text stringlengths 83 79.5k |
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H: On $n\times n$ matrices $A$ with trace of the powers equal to $0$
Let $R$ be a commutative ring with identity and let $A \in M_n(R)$ be such that $$\mbox{tr}A = \mbox{tr}A^2 = \cdots = \mbox{tr}A^n = 0 .$$ I want to show that $n!A^n= 0$.
Any suggestion or reference would be helpful.
P.S.: When $R$ is a field o... |
H: Chain homotopy and compositions of morphisms.
Show that if $\alpha_1 \sim \beta_1$ and $\alpha_2 \sim \beta_2$ , then (whenever composition makes sense) $\alpha_1 \circ \alpha_2 \sim \beta_1 \circ \beta_2$.
I have two questions.
So are these morphisms ($\alpha_1$, $\alpha_2$, $\beta_1$, and $\beta_2$) maps from $... |
H: Exponentiation by squaring
I need to calculate $7^{2012} \mod {13}$ by hand using exponentiation by squaring, but I cant seem to figure it out.
I started with this but I don't know for sure if its correct or where it's going.
\begin{align}
(7^2)^{1006} &\equiv 49^{1006} \pmod {13} \\
(10^2)^{503} &\equiv 100^{503}... |
H: Show that a polynomial has at least one positive solution/root.
Let: $P(x) = a_nx^n + a_{n-1}x^{n-1} + ..... + a_1x + a_0$ where $a_0a_n < 0$
I have to prove that the Polynomial $P(x)$ has at least one positive root
how can I prove it? Any ideas?
AI: For $x$ very large, $P(x)\approx a_n x^n$ and so has a sign diffe... |
H: Tensor Product and Direct Sum
Let $R$ be a commutative ring with identity and let $\{M_\alpha\}$ be a family of $R$-modules and $N$ another $R$-module. I've tried to show that
$$\left(\bigoplus_\alpha M_\alpha\right)\otimes N\simeq \bigoplus_\alpha (M_\alpha\otimes N).$$
I've tried to simply generalize the proof I'... |
H: prove or disprove about the statement for a limit of a sequence
Let $(a_n)$ be a sequence of numbers such that $\displaystyle\lim_{n\rightarrow\infty}a_{n+1}=L$. Does this imply that $\displaystyle\lim_{n\rightarrow\infty}a_{n}=L$?
AI: Yes, a limit of a sequence is determined by the "infinite tail" of the sequence.... |
H: Help solving the limit $\lim_{n \to \infty} \frac{2^{\ln n}}{n^3}$
I want to evaluate the following limit: $$\displaystyle\lim _{n\rightarrow\infty}\frac{2^{\ln n}}{n^3}.$$ Can someone help me how to solve it? I don't want the result but the solution.
AI: Note that $2^{\ln n} = n^{\ln 2}$ and so
$$
\lim _{n\rightar... |
H: Is $\epsilon$ in every alphabet?
Given a $\Sigma$ an alphabet, is $\epsilon$ in it logically?
For example, if I have a function $ f : \Sigma \to \Sigma $, can I define it for example $ f(\sigma) = \epsilon$? even if my alphabet is for example only $\Sigma = \{a,b,c\}$?
AI: As Arthur Fischer wrote in the comments: $... |
H: Expected Hamming distance
I choose two code words independently at random from $\mathbb F_2^n$ where each string has $n$ binary digits equally likely. $\mathbb F_2$ represens the binary digits.
The Hamming distance is between two vectors $x,y\in\mathbb F_2^n$ is the number of places where they differ: $d(x,y)=|\{j... |
H: Is $(f(x_2 )-f(x_1 ))/(x_2-x_1 )=f'(c)$ for $f$ is continuous derivative on $(a,b)$, $x_1,x_2,c∈(a,b)$
Suppose that $f$ is continuous derivative on $(a,b)$, and let $c∈(a,b)$. Prove or disprove: there exist points $x_1,x_2∈(a,b)$ such that
$(f(x_2 )-f(x_1 ))/(x_2-x_1 )=f'(c) $
This seem true, but I still have the ... |
H: Rewriting algebra
I'm working on calculations on polynomials and a paper gives the following algebra step:
$\frac{x^{15} - 1}{x^3 - 1} = x^{12} + x^9 + x^6 + x^3 + 1$
They do not explain how they get to this result, and I can not follow how they did this.
Could someone tell me what step i'm missing here?
AI: If we ... |
H: I'm not managing to prove that, $\left( \frac{1}{\sqrt{n}}\sum_{i=1}^na_i\right)^2\leq\sum_{i=1}^na_i^2$
I'm not managing to prove that, $$\left( \frac{1}{\sqrt{n}}\sum_{i=1}^na_i\right)^2\leq\sum_{i=1}^na_i^2$$I think it should be used something like domestic product, but do not know how to use.
AI: $$\left(\frac1... |
H: Uniform convergence with Lp functions
I have a convergence question:
Say we have a sequence of functions $\{u_{m}\}_{m=1}^{\infty}$ where $\{u_{m}\}_{1}^{\infty} \subset L^{p}(U)$ and where $U$ is bounded. Consider $u^{\epsilon} := \eta_{\epsilon}\ast u_{m}$ where $\eta_{\epsilon}$ is the usual mollifier.
If I was... |
H: Non-isomorphic $\mathbb{C}$-algebras
The question is as follows:
Show that the $\mathbb{C}$-algebras: $A=\mathbb{C}[x,y]/(x^2y-xy)$, $B=\mathbb{C}[x,y]/(x^2y+xy^2)$, $C=\mathbb{C}[x,y,z]/(xy, yz, zx)$, and $D=\mathbb{C}[x,y]/(x^2y+xy^2+x^4+y^4)$ are pairwise non-isomorphic.
Is there a particularly elegant way to a... |
H: Connected and irreducible topological spaces
A topological space is called connected if any presentation of $X$ as $X = V_1 \uplus V_2$ by disjoint open subsets implies that one of them is trivial ($V_1 = X$ or $V_2 = X$). By taking complement one can replace the word "open" by "close".
A topological space is calle... |
H: Find a non-principal ideal (if one exists) in $\mathbb Z[x]$ and $\mathbb Q[x,y]$
I know that $\mathbb Z$ is not a field so this doesn't rule out non-principal ideals. I don't know how to find them though besides with guessing, which could take forever.
As for $\mathbb Q[x,y]$ I know $\mathbb Q$ is a field which wo... |
H: Is exponential of a concave function concave?
is this function:
$$\exp\Big(-||Ax||^2\Big)$$
concave in A??
I know that exponential of a convex function is convex, but is exponential of a concave function concave??
AI: Did's example
Let $f(x)=e^{-x^2}$.
Then $f'(x)=-2xe^{-x^2}$, so $f''(x)=4x^2e^{-x^2}-2e^{-x^2}=(4x... |
H: Definition of infinite limit
Following is an exercise I solved, it is not homework so I don't know how to check my solution. Can you please check the solution for me? The exercise is this:
Modify the definition of convergent sequence to obtain a definition of convergence to $\infty$ and then use your definition to ... |
H: Differential Equation with Given Initial Condition
Given is the differential equation
$N(t)' = 2 * \sqrt{N(t)}$
We have to show that the constant function N(t) = 0 is a solution for the initial condition N(0) = 0, and that the function N(t) = $t^2$ is a solution again for the same initial condition.
So I would plug... |
H: Why is $2^n$ the maximum number of subsets of a set of size $n$?
There is a set with $n$ elements. Why is the maximum number of subsets that can be formed out of it $2^n$?
AI: We must show that $${n\choose 0}+{n\choose 1}+{n\choose 2}+\cdots +{n\choose n}=2^n$$ is the number of subsets of an $n$-element set $S$ whe... |
H: beta reduction bascis
Hi I get the basics of beta reduction e.g.
$$(\lambda var.body)arg $$
you just replace the occurrences of var with arg in body.
However what happens here?
$$(\lambda x.xx)(\lambda x.xx) \rightsquigarrow_\beta (\lambda x.xx)(\lambda x.xx)$$
Let's call them $A$ and $B$, so I replace all occurre... |
H: Show that if $ f $ is a symmetric bilinear form such that $f(u,u)=0 $, $ \forall u \in V$ so $f=0$.
Let $V$ a vector space of demension $n$ over a field $\mathbb{K}$. Show that if $ f $ is a symmetric bilinear form such that $f(u,u)=0 $, $ \forall u \in V$ so $f=0$.
I could answer this question by using the matrix ... |
H: Is there any number which $n!$ is lower than $2^n$ or same?
I interested in this question.
how many numbers meet this condition?
I think a few of them meet this but I want a proof for this.
also I'm not very pro in mathematics.
AI: Observe that $n!<2^n$ if $n<4$ and $n!>2^n$ for $n=4$
Let $m!>2^m$ for $m\ge 4$
The... |
H: Difference between NFA and DFA
In very simple terms please, all resources I'm finding are talking about tuples and stuff and I just need a simple explanation that I can remember easily because I keep getting them mixed up.
AI: Each input to a DFA or NFA affects the state of the automaton: if it was in state $q$ imm... |
H: undetermined coefficients. What am I doing wrong?
I am having some trouble to solve the following differential equation for the undetermined coefficient:
$$
y''+2y'+y=xe^{-x}
$$
I have been watching some videos on youtube and done some reading but still do not fully understand what must be done to solve it. I was a... |
H: Ratio of angles in a triangle, given lengths of triangle's sides.
If I have a triangle $\,\triangle ABC,\,$ with sides of lengths $\,AB=6, \;BC=4, \;CA=5,\,$
then what can I know about the ratio of $\,\dfrac{\angle ACB}{\angle BAC}\,$?
AI: You can use the Law of Sines to find the ratio of the sines of your two ang... |
H: Question on Polynomial function
Suppose $f(x)$ is a polynomial in $x$ having integer coefficients and $13 < a < b < c$ are integers such that $f(13) = f(a) = 13$ and $f(b) = f(c) = 19$: Determine the possible values of $a,b,c$.
Could someone give me an idea what type of $f(x)$ I should assume? The term polynomia... |
H: Geometry GRE question
This is a GRE quesrtion, and I could not find the length to save my life, please help!
A circle with diameter PQ of length 10is internally tangent at P to a circle of radius 20. A sqare ABCD is constructed from A, B on the larger circle, CD tangent at Q for the smaller circle. and the smaller ... |
H: Functions from ordinals to ordinals
I'm trying to solve this problem, which appears in Schimmerling's "A Course in Set Theory."
Problem. Find two functions
$$f:\omega\rightarrow\omega\cdot2$$
and
$$g:\omega\cdot2\rightarrow\omega\cdot3$$
such that $\sup\{f(\omega)\}=\omega\cdot2$ and $\sup\{g(\omega\cdot2)\}=\om... |
H: On simple extension
Show that $\mathbb{R}$
is not a simple extension of $\mathbb{Q}$
as follow:
a. $\mathbb{Q}$
is countable.
b. Any simple extension of a countable field is countable.
c. $\mathbb{R}$
is not countable.
I 've done a. and c. Can anyone help me a hint to prove b.?
AI: Let $F$ be a countable fi... |
H: Kleene Star operation on sets
I have the following question, and do not understand the Kleene star operation in the context of relations.
Let R be the relation $R=\{(0,1),(0,2),(1,4),(1,5),(2,3),(2,4),(2,5)\}^*$ on the set $A=\{0,1,2,3,4,5\}$. Find all minimal, maximal, smallest and largest elements, if possible, o... |
H: Proving that a polynomial about the volume of a tetrahedron is irreducible
We know that the volume of a tetrahedron $ABCD$ can be represented as
$$144V^2=(a^2b^2d^2+b^2c^2e^2+c^2a^2f^2+b^2a^2e^2+c^2b^2f^2+a^2c^2d^2+c^2e^2f^2+a^2f^2d^2+b^2d^2e^2+c^2d^2f^2+a^2e^2d^2+b^2f^2e^2)-(a^2b^2c^2+a^2e^2f^2+b^2f^2d^2+c^2d^2e^2... |
H: Properties of the $\text{lcm}(1,2 ,... n)$ function
I was thinking the other day about the following function - a sort of prime factorial:
$$f(n) = \text{lcm}(1,2,\cdots,n) $$
Does this function have a name? Does it have any interesting properties analagous to $n!$ (e.g. a version of Sterling's formula?)
EDIT A pre... |
H: If $u \in K$ is transcendental over $F$, then $F(u) \cong F(x)$
This isn't for homework, but I would just like a small hint please. The question asks
Let $K$ be an extension field of $F$. If $u \in K$ is transcendental over $F$, then $F(u) \cong F(x)$ (where $F(x)$ is the field of quotients of $F[x]$).
I should... |
H: On simple extension on Q
Prove that $\mathbb{Q}\left(\sqrt{5},\sqrt{7}\right)$ is a simple extension on $\mathbb{Q}$. More precisely, show that $\mathbb{Q}\left(\sqrt{5},\sqrt{7}\right)=\mathbb{Q}\left(\sqrt{5}+\sqrt{7}\right)$
AI: Clearly, $\mathbb{Q}(\sqrt{5}+\sqrt{7})\subset\mathbb{Q}(\sqrt{5},\sqrt{7})$ as $\sq... |
H: Probability of a three-pairs in 6-card poker
Suppose out of a deck of 52 regular playing cards, you are dealt a hand of 6 cards. What is the probability of you getting a three-pair hand? That is, three faces that each appear twice.
I've worked out $${\binom{13}{1}\binom{4}{2}\cdot\binom{12}{1}\binom{4}{2}\cdot\bi... |
H: Convergence of iterative method
Assume that iterative method:
$x_{k+1}=F(x_{k})$
where $(k=0,\,1,\,2,\,...)$
converges to $\alpha$
which is root of $f(x)=0$
equation.
Prove that if $F(\alpha)=\alpha$, $
F'(\alpha)=F''(\alpha)=...=F^{(p-1)}(\alpha)=0
$ and $ F^{(p)}(\alpha)\neq0$
then convergence of that... |
H: Given the index of an element in a triangular array, how do I find its row?
Consider a triangular array(numbers laid out in rows, where the r-th row contains r elements). Given the index i of an element in this array (assuming the numbers are laid out at indices 1, 2, 3, etc. starting from the top and moving down),... |
H: A nonlinear first-order differential equation
How do we solve the following differential equation?
$$(y{}')^{2}+p(x)(1+y^2)^{3}=0$$
AI: Here is one approach.
Given:
$$(y')^{2}+p(x)(y^2+1)^{3}=0$$
We can solve for $y'$ by taking square roots, yielding:
$$y' = \pm ~ i ~ \sqrt{p(x)}~(y^2+1)^{3/2}$$
We can now separate... |
H: General formula for roots for cubic equation
I have the following cubic equation with $ \beta \in [0,1]$ and $ \delta\in [0,1]$ are 2 parameters. Is it possible to use software to get the explicit expression of the solution.
\begin{equation*}
U^{3}\left( -\left( \frac{\delta^{2}}{2} + \beta\right)+\delta-\frac{1}... |
H: Finding the limit of $\lim_{v\to180}\frac{360\cos\left(\frac{v}{2}\right)}{180-v}$
I need to find this limit:
$\displaystyle\lim_{v\to180}\frac{360\cos\left(\dfrac{v}{2}\right)}{180-v}$, with $v$ in degrees.
I have tried to do this:
$\displaystyle\lim_{v\to180}\frac{360\cos\left(\dfrac{v}{2}\right)}{180-v}=^{L'H}\d... |
H: Identify the singularity in this ..
$\frac {\sin^2z}{z^2} $
What kind of singularity is present in this ?
My take on this is that, the limit at $z = \infty$ , is $0\,\,$.So, limit is finite.Thus, there should be no essential singularity at $z = \infty$. But,the answer says there is an essential singularity at th... |
H: Find a minium value of a function
Given x,y,z are positive real numbers such that
$$
x^2+y^2+6z^2=4z(x+y).
$$
Find the minimum value of the following function
$$
P=\frac{x^3}{y(x+z)^2}+\frac{y^3}{x(y+z)^2}+\frac{\sqrt{x^2+y^2}}{z}
$$
AI: let $$a=\dfrac{x}{z},b=\dfrac{y}{z}$$
then $$a^2+b^2+6=4(a+b)$$
and we only fi... |
H: Can there exist an injective function from $\mathbb R$ to $(0,1)$?
I've been trying high and low to find an injective function from $\mathbb R$ to $(0,1)$, but to no avail. I've tried all sorts of polynomial functions, exponential functions, etc. but I've had no luck so far. Can anyone provide an example of such a ... |
H: Finding a point in a 2D space
After sitting with this one for half a day, I'm not sure it is possible anymore, given my scenario.
The idea is to get a location on a map (coordinates) from some wifi data.
I have this wonderful system that on demand gives me my estimated position in a building. These positions are li... |
H: Prove that if $ac + bc + c^2 < 0$ then equation (usual notation) has two roots
We have $a, b, c$ real parameters, $a ≠ 0$.
Prove that $ax^2 + bx + c = 0$ has two different roots ($b^2-4ac > 0$), if $ac + bc + c^2 < 0$
AI: If $P(x) = ax^2+bx+c$, from $c(a+b+c)<0$, we have $P(0)P(1) < 0$. So there is a root in $(0, 1... |
H: Spectral radius of a compact operator and convergence
Let $T$ be a compact operator such that the spectral radius $\rho(T)<1$
does it then follow that $||T^n|| \rightarrow 0$ as $n \rightarrow \infty$?
AI: You need the spectral radius formula
$$
\rho(T) = \lim \|T^n\|^{1/n}
$$
Fix $\delta > 0$. Suppose $\rho(T) < 1... |
H: Centre manifold non-hyperbolic fixed points
I have the following dynamical system $\dot x=-x^3, \dot y=-y$.
I would like to prove that there are an infinite number of trajectories that become tangent to the line $y=0$ for $x\rightarrow 0$. Each of the trajectories is a centre manifold and they satisfy $y=h(x)$ whe... |
H: Clarification on $\frac{0^n}{0}$ when $n>0$
If this is a duplicate, I will gladly delete this if there is a duplicate but I've had difficulty finding one. I had, until recently, believed that we don't define $\frac 0 0$ as the limits coming from different directions vary widely and so no value works for all cases.... |
H: $\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$ if $(a,b)=1$
I hope to show that
$$\mathbb{Z}[i]/(a+bi) \cong \mathbb{Z}_{a^2+b^2}$$
for $(a,b)=1$.
I made an effort to find a homomorphism but I failed.
Can you give a hint?
AI: Hint: $(a,b)=1$ if and only if there exist integer $m,n$ such that $an+bm=-1$.
Incident... |
H: Can you please check my limit proof
I proved the following:
If $x_n \to x$ then $\sqrt{x_n} \to \sqrt{x}$
How I proved it: First I proved it for $x=0$. Then: Let $\varepsilon > 0$. If $x_n \to x$ then $(x_n - x) \to 0$. Since we have shown for $x=0$ we may assume that $x \neq 0$. Then
$$ x_n - x = (\sqrt{x_n} - \... |
H: Having trouble with a Lim (infinity-infinity)
so I am learning about Limits at the moment and I am having some trouble with this Lim..
$\lim_{x\to ∞}=\sqrt{x}(\sqrt{16x^3+4x-1}-\sqrt{16x^3+3}) $
I just can't get the right answer, which is $ \frac 12 $ ,
I know how to do the -> $\lim_{x\to ∞}=(\sqrt{16x^3+4x-1}-\sqr... |
H: Leisure reading for an undergraduate student
I am a freshman at a local university. I never really had much passion for math, but I always did well in math exams . I attribute this lack of passion to rote learning/emphasis on methods/formulas than philosophy behind those methods that is so common in Indian educatio... |
H: Complex number - how to find the angle between the imaginary axis and real axis?
Assume I have complex number $z = a + ib$.
$z$ can be represented by a polar representation as $r(\cos \theta+i\sin \theta)$,
when $r$ is the absolute value of $z$, $\sqrt{a^2 + b^2}$.
But how can I find $\theta$?
AI: Consider the fo... |
H: Express $\cos2\theta$ in terms of $\cos$ and $\sin$ (De Moivre's Theorem)
Use De Moivre's to express $\cos2\theta$ in terms of powers of $\sin$ and $\cos$.
What I have is:
$\cos2\theta + i\sin2\theta\\
= (\cos\theta + i \sin\theta)^2\\
= \cos^2\theta + 2 \cos\theta ~i \sin\theta + (i \sin)^2\theta\\
= \cos^2\theta ... |
H: Please explain to me why $ \int_b p(a|b) db \neq p(a) $
I have one question that bugs me. How is it that:
$
\int_a p(a|b) da = \int_a \frac{p(a,b)}{p(b)} da = 1
$
but
$
\int_b p(a|b) db = \int_b \frac{p(a,b)}{p(b)} db \neq p(a)
$
I don't understand this because $ p(a,b) = p(b,a) $ and therefore
$
p(a|b) = \frac{p(a... |
H: Simple Derivative
I am wanting trying to remember how to solve a derivative of this nature:
$$ \frac{dM}{dt} = rM(t)$$
$$ dM = rM(t)dt $$
Solving when t = 0 equals 1 we can get the solution as
$$ M(t)= M(0)e^{rt} $$
AI: divide both side by $M(t)$,we get $dM/M(t)=rdt$ now know that integral of $1/t$ is $ln(t)$,c... |
H: Limit of $\frac{\sqrt{mx^2}}{\sqrt{\sin(m+1)x^2}}$
I know it's an easy question, but I could use your help here, anyway.
Could you tell me how to prove that $ \lim _{x \rightarrow 0}\frac{\sqrt{mx^2}}{\sqrt{\sin(m+1)x^2}} = \sqrt{\frac{m}{m+1}}$? I know I should use the fact that $\lim _{x \rightarrow 0} \frac{\si... |
H: Is this a poset?
Is $(S, R)$ a poset where $S$ is the set of all people in the world and $(a, b) \in R$, where $a$ and $b$ are people if $a$ is no shorter than $b$?
My attempt:
$a$ is no shorter than $a$. This is not reflexive because $a$ can be taller than $a$. Contradiction
True?
AI: Are you shorter than yourself... |
H: How to prove sin(sin(sin...(sin(1))) converges to 0 without using continuity?
Using continuity I was able to show the sequence $x_0 = 1$, $x_{n+1} = sin(x_n)$ converges to 0, but I was wondering if there was a way to prove it using only properties and theorems related to sequences and series, without using continui... |
H: System of equations with multiplication
I want to find all $x,y,z\in\mathbb{R}$ such that $(x+1)yz=12, (y+1)zx=4, (z+1)xy=4$.
I can multiply all three equations to get $(x+1)(y+1)(z+1)x^2y^2z^2=192$.
I can divide the first equation by the second to get $\dfrac{(x+1)y}{(y+1)x}=4$, which simplifies to $3xy+4x-y=0$.
N... |
H: Removing logs from equation
I have a simple question that I need clarification on:
If
$$\log(a) = \log(b) + c$$
is it true that
$$a = b + \exp(c)$$
Is this correct or am I missing something really basic that I cant remember from maths class?
AI: You’re missing something, namely, one of the laws of exponents: if $... |
H: Prove using Jensen's Inequality
Let $\alpha_1, \alpha_2, . . . , \alpha_n$ be the interior angles of a convex (but not necessarily regular) n-gon. Prove, that for all integers $n\geq3$:
$$\cos \alpha_1 + \cos \alpha_2 + \cdots + \cos \alpha_n + n \cos\left(\dfrac{2\pi}{n}\right) \leq0$$
The prof said that I need ... |
H: Third-degree cosine inequality for obtuse triangle
Suppose $\triangle ABC$ is an obtuse triangle with side lengths $a=BC, b=CA, c=AB$. I want to show that $$a^3\cos A+b^3\cos B+c^3\cos C<abc.$$
My idea is to use the cosine rule. I have $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$, etc. Plugging into the inequality I get
$$a^4... |
H: Find asymptotes $e^{-x}-e^{-2x}$
I should find the asymptotes for $e^{-x}-e^{-2x}$. So I will take limits.
My attempt goes like this:
1) when x → $+\infty$ then $e^{-x}$ → $0$ and so does $e^{-2x}$. Thus: $0$. Quite easy.
2) when x → $-\infty$ then $e^{-x}$ → $\infty$ and so does $e^{-2x}$. So we have $\infty - \in... |
H: Power series help please!
$$ \frac{2}{3-x} $$
I need to find a power series representation for this. I figured i'd pull the two out, but I can't figure out what to do with the three.
AI: Recall that for $|x|<1$:
$$\frac{1}{1-x} = \sum_{n=0}^\infty x^n$$
So you'd like to bring your function in a similar form. You c... |
H: solution of $y^2 - x = 15$ and $x^2 -xy = 2009$
Find all the integer solutions to the equations:
\begin{eqnarray}
y^2 - x &=& 15 \\
x^2 -xy &=& 2009
\end{eqnarray}
Not sure how to solve this :/, tried the usual algebra way (solving for something and substetuting) but didn't really work out
$x+15$ must be a square... |
H: How can I express the non-intersecting sections of multiple sets with a single set operation?
I don't have a lot of experience with set theory, as I suspect this question will make clear!
As the title says, I'm interested in expressing the non-intersecting sections of three sets using a single set operation.
What I... |
H: Solving recursion formula with sum
I am trying to solve the following recurrence, but i am stuck...
$$t(n)=n + \sum_{j=1}^n t(n-j)$$
I really appreciate your help,
Tarcísio.
AI: Notice that as $j$ runs from $1$ to $n$ in the summation, $n-j$ runs from $n-1$ down to $0$, so we can rewrite the recurrence as
$$t(n)=n+... |
H: Why factorials when divided by factorials less than the number have a remainder 0?
Lets take the example, if we take the expression $\frac{X!}{y_1!\cdot y_2!\cdots y_n!} $as long as summation $S=y_1+y_2+...y_n$ is less than or equals $X$, the remainder is always $0$. Thats How the permutation of $X$ things where th... |
H: Theoretical impossibility? Deviation from normality with a sample greater than 300?
Huge thanks in advance!
I've been lead to believe that the following is a theoretical impossibility: a population larger than 300 records without an approximation of a normal distribution. The dataset I used is a set of amounts of f... |
H: Equation $\sqrt{x}+\sqrt{y}=\sqrt{2013}$ in rationals
Can we find all rational numbers $x,y$ such that $\sqrt{x}+\sqrt{y}=\sqrt{2013}$?
Certainly possible answers are $(2013,0)$ and $(0,2013)$.
If we square the equation, we get $x+y+2\sqrt{xy}=2013$, so $\sqrt{xy}$ must be rational.
AI: if $x\neq 0$ then $x,\sqrt{x... |
H: Separable set, the real [0,1] interval and measure
I am having a hard time understanding exactly what "separable" means, and I am trying to relate that to the measure of the real [0,1] segment (which is 1, right?).
My confusion started when studying measure for Lebesgue integration. The set of rationals in [0,1] is... |
H: about completion of a metric space
Is the following proposition true? If yes, how would you prove this?
Proposition :
Let $(X,d)$ a metric space and let $(\widetilde{X},\widetilde{d})$ be a completion of $(X,d)$
then if : $Y\subset \widetilde{X}$ and $\overline{Y}=\widetilde{X} \Rightarrow X=Y.$
AI: It’s clearly fa... |
H: List of preferences for four ladies and four gentleman where no one obtains his or her first pick.
How can I determine a list of preferences for four ladies and four gentlemen where no one obtains his or her first choice in a stable matching determined by the Gale-Shapely algorithm, regardless of who proposes.
I un... |
H: Examples of rare, meager and nonmeager sets in $\mathbb{R}$
Kreyszig Functional Analysis book presents the following definition.
I'm trying to get some examples.
(a) The cantor set $K$ is rare in $\mathbb{R}$ because it's closed and has empty interior so that $$\operatorname{int}\left(\overline{K}\right)=\operator... |
H: Order of Automorphism group of a group of prime order.
An easy question I am having a difficult time finding a straightforward answer to.
The quotient $\frac{G}{C_G(P)}$ is isomorphic to a subgroup of $Aut(P)$ where $P$ is a group of prime order.
Is $|Aut(P)|=p-1$?
AI: First of all, let's try to demonstrate that ... |
H: Find the number $c > 0$ such that the region bounded by the curves $y = x$, $y = -2x$, $x=c$ has area 6.
My intuition leads me to believe that I need to separate this into two regions and use a definite integral to compute the area. I am having troubles associating $x=c$ and each of the given functions.
AI: Conside... |
H: I want to prove that these definitions of expected value hold
Let $(\Omega,\mathcal B,P)$ be a probability space. I have two (related) questions. Assuming that $g:\mathbb{R}\to\mathbb{R}$ is Borel measurable, and understanding that
$$E(g(X)) = \int_{\Omega}g(X(\omega))dP(\omega),$$
how do I prove that these equali... |
H: Under what conditions is this true? A finitely generated ideal minus a generator equals an ideal containing that.
Let $(r_1, \dots, r_{s-1}) \subset I \subset J = (r_1, \dots, r_s)$ be ideals in a commutative ring $R$. In other words $r_s \notin I$. Also, let $\{r_1, \dots, r_s\}$ be a minimal generating set for... |
H: Understanding the last few lines in a proof by Royden in Real Analysis.
The book states:
$\textbf{Proposition 2}$: Let $C$ be a countable subset of the open interval $(a, b)$. Then there is an increasing function on $(a, b)$ that is continuous only at points in $(a, b) \setminus C$
$\textbf{Proof}$ If $C$ is finite... |
H: Finding a matrix where the column space is a subset of the null space
Let $A_{3x3}$ be a matrix $ \ne 0$ such that the column space of $A$ is a subset of the null space of $A$. I need to find $A$.
Here's my process so far:
let $v_1, v_2, v_3$ be the column vectors of $A$
$Col(A)=c_1 v_1 + c_2 v_2 + c_3 v_3$ is a su... |
H: Integration problem in real analysis
Let $f: [a,b]\rightarrow \mathbb{R}$ be a function, and let $m$=inf{$f(x)|x\in [a,b]$}, $M$=sup{$f(x)|x\in[a,b]$}.
Prove that for any partition $P$ and sample set $S$, $m(b-a)\leq RS(f,P,S)\leq M(b-a)$.
My attempt: $RS(f,P,S)=f(s_1)\Delta x_1 + ... + f(s_n)\Delta x_n$ ; $m(\Delt... |
H: Show that a progression converges and assess the limit.
$a_{n} = \sqrt{9n^2+2n+1} -3n$
I tried to simplify the term to show that the limit is $\frac{1}{3}$ with the binominal formula for example $\sqrt{9n^2+6n+1-4n} = \sqrt{(3n+1)^2-4n}$ or playing around with the $9n^2$ but because of the $-3n$ after the radical, ... |
H: Help with limit of radical function
$$\lim_{x \to \infty} \frac{\sqrt{4x^{4}+3}}{5x^2+3}$$
$$= \lim_{x \to \infty} \frac{(4x^{4}+3)^{1/2}}{5x^2+3}$$
$$= \lim_{x \to \infty} \frac{(\frac{4x^{4}}{x^{1/2}} +\frac{3}{x^{1/2}})^{1/2} }{5+\frac{3}{x^2}}$$
$$= \lim_{x \to \infty} \frac{2x+\frac{\sqrt{3}}{x}}{5+\frac... |
H: semisimplicity of Lie algebra
Let $L$ be a lie algebra. Then if $L$ is semisimple, we have $L = L_1 \oplus \cdots\oplus L_n$ for some simple ideals $L_i$. But we can also consider the adjoint representation. In this representation, each $L_i$ will be an irreducible submodule. So we get that the representation is co... |
H: Open and Closed Quotient Maps.
a) Find a subset $A$ of $\Bbb{R}$ such that the quotient map $p: \Bbb{R} \rightarrow \Bbb{R}/A$ is not open.
If we let $A= \Bbb{Q}$, then we can see that $(0,1)$ is open in $\Bbb{R}$. But if we pick a rational number $p/q \in (0,1)$ and draw a ball around it in $\Bbb{R}/\Bbb{Q}$, we... |
H: How can I compute $\int\frac1{e^{ct}} dt $
$$ \displaystyle \int \dfrac{1}{e^{ct}} dt $$
How do I do this, I've tried substitution. Rewrite the above integral as $\displaystyle \int e^{-ct} dt $, then we have $u=-ct$, but at this point derivatives and integrals are confusing me. If we want $[e^u]'$, we just have $[... |
H: Basic induction proof that all natural numbers can be written in the form $2a + 3b$
The theorem given is:
If $n$ is a natural number then $n$ can be written in the form $2a + 3b$ for some integers $a$ and $b$.
How would I prove this by induction? I've had a go at proving this but I don't know if my technique is s... |
H: The rule $\phi \leadsto \phi\land\psi$ is not sound
I have been asked to show that this rule is not sound: $$\frac{\varphi}{\varphi\wedge\psi}\wedge I'$$
Any help with this would be greatly appreciated.
AI: Work in propositional logic with variable letters $A$ and $B$. Let $\phi = A \lor \lnot A$ and let $\psi = B$... |
H: Proving there is a max,with two limits given.
So I need help with this exercise. If $f$ is a positive and continuous function with
$$\lim_{x \to -\infty} f(x) = \lim_{x \to +\infty} f(x) = 0 $$
Prove that $f(x)$ has a maximum. Thanks in advance :)
AI: Well, go back to the definition of a limit that $\forall\; \va... |
H: Model and countermodel to $\exists x.\forall y. x
Can someone please help me with this question. I have been struggling with it for ages and can't quite seem to work it out:
Let $<$ be a binary relation symbol that we will write infix. Let $\phi = \exists x.\forall y.x<y$. Define structures $A,B$ such that $A \mod... |
H: Find a matrix with the null space equal to the column space of that matrix
I know this question has been asked and answered here: Can a matrix have a null space that is equal to its column space?. However, I'm not clear on the mechanism used to find an actual matrix so I figured I would make a new question. If this... |
H: Variable Endpoint of definite integral.
My book asks to find the number $c$ such that the region bounded by the curves $y = x^2$, and $y = c$ has area 36. I understand the a definite integral will solve the question but am having a hard time applying the right interval. Right now I have the proper integral with con... |
H: ordered topology on integers and local compactness
Hi I am trying to make sure my logic is sound,
Let's suppose that we declared the discrete topology on $\mathbb{Z}$. Let us consider the set $\{1,2,3\}$. This set is open. However, this set is also closed because it has no limit points (the open set $\{3\}$ contai... |
H: Problems with definition of almost surely
Let $(\Omega,\mathcal F,\mathbb P)$ be a probability space. In probability theory one says that and event $F\in\mathcal F$ happens $\mathbb P$-almost surely, if $\mathbb P(E)=1.$ Intuitively, as a beginner one thinks that this means that there exists an event $N\in\mathcal ... |
H: Monotonicity of $2^x-x-1$
I'm trying to prove that the function $g(x) := 2^x-x-1$ is strictly increasing on $(1, \infty)$. Using its derivative I was able to conclude that it is increasing on that interval. So, the problem is now reduced to showing that $g$ is injective.
So far I haven't been able to prove this. It... |
H: Help with limit of radical expression
$$\lim_{x \to \infty} (\sqrt{x^2-49}-\sqrt{x^2-16} ) $$
I multiplied by the conjugate radical expression:
$$=(\sqrt{x^2-49}-\sqrt{x^2-16}) \times (\sqrt{x^2-49}+\sqrt{x^2-16}) $$
$$= x^2-49-(x^2-16)=x^2-49-x^2+16=-33$$
$$\lim_{x \to \infty}f(x) = -33$$
This is wrong. ... |
H: Find CFG for $a^i b^j c^k$ with $!(i=j=k)$
I need to show that the following language is context free:
It contains all words $a^i b^j c^k$ with the condition that there're not exactly as many a's and b's and c's.
I would write it like this:
$$\{a^i b^j c^k\; : \;\;!(i=j=k) \;\}$$
Here's what I've come up with:
I sp... |
H: Flatness, Hilbert polynomial and reduced schemes.
Let $f:X \to S$ be a projective morphism of schemes and $F$ a coherent $O_X$-module. We have that if $F$ is $S$-flat then the Hilbert polynomial $P(F_s)$ is locally costant as a function of $s \in S$. (for reference http://carlossicoli.free.fr/H/Huybrechts_D.,_Lehn_... |
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