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H: Vector spaces without an inner product
Many simple vector spaces such as $\mathbb{R}^m$ have natural inner products, in this case the dot product. Even function spaces such as $C[0,1]$, continuous real-valued functions on the unit interval, lend themselves to nice inner products, e.g.
$$
\langle f,g \rangle := \int... |
H: Why is [(4,4), (3)] not a descending plane partition of order 4?
I am trying to understand descending plane partitions.
According to Wolfram MathWorld,
A descending plane partition of order n is a two-dimensional array (possibly empty) of positive integers less than or equal to n such
that the left-hand edges a... |
H: Counting endofunctions with a certain recurrent condition.
Problem: Let $\phi:X\rightarrow X$ be an endofunction. A vertex $v\in X$ is recurrent if there is some positive integer $k\geq 1$ such that $\phi^k(v)=v$. Let $\mathcal{Q}$ be the species of endofunctions $\phi:X\rightarrow X$ such that if $v\in X$ has $|\p... |
H: Derivative axiom
This is confusing me very much... Is there (rigorous) proof that slope of secant line "goes to" slope of tangent line on some point when $\Delta x \rightarrow 0$? This is actually not obvious at all.
Intuitive problem: What guarantees me that function will not be all messy at infinite small input ... |
H: Find solutions to $g : \mathbb{R} \to \mathbb{R}$ where $g$ is additive and satisfies $g(x^z) = g(x)^z$
Find solutions to $g : \mathbb{R} \to \mathbb{R}$ such that $g(x+y) = g(x) + g(y)$ and $g(x^z) = g(x)^z$ for $z \in \mathbb{R} \backslash \{0,1\}$. $z$ is a fixed number and not a variable.
Note : Please do not i... |
H: Partial Derivatives on Manifolds as Derivations
first time poster, finally decided to take the plunge and not just lurk anonymously. I'm just an experimental physicist with a desire to know some math beyond my few semesters of abstract algebra.
Just a quick question regarding the set of point derivations $ D: C_p^... |
H: Matrix inner product, how can I prove this question
$\langle u,v\rangle$ is the Euclidean inner product on $\mathbb R^n$ and $A$ is an $n \times n$ matrix, then
how can I prove $\langle u,Av\rangle = \langle A^Tu,v\rangle$ ?
AI: Using the definition of inner product $⟨u,v⟩:=u^Tv$. Then
$$⟨u,Av⟩=u^T(A^T)^Tv=(A^Tu)^T... |
H: (Proof verification) For each $s \in S$, show that $\sum_{t \in Gs} \frac{1}{|Gt|}=1$
This is an exercise from Lang's Algebra, Chapter 1. I got my solution but I'm not sure that it's correct. Feel free to point out what's wrong with me.
Let $G$ be a finite group operating on a finite set $S$. For each $s \in S$, s... |
H: Complex inner product spaces: are $A^*A$ and $AA^*$ always unitarily equivalent?
Problem 2(b) from Sec 79, pg 158 of PR Halmos's Finite-Dimensional Vector Spaces:
If $A$ is an arbitrary linear transformation on a complex inner product space $V$ (not given to be finite-dimensional), does it follow that $A^*A$ and $A... |
H: Let $a$ be an element of a group. Show that $a$ and $a^{-1}$ have the same order
I think one way to solve it is by considering the conjugation of $a$ and $a^{-1}$ using $g \in G$. That is, $a^g=gag^{-1}$ and $(a^{-1})^g= ga^{-1}g^{-1}$. Then, if $n$ is the order of $a$ we have $(a^g)^n=e$; since,
$(a^g)^n=gag^{-1}... |
H: Understanding the definition of the index of $H$ in $G$, i.e., $[G:H]$
Let $H \leq G$. Then,
the number of left cosets of $H$ in $G$ is the index $[G:H]$ of $H$ in $G$.
Now, I read that $[G:H]$ can either be finite or infinite. My question: if I know that $[G:H]$ is finite, would it be valid to conclude that $G... |
H: Is this justification to prove this is a continuous function correct?
Given the following function:
$$
f(x,y)=\frac{\sin xy}{e^x-y^2}\;,
$$
is the following justification correct?
As sin is a continuous function, the exponential is a continuous function, the polynomials are continuous functions and the subtraction ... |
H: Complete orhonormal set is basis?
Do I understand correctly that a "complete orthonormal set" is the same as basis?
Complete meaning that every Cauchy sequence in the set converges to an element in the set.
But then if you consider $l^2$, can't I choose the set containing $e_1,e_3,e_4,...$ e.g. skipping over the se... |
H: Explain the value of $(0,0)$ for $f(x,y)=1+x\frac{x^2-y^2}{x^2+y^2}$
Let $f:\mathbb R^2\to\mathbb R$ (where $\mathbb R$ is the real numbers) continuous in $\mathbb R^2$, such that
$$f(x,y)=1+x\frac{x^2-y^2}{x^2+y^2}$$ for $(x,y)\neq(0,0)$
Justify the value of $f(0,0)$.
The function is continuous in its domain,... |
H: Consider the set $A=\{1,2,3,4,…99,100\}$, maximum number of elements that can be chosen making sure that their sum does not exceed 1000?.
Consider the set $A = \{1,2,3,4,…99,100\}$, i.e. the set of natural numbers less or equal to $100$.
Elements are chosen at random from this set without repetition of elements.
Wh... |
H: Why is the derivative of the binormal vector parallel to the normal vector?
In my notes, it says to consider an arc length parameterization r(s). Then we can show that B' points in the direction of N such that we can write B'=-N. I understand why ||B'||= but am unsure how to show why B' is parallel to N? Also, is t... |
H: Estimating the remainder of a converges series
Is there a way in general to estimate how fast the remainder of a convergent series decreases?
Given a sequence $\{a_n\}$ s.t. $\sum_{n=1}^\infty a_n$ converges, find how fast $\sum_{n=j}^\infty a_n$ decreases.
This is frequently needed in large number (probability t... |
H: How to notate this kind of function?
I have a function $f$ whose domain is $\mathbb{N}$ and range is a set of another functions. The domain of $f(n)$ is $\mathbb{N}_0 \cap [0,\frac{n}{2})$ and its codomain is $\mathbb{Z}$. How can I notate the codomain of $f$?
AI: You can use either
$$f:\Bbb N\to\bigcup_{n\in\Bbb N... |
H: For ordinals $\delta$ and $\beta$, if $\delta\in$ or $=\beta$, then there exists $\gamma\in$ or $=\beta$ such that $\beta=\delta+\gamma$
Prove that for ordinals $\delta$ and $\beta$, if $\delta \in$ or $= \beta$ then there exists an ordinal $\gamma \in$ or $= \beta$ such that $\beta = \delta + \gamma$.
I tried u... |
H: Proving, with Dedekind cuts, that $\mathbb{R}$ has the lub property
I am trying to follow Rudin's proof that $\mathbb{R} = \{\text{set ot cuts}\}$ has the least upper bound property. Here is the set up:
$A$ is a nonempty subset of $\mathbb{R}$ (so it's some set of cuts) and $\beta$ is an upper bound of $A$. We def... |
H: Determining all $f : \mathbb R^+ \to \mathbb R^+$ that satisfy $f(x + y) = f\left(x^2 + y^2\right)$
Determine all $f : \mathbb R^+ \to \mathbb R^+$ that satisfy
$$f(x + y) = f(x^2 + y^2)\ \forall x,y \in \mathbb R^+.$$
My Proof:
I define $g(x) + g(y) = f(x + y)$.
Then, we get $g\left(x^2\right) - g(x) = c$; $c \i... |
H: Calcuating an Integral via Residues
I am trying to solve the following integral:
$$\int_0^\infty\frac{dx*x}{(x^2+1)(2+x)}$$
What I have done is an analytical extenstion to the upper half plane only by recognising that the above integral is just 1/2 the integral over the entire real line, then only considered poles ... |
H: Measure "uniform distribution" or "how equally distributed" is an amount represented with a group of numbers
To code a program I'm looking for an algorithm that returns 0 when a set is evenly distributed and 1 when it's totally "unfairly" distributed (or the other way around, it does not matter). For example:
Let's... |
H: Prove Lissajous curves are closed given the ratio between a and b is rational
Lissajous curves are curves described by the parametric equations:
$$x(t)=A\sin(at+k), \ \ y(t)=B\sin(bt)$$
According to wikipedia, the curve is only closed iff the ratio between $a$ and $b$ is rational. I was wondering if someone could ... |
H: I cannot follow the logic of the proof of 3-5 Theorem on p.51 in "Calculus on Manifolds" by Michael Spivak. Please explain me.
I am reading "Calculus on Manifolds" by Michael Spivak.
I cannot follow the logic of the proof of the above theorem.
Then each $v(U_i)$ is the sum of certain $t_j - t_{j-1}$. Moreover, ... |
H: Infinite union clarification
I am 99% sure but still I want some clarification. Is it true that:
$$\bigcup_{n\in\mathbb N}[0,\frac{n-1}{n})=[0,1]$$
AI: Unfortunately, this is false; it's actually $[0,1)$, with an open endpoint at $1$. This is because $1$ is not in any set $[0, \frac{n-1}{n})$. |
H: Asymmetric Random Walk on $\mathbb{Z}$
Suppose we have an asymmetric random walk on $\mathbb{Z}$ starting at $0$, with transition probabilities $p(x,x+1)=\frac{1}{3}$ and $p(x, x-1)=\frac{2}{3}$. What is the probability that this random walk ever reaches some positive integer $n$?
I see that this random walk is as... |
H: Finding the circumradius-to-inradius ratio for the regular pentagon
Find the value of $R/r$:
I go with co-ordinate geometry, considering the centre of the circles is at the origin, then the equation of the circle becomes as
$$ x^2 + y^2 = R^2 $$
$$ x^2 + y^2 = r^2 $$
After this I not able to solve this.
2nd Att... |
H: Diagram shows quadrilateral ABCD, with OA = (-6,3) , OB = (5,5), OC = (7,-2) , OD = (-4,-6). Show that midpoints P, Q, R & S form a parallelogram.
The diagram below shows the quadrilateral ABCD, with OA = (-6,3) ,
OB = (5,5), OC = (7,-2) , OD = (-4,-6).
Show that the midpoints P, Q, R & S form a parallelogram.
... |
H: A beginner’s explanation for PCA on a multivariate time series
This is very much a beginner’s question.
Say you have a 10 dimensional vector for every day in a time series of 100 days. I was reading about using PCA to reduce this to a one dimensional time series. If the time series is represented by a 10 by 100 ma... |
H: Contour integral around an essential singularity
The complex function
$$f(z)=\frac{1}{e^{1/z}-1}$$
has an essential singularity at $z=0$, and an infinite quantity of poles inside every open neighborhood containing it.
Let $\mathbb{R}\ni\epsilon>0$. What is the value of
$$\int_{|z|=\epsilon} \frac{dz}{e^{1/z}-1}=?$$... |
H: Why is $a$ not coercive when it is defined on $H^1(\Omega)$?
Given the Poisson's equation with homogeneous Neumann boundary conditions and the associated bilinear form
$$a(u,v) = \int_{\Omega}\nabla u \cdot \nabla v \, dx$$ on $H^1(\Omega)$, why is $a$ not coercive? Whereever I look it says we need $H_0^1(\Omega)$ ... |
H: Is the sequence $f_n(x)=\frac{x^{2n}}{1+x^{2n}}$ uniformly convergent?
Given $R\in\mathbb{R}$, $R>1$ investigate the sequence of functions $(f_n)_{n\in\mathbb{N}}$, given by
$$f_n(x)=\frac{x^{2n}}{1+x^{2n}}, \qquad x \in\ [R,\infty)$$
with regard to uniform convergence.
Could someone help me with this probl... |
H: How to find an argument of $ z=1-\cos\varphi-i\sin\varphi $?
How to find an argument of this complex number?
$$
z=1-e^{i\varphi}=1-\cos\varphi-i\sin\varphi
$$
I know I should try to make $z$ equal to $r(\cos\psi+i\sin\psi)$. And then $Arg(z)=\psi$. But I don't really understand how to do that.
I tried something... |
H: Multivariable polynomial in $R[X_1,..,X_n]$ whose roots are all elements in $R^n$ is zero.
Suppose that the roots of $p\in R[x_1,...,x_n]$, where R is algebraically closed and $n\geq 1 $, is the entire set $R^n$, is $p$ necessarily the zero polynomial. I think it might be true but I cant prove it. I know that it ho... |
H: Is ring $\mathbb{Z}_5[x]/I$ is a field, when: $I=(3x^3+2x+1)\mathbb{Z}_5[x]$?
Is ring $\mathbb{Z}_5[x]/I$ is field, when: $I=(3x^3+2x+1)\mathbb{Z}_5[x]$?
Can you show me a step by step instruction how to solve this problem?
AI: Since $\mathbb{Z}_5$ is a field, we know $\mathbb{Z}_5[x]$ is a PID. Note that $3x^3+2x+... |
H: Distributive law for subgroups
I believe that for subgroups $H,K,L\le G$ distributivity does not apply, but could someone give an example to illustrate this:
$$
\langle H\cup K\rangle\cap L\neq\langle (H\cap L)\cup (K\cap L)\rangle
$$
Apparently with subgroups $n\mathbb{Z}$ of $\mathbb{Z}$ equality is fulfilled.
An... |
H: Proving that PDF of a random variable is given by $\frac{l}{\pi(l^2+t^2)}$
I'm solving the following exercise in "Problems of probability theory by Sveshnikov".
A straight line is drawn at random through the point $(0,l$) in the plane. Let $X$ be the random variable denoting the distance between the $x$-intercept o... |
H: Angle between the diagonal and edge of a cube (application of dot product)
Question: Find the angle between the diagonal of a cube and one of its edges.
I know how to do this question but in particular I do not understand why $\vec b=<1,0,0>$ rather than $<0,0,0>$
For this question, I assumed that the cube has sid... |
H: Continuity of map from $M(n,\mathbb{R})$ to itself.
I am self studying topology of metric spaces and i encountered an exercise which requires to prove the map going from $M(n,\mathbb{R})$ to $M(n,\mathbb{R})$ defined by $A \mapsto A^{k}$ where k is any positive integer and $M(n,\mathbb{R})$ is given with euclidean... |
H: Mistake in the proof of Binet's formula
I want to prove Binet's formula $F_{k} = \dfrac{1}{\sqrt{5}}\times\left[\left(\dfrac{1+\sqrt{5}}{2}\right)^k-\left(\dfrac{1-\sqrt{5}}{2}\right)^k\right]$ for the $k_{th}$ fibonacci number.
I did as follows -
$F_{0}=0, F_{1} = 1$ .....
$F_{k} = F_{k-1}+F_{k-2}$
$F_kx^k=xF_{k... |
H: Auto-Homeomorphism of R making polynomials discontinuous
Consider $\mathbb{R}$ with its usual topology and let $f(x)=x^2$ and $g(x)=\max\{1,x\}$. Does there exist a homeomorphism $\phi:\mathbb{R}\to \mathbb{R}$ such that $\phi\circ f = g$? Obviously no such diffeomorphism can exist but it's not clear that this is... |
H: Approximation of an integrable function by continuous function with an extra criteria.
Let $f :[0,1]\rightarrow \mathbb{R}$ satisfying
$$\int_0^1|f(t)|dt<\infty .$$
We know that for given $\epsilon >0$ there exists a continuous function $g$ on $[0,1]$ such that $$\|f-g\|_{L^1([0,1])}<\epsilon .$$
Question: Can we ... |
H: Probability: Indicators function and random variable
Given $A,B,C$ three events of a sample space, with their indictaros functions, $I_A$, $I_B$ and $I_C$. Prove that the application defined by $$Z=(I_A-I_B)*(I_C-I_B)$$ is a random variable.
AI: Hint:
$(\mathsf1_A-\mathsf1_B)(\mathsf1_C-\mathsf1_B)$ is also an indi... |
H: Conditional distribution of a Bernoulli given the relative binomial
Let $X_1,\dots,X_n$ be independent random variables with $X_i \sim $ Ber($p$), and let $S_n=X_1+\dots+X_n$ be the relative binomial distribution with parameters $n,p$. Then, what is the conditional distribution of $X_i$ given $S_n=r$? I have that
$... |
H: Prove that solution of equation $x^p=a$ exists, where $a$ is a fixed element in a finite field $F$ and $charF=p.$
I have to prove that solution of equation $x^p=a$ exists and it's unique, where $a$ is a fixed element in a finite field $(F,+,\cdot,0,1)$ and $charF=p.$
I know how to prove that $p$ is a prime number a... |
H: Discriminant of $x^4 - 7$
According to the formula on the wikipedia page, the discriminant of $x^4 - 7$ should be 0. But the polynomial has roots $\pm \sqrt[4]{7}$ and $\pm i\sqrt[4]{7}$ so it is separable, so it shouldn't have zero discriminant. What's going on?
AI: There is the term $256a^3e^3$, which is nonzero,... |
H: Why z=0 is a simple pole of $\frac{z}{1-e^{z^2}}$
Why $z=0$ is a simple pole of $\frac{z}{1-e^{z^2}}$
I understood why for $z\neq 0 $ it is a simple pole but for the other part I didn't an explanation will be nice :)
AI: It suffices to show that $\lim_{z \to 0} z\left(\frac{z}{1 - e^{z^2}}\right) = L \neq 0$ (i.e. ... |
H: Easy example of a bijective continuous self mapping whose inverse is discontinuous
Let $f : X \to X$ be a continuous bijective mapping from a metric space onto itself.
Is $f^{-1}$ continuous too? I don't think so, but I'm struggling to find a counterexample.
I've read that if $X=\mathbb{R}^n$ or $X$ a compact Hausd... |
H: What is the $\mathbb{P} [S_n=0 \mbox{ for infinitely many $n$}]$?
For random variables $X_i$ and define the sum $S_n=\sum_{i=1}^n X_i$.
$$\limsup_{n} \mathbb{P} [S_n=0]?$$
AI: In general you cannot express $P[S_n=0 \, \text {for infinitely many n} \, ] $ in terms of the numbers $P[S_n=0]$. What you can say is $P[S_... |
H: Square root in $\mathbb F_{2^n}$
Let $\mathbb F_{2^n}$ be a finite field with $2^n$ elements.
I am just wondering if it is true that for all $n\in \mathbb N$ all elements of $\mathbb F_{2^n}$ have square roots, i.e for all $a\in \mathbb F_{2^n}$ there is an element $b\in \mathbb F_{2^n}$, with $a=b^2$?
It may be v... |
H: For $\pi<\alpha<\frac{3\pi}{2}$ what is the value of $\sqrt{4\sin^4\alpha + \sin^2 2\alpha} + 4\cos^2\left(\frac{\pi}{4} - \frac{\alpha}{2}\right)$?
I solved part of it this way:
$$\sqrt{4\sin^4\alpha + \sin^2 2\alpha} + 4\cos^2\left(\frac{\pi}{4} - \frac{\alpha}{2}\right)$$
$$= \sqrt{4\sin^4\alpha + 4\sin^2\alpha ... |
H: Chain rule for matrix-vector composition
Suppose $A(t,x)$ is a $n\times n$ matrix that depends on a parameter $t$ and a variable $x$, and let $f(t,x)$ be such that $f(t,\cdot)\colon \mathbb{R}^n \to \mathbb{R}^n$.
Is there a chain rule for $$\frac{d}{dt} A(t,f(t,x))?$$
It should be something like $A_t(t,f(t,x)) + .... |
H: Let $f$ holomorphic at $D=\{z:0<|z|< 1\}$ Prove that exist a unique $c\in\mathbb{C}$ s.t $f(z)-\frac{c}{z}$ has primitive function.
Let $f$ holomorphic at $D=\{z:0< |z|< 1\}$ Prove that exist a unique $c\in\mathbb{C}$ s.t $f(z)-\frac{c}{z}$ has primitive function in D.
So my thoughts were that we can write a laure... |
H: Extending the definition of stochastic integral from simple processes
I am reading stochastic integration from Brownian Motion And Stochastic Calculus by Karatzas and Shreve. In the course of extending the definition of the stochastic integral from simple processes to other measurable and adapted processes, they fi... |
H: People choosing numbers - uniform distribution
there's a group of 100 people. each one chose a number that distributes uniformly $Ud~(1,50)$.
$X$ is all the numbers that got chosen. What is the $E(x)$?
So all I got to do is $(1+50)/2$ and to multiply it by $100$? or is there a trick?
AI: It is correct, there is no ... |
H: Let $f_n:[1,2]\to[0,1]$ be given by $f_n(x)=(2-x)^n$ for all non-negative integers $n$. Let $f(x)=\lim_{n\to \infty}f_n(x)$ for $1\le x\le 2$.
Let $f_n:[1,2]\to[0,1]$ be given by $f_n(x)=(2-x)^n$ for all non-negative integers $n$. Let $f(x)=\lim_{n\to \infty}f_n(x)$ for $1\le x\le 2$. Then which of the following i... |
H: How can I solve $f(x) = \int \frac{\cos{x}(1+4\cos{2x})}{\sin{x}(1+4\cos^2{x})}dx$?
$$f(x) = \int \frac{\cos{x}(1+4\cos{2x})}{\sin{x}(1+4\cos^2{x})}dx$$
I have been up on this problem for an hour, but without any clues.
Can someone please help me solving this?
AI: It is quite straight forward after rewriting the i... |
H: Scaling property of Laplace Transform not working
I know that $L(f(t)) = F(s) \implies L(f(at))= \frac{1}{a}F(s/a)$ and if $L(1) = 1/s$
$L(2 \cdot 1) = 1/2 \cdot F(s/2) = 1/2 \cdot 1/(s/2) = 1/s$ but
$L(2 \cdot 1) = 2 \cdot L(1) = 2 \cdot 1/s = 2/s$
Why is there a contradiction? What am I missing?
AI: In both c... |
H: Limits of two variables
Using $\delta-\epsilon$ approach can someone help me prove that the limit is equal to 1?
$$\lim_{(x,y)\rightarrow (0,0)}\frac{e^{xy}}{1+x^2}$$
AI: Pick $\varepsilon>0$, and suppose $x^2+y^2<\delta$ where $\delta=f(\varepsilon)$ will be chosen later.
Then $|xy|\le\frac{x^2+y^2}{2}<x^2+y^2$ s... |
H: Upper bound of $\int^{\pi}_{0} \frac{\operatorname dx}{(\sqrt{2-2\cos(x)})^\alpha}$ for $\alpha\in(0,1)$
I have the following integral:
$$\int^{\pi}_{0} \frac{\operatorname dx}{(\sqrt{2-2\cos(x)})^\alpha} \qquad \text{for $\alpha\in(0,1)$}$$
Now, I know that I can use the fact that $1-\cos(x)\ge\frac{x^2}{9}$ to ap... |
H: Proposed proof for:$b! \equiv 0 \pmod a$ $\Rightarrow$ $a \le b$
Let a,b $\in$ $\mathbb{Z}$. Obviously, a counterexample can disprove this statement. I tried this approach to seek another method (possibly proof by contradiction):
$a|b!$
$a|[b(b-1)!]$
If $gcd(a,(b-1)!)=1$, then $a|b$ and hence: $a \le b$. But I ... |
H: $(f_n)$ integrable sequence of functions that converges uniformly to $f$, then $f$ is integrable
I'm reading Abbot's Understanding Analysis, and have stumbled across this problem, but there is a step that I don't fully understand.
The problem is:
Assume that for each $n$, $f_n$ is an (Riemann) integrable function ... |
H: Supremum and maximum
I'm asked to tell whether the supremum and maximum of the following functions exist and to derive them when $x \in B$:
$1- B = [0, 4 \pi)~\text{and}~g(x)=\cos x$
$2- B = [0, 4 \pi)~\text{and}~g(x)=\sin^2x+\cos^2x$
For the first one, I found that $\max f(x)_{x \in B}$ doesn't exist and $\sup f(x... |
H: Thinker with no maths knowledge - checking primes
My route for checking whether a number is a prime (>3) is if
n-5÷6= whole number
or
n-7÷6= whole number
Is this wrong please?
Sue
AI: This is wrong: it is a very elementary sieve which relies on the fact that any prime greater than $3$ is of the form $6n\pm 1$ or al... |
H: Prove the following measure statement
Prove: An algebra $A$ is a $\sigma$-algebra if and only if it is close for disjoint numerable unions.
Well my doubt is in the right direction because I don't know how to start, the left direction is okay.
After thinking about it I did the following development:
I write $\bigsq... |
H: Existence of linear combination
My book proves the following theorems:
Three distinct points $A,B,C$ are collinear if and only if there exist 3 numbers, $\lambda_1,\lambda_2,\lambda_3$, all different from zero, such that $$\lambda_1 \boldsymbol a +\lambda_2 \boldsymbol b +\lambda_3 \boldsymbol c =0,\space \space ... |
H: Convergence of $\sqrt{\frac{2N}{d}}\frac{\Gamma(\frac{d+1}{2})}{\Gamma(\frac{d}{2})}$? (expected value from random walks)
I saw from this post (stack) that the expected distance from the origin after $N$ steps in $d-$dimensional space is
$$ \sqrt{\frac{2N}{d}}\frac{\Gamma(\frac{d+1}{2})}{\Gamma(\frac{d}{2})}.
$$
I... |
H: Examine the convergence: $\sum_{n=1}^{\infty}(-1)^{n-1}\sin\frac{a}{n}(a\gt 0)$ (I solved but not sure if it is correct)
I have a question that I don't know if I solved it correct.
Can you check and show me the right way?
Question:
Examine the convergence of series given below:
$$\sum_{n=1}^{\infty}(-1)^{n-1}\sin\f... |
H: Problem related to output in python programming (not gives expected result)
I'm very beginner at python. Given code gives error with output. Want to correct it.
The mathematics behind the code is :
Let denote $S_m(n)=1^m+2^m+...+n^m$ and denote $D_x(y)$ is sum of digits of $y$ in base $x$.
Example $S_2(3)=14$ and... |
H: Show that each local max/min point is a critical point
I have that $X$ is a normed vector spaces over $\mathbb{R}$, $U\subseteq X$ is an open set and $F:U\rightarrow \mathbb{R}$ is a differentiable function. Need to show that each local max/min is also a critical point. I figured one could somehow attempt to combin... |
H: Partial derivatives of random variables
I have a question about partial derivatives and random variables, say we have two random variables, $r$ and $s$, and they are related by the expression $s=e^r$. Now, $r$ has an explicit dependency on the variable $t$, $r(t)$. Hence, $s$ doesn't explicitly depend on the variab... |
H: Do all diagonalizable matrices $A\in M_n(\mathbb C)$ have square root in $\mathbb C$?
Let $A\in M_n(\mathbb C)$ be diagonalizable matrix and $\mathbb F = \mathbb C$.
Does it mean this matrix has root?
I think this is correct, and thought on proving it with the fact that every polynomial has root in the complex numb... |
H: Confusion on how to find derivative.
I am confused about the following, why do I get different results when I change the point where I use the substitution: (ie where am I making a silly mistake)
In both I use the substitution $$x = \cos\theta$$ so $$\frac{dy}{d\theta} = -\sin\theta\frac{dy}{dx}:$$
$$\frac{1}{\sin\... |
H: Be $E\supset F$ galois extension and $\alpha \in E$, $O$ the orbit of $\alpha$ under $Aut(E/F)$. $|O|=|F(\alpha)/F|$?
Be $E\supset F$ galois extension and $\alpha \in E$. Be $O$ the orbit of $\alpha$ under $Aut(E/F)$.
I know that $|O|=|F(\alpha)/F|$ if $F(\alpha)\supset F$ is a galois extension. My question is: th... |
H: Minimal polynomial of finite Galois extensions
Let $L/K$ be a finite Galois extension. In this context, I heard people talk about the minimal polynomial of $L/K$. I want to understand what is exactly meant by that phrase.
What I suppose: Since $L/K$ is Galois, the extension is separable. Therefore, there is an elem... |
H: Spectrum and resolvent set of operator $ (Tx)(t) = (1+t^2) x(t) $ on $C[0,1]$
Operator $T : C[0,1] \rightarrow C[0,1]$ is defined by: $ (Tx)(t) = (1+t^2) x(t) , \forall x \in C [0,1]$. We assume that norm on $C[0,1] $ is standard supremum norm. I have to find $\rho(T), \sigma_{p}(T), \sigma_{c}(T) $and $\sigm... |
H: Manifold with a nilpotent fundamental group.
I am looking for a manifold with a nonabelian nilpotent fundamental group. I know the above terms, but I couldn't find out an example of that.
AI: The Klein bottle has a fundamental group with a normal $\mathbb Z \oplus \mathbb Z$ subgroup and a $\mathbb Z / 2 \mathbb Z$... |
H: Complex number square roots
I am very confused how to do this problem and I would appreciate if someone could explain. Cheers
Suppose u+vi is a square root of x+yi. Express the following in terms of x and/or y only.
u^2 - v^2 = ?
2uv = ?
u^2 + v^2 = ?
Hence find the square root of 8-15i
AI: As $u+vi$ is a square ro... |
H: What is meaning of $X/P$? ($X$ is a set and $P$ is a partition)
The definition of $x/E$ when $E$ is an equivalence relation is :
$$x/E = \{y\in X \mid (y,x)\in E \},$$
and the definition of $X/E$:
$$X/E = \{x/E\ \mid x\in X\}.$$
Now, what is $X/P$ when $P$ is a non-empty partition of X?
AI: A partition $P$ defines ... |
H: Homeomorphism on (a,b) an R in terms of discrete topology
Are (a,b) and R homeomorphic, if they both have discrete topology?
AI: Any bijection between spaces that have the discrete topology is a homeomorphism (all maps on a discrete space are continuous). |
H: Positive solutions to this exponential diophantine
Find positive integral solutions to :
$2013^p$ $+$ $2014^q$ = $2015^r$
Attempt:
$2013^p$ $+$ $(2013 + 1)^q$ $=$ $($2013$ + $2$)^r$
Expanding,
$2013^p$ + $2013^q$ + $1$ + $2013Q$ = $2013^r$ + $2^r$ + $2013R$
Here R and Q are some constants, via binomial theorem expa... |
H: Sufficient conditions on $f$ to obtain that $\arg\max_{x \in K}f(x) \cap K' \neq \emptyset$ ($K$, $K'$ compact)
Let $f : \mathbb{R}^n \rightarrow \mathbb{R}$ (for some $n \in \mathbb{N}^\ast$) be a continuous function and $K, K'$ be two compact subspaces of $\mathbb{R}^n$ such that $K' \subset K$.
I was wondering i... |
H: Joint Default Probability Range
There is a 50% probability that bond A will default next year and a
30% probability that bond B will default. What is the probability
range for event that at least one bond defaults?
The author then goes on with:
To have the largest probability we can assume whenever A default... |
H: The infinite union of singletons in the Lebesgue measure in $\mathbb{R}$.
I know that the Lebesgue measure of a countable infinite number of singletons has a zero measure, but what about the Lebesgue measure of an infinite number of singletons, which is not countable. Can we generalize on this particular measure sp... |
H: Asymptotic variance
$\text{Let}~(X_i)_{i=1}^n~ \text{be an i.i.d. sample of}$
$\text{n observations, with} ~ E(X_i)=\mu\in\mathbb{R}~ \text{and} ~ Var(X_i)=\sigma^2 \in (0, \infty)$
$\text{I'm asked to propose estimator}$
$\text{for the asymptotic variance of the}$ $\text{following distribution and to prove}$
$\tex... |
H: Prove that is a vectorial subspace
Given $I\subset \mathbb{R}$ an interval. I want to prove that $C_0(I)$, the set of continuous function $f:I\to \mathbb{R}$ that, for all $\epsilon>0$, $\{x\in I, |f(x)|\geq \epsilon\}$ is compact and that it is a vectorial subspace of $C_b(I)=\{f:I\to \mathbb{R}, \text{continuous}... |
H: Selecting a ball probability
There are 11 white and 3 black balls in the urn. Balls are randomly removed from the urn until it is empty. What is the probability that the 8th ball will be taken a black ball?
Kind of stuck with this question, is Bayes' Theorem required here?
AI: We have 3 Black Balls on 14 total. The... |
H: How do error bars change for variables squared?
Let's say I have some variable $\mu$ with an uncertainty estimate:
$$\mu = 2 \pm .5$$
Let's say I have another variable $\nu = \mu^2$. Is the uncertainty estimate in $\nu$ equal to the the uncertainty in $\mu$ squared, such that
$$\nu = 4 \pm .25$$
This does not seem... |
H: discrete metric converges iff it is eventually constant
I have read previous answers about the proof but there is a small point I want to be sure of.
First if we assume that the sequence converges $x_n \to x$ there for all $N\leq n$ there is $\varepsilon >0$ s.t $d(x_n,x)< \varepsilon$
Let $\varepsilon = \frac{1}{2... |
H: How to find a basis for an eigenspace?
Consider the real vector space $P_2(\mathbb{R})$ of real polynomials of grade $\leq 1$. Consider the inner product defined as
$$
\langle p,q \rangle = p(0)q(0)+p(1)q(1)
$$
and the linear operator
$$
L: P_2(\mathbb{R}) \rightarrow P_2(\mathbb{R})
$$
defined as
$$
L(\alpha + \be... |
H: How to prove $\lim_{x\rightarrow\infty}-x^2+\log(x^2)=-\infty$
I'm proving the following
$$\lim_{x\rightarrow\infty} \bigg(-\frac{x^2}{2}+\frac{7}{2}\log(x^2+2\sqrt{2})\bigg)=-\infty$$
but how to state that $-x^2$ is going faster to $-\infty$ than $\log(x^2)$ is going to $\infty$?
This is a part of a proof for conv... |
H: Alternate approach on probability proof
Let $A_1,A_2,...$ be an infinite sequence of events which is monotonically increasing, $A_n \subset A_{n+1}$ for every $n$. Let $A=\cup_{n=1}^{\infty} A_n$. Show that $P(A)=\lim_{n \rightarrow \infty} P(A_n)$.
My approach : As the $A_i$'s are monotonically increasing, their r... |
H: How can I convert these Euler angles to a 3x3 rotation matrix
I'm working with a set of stereo cameras and having trouble with the math for the rotation calibration in openCV. Each set of cameras comes with a set of calibration data that includes three Euler angles to describe rotation between the left camera to t... |
H: Convergence of series in Hilbert space
I want to show that for $(a_n)$ a bounded series in $\mathbb{R}$, $x_n := \frac{1}{n} \sum_{k= 0}^n a_k e_k$ converges in norm (induced by the inner product) to 0, where $e_n$ is an orthonormal basis of the Hilbert space.
If $c$ is the upper bound for $(a_n)$, I can show that
... |
H: $p$-Norm of Block Diagonal Matrix
Let $A\in \mathbb{K}^{r\times r}$, $B\in \mathbb{K}^{(n-r)\times (n-r)}$ and $C = \mathbb{K}^{n\times n}$ such that
\begin{equation}
C =
\begin{pmatrix}
A & 0_{r\times (n-r)}\\
0_{(n-r)\times r} & B\\
\end{pmatrix}.
\end{equation}
I want to prove $||C||_p = \max\{||A||_p, ||B||_p\}... |
H: Finding the order of elements in $\mathbb{Z}_4 \oplus \mathbb{Z}_2$
For the solution in part (b), I am confused about how they got that $(0,1)$ has order $2$. For the others, such as say $(2,1)$, isn't is just lcm$(|2|,|1|)=2$ for the order? I'm confused about how this works.
AI: $(0,1)$ has order $2$ in $\mathbb ... |
H: Radius of convergence $\Sigma_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$ is my method right?
I am trying to find the radius of convergence of the power series $$\Sigma_{n=0}^{\infty}\frac{(-1)^n}{(2n)!}x^{2n}$$
is it true to mark $x^2=t$ than substitude in the original series and get $\Sigma_{n=0}^{\infty}\frac{(-1... |
H: Order of an element in $U(13)$ and $U(7)$.
It is a fact that in $U(13)$, the order of $12$ is $|12|=2$ and in $U(7)$, the order of $6$ is $|6|=2$.
But I don't understand why.
What I am trying is this:
$(12)(12) = 144 \equiv_{13} 1$ so the order should be 1... can anyone explain this to me?
AI: The order of an elem... |
H: In how many permutations of the integers 1,2,...,7 does 1 appear between 2 and 3?
I am missing something in this and I need to discuss it .
Step 1) We can arrange numbers [4 ,7] with $4!$ , different ways .
Step 2) There are 4 dinstinct cases:
a) We are going to put '2' , '1' and '3' in 3 of the 5 available
space... |
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