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H: Non-square tensors?
I learnt tensor algebra for physics and I never saw a non-square (or non-cubic...) tensor. But, from a mathematical point of view, do non-square tensors exist? And if so, are they used in some area in physics?
AI: I don't know much about the physical uses of a non-square tensor, but there is no ... |
H: A $p$-group of order $p^n$ has a normal subgroup of order $p^k$ for each $0\le k \le n$
This is problem 3 from Hungerford's section about the Sylow theorems.
I have already read hints saying to use induction and that $p$-groups always have non-trivial centres, but I'm still confused. This is what I have so far:
Sup... |
H: Bijection between column space and row space
Suppose that $A_{mn}$ is a matrix over some field, and that $C, R$ is its column space and row space, without using the fact that $rank(C) = rank(R)$, can we show that, there exists a bijection between $C$ and $R$?
AI: Hint: For $x\in R$, consider $Ax\in C$. Show this ma... |
H: Function that converge uniformly
Does the sequence $x, ..., x^n $ of functions converge uniformly on the interval $[0,k]$ for $k\in(0,1)$? If it does, prove it. How about on the interval $[0,1]?$
Can you help me complete the proof?
For the interval $[0,k]$ I got it does and for $[0,1]$ it doesn't. So let $\{f_n\} =... |
H: Showing that the sequence converges knowing that three other sequences converge
I have a question in Analysis. Knowing that $x_{2n}$, $x_{2n-1}$, $x_{3n}$ converge, how can I show that $x_{n}$ converges?
AI: Hint: 1) $x_{2n}$ and $x_{3n}$ should converge to the same limit (by looking at a profitable subsequence of ... |
H: Proving Vector Space with Scalar Multiplication
I'm trying to prove a vector space with the following definitions:
$$[x,y]+[a,b]=[x+a+1,y+b]$$
$$r[x,y]=[rx+r-1,ry]$$
I'm working on the distributive law but I'm running into a problem. It's not coming out correctly for reason. Here's my work:
$$r([x,y]+[a,b])=r[x+a+1... |
H: How to find the points of tangency of a parabola using Calculus?
How can someone find the points of tangency of a parabola in this situation?
I need to find two points of tangency so that the triangle formed by the two tangent lines at those points and the x axis is an equilateral triangle.
What approach should I ... |
H: Connected Set and connected subset
Question: Let $Y \subset X$; let $X$ and $Y$ be connected. Show that if $A$ and $B$ form a separation of $X-Y$, then $Y \cup A$ and $Y \cup B$ are connected.
Attempt at an answer: since $A,B$ form a separation of $X-Y$ then $X-Y = A \cup B$ and $A \cap B = \emptyset$. Then $X = A... |
H: Classifying complex $2\times 2$ matrices up to similarity
I would like to prove the following proposition, which is given as an exercise in Hoffman and Kunze:
If $A$ is a $2\times 2$ matrix with coefficients in $\mathbb{C}$, then $A$ is similar either to a matrix of the form $\begin{pmatrix} a & 0 \\ 0 & b \end{pm... |
H: Symmetric System of Equations
I'm new on studying Systems of equations. I just want to know the number of real solutions of this system of equations:
$$x^2-y^2=z$$
$$y^2-z^2=x$$
$$z^2-x^2=y$$
I also want to know how was your solution and explanation on how did you find your answer.
AI: Add $x^2 - y^2 = z$ and $y^... |
H: Is it possible to solve this nonlinear equation analytically?
Is it possible to solve the following equation analytically?
$B_1\exp(\beta_1 x) + B_2\exp(\beta_2 x) = C_1\exp(\alpha_1 x) + C_2\exp(\alpha_2 x)$
where, $B_1$, $B_2$, $C_1$, $C_2$, $\beta_1$, $\beta_2$, $\alpha_1$ and $\alpha_2$ are constants. And $x$ i... |
H: Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots.
Prove that the polynomial $x^6+x^4-5x^2+1$ has at least four real roots.
Talking analysis here, using the definition of continuity, intermediate value theorem, and extreme value theorem.
AI: Giving your polynomial the name $f$,
$f(-2)$ is po... |
H: Using residue to compute real fractional integral
Compute the integral $$\int_{-\infty}^\infty \dfrac{t-1}{t^5-1}dt$$ The hint is to use residues. I tried taking a look at the residue theorem, but I don't know which curve in the complex plane I should be integrating over, and what the complex function should be (it... |
H: Proof for the property of a fixed point of $f$.
Suppose that $f \colon [a, b] \to [a, b]$ is continuous. (Note that the range of $f$ is a subset of $[a, b]$)
Prove that there exists at least one point $x \in [a, b]$ such that $f(x) = x$. A point with this property is known as a fixed point of $f$.
AI: If $f(a)=a$ ... |
H: Zeros of polynomials are continuous
For two sets $A,B$, let $d(A,B)=\sup_{x\in A}\inf_{y\in B}|x-y|+\sup_{y\in B}\inf_{x\in A}|x-y|$.
Let $p(z)=a_nz^n+\ldots+a_0$, and let $\epsilon>0$. Show that there exists $\delta>0$ such that for any $q(z)=b_nz^n+\ldots+b_0$ such that $\max_k|a_k-b_k|<\delta$, we have $d(Z_p,Z... |
H: What's the exact meaning of this sentence from George Peacock?
I am reading the book "A history of abstract algebra" by Israel Kleiner. The following sentence is said by George Peacock. I am not a native English speaker. So could someone translate it into plain English?
In symbolical algebra, the rules determine t... |
H: Linear homogeneous recursive sequence of constant sign
Let recursive sequence be defined by the formula
$$
s_{j+1}=as_j-s_{j-1},
$$
where $a>1$ is some integer number. Is it true that $s_0<0$, $s_1<0$ implies $s_j<0$ for $j \geq 0$?
Edit: No, its wrong. Under what conditions on $s_0$ and $s_1$ property $s_j<0$ hol... |
H: Calculating the expected profit with Probability A level maths CIE
Company sets up display of 20 fireworks! for each firework, the probability that it fails is 0.05,independently of other fireworks
the probability that more than 1 firework fails is 0.264
the 20 firework cost company 24 dollars each .450 pay the ... |
H: Evaluating a Real Improper Integral by Residues
I am having trouble evaluating this improper integral due to its integrand and the singularities that are present. The question reads as
Show that $\int_{-\infty}^{\infty}\frac{dx}{x^4+1}=\frac{\pi}{\sqrt{2}}$.
The contour is assumed to be the boundary of the half ... |
H: Is there an example where the maximal and maximum elements are different?
I know by heart the definitions of both maximal and maximum elements but cannot grasp examples when these dont coincide. Usually with numbers it is easy to see that maximum and maximal coincide. But what about when they dont coincide?
I recen... |
H: Prove a function is not uniformly continuous.
Use the definition of uniform continuity to prove the function G(x) = x^3 is not uniformly on [0, infinity).
AI: Equivalent sequential Criterion for Uniform continuity is violated by taking
$x_n=n+{1\over n},y_n=n, \text{Then} (x_n-y_n)\to 0\text {but } f(x_n)-f(y_n)\n... |
H: Orthogonal projection on vector space convergence in $L^p$
Let $R_N$ be the set of $2^N$ intervals $$\left\{\left[0,\frac{1}{2^N}\right), \left[\frac{1}{2^N}, \frac{2}{2^N}\right),\ldots,\left[\frac{2^N-1}{2^N}, 1\right)\right\}.$$ Let $$V_N=\operatorname{span}\{1_I\mid I\in R^N\}$$ Let $P_N:L^2([0,1])\rightarrow ... |
H: How to find 'k' from this equation
I have a problem to calculate 'k' from this equation :
$$X = \frac{\left(\rho-\rho^{k+1}\right)\left(1-\frac{\gamma}{2}\rho\right)^{2}-k\rho^{k}\left(1-\frac{\gamma}{2}\rho\right)}{\lambda\left(1-\rho\right)\left(1-\rho^{k}\right)}$$
When I expand this equation, it will be :
$\rho... |
H: Borel sets on the plane
Let's say we have two sigma algebras $D_1$ and $D_2$ both of which contain open intervals. We know that the Borel sigma algebra $B(R)\subset D_{1}\cap D_{2}$. I'm having difficulty proving that $B(R)\otimes B(R)\subset D_{1}\otimes D_{2}$. Any help will be greatly appreciated.
Thanks.
AI: Ev... |
H: Logarithm of singular matrix
How do we define logarithm of a singular matrix(Say it is real square symmetric and has distinct eigen values). I tried searching online but could not find much information(Something that someone as dumb as me could understand).
MATLAB logm help says something about principal and non pr... |
H: Hackenbush game strategy for stalk
There are some piles of numbers. Numbers are divided in 2 groups, A and B. Player x plays with group A and player y plays with group B. x makes the move first. On each step a player chooses a pile and chooses a number of his group and remove all numbers smaller and equal to the nu... |
H: Evaluating $\int_0^\infty \frac{\log t}{1+t^2}\,\mathrm dt$ using residues
I want to integrate $$\int_0^\infty \dfrac{\log t}{1+t^2}\,\mathrm dt$$ using the residue theorem. The poles are at $i,-i$. If the integral were from $-\infty$ to $\infty$, I would consider integrating along the semicircle with large radius ... |
H: How is it possible to have a transitive yet not complete relation?
How can we say we cannot compare between pairs (no completeness) and yet we can have transitivity?
Let's assume the relation is a preference relation.
For instance if my set has: $(a, b, c)$ How can I say that a is preferred to b, b is preferred to ... |
H: How find the $\sum_{n=1}^{\infty}\frac{1}{n^p}\left(1-\frac{x\ln{n}}{n}\right)^n$ convergent
Question:
Study the series
$$\sum_{n=1}^{\infty}\dfrac{1}{n^p}\left(1-\dfrac{x\ln{n}}{n}\right)^n$$
convergence,when $p$ and $x$ such what conditions?
My try:
since
$$\dfrac{1}{n^p}\left(1-\dfrac{x\ln{n}}{n}\right)^n=\dfrac... |
H: Linear System of ODE
Would you mind telling me how do we use Matrix Algebra to get a general solution of the system:
$$x'=-{\delta}^{2}x+y+\delta z$$
$$y'=-x-{\delta}^{2}y+\delta z$$
$$z'=-\delta z$$
where $\delta$ is a parameter.
AI: We rewrite this as:
$$x' = A x = \begin{bmatrix} -\delta^2 & 1 & \delta \\ -1 & -... |
H: Sum $\binom{n}{k}+\binom{n+1}{k}+\binom{n+2}{k}+...+\binom{n+m}{k}$
Evaluate the following series sum which n, m, k are nonnegative integers.
$$\binom{n}{k}+\binom{n+1}{k}+\binom{n+2}{k}+...+\binom{n+m}{k}$$
I have no idea about it@@
AI: I don't know if this is the kind of thing you are looking for, but here goes :... |
H: How to bound away integral over complex rectangle?
I am integrating the following integral
$$\int_{-\infty}^\infty\frac{\cos t}{e^t+e^{-t}}dt$$
by computing residues inside the rectangle with vertices $-R,R,-R+\pi i,R+\pi i$. On the left and right side of the rectangles, $\cos z$ is bounded, while $e^z+e^{-z}$ ge... |
H: Is multiplication by zero in an equation allowed?
If we have equal quantities, we cannot divide with zero. But, we can multiply both sides with zero. But, my friend said, even multiplication with zero also wrong it seems. Unfortunately, he is not explaining the why wrong, if multiplication with zero on both sides? ... |
H: How to find $f$ if $f(f(x))=\frac{x+1}{x+2}$
let $f:\mathbb R\to \mathbb R$,and such
$$f(f(x))=\dfrac{x+1}{x+2}$$
Find the $f(x)$
My try
I found $f(x)=\dfrac{1}{x+1}$
because when $f(x)=\dfrac{1}{x+1}$,then
$$f(f(x))=f\left(\dfrac{1}{x+1}\right)=\dfrac{1}{\dfrac{1}{x+1}+1}=\dfrac{x+1}{x+2}$$
so $f(x)=\dfra... |
H: Polynomial functions of odd degree are surjective
Prove if the function $f: \mathbb{R} \to \mathbb{R}$ is a polynomial function of odd degree, then $f(\mathbb{R}) = \mathbb{R}.$
We know a polynomial, $f(x)=a_nx^n +a_{n−1}x^{n−1} ...a_1x+a_0$ with real coefficients is continuous. Also, $\mathbb{R}$ is connected now ... |
H: Show $ n + \operatorname{lcm}(a,b)\mathbb{Z} \subseteq (n + a \mathbb{Z}) \cap (n + b \mathbb{Z}) $
let $n \in \mathbb{Z}$. I want to show
$$ n + \operatorname{lcm}(a,b)\mathbb{Z} \subseteq (n + a \mathbb{Z}) \cap (n + b \mathbb{Z}) $$
for integers $a,b$
TRY:
Since $lcm(a,b)$ is multiple of $a,b$, then $lcm(a,b) = ... |
H: Finding value(s) of a for which f is continuous
A function $f$ is defined as follows: $$f(x)=\begin{cases}\sin x&\text{if }\;x\leq c,\\ax+b&\text{if }\;x>c,\end{cases}$$ where $a,b,c$ are constants. If $b$ and $c$ are given, find all values of $a$ (if any exist) for which $f$ is continuous at the point $x=c$.
So,... |
H: Representing functions as power series: Homework questions
Completed all my homework exept for this problem. Our teacher likes to give us a problem at the end that is for a future lesson. That is this question above. So it is kind of tricky...
I don't have any written solution so far. I believe there is something... |
H: Existance of Hamiltonian cycle in the connected graph.
I know the fact, that if a graph is connected and each of its vertices has a degree of $2$, then graph is a cycle graph and it has a Hamiltonian path. From that I easily conclude, that, if graph with n vertices is connected and each of its vertices has a degree... |
H: Finding the inverse function
The question is to find the inverse function of
$$f(x)=x-(2\sqrt{x})+1$$
I first found that the domain of definition is $\,x\ge 0$
Then studied the variation of the function and it is decreasing between $0$ and $1$ and increasing otherwise. Thus there are $2$ inverse functions to be f... |
H: Example of smooth function without compact support on open real interval
What is an explicit example of a smooth function on a real open interval that does not have compact support, i.e. for given $a,b \in \mathbb{R}$, a function (in common notation)
$$f \in C^{\infty}((a,b))\setminus C_0^{\infty}((a,b))$$
More pre... |
H: does the max function holds the triangle inequality?
I need to prove if the following is a norm:
$$||f||:=\max_{-1<x<0}|f| + \max_{0<x<1}|f|$$
when $f$ is a continuous on $[-1,1]$.
The only problem I have is with showing it holds the triangle inequality.
My question is, what I can say about the max function that ca... |
H: Countable disjoint union of non-measurable sets
Can a countable union of non-measurable sets of reals be measurable?
For instance, can we partition $\mathbb{C}$ into countably many disjoint non-measurable sets?
AI: Sure. Just take any non-measurable set and its complement. For example, let $V$ be a Vitali set, whic... |
H: special parameter integral
Does anyone know a proof for the following formula ?
$$\int_{0}^{\infty} \frac {1}{x^y+1} dx=\frac{\frac{\pi}{y}}{\sin(\frac{\pi}{y})}$$
for $y>1$?
If $y$ is an even positive integer than the integral can be calculated using the
residue theorem. But even the case, that $y$ is an odd posit... |
H: A mean square derivative
I'm doing an exercise where I have to check some properties about these two stochastical processes:
$X(t)=At+B\;\;$ and $\;\;Y(t)=\frac{1}{t}\displaystyle\int_{0}^{t}X(\tau)\;d\tau$, $t>0$.
Where $A$ and $B$ are uncorrelated 2-random variables with $\mathbb{E}[A]=m_A$, $\mathbb{E}[B]=m_B$... |
H: Difference $\Delta P_t$ approaches 0, then its relative difference $\Delta P_t / P_(t-1) \approx ln(P_t / P_(t-1))$.
When difference $\Delta P_t$ approaches 0, its relative difference $\dfrac{\Delta P_t}{P_{t-1}} \approx \ln(\dfrac{P_t}{P_{t-1}})$.
I know that it can be shown somehow with Taylor series: $\ln(1+x)=x... |
H: How to prove properties of the family of closed sets in a metric space
I know that is true:
Let $(X, d)$ a metric space. The family $\mathcal {U}$ of all open subsets of $X$ has these properties:
$1)$ $\phi, X\in \mathcal {U}$;
$2)$ $U_1, U_2 \in \mathcal {U}\Rightarrow U_1\cap U_2 \in \mathcal {U}$;
$3)$ $\{U_i\}_... |
H: Two complex integrals of the function $1/z$
Let $a,b \in \mathbb{R}^{*}$ and $\alpha, \beta : [0,1]\rightarrow\mathbb{C}$ be defined by:
$$\alpha(t):=a\cos(2\pi t)+ia\sin(2\pi t)$$
$$\beta(t):=a\cos(2\pi t)+ib\sin(2\pi t)$$
Show that $\int_\alpha 1/z \,dz = \int_\beta 1/z \,dz$.
Progress
I thought that the first ... |
H: Find the Laurent series for $f(z) = (z^2 - 4)/(z-1)^2 $ for $z=1$
What I understand is that we have to expand $f(z$) in the positive and negative powers of $(z-1)$.
Hence I tried factorizing the numerator $(z^2-4)=(z+2)(z-2)$ , which can then be written in terms of $(z-1)$ as:
$(z-1-1)(z-1+3)/(z-1)^2$ . however i c... |
H: Proving that $R$ is antisymmetric if and only if $G \cap G^{-1} \subseteq D$
I found this relations exercise that I couldn't finish. It involves a set's diagonal. I got it almost done - had trouble wording some parts related to the diagonal.
Is the procedure correct?
How can I better word the indicated parts?
Did ... |
H: Does inclusion of an affine open into an affine scheme correspond to restriction?
Let $X$ be an affine scheme and $U\subset X$ be an affine open. Let $i:U\to X$ be the inclusion, and let $\phi:\mathcal{O}_X(X) \to \mathcal{O}_X(U)$ be the induced morphism of rings.
Is $\phi$ the restriction map?
AI: Yes. But this c... |
H: How find this $n(n+1)a_{n+1}=n(n-1)a_{n}-(n-2)a_{n-1}$
Suppose
$$n(n+1)a_{n+1}=n(n-1)a_{n}-(n-2)a_{n-1}$$
for every postive integers $n\ge 1$,Give that $a_{0}=1,a_{1}=2$
find the $a_{n}$
My try:
$$a_{2}=\dfrac{1}{2}=\dfrac{1}{2!},a_{3}=\dfrac{1}{6}=\dfrac{1}{3!},\cdots$$
so I guess
$$a_{n}=\dfrac{1}{n!}... |
H: Does this reduce to finding PDF of a function of a random variable?
In the below question in image, there is a deterministic non-service period ($\tau$) between serving customers, and then time to serve a customer is given to be $t \sim \varepsilon(\lambda)$. I need to find the PDF of time between consecutive custo... |
H: Cantor's Theorem for $\Bbb N$.
Hi I would like to prove this statement.
Show that there is no one-to-one correspondence from
the set of positive integers to the power set of the set of
positive integers.
[Hint: Assume that there is such a one-
to-one correspondence. Represent a subset of the set of
positive integer... |
H: Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the induced subspace topology, then $X$ is also connected
PROBLEM
Let $A$ be a dense subset of $X$ and suppose $A$ is connected in the
induced subspace topology, then $X$ is also connected
ATTEMPT
Suppose $X = U \cup V $ for open, nonempty, disjoint... |
H: Euler Totient clarification
I'm asked to determine what $\varphi{(p^k)}$ is for an arbitrary prime $p$. By definition, $\varphi{(p^k)}=p^k\left(1-\frac1{p}\right)=p^k\left(\frac{p-1}{p}\right)=p^{k-1}(p-1)$. But I thought that since the Totient function was multiplicative that $\varphi{(p^k)}=\varphi{(p)}^k=(p-1)... |
H: Why does an algebraic curve over an algebraic closed field have smooth points?
Is there an easy way to see this fact? I could try to show that the differential of the defining polynomial cannot vanish at all the zeros. However, I don't see how this could be done. Also there should be a more sophisticated way.
Thank... |
H: Proving that $A \cap (A \cup B) = A$ . Please check solution
For homework I need to prove the folloving:
$$
A \cap (A \cup B) = A
$$
I did that in the following manner:
$$
A \cap (A \cup B)\\
x \in A \land (x \in A \lor x \in B)\\
(x \in A\ \land x \in A) \lor (x \in A \land x \in B)\\
x \in A \lor (x \in A \land x... |
H: How to determine Depedent and Span of matrices?
$ \displaystyle s= (2,4,6)^T ,(0,0,0)^T ,(0,1,1)^T \in R^3 $
Does S are dependent linear?
Does S are Span of $R^3$ ?
AI: A set of vectors $\mathbf{v}_1,\mathbf{v}_2,\ldots,\mathbf{v}_n$ is said to be linearly independent if
$$\alpha_1\mathbf{v}_1+\alpha_2\mathbf{v}... |
H: chain of compact subspaces must be nonempty
Suppose $X$ is a compact space and suppose $\{ F_i \} $ is a collection of closed subsets of $X$ such that $F_{i+1} \subseteq F_i $ for all $i$. , then we must have $\bigcap_i F_i \neq \varnothing $.
MY try: Suppose $\bigcap_i F_i = \varnothing $. We know the $\{ F_i \} ... |
H: How to take this limit? $\lim_{x \to \infty}(x-ln^3(x))$
Need to take the limit:
$$\lim_{x \to \infty}(x-ln^3(x))$$
My idea is to come to indeterminate form $(\frac{0}{0})$, then use L'Hospital's rule:
$$\lim_{x \to \infty}(x-ln^3(x))=ln\lim_{x \to \infty}\frac{e^x}{e^{ln^3(x)}}=\left(\frac{0}{0}\right) = ln\left( ... |
H: Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group
Show that $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ is not a cyclic group. This question is from the book 'Of Abstract Algebra' by Pinter. Now $\mathbb{Z}_{2} \times \mathbb{Z}_{4}$ containt 8 elements. I found them to be as follows:
$$(0,0),(0,1),... |
H: Some basic questions about vectors
I've got two quite basic questions about vectors. I'm sorry if it isn't right to put two questions at the same thread. I'm quite confused about the technique of solving such problems.
Let $\vec v=(3,-4)$, $\vec u=(1,2)$. Find two vectors $\vec w_1, \vec w_2$ so that:
(i) $\vec ... |
H: Denesting Phi, Denesting Cube Roots
I have been looking into denesting square roots but I have found that $\sqrt[3]{2+\sqrt{5}}$ equals $(1+\sqrt{5})/2$. The same is true for $\sqrt[3]{2-\sqrt{5}}$ and $(1-\sqrt{5})/2$. I cannot figure how this is true. I proved this by setting both equal to x and forming polynomi... |
H: Integrate $\frac{x}{1+x^4}$
How do I integrate something like this? $$\int \frac{x}{1+x^4}\mathrm{d}x$$
I've tried trig substitutions but none have worked.
AI: Let $u = x^2$. Then $\,du = 2x \,dx $.
$$\int \dfrac x{1 + x^4} \,dx = \dfrac 12 \int \dfrac{2x\,dx}{1 + (x^2)^2} = \dfrac 12 \int \dfrac{\,du}{1 + u^2}$$
... |
H: $f$ uniformly continuous , $a_n$ Cauchy $\Rightarrow f(a_n)$ is Cauchy
Let $I$ be an interval and let $f: I\to \mathbb{ R}$ be uniformly continuous on I. Suppose that $\{a_n\}$ is a Cauchy sequence in $I$. Prove that $\{f(a_n)\} $is a Cauchy sequence.
AI: Fix $\epsilon>0$, Since $f$ is uniformly continuous,$\exists... |
H: evaluation of $\int_{0}^{1}\frac{1-x^2}{\left(1+x^2\right)\sqrt{1+x^4}}dx$
$\displaystyle \int_{0}^{1}\frac{1-x^2}{\left(1+x^2\right)\sqrt{1+x^4}}dx$
By Using Substution $\sqrt{1+x^4} = (1+x^2)\cdot \cos \theta$
I have Tried it without using the given substution.
$\bf{My\; Try}::$ $\displaystyle \int_{0}^{1}\frac{1... |
H: How find this $\lim_{n\to\infty}n^2\left(\frac{1^k+2^k+\cdots+n^k}{n^{k+1}}-\frac{1}{k+1}-\frac{1}{2n}\right)$
Find this limit
$$\lim_{n\to\infty}n^2\left(\dfrac{1^k+2^k+\cdots+n^k}{n^{k+1}}-\dfrac{1}{k+1}-\dfrac{1}{2n}\right)\tag{1}$$
I can only solve this limit
$$I=\lim_{n\to\infty}\left(\dfrac{1^k+2^k+\cdots... |
H: Understanding issue of basic sets
I have some answers here, that I can barely understand.
1) $\{a,b,\{a,b\}\} - \{a,b\} = \{a,b\}$. The answer indicates it is wrong. I think it is correct, what is it that I cant see? (very confused here).
2) $\{a\} \subseteq \{a,b,\{\{\{a\},b\}\}\}$. Correct. Mr. obvious (?) ...
3)... |
H: Let $G$ be a group, then let $f :G\to G$ via $f(x) = x^2$. Now, is $f$ injective and/or surjective?
Let $G$ be a group, then let $f :G\to G$ via $f(x) = x^2$. Now, is $f$ injective and/or surjective? To this end let $f(x)=f(y)$. Then
$$x^2 =y^2 $$
If we take a look at the group $( \Bbb Q \setminus\{0\} , \cdot )$,... |
H: Prove that every manifold is regular.
Prove that every manifold is regular and hence metrizable.
AI: HINT: A space $X$ is regular if and only for each point $x\in X$ and each open nbhd $U$ of $x$ there is an open set $V$ such that $x\in V\subseteq\operatorname{cl}V\subseteq U$. Use the fact that a manifold is local... |
H: Solving partial fraction expansion with all variables
Okay so I have an equation in my book which is as follows..
$$
\frac {a}{s(s+a)}
$$
it says "using partial fractions this can be expanded to
$$
\frac {1}{s} + \frac {-1}{s+a}
$$
My usual method would be to cross multiply and do something like this
$$
\frac {a}{... |
H: subsequence of unitary matrices has two limit
Consider the sequence of unitary matrices $U_k=\left(\begin{array}{cc} 0 & 1 \\
1 & 0 \end{array}
\right)^k,k=1,2,\dots$
I am not able to show that there are two possible limits of subsequences
Could anyone help me how?
AI: HINT: Calculate a few powers of $U$ to see w... |
H: How to find the biggest m that a function is in O($h^m$)
i try to find the biggest m that
$$f(h) = h^{-2}(\sin(1+h)-2\sin(1)+\sin(1-h))+\sin(1)$$
is $\in O(h^m)$ ($h \to 0, h > 0$)
I tried this:
$$ f(h) = h^{-2} (((1+h)-\frac{(1+h)^3}{3!}+\frac{(1+h)^5}{5!} \mp ...) -2(1-\frac{1}{3!}+\frac{1}{5!} \mp...) +... |
H: If 2 vectors form a basis for $\mathbb{R}^2$, must these 2 vectors always be orthogonal to each other?
If 2 vectors form a basis for $\mathbb{R}^2$, must these 2 vectors always be orthogonal to each other?
For instance, the standard bases in $\mathbb{R}^2$ are definitely orthogonal (easily drawn). How about other b... |
H: Not injective given cardinalities of sets
How do I prove that a function $f:G \rightarrow H$ is not one-to-one if $|G|=20$ and $|H|=24$?
AI: You can't prove that $f$ is not one-to-one. It may very well be.
What you can prove is that $f$ is not onto (surjective).
Aside:
If you are working with a group homomorphism ... |
H: Degree of continuous maps from S1 to S1 - Two equivalent properties
I understand what is meant by the degree of a continuous map $f$ from $S^1$ to $S^1$. If we let $[S^1, S^1]$ denote the set of homotopy classes of continuous maps from $S^1$ to $S^1$, it turns out that the degree map gives a bijection from $[S^1, ... |
H: CF grammar on this language
I'm trying to write a context-free grammar for this language:
$L = \{a^n b a^m (bb)^n : m \ge 1, n \ge 0\}$
I was getting lost with maintaining $n$ number of $a$'s and $(bb)$'s and I'm not sure how to fix it. The terminals don't seem to be working out because I think splitting it up bet... |
H: Is the empty set always part of the result of an intersection?
From very basic set theory we have that:
"The empty set is inevitably an element of every set."
Then, is it correct to assume that the intersection of
$A = \{1, 2, 3\}$ and $B = \{3, 4, 5\}$ is actually $\{\emptyset, 3\}$, and not just $\{3\}$?
Thank ... |
H: Infinite sum $\sum_{n=1}^\infty{\frac{1}{n2^n}}$
How do I evaluate this sum:
$$\sum_{n=1}^\infty{\frac{1}{n2^n}}$$
AI: Another way is
$$\sum_{n=1}^\infty{\frac{1}{n2^n}}=\sum_{n=1}^\infty{\int_0^{1/2}x^{n-1}\,\mathrm{d}x=\int_0^{1/2}\sum_{n=1}^\infty x^{n-1}\,\mathrm{d}x=\int_0^{1/2}{\frac{1}{1-x}}}\,\mathrm{d}x=\l... |
H: How to determine basis by Reducing a sets
How to do Reducing set of $(x_1,x_2,x_3,x_4)$
So that form basis for $\mathbb{R}^3$
for vectors:
$\displaystyle x_1 = \begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}x_2=\begin{bmatrix} -3 \\ 2 \\ 1 \end{bmatrix}x_3=\begin{bmatrix} 3\\ 2\\ -1 \end{bmatrix}
x_4=\begin{bmatrix} -2\\... |
H: Using limit of sinx/x as x approaches 0 to simplify the equation
So, I know that I'm supposed to use the rule that the limit of $\sin(x)/x$ as x approaches 0 is equal to 1 to simplify the following:
$$\lim_{x\to0}\frac{\sin(x)\cos(4x)}{x+x\cos(5x)} $$
However, I'm not sure where to simplify. Which $x$ on the denom... |
H: $H_0 ( X )$ is free abelian on the path components of $X$ .
I do not quite understand this statement. Does this mean if there is one path-components, then $H_0 ( X )$ is generated by one generator, or cyclic. And if there is two path-components, then $H_0 ( X )$ is generated by two generator, but of the form $a^i b... |
H: find a generator for the group $G =\{ f(x) = x+n\mid n\in \Bbb Z \}$ with the group operation being composition.
Another question from 'A book of Abstract Algebra' by Pinter.
For each $n\in \Bbb Z$ define $f_n = x+n$. Then $f_n\in S_{\Bbb R}$, the symmetric set on $\Bbb R$. The group operation being composition. No... |
H: divisibility of $n^{2}+n+1$ by $6k-1$ when $n,k$ are integers.
I wanted to prove that for any $n\in\mathbb{Z}$, the integer $n^{2}+n+1$ does not have
any divisors of the form $6k-1$. So far, I proved $n^{2}+n+1\neq-1 \pmod 6$ and an integer which is $-1\pmod6$ should have a prime factor which is also $-1 \pmod6$. ... |
H: Negation of statement and determining truth
a,b $\in$ $\mathbb{R}$
Original statement: $\exists a$ such that $\forall b$, $a+b>0$
My negation: $\forall a$, $\exists b$ such that $a+b \leq 0$
Is my negation correct? If it is, is the negation true whereas the original statement is false?
I drew this conclusion becaus... |
H: Proving using vectors, that if a median is also a height, then the triangle is isosceles.
Proving using vectors, that if a median is also a height, then the triangle is isosceles.
*Better wording would be very helpful.
Thanks in advance for any help.
AI: Let the vertices be the origin $O$, and the points $A$ and $B... |
H: $5^a - 5^b$ is divisible by $n$ (prove)
Prove that for every n natural number exist natural numbers $a,b \leq 4n, a\not= b $, which accomplish, that number $ 5^a - 5^b $ is divisible by n. How many of these pairs exist?
Help please, I'm stuck with this problem. Any help is appreciated.
AI: Let $n=5^km$ with $gcd(5... |
H: Show that interval $(a, b)$ is not open in $\mathbb{R}^2$
I know that interval $(a, b)$ is open in $\mathbb{R}$. To show that interval $(a,b)$ is open in $\mathbb{R}$, I have done so: Let it be $x\in (a,b)$. Enough to find an open ball containing the point $x$, and that is included in the interval $(a,b)$. Suffice ... |
H: Computing $\int_{0}^\infty\frac{t^a}{1+t^2}dt$ for $-1
I am integrating the following integral
$$\int_{0}^\infty\frac{t^a}{1+t^2}dt$$ for $-1<a<0$.
by computing residues inside some contour. But I'm not sure what contour to use here, since $z^{a}$ gets large when $z$ is small, so I have to be careful about the p... |
H: $Y$ is a function of $X$: making an inference based on the markovity of $ X$
In the information theory book by Cover and Thomas it is written:
if $X$ is markov and $Y$ is a function of $X$ then:
$H(Y_n|Y_{n-1},Y_{n-2},...,Y_1,X_1)=H(Y_n|Y_{n-1},Y_{n-2},...,Y_1,X_1,X_0,X_{-1},...,X_{-k})$ because $X$ is markov.
Can ... |
H: Does $\text{Aut}(A)\cong \text{Aut}(B)\cong \text{Aut}(A\bigcap B)$ imply $A=B$?
The question is in the title: given two mathematical structures $A$ and $B$ that we conjecture to be equal, is it sufficient to prove that $\text{Aut}(A)\cong \text{Aut}(B)\cong \text{Aut}(A\bigcap B)$, where $\text{Aut}(X)$ is the aut... |
H: What does this huge X mean that is written like the sigma notation?
I hope that you will not mind if I do not explain the background of this formula. The problem which I encounter is probably a simple one: What does the huge $\large \times$ mean and why is it written like a sigma? I had no clue how to search for th... |
H: Order of a homomorphism of groups
Let $f: G \to H$ be a homomorphism of groups. Assume that $a\in G$ and $\operatorname{ord}(a)=n$. Prove that the order of $f(a)$ is a divisor of $n$.
I know that if $H$ is a subgroup of $G$, then $$|G|=|H|\cdot|\text{distinct cosets of $H$}|$$ so $|H|$ must be divisor of $|G|$, but... |
H: Locus in complex z-plane given eqn
I have the question:
$
\text{Find the locus in the complex }z\text{-plane that satisfies the equation: }
z-c=\rho\dfrac{1+it}{1-it}, \text{where }c\text{ is complex, }\rho\text{ is real, and }t\text{ is a real parameter that varies in the range }-\infty<t<\infty.
$
[1]
But I am un... |
H: Let $A$ be an infinite set, $B\subset A$, let $G \subset S_A$ where $f$ in $G$ implies $x\in B\implies f(x)\in B$. find example where $G$ not a group
Let $A$ be a finite set and $B$ a subset. Let $G$ be the subset of $S_A$ (the symmetric group on $A$) consisting of all the permutations $f$ on $A$ such that $f(x)\in... |
H: Decide with proof if the groups are isomorphic.
Decide with proof if the groups are isomorphic.
$(a)\quad (\mathbb Z, +)$
$(b)\quad (3\mathbb Z, +)$
$(c) \quad$ The additive group of the rational numbers.
I've just done questions of the form groups "$C_2\times C_2$" etc but not really sure how to go about this.
Tha... |
H: Why $\arctan x \to \pi/2$ when $x \to \infty$
The limit of $\arctan x$ as $x \to +\infty$ is $\pi/2$. Maybe I don't understand the arctan enough, because by looking at the graph I see that indeed it approaches pi. But: why? How can I understand that?
AI: $\tan{x}$ is the ratio of the opposite side to the adjacent s... |
H: The intervals at which $f(x) = \cos x + \sin x$ is concave up and down and $f$'s inflection point(s)
Describe the intervals at which $f(x) = \cos x + \sin x$ is concave up and down and $f$'s inflection point(s).
I totally went blank on this once I hit the analysis of second derivative, $f''(x)=-\cos x-\sin x$. I fo... |
H: Probability of Combinations which have different probabilties.
I cannot figure this out. The problem goes
A boy has a bag filled with balls:
1 red
2 green
4 Blue
7 White
A child plays a game in which he withdraws one ball, writes down its color, and then replaces the ball. In order to win the game he must write ... |
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