-1}dx$$
Initially I thought it was straight forward and I could just use residues. However, the Residue Theorem requires the poles to be in the upper plane ($y > 0$), and in this... |
H: The group of roots of unity in an algebraic number field
Is the following proposition true? If yes, how would you prove this?
Proposition.
Let $K$ be an algebraic number field.
The group of roots of unity in $K$ is finite.
In other words, the torsion subgroup of $K^*$ is finite.
Motivation.
Let $A$ be the ring of a... |
H: Limit of a complex valued integral
The question is:
Compute $$\lim_{p\rightarrow0^{+}}\int_{C_p}\frac{e^{3iz}}{z^{2}-1}dz$$
Where $$C_p: z = 1 + pe^{i\theta}$$
My initial thought was to use residues, yet the poles are -1 and 1, so they're on the real line (thus the Residue Theorem does not apply). My next thought w... |
H: Solve system of nonlinear differential equations
I am trying to solve a large system of differential equations. Ideally, I would like to solve it exactly, but if not, can anyone suggest me a numerical method?
In all its generality, the system I am trying to solve is like this: (here, $x = x(t) \in R^n$, and $\dot x... |
H: Gradient And Hessian Of General 2-Norm
Given $f(\mathbf{x}) = \|\mathbf{Ax}\|_2 = (\mathbf{x}^\mathrm{T} \mathbf{A}^\mathrm{T} \mathbf{Ax} )^{1/2}$,
$\nabla f(\mathbf{x}) = \frac {\mathbf{A}^\mathrm{T} \mathbf{Ax}} {\|\mathbf{Ax}\|_2} = \frac {\mathbf{A}^\mathrm{T} \mathbf{Ax}} {(\mathbf{x}^\mathrm{T} \mathbf{A}^\m... |
H: Vectors in Clifford Algebra
I'm studying Clifford Algebra $\mathcal{Cl}_2$ and got stuck in an exercise:
Let $\mathbf{a}=e_2+e_{12},\quad \mathbf{b}=(1/2)(1+e_1).$ Compute $\mathbf{ab}$.
The answer is zero, but I can't get to it and I'm having trouble putting $\mathbf{b}$ in the form $\mathbf{b}=b_1e_1+b_2e_2$.
Imp... |
H: Matrix Multiplication in 3 Dimensions
Possible Duplicate:
Is there a 3-dimensional “matrix” by “matrix” product?
Is matrix multiplication of 3-dimensional matrices defined? I cannot wrap my mind around how it would even work. Equivalently, is matrix multiplication only defined for 2-dimensional matrices?
Edit:... |
H: evaluate the integral: $\int{\frac{8y}{4-y^2}dy}$
$$\int{\frac{8y}{4-y^2}dy}$$
The answer isn't in the back of my book, so I have no way to see if I'm right! (I'm about 99% sure I'm wrong though)
AI: $$\int\frac{8y}{4-y^2}\,dy=-4\int\frac{d(4-y^2)}{4-y^2}=-4\log|4-y^2|+K\,\,(constant)$$ |
H: Area/Arc Length of a Hyperbolic Segment
I'm given a hyperbolic segment, similar to the parabolic segment shown here: http://mathworld.wolfram.com/ParabolicSegment.html
I know the height of the segment ("h" in the wolfram article), and the length of the line segment joining the endpoints of the hyperbola ("2a" in th... |
H: About finite dimensional vector spaces
Would you tell me why the statement below holds?
A vector space $V$ has a basis if and only if $0 < \dim V < \infty.$
AI: The claim is false, whether or not one accepts the Axiom of Choice.
Consider the vector space $V$ (over the reals) of all polynomials $P(x)$ with real co... |
H: evaluate the integral: $\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$
$$\int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dx}$$
I know this is a simple problem, but I don't have the answer for it and I just want to make sure that I'm correct!
AI: $$ \int{(3\csc(x)\cot(x) - 5x^7 +\frac{4}{x} + 3)dy} = y(3\csc(x)\co... |
H: How many elements in the finite field $F_{256}$ satisfy $x^{103}=x$?
How many elements of the finite field $\mathbb{F}_{256}$ with 256 elements satisfy $x^{103}=x$?
AI: We will remember about the solution $x=0$ at the end. For the others, we want to solve $x^{102}=1$. The multiplicative group of non-zero elements ... |
H: evaluate the integral: $\int_0 ^\sqrt5 \frac{4x}{\sqrt{x^2+4}}dx$
$$\int_0 ^\sqrt5 \frac{4x}{\sqrt{x^2+4}}dx$$
I got $4(5^\frac{1}{4})$ but I'm not sure if thats right
AI: Your computation is incorrect.
The substitution $u=x^2+4$ gives $du = 2x\,dx$. When $x=0$, we have $u=4$; when $x=\sqrt{5}$, we get $u=9$. So
$... |
H: Integral inequality - does this look correct?
I have been studying for an upcoming exam and this question was on a previous exam:
Prove that for $1 < p < \infty, \alpha > 1/p$
$$ \int_0^{\infty} x^{-\alpha p} \left| \int_0^x f(t)dt\right|^pdx \leq C_p \int_0^{\infty} |f(x)x^{1-\alpha}|^p dx. $$
So this is what I wr... |
H: Lipschitz Functions
Does uniform convergence on a closed and bounded interval preserve Lipschitz functions?
(Assume that the sequence of functions has a common Lipschitz constant $K$).
AI: It's late at night where I am, so maybe I'm missing something obvious, but....
If $f_n\colon [a,b] \to \mathbb{R}$ each satisfy... |
H: $\sum a_n$ converges absolutely, does $\sum (a_n + \cdots + a_n^n)$ converge
Suppose $\sum_{n=1}^\infty a_n$ converges absolutely. Does this imply that the series $$\sum_{n=1}^\infty (a_n + \cdots + a_n^n)$$ converges?
I believe the answer is yes, but I can't figure out how to prove it. Any help would be apprecia... |
H: The Star Trek Problem in Williams's Book
This problem is from the book Probability with martingales by Williams. It's numbered as Exercise 12.3 on page 236. It can be stated as follows:
The control system on the starship has gone wonky. All that one can do is to set a distance to be traveled. The spaceship will th... |
H: Tracing diagonal numbers on a 2D grid or matrix.
I need to test a cell on a 2D grid/matrix to see if it's diagonal numbers have a power of 2 in them.
Example:
C0 C1 C2 C3 C4 C5 C6 C7 C8 C9
R0: 00 01 02 03 04 05 06 07 08 09
R1: 10 11 12 13 14 15 16 17 18 19
R2: 20 21 22 23 24 25 26 27 28 29
R3: 30 31 32 33 34 35... |
H: Possibilities of x in a right angle tringle
I am stumped on the following question:
Which of the following could be the value of x in the diagram a)10 b)20 c)30 d)40 e)50 (Ans b and c)
Any suggestions relating to solving this problem ?
AI: The angle must be between $90$ and $180$ degrees and is equal to $5x.$ ... |
H: Basic question about fractions
I'm solving some exercises about fields and am trying to find the inverse for $a_1 + \sqrt{2}b_1$, i.e. $\frac{1}{a_1 + \sqrt{2}b_1}$. This means I need to split the fraction into something of the form $x_1 + \sqrt{2}x_2$ but I can't seem to remember how to do such a basic thing! Can ... |
H: $p$-adic completion of integers
I'm trying to do the following exercise:
Let $p$ be a prime and for $n\geq 1$ let $\alpha_n :\mathbb Z/p \mathbb Z \to \mathbb Z/p^n \mathbb Z$ be the injection of abelian groups given by $1 \mapsto p^{n−1}$. Consider the direct sum $\alpha : A \to B$ of these maps where $A$ is a ... |
H: How follows the Strong Law of Large Numbers from Birkhoff's Ergodic Theorem?
We want to prove the strong law of large numbers with Birkhoff's ergodic theorem.
Let $X_k$ be an i.i.d. sequence of $\mathcal{L}^1$ random variables. This is a stochastic process with measure-preserving operation $\theta$ (the shift opera... |
H: Uniform convergence of functions, Spring 2002
The question I have in mind is (see here, page 60, the solution is at page 297):
Assume $f_{n}$ is a sequence of functions from a metric space $X$ to $Y$. Suppose $f_{n}\rightarrow f$ uniformly and has inverse $g_{n}$. Now assume $f$'s inverse $g$ is uniformly continuou... |
H: Convergence of a series of reciprocal prime numbers
If $p$ is a prime number, and $q$ is its twin prime, the sum of the reciprocal twin numbers is convergent and the value of the sum of the series is the Brun constant. Now, if we consider the prime numbers $x=p+\alpha$ where $\alpha$ is a constant such that $x$ is ... |
H: what is the behaviour of moving dot with 50% chance to go left or right?
If a dot is moving (from zero) left or right, by one, with 50% chance to go left or right - is it going to go to the +inf or -inf when it has infinite moves?
AI: This is the simple symmetric random walk on the integers. It is well known that t... |
H: Finite family of infinite sets / A.C.
Let $\{A_i\mid i\in n\}$ be a finite family of infinite sets. ( That is, $A_i$ is infinite for every $i\in n$ and $n\in \mathbb{N}$)
Here, we can choose representative $a_i$ from each $A_i$ and construct $\{a_i\mid i\in n\}$.
This process really doesn't use Axiom of Choice?
How... |
H: Higher variance implies larger spread?
Suppose a configuration of points is given in $X\in\mathbb{R}^{n\times 2}$. Given that the configuration has zero column means, the variance of each axis, $x$ and $y$, can be expressed as
$$\mathrm{var}(x)=\frac{1}{n}x^Ty\qquad\text{and}\qquad\mathrm{var}(y)=\frac{1}{n}y^Ty.$... |
H: P[random x is composite | $2^{x-1}$ mod $x = 1$ ]?
Select a uniformly random integer $n$ between $2^{1024}$ and $2^{1025}$
(Q) What is the probability that n is composite given that $2^{n-1}$ mod $n = 1$ ?
How did you calculate this?
More info:
One way to calculate this would be if you had the following two variabl... |
H: How can I prove $\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $?
For continuous $f$, $f \in L^2$, prove that $$\lim_{n \to \infty} \int_{0}^{\pi/2} f(x) \sin ((2n+1) x) dx =0 $$
AI: Show it when $f$ is a polynomial.
Argue by density, using Stone-Weierstrass theorem. |
H: Will this problem be solved using Thales theorem for triangles
I am stumped on the following question:
In triangle ABC , AD=DB , DE is parallel to BC. The area of Triangle ABC is 40. What is the area of triangle ADE
I know Thales theorem must be applied here . But I cant figure how to ? Any suggestions ?
AI:... |
H: Structures on torus
Quotienting $\mathbb R^2$ by different lattices isomorphic to $\mathbb Z^2$, we get different tori.
Somehow I think of the tori as having different "structures", but thinking more about it, I am not quite sure what different structures I am really thinking of. Two structures I am guessing at are... |
H: About the orthonormal decomposition of $L^2 (-\pi , \pi) $
For any $f \in L^2 (-\pi, \pi)$, prove that there exists unique orthonormal decomposition with even functions and odd functions : $$ L^2 ( -\pi , \pi) = L^2 _{odd} (-\pi , \pi ) \oplus L^2_{even} (-\pi , \pi).$$
AI: You can also think this way. Suppose the ... |
H: Probability with card game
I need help calculating the chances of winning this strange game that I'm going to explain right now:
You have a deck of 52 cards(4 suits,Ace to King). All you have to do is turning
the cards one by one counting one,two,three while you turn them. If you get an
Ace when you count one o... |
H: How to simplify an expression like this: $(x^2+x^{-2}-2)^{1/2}$
Sorry, I am not sure how to do the maths mark-up on this site but hopefully the question will make sense. I should know how to do this, but I have got myself stuck! Can anyone help?
$(x^2+x^{-2}-2)^{1/2}$
AI: $$\left(x-\frac{1}{x}\right)^2= \dots?$$ |
H: A proof about complex number
If $a, b, c\in \mathbb{C}$, and if $\left \| a \right \|=\left \| b \right \|=\left \| c \right \|=1$, prove $(a+b)(b+c)(c+a)/(abc)\in \mathbb{R}$.
I have thought this Q for a long time, but I can only get something long and troublesome but not the answer. Can anyone help me please? THA... |
H: Proving that the general linear group is a differentiable manifold
We know that the the general linear group is defined as the set $\{A\in M_n(R): \det A \neq 0\}$. I have a homework on how to prove that it is a smooth manifold. So far my only idea is that we can think of each matrix, say $A$, in that group as an $... |
H: measuring distance between probability measures only at the tail
Is there any official (i.e., to be found in probability books) metric for the distance between two probability measures, defined only on a subset of their support?
Take, for example, the total variation distance:
$$TV(\mu,\nu)=\sup_{A\in\mathcal{F}}|\... |
H: About completeness of the Fourier series.
The Fourier series of a function is given by $$ \frac{a_0}{2} + \sum_{n=1}^\infty a_n \cos n \theta + \sum_{n=1}^\infty b_n \sin n \theta . $$ Here what does the statement " $\sum_{n=1}^\infty b_n \sin n \theta $ is complete" mean? And would you tell me how can I prove t... |
H: Presence of Identity Element and abelian groups
I.N.Herstein in Topics of Algebra defines a group as a set having a special element $i$ such that:
$a,i\in A(S) $ which satisfies $i\cdot a = a\cdot i = a$
In this way, this follows commutativity holds true for identity element, then doesnt it run contradictory to th... |
H: Application of the Schwarz Lemma, what if $f(f(z))=cz$?
I have an analytic function $f$ mapping the (unit) disc to the (unit) disc, with $f(0)=b$ in $D$, and $f(b)=0$.
Part (a) was to show that $|f'(b)f'(0)|\leq 1$, which I have already done by applying the Schwarz lemma to $f(f(z))$.
Next, part (b) states that ... |
H: What is the formula for the difference between CI and SI?
If principal, time and rate are given how,do I find the difference between Compound interest and Simple Interest?
p=12,000
n=1 and a 1/2 yrs.
r=10% per year
Formulae that I know:
CI - SI for 2 years = P(R/100)^2
CI-SI for 3 years = P(R/100)^2 (R/100 + 3)
... |
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