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H: Is it possible to use the Central limit theorem for standard Cauchy distribution?
I have a simple question about CTL (Central limit theorem)
Is it possible to use the Central limit theorem for standard Cauchy distribution?
I think that it´s not possible because the Mean of the standard Cauchy distribution is undefi... |
H: Recurrence relations problem (linear, 2nd order, constant coeff, homogeneous)
I'm currently stonewalled on this problem, in which I have to solve the following recurrence relation and prove my answers satisfy the recurrence.
My boundary conditions are $a_0 = 1$ and $a_1 = 9$.
The problem is $\forall\ n \in N$, $a_{... |
H: Probability Density Function of a Minimum Function
Suppose $U_1, U_2, \dots, U_5$ are independent $\operatorname{unif}(0,1)$ random variables.
Suppose $T = \min(U_1, U_2, \dots, U_5)$.
How would I find the p.d.f. of $T$? I know how to do regular cases like $T = A + B$ given the distribution of $A$ and $B$ but how d... |
H: Interval of convergence of $\sum\limits_{n\geq0} \binom{2n}{n} x^n$
We consider the power series $\displaystyle{\sum_{n\geq0} {2n \choose n} x^n}$. By Ratio Test, the radius of convergence is easily shown to be $R=\frac{1}{4}$.
For $x=\frac{1}{4}$, Stirling equivalent and Ratio Test imply that the series is diverge... |
H: Uniform convergence of functions and intervals
We define $f_n:\mathbb{R}\to\mathbb{R}$ by $f_n(x)=\dfrac{x}{1+nx^2}$ for each $n\ge 1$.
I compute that $f(x):= \displaystyle\lim_{n\to \infty}f_n(x) = 0$ for each $x\in\mathbb{R}$.
Now, I want to know in which intervals $I\subseteq \mathbb{R}$ the convergence is unifo... |
H: Why do we use the inverse conversion formula to convert slope per radians to slope per degrees
This is a contribution question I'm making in hopes that others may benefit. I will provide my answer underneath. Initially I wanted to ask this question, but I solved it myself and I'd like to give back for the question ... |
H: Is the curl of every non-conservative vector field nonzero at some point?
Counterexamples? Intuitively, why?
Thanks for any answers.
As a side note, in what math class are gradient, divergence and curl taught typically?
AI: The question to ask is:
If there is a smooth vector field $\mathbf{v}$ such that
$$\nabl... |
H: Question on $\mbox{Ext}^1$
I have 2 questions, one of them concerning the isomorphicity of quotient groups (rings), and the other is on $\mbox{Ext}^1$. It's pretty long, but somehow related to each other. So I just kinda put everything here, instead of dividing it into different posts.
Question 1
I know that given ... |
H: Exercise 28 of Chapter 1 in Stein-Shakarchi's Real Analysis
I am doing an exercise [similar to] Exercise 28 in Chapter 1 of Stein-Shakarchi's Real Analysis (page 44):
For any $E \subseteq \Bbb{R}^d$, $0 < \alpha <1$, we can find an open set $O \supseteq E$ such that $m_\ast(E) > \alpha m_\ast (O)$ where $\ast$ mea... |
H: Infinitely many zeros of a nonconstant continuous function?
Let $f:[0,1]\to\mathbb{R}$ be a nonconstant continuous function. Is $S=\{x: f(x)=0\}$ finite?
I have thought of a function with countably many $0$'s like lots of triangular bumps at each point $\{1/n\}$, I mean lots of $W/M$ shapes on $[0,1]$. Is it okay?... |
H: $\{\cos n+\sin n\}$ have a convergent subsequence?
Does the $\{\cos n+\sin n\}$ have a convergent subsequence?
I am totally clueless.
AI: HINT: It’s a bounded sequence: $-2\le\cos n+\sin n\le 2$ for all $n$. That should be enough, but if not, a further hint is spoiler-protected below.
See the Bolzano-Weierstrass ... |
H: If $f \circ g = f$, prove that $f$ is a constant function.
Suppose $A$ is a nonempty set and $f: A \rightarrow A$ and for all $g:A \rightarrow A,$ $f \circ g = f$. Prove that $f$ is a constant function.
This result seems obvious, but I can't seem to find a way to prove it. The book I got this problem from hinted ... |
H: two sequences asymptotic, then their differences go to zero
One last problem before I go to sleep, I must be too tired to see this one. This is another qual type question.
Let $\{x_{n}\}$ and $\{y_{n}\}$ be sequences of real numbers such that $\forall n \in \mathbb{N}$
$y_{n} \neq 0$ and $\frac{x_{n}}{y_{n}} \righ... |
H: Computing intersection of two subspaces of $C^{\infty}_{2\pi}(\mathbb{R},\mathbb{R})$.
I've been thinking of the following two subspaces of $C^{\infty}_{2\pi}(\mathbb{R},\mathbb{R})$:
$$
A=\{a_1\sin(t)+a_2\sin(2t)+a_3\sin(3t):a_1,a_2,a_3\in\mathbb{R}\}
$$
and
$$
B=\{b_1\sin(t)+b_2\sin^2(t)+b_3\sin^3(t):b_1,b_2,b_3\... |
H: Application of Green's Theorem
I know this is a really basic question, but I seem to be kind of rusty.
$C$ is the boundary of the circle $x^2+y^2=4$
$$\int_C y^3dx-x^3 dy = \int_A -3x^2-3y^2 dA= \int_0^{2 \pi} \int_0^2 -3 r^2 r dr d \theta = -12 \pi$$
Did I make a mistake? My book says it's $-24 \pi$
AI: As noted ... |
H: does $\intop_{1}^{\infty}x\sin(x^{3})dx$ really converge?
I'm trying to find a continuous function $f(x)$ on $[0,\infty)$ such that:
$\intop_{1}^{\infty}f(x)dx$ converges while $f(x)$ isn't bounded.
I came up with $f(x)=x\sin(x^{3})dx$, as a function which oscillates like crazy when x tends to infinity, and much ... |
H: Does this function have a (global) minimum?
A good day to everyone.
Does the following function have a (global) minimum:
$$1 + \frac{1}{x} + {\left(1 + \frac{1}{x}\right)}^\theta,~~x\in\mathbb R$$
where
$$\theta = {\displaystyle\frac{3\log 2 - \log 5}{2(\log 5 - 2\log 2)}} > 1?$$
WolframAlpha says it has none.
AI: ... |
H: Modules decomposition into indecomposables
I think it's not true that every module (over arbitrary ring) is the sum of indecomposable modules, but I can't find counterexample and literature about this problem. Can anyone help me?
Also I have similar question: is every abelian group is quasiisomorphic to the direct ... |
H: Prove that If $f$ is polynomial function of even degree $n$ with always $f\geq0$ then $f+f'+f''+\cdots+f^{(n)}\geq 0$.
I can't solve this problem:
Suppose $f$ is polynomial function of even degree $n$ with always $f\geq0$.
Prove that
$f+f'+f''+\cdots+f^{(n)}\geq 0$.
AI: Let $g=f+f'+\cdots+f^{(n)}$ and let $h(x)=e^{... |
H: Proof using Möbius transformation
Let D be the open unit disc, and $f:D\to D$ an analitic function.
How can I prove that $|f'(0)|\le1$?
AI: For every $R\in (0,1)$ we have by Cauchy's Integral formula: $$|f'(0)|=\left|\frac{1}{2\pi i}\int_{|z|=R} \frac{f}{(z-0)^2} dz\right|\le \frac{1}{2\pi} \int_0^{2\pi} \frac{\max... |
H: Every $v \in V - \{ 0 \}$ is cyclic iff the characteristic polynomial of $T : V \to V$ is irreducible over $F$
Let $V\neq \{0\}$ be a vector space over $F$, and $T$ a linear operator on $V$. Prove that every $0\neq v \in V$ is a cyclic vector if and only if the characteristic polynomial of $T$ is irreducible over $... |
H: Let $I$ be an ideal generated by a polynomial in $\mathbb Q[x]$. When is $\mathbb Q[x] / I$ a field?
I was looking at my old exam papers and I was stuck on the following problem:
Let $I_1$ be the ideal generated by $x^4+3x^2+2$ and $I_2$ be the ideal generated by $x^3+1$ in $\mathbb Q[x]$. If $F_1=\mathbb Q[x]/I... |
H: Why does this inequality hold: $4n+2\le4n\log{n}+2n\log{n}$
Why is the following true? (I came across this in an algorithm analysis book but this inequality is not related to algorithm analysis)
$$
4n+2\le4n\log{n}+2n\log{n}
$$
AI: Well based on the comments, it appears there is no mistake and the OP wants to see... |
H: Integral function
This exercise asks me to calculate the integral function with starting point $x=0$ of the following function:
\begin{equation}
y=
\begin{cases}
2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x<0\\
x+2\ \ \ \ \ \ \ \ 0\leq x\leq 2\\
4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x>2\\
\end{cases}
\end{equation}
Is my solution cor... |
H: Angle between matrices
This is the problem from my homework:
If $A$ is antisymmetric matrix, and $S$ is symmetric matrix where $A,S \in M_n (\mathbb{R})$, determine the angle between them according to the inner product defined as $(A,B)=\text{tr}(AB^T)$.
I have tried calculating the angle between two pairs of con... |
H: Does the following number-theoretic equation have any solutions?
Does the following number-theoretic equation have any solutions for prime $p$ and positive $x$?
$$\frac{\sigma(p^x)}{p^x} = \frac{10}{9}$$
It does not have any for $x = 1$. How about for $x > 1$?
(Note that $\sigma = \sigma_1$ is the sum-of-divisors ... |
H: Every orthogonal operator is diagonalizable?
Answer is false and the rotation is a counterexample.
But I can't understand well.
Let $A=\begin {pmatrix}0&-1\\1&0\end{pmatrix}$ then it is rotation and also orthogonal operator. I think it can be diagonalizable by eigenvalue $i,-i$.
AI: They (implicitly) meant "diagona... |
H: Linear vs nonlinear differential equation
How to distinguish linear differential equations from nonlinear ones?
I know, that e.g.:
$$
y''-2y = \ln(x)
$$
is linear, but
$$
3+ yy'= x - y
$$
is nonlinear.
Why?
AI: Linear differential equations are those which can be reduced to the form $Ly = f$, where $L$ is some line... |
H: proving congruence of a number modulo 17
We need to prove that $3^{32}-2^{32}\equiv0\pmod{ 17}$.How can we do that?
I tried to express them modulo $17$ in such a way that both cancel out.Really has not helped much.A little hint will be appreciated.
AI: HINT:
Using Fermat's Little theorem, $$a^{16}\equiv1\pmod{17}\... |
H: Spectral radius and operator norm
Consider a FINITE endomorphism $A$ , then I was wondering whether the relation between the operator norm and the spectral radius $\rho$, given by:
$\|A\| \ge \rho(A)$ is true for all operator norms or only the 2-norm?
AI: It is true, not only for all operator norms, but also for al... |
H: Geometrically distributed RV's - what's wrong with my reasoning?
Let $X$ and $Y$ be iid random variables distributed geometrically with probability of success $p$ and support $\mathbb{N}=\{0,1,2,\cdots\}.$ So in particular, letting $q=1-p$, we have that $$\mathbf{P}(X=k)=\mathbf{P}(Y=k)=pq^k\quad\forall k \in \math... |
H: The principal ideal $(x(x^2+1))$ equals its radical.
Let $\mathbb R$ be the reals and $\mathbb R[x]$ be the polynomial ring of one variable with real coefficients. Let $I$ be the principal ideal $(x(x^2+1))$. I want to prove that the ideal of the ideals variety is not the same as its radical, that is, $I(V(I))\not=... |
H: Understanding examples of subfield and prime subfield of a finite field
I have already taken a look at this answer. Somehow it did not answer my question.
As I can find, in various literatures,
A lecture note, Definition 4.1:
Let
$F$
be a field. A subset
$K$
that is itself a field under the operations of
$F$
is ca... |
H: Uniform convergence of a power series
I'm new to this subject and would very much appreciate your help with this question. I'm not really sure about how to approach this.
$$f(x) = \sum_1^\infty\frac{1}{n}x^n$$
If I'm not mistaken the domain of convergence of this function is $x=[-1,1)$.
I need to check if it con... |
H: How to solve a equation with floor in it?
I tried to do everything I could, but I don't know what to do with that floor.
$58 = y\cdot\left[\frac{80}{y}\right]$
Where $[x]$ is floor function.
AI: HINT:
So, $\frac{58}y= \left[\frac{80}{y}\right] $ which is an integer
$\implies y$ must divide $58$ |
H: Ring without zero divisors that has positive characteristic must have prime characteristic
Let $R$ be an integral domain and suppose $R$ has characteristic $n > 0$.
Prove that $n$ must be prime.
I just proved this exercise, but I think it needs extra conditions. We can prove the statement if $R$ is ring without... |
H: Study regimen for discrete mathematics? - Lack high-school maths...
I have just gotten into college, and will be studying mathematics from next semester. (this course)
Unfortunately I did not study mathematics for the last 2-3 years of high-school mathematics.
What should I study for the next 50 days in preparation... |
H: Switch place of 2 infinite summations
In our course about queuing theory sometimes this rule is used:
$$\sum_{n=0}^\infty \sum_{k=n+1}^\infty n\cdot k = \sum_{k=1}^\infty \sum_{n=0}^{k-1} n\cdot k$$
Another example (but without infinity):
$$\sum_{n=0}^m n \sum_{i=0}^{n-1} i = \sum_{i=0}^{m-1} i \sum_{n=i+1}^{m} n... |
H: Does this inequality have any solutions in $\mathbb{N}$?
Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$?
$$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$
Notice that we necessarily have $x > 1$.
AI: Yes; in fact, almost any deficient number will do. Let us show that this is the case for t... |
H: Concurrent chances calculation
I have a question I don't know how to solve correctly, hope you can help me.
Let's say I have something bad, a cancer or something, and I have each year 1 chance between 750 of dying. The more years I live with the cancer, the more years I am exposed to death.
I want to calculate what... |
H: Complement of a Topology
Let $(X, \tau)$ topology, I was wondering, if given $$\tau'=\{A^C \mid A \in \tau\}$$ did $\tau'$ also a topology on $X$? If so, why? Thank you.
AI: Not in general as closed sets aren't usually stable under infinite unions. If $(X,\tau)$ has this property, we say that $(X,\tau)$ is an Alexa... |
H: Can the inverse of an element of a countable set approach infinity?
Say we have a countable set $A$.
$f:\mathbb{N}\to A$.
Can we say that there exists at least one element $a\in A$ such that $f^{-1}(a)$ is greater than any number $n\in \mathbb{N}$ we choose. Remember that $a$ is an element we explicitly know.
AI:... |
H: Evaluate $\int_0^\infty \frac{\log(1+x^3)}{(1+x^2)^2}dx$ and $\int_0^\infty \frac{\log(1+x^4)}{(1+x^2)^2}dx$
Background: Evaluation of $\int_0^\infty \frac{\log(1+x^2)}{(1+x^2)^2}dx$
We can prove using the Beta-Function identity that
$$\int_0^\infty \frac{1}{(1+x^2)^\lambda}dx=\sqrt{\pi}\frac{\Gamma \left(\lambda-\... |
H: Eigenvectors of inverse complex matrix
For a non-singular matrix, its pretty straightforward to prove that $\lambda$ is eigenvalue of $A$ if and only if $\frac{1}{\lambda}$ is eigenvalue of $A^{-1}$. Let $A$ be a non-singular matrix, $x$ an eigenvector of $A$ and $\lambda \neq 0$ its eigenvalue :
$$Ax = \lambda x \... |
H: Show that $\int^{\infty}_{0}\left(\frac{\sin(x)}{x}\right)^2 < 2$
I`m trying to show that this integral is converges and $<2$
$$\int^{\infty}_{0}\left(\frac{\sin(x)}{x}\right)^2dx < 2$$
What I did is to show this expression:
$$\int^{1}_{0}\left(\frac{\sin(x)}{x}\right)^2dx + \int^{\infty}_{1}\left(\frac{\sin(x)}{x... |
H: Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?
I understand "transform methods" as recipes, but beyond this they are a big mystery to me.
There are two aspects of them I find bewildering.
One is the sheer number of them. Is there a unified framework that includ... |
H: Function with continuous derivative is continuous?
Is it true that if $\frac{d}{dx}f(x)$ is continuous, then $f(x)$ is continuous too?
If not, can you give a counterexample?
AI: Just the fact that your function $f(x)$ is differentiable is enough to prove that it is continuous. The derivative $\frac{d}{dx}f(x)$, ne... |
H: calculate $ F(x)= \int_{0}^{\sin x}\sqrt{1-t^2}\,dt $
Calculate $F'(x)$
I have this exercise in my worksheet, I am having a problem obtaining the correct answer which is as listed on the answer sheet $-\cos^2(x)$.
AI: You have $F(x)=G(\sin x)$ where $G(y)=\int_0^y\sqrt{1-t^2}dt$ satisfies, by the fundamental theo... |
H: An equation about a rectangle with given perimeter
I am doing a revision calculator paper and am stuck on an algebra question.
There is a picture of a rectangle. One side is $x-2,$ another side is $2x +1.$
The question is. Setup and solve an equation to work out the value of $x.$
The perimeter of this rectangle i... |
H: Bayes' Theorem problem - Enough information?
There are 5000 British people. A British is either an English, a Scott
or a Welsh. 30% of the British are Scottish, and the English are six
times the Welsh. The probability that a British has red hair is 0.25,
while the probability that a non-Welsh British has red... |
H: Prove that $E_1E_2= E_2E_1 = E_2$
I have this problem about projections I don't understand, Can somebody help me please?
Let $V$ be a vector space over the field $F$ and let $E_1$ and $E_2$ are projections of V with image $R_1$ and $R_2$ and nullspaces $N_1$ and $N_2$ respectively. Suposse that $R_2\subset{R_1}$ a... |
H: Integral of $\int^{\infty}_0\frac{x}{x^4+1}\,dx$
I want to evaluate this integral and trying to figure how to do that.
$$\int^{\infty}_0\frac{x}{x^4+1}dx$$
What I did is:
I`m abandons the limits for now and do the following step:
$$\int^{\infty}_0\frac{x}{x^4+1}dx = \frac{1}{2}\int\frac{tdt}{t^2+1} $$
I can use th... |
H: What is the use of the Dot Product of two vectors?
Suppose you have two vectors a and b that you want to take the dot product of, now this is done quite simply by taking each corresponding coordinate of each vector, multiplying them and then adding the result together. At the end of performing our operation we are ... |
H: For which values of $a$ the function $f(x) = \max(x^2+2x,a) $ can be differentiate
I want to find for what values of $a$ the function can be differentiate.
$$f(x) = \max(x^2+2x,a) $$
What I tried to do is:
$$f(x) = \begin{cases} x^2+2x &\text{if } a>\dots \\
a& \text{if } a<\dots \end{cases} $$
the derivative of... |
H: On the fundamental theorem of field extensions
I'm re-reading the fundamental theorem of field extensions. (K is normal $\iff$ K is a factorization field.)
Assume $K=F(\alpha_1, \dots , \alpha_n)$, is the factorization field of $f\in F[x]$, over the field $F$. Then for each $\alpha\in K, \alpha = g(\alpha_1,\dots,\... |
H: rank of a diagonal matrix + rank-one perturbation
Let $D$ be a $n \times n$ diagonal matrix, and $A$ is a $n \times n$ rank-one matrix that can be rewritten as $A=a\cdot b^T$, where $a$ and $b$ are $n \times 1$ vectors. Now what is the lower bound for the rank of the matrix $D+A$ ? Any suggestions are welcome.
AI: ... |
H: Limit of $\lim\limits_{n\to \infty}\left(\frac{1^\frac{1}{3}+2^\frac{1}{3}+3^\frac{1}{3}+\dots+n^\frac{1}{3}}{n\cdot n^\frac{1}{3}} \right)$
I want to evaluate this limit.
$$\lim_{n\to \infty}\left(\frac{1^\frac{1}{3}+2^\frac{1}{3}+3^\frac{1}{3}+\dots+n^\frac{1}{3}}{n\cdot n^\frac{1}{3}} \right)$$
What I did is:
se... |
H: finding the maximum area of 2 circles
An equilateral triangle with height $h$ has 2 different incircles.
the bottom circle is tangent to the base of the triangle at the middle point of the base.
what should be the radius of the upper circle so the sum of the area of the circles will be maximum?
i tried to to find ... |
H: Distribution of Modular Expressions
This is really a programming question that I'm unable to solve because of a math question. I'm having a problem understanding the rules of distribution with modular arithmetic.
I have two expressions:
a = q mod c + (n - 1) mod c
b = q mod c - (n - 1) mod c
I made the mistake of ... |
H: Generalizing the central series to ordinal length
One can generalize the ascending and descending central series by transfinite induction, setting $G _{\alpha +1}=[G_\alpha, G]$ and $ G_\beta= \cap _{\alpha <\beta} G_\alpha $ (and analogously for the upper centers).
What is known about these series? Above all, if o... |
H: Question on linear maps defined in Khovanov homology
There are two linear maps $m:V \otimes V \rightarrow V$ and $\Delta:V \rightarrow V\otimes V$ in the definition of the differential of Khovanov homology. So my question is why do they map elements as below?
$$m:v_+ \otimes v_- \rightarrow v_-$$
$$m:v_+ \otimes v_... |
H: Rouche's theorem problem
Prove that the polynomial $z^n + nz-1$ has $n$ zeroes inside the circle with centre at $0$ and radius $1+\sqrt{2/(n-1)}$ for $n=3,4,\dotsc$
Please give me some hints as to how to apply Rouche's nicely without expanding binomials with square roots all over the place? Thanks!
AI: Hint:Do you ... |
H: There are no integers $x,y$ such that $x^2-6y^2=7$
How to show that there is no here are no integers $x,y$ such that $x^2-6y^2=7$?
Help me. I'm clueless.
AI: Hint: Show that, if there are any solutions $x$ and $y$, then it cannot be the case that both $x$ and $y$ are multiples of $7$. Then work modulo $7$. Which of... |
H: Diophantine: $x^3+y^3=z^3 \pm 1$
How many nontrivial integer solutions does $x^3+y^3=z^3 \pm 1$ have?
The trivial solutions are $(\pm 1,z,z)$ and $(z,\pm 1,z)$.
AI: There are infinitely many solutions.
Parametric subset of solutions for $x^3+y^3=z^3+1$:
$x=9n^3+1$,
$y=9n^4$, $\qquad\qquad\qquad$ $(n\in \mathbb{N... |
H: Gauss Lemma for Polynomials and Divisibility in $\mathbb Z$ and $\mathbb Q$.
I am working through Gauss Lemma and various corollaries of it. In the book Algebra of Michael Artin, I have a question to the proof of the following Theorem:
Theorem.
(a) Let $f,g$ be polynomials in $\mathbb Q[x]$, and let $f_0, g_0$ be t... |
H: Integral substitutions.
For integrals of the form:
$$\intop_a^bg(t)dt,$$
we can apply the $tanh$-substitution to transform the integral into a doubly infinite integral, i.e:
$$\intop_a^bg(t)dt = \frac{b-a}{2}\intop_{-\infty}^\infty g \left( \frac{b+a}{2} \frac{b-a}{2}\tanh(u) \right)\mathrm{sech^2(u)} du.$$
My ques... |
H: Factoring a third degree polynomial
I'm trying to find all solutions for $36x^3-127x+91=0$ with $x \in \mathbb{R}$. So, I tried to factor this polynomial. It can be written in the following way:
$$
(ax^2+bx+c)\cdot(dx+e)\quad (a,b,c,d,e \in \mathbb{Z})
$$
with
\begin{cases}
a \cdot d = 36 = 2^2 \cdot 3^2\\
a \cdot ... |
H: How to evaluate binomial coefficients when $k=0$ and $1\geq|n|\geq0$
So I am doing some taylor series which boil down to the geometric series, for which I then need to evaluate various binomial coefficients. I have always used $n\choose k$ $=\frac{n!}{(n-k)!k!}$ to do these, this however becomes troublesome for non... |
H: when Fourier transform function in $\mathbb C$?
The Fourier transform of a function $f\in\mathscr L^1(\mathbb R)$ is
$$\widehat f\colon\mathbb R\rightarrow\mathbb C, x\mapsto\int_{-\infty}^\infty f(t)\exp(-ixt)\,\textrm{d}t$$
When is this indeed a function in $\mathbb C$? Most of calculations you get functions in $... |
H: span of $AA^T$ is the same as $A$?
Suppose $A$ is an $m$ by $n$ real matrix.
How do you prove that the span of columns of $AA^T$ is the same as columns of $A$?
AI: Note: I am using repeatedly the following property: $b \in Col(B)$ if and only if $Bx=b$ is consistent.
$Col(AA^T) \subset Col(A)$ is clear. We prove ... |
H: X is infinite if and only if X is equivalent to a proper subset of itself.
Prove that a set $X$ is infinite if and only if $X$ is equivalent to a proper subset of itself.
If $X$ is finite, then suppose $|X|=n$. Any proper subset $Y$ of $X$ has size $m<n$, and so there cannot be any bijective mapping between $Y$ a... |
H: Solving Lagrange multipliers system
I need help solving this system:
$$
\begin{cases} 2(x-1) = \lambda2x \\ 2(y-2) = \lambda2y \\ 2(z-2) = \lambda2z \\x^2 + y^2+z^2 = 1 \end{cases}
$$
I can find $$ \lambda = (x-1)/x $$ but can't go further.
Any help?
AI: First three equations lead to:
$$\lambda = \frac{x-1}{x} = \... |
H: Using Newton's Generalized Binomial Theorem
I am trying to use Newton's Theorem in a proof with some inequalities. I have something of the form $(a+b)^c$ in my denominator, where $0<c<1$, and I'd like to find a sharp expression less than or equal to this with at least 2 terms (exactly 2 would be nice). Since it is ... |
H: about the rank of block matrix
Let $m,n,$ and $k$ be positive integers and assume that $A\in {\mathbb{R}^{m\times n}}$,$B\in {\mathbb{R}^{k\times n}}$,( that is, A and B are matrices with real entries of sizes $m\times n$ and $k\times n$, respectively). Define $C=\left[\frac{A}{B}\right]$. Prove that $rank(C)\ge ra... |
H: Exterior Measure and Non-Measurable Sets
Can you help me to prove that a set $E\subset\mathbb{R}$ has non-measurable subsets (Lebesgue Measure) if and only if $m^{*}(E)>0$ ?
Somehow i couldn't generalize Vitali Set to prove $\Leftarrow$ direction. The other direction is not a problem !
Thank you :)
AI: Try this ins... |
H: Does the triangle inequality follow from the rest of the properties of a subfield-valued absolute value?
(This is a much more specific version of my earlier question from over a year ago.)
Let $F$ be a field, let $E$ be an ordered subfield of $F$, and let $\;\; |\hspace{-0.03 in}\cdot\hspace{-0.03 in}| \: : \: F \:... |
H: How should I numerically solve this PDE?
I am hoping to figure out the function $u(x,y,t)$ for some integer arguments when $u(x,y,0)$ is given (by figuring out I mean generating some images in MatLab), also time $t \ge 0$.
$$\frac{\partial u}{\partial t} = -(\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\... |
H: How can this implication be equal to the set of all possible evaluations?
I am having a hard time with this (simple) excercise in logic:
First, let us define the set $M$ which contains all possible evaluations.
Then, for each proposition sentence $A$, we define a set $[A] = \{ v \in M | v(A) = 1 \}$
So, the task is... |
H: Order type of the real algebraic numbers
As a countable, everywhere-dense, totally ordered set without minimal or maximal elements, $\Bbb{A}$, the set of real algebraic numbers, must be order isomorphic to $\Bbb{Q}$. I'm wondering how "nice" such an isomorphism can b made. Does it admit any reasonably explicit desc... |
H: Markov Chain Converging in Single Step
I have a markov kernel K. From this I find the invariant probability $\pi$. The question is to design a "dream" matrix K*, that converges in one step. Such that $\lambda_{SLEM}=0$ (second largest eigen-value modulus). I am not sure how to go about designing the dream matrix. A... |
H: Terminologies related to "compact?"
A set can be either open or closed, and there can either be a finite or infinite number of them.
A "compact" set is one where every open cover has finite subcover.
Is there such a thing as a set that is covered by an infinite cover of open subsets, and what would it be called?
H... |
H: Why any field is a principal ideal domain?
Why any field is a principal ideal domain?
According to the definition of P.I.D, first, a ring's ideal can be generated from a single element; second, this ring has no zero-divisor. This two conditions make a ring P.I.D.
But how to prove any field is P.I.D?
AI: Let $F$ be ... |
H: Computing limit of $(1+1/n)^{n^2}$
How can I compute the limit $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n^2}$? Of course $\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}\right)^{n} = e$, and then $\left(1+\frac{1}{n}\right)^{n^2} = \left(\left(1+\frac{1}{n}\right)^n\right)^n$. Since the term inside conver... |
H: distributing z different objects among k people almost evenly
We have z objects (all different), and we want to distribute them among k people ( k < = z ) so that the distribution is almost even.
i.e. the difference between the number of articles given to the person with maximum articles, and the one with minimum a... |
H: Proving every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs
I am going through the proof of
Every open subset $H$ of the plane $\mathbb{R^2}$ is the union of open discs
in a book (open disc is the standard open Euclidean disc or open ball). It goes like this:
Since $H$ is open, for each... |
H: Find the value of each number which derives an average given a starting point, end point and total of numbers used.
Given an average, a low number, a high number and the total amount of numbers used (stated) to derive the average, is there a formula to determine what the value of each number was, that amounted to t... |
H: Question for mathematicians who started before the computer era: what constants did you have memorized, in what form, and why?
A former department chair at BYU, Wayne Barrett, would always amaze grad students by his vast knowledge of mathematical constants, like the radical form of $\cos(2\pi/5)$. I've never memori... |
H: Basic discrete math question regarding translation of logic ↔ English
I just started Discrete Mathematics, and am having a little bit of trouble in understanding the conversions of English ↔ logic.
$p$: "you get an A on the final exam."
$q$: "you do every exercise in the book."
$r$: "you get an A in the class."
... |
H: Computing limit of $(1+1/n^2)^n$
I want to compute the limit $\left(1+\frac{1}{n^2}\right)^n$. One way to do this is to take logs. So $$x_n=\left(1+\frac{1}{n^2}\right)^n$$ Then $$\log x_n = n\log\left(1+\frac{1}{n^2}\right) = n\left(\frac{1}{n^2}-\frac{1}{2n^4}+O\left(\frac{1}{n^6}\right)\right)$$So $\log x_n$ con... |
H: Limit of recursive sequence $a_n=\frac{(x/a_{n-1})+a_{n-1}}{2}$
Let $x$ and $y$ be positive numbers. Let $a_0=y$, and let $$a_n=\frac{(x/a_{n-1})+a_{n-1}}{2}$$Prove that the sequence $\{a_n\}$ has limit $\sqrt{x}$.
I rearranged the equation to be $a_n-\sqrt{x}=\dfrac{(a_{n-1}-\sqrt{x})^2}{2a_{n-1}}$. I think I sh... |
H: Is there a problem when defining exponential with negative base?
Well, this question may seem silly at first, but I'll make my point clear. Suppose $n \in \Bbb N$ and suppose $a \in \Bbb R$ is any number. Then the definition of $a^n$ is clear for any $a$ we choose. Indeed we define:
$$a^n = \prod_{k=1}^na$$
And eve... |
H: Interpolation between iterated logarithms
I am investigating the family of functions $$\log_{(n)}(x):=\log\circ \cdots \circ \log(x)$$
Is there a known smooth interpolation function $H(\alpha, x)$ such that $H(n,x)=\log_{(n)}(x)$ for $n\in\mathbb{N}$?
AI: Your function $H(\alpha,x)$ is the same as $\exp^{-\alpha}(x... |
H: This is correct? it is my project
Integral definite, The area of [a,b] where a=-2.5 and b=2.5, by rectangles I have a similar area, is the arch of a bridge(model) .
$$ y=-\frac{4}{25}x^2+1 $$
$$ \int_{-2.5}^{2.5}(-\frac{4}{25}x^2+1)dx$$
$$-\int_{-2.5}^{2.5}\frac{4}{25}x^2dx+\int_{-2.5}^{2.5}dx$$
$$-\frac{4}{25}\fra... |
H: Gaussian linking coefficient definition
As I read from the wikipedia page of the linking number, it says that the linking number of two curves $\gamma_1$ and $\gamma_2$ in space can be found using the integral
$$\,\frac{1}{4\pi}
\oint_{\gamma_1}\oint_{\gamma_2}
\frac{\mathbf{r}_1 - \mathbf{r}_2}{|\mathbf{r}_1 - \m... |
H: Limit $\lim\limits_{n\to\infty}\left(1+\frac{1}{n}+a_n\right)^n=e$ if $\lim\limits_{n\to\infty}na_n=0$
Let $\{a_n\}$ be any sequence of real numbers such that $\lim_{n\rightarrow\infty}na_n=0$. Prove that $$\lim_{n\rightarrow\infty}\left(1+\frac{1}{n}+a_n\right)^n=e$$
I thought about using binomial theorem. So $\... |
H: limit $a_{n+1}/a_n$ for recurrence $a_{n+2}=a_{n+1}+a_n$
Let $\{a_n\}$ be a positive sequence which satisfies $a_{n+2}=a_{n+1}+a_n$ for $n=1,2,\ldots$. Let $z_n=a_{n+1}/a_n$. How can I prove that $\lim_{n\rightarrow\infty}z_n$ exists?
I looked at $z_{n+1}=1+1/z_n$, but I still don't know how to go from here.
AI: C... |
H: The identifications of $R$ in its ring of fractions $S^{-1}R$
If $R$ is an integral domain, we can identify the elements $r\in R$ as elements $rs/s$ of the ring of fractions $S^{-1}R$. In this way, we can identify $r\in R$ as $r/1_R$. I've seen in somewhere that we can have a more general case identifying $r$ as $r... |
H: Proper map on from compact manifolds
I'm stuck on this statement. Could anyone please help me out?
Let $X$ be a compact manifold, every map $f: X \longrightarrow Y$ is proper.
The definition of proper: a smooth map between manifolds is called proper if inverse images of compact subsets are compact.
I know that con... |
H: Fair and Unfair coin Probability
I am stuck on this question.
A coin with $P(H) = \frac{1}{2}$ is flipped $4$ times and then a coin with $P(H) = \frac{2}{3}$ is tossed twice. What is the probability that a total of $5$ heads occurs?
I keep getting $\frac{1}{6}$ but the answer is $\frac{5}{36}$.
Attempt: $P($all... |
H: computing recursive functions
I have a function $\alpha : \mathbb{N}\times\mathbb{N} \rightarrow\mathbb{N}$, defined recursively, as below:
$\forall n \in\mathbb{N}, \alpha(n,10) := \begin{cases} \alpha(n-1-9, 10) + 9 &\text{if}\ n \ge 10,\\ 0 &\text{if}\ n \lt 10 \end{cases}$.
When I compute $\alpha(10, 10)$ my ... |
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