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H: Is the following True of False?
Provide a proof if true or a counterexample if false:
Let a,b be two integers (not both zero), then the gcd(a,b) divides ay+bx for all for x,y ∈ Z.
I tried with several cases such as gcd(5,10) = 5 and then multiplied by various integers and could not find a counterexamaple. I do not ... |
H: Stuck at Extended Euclidean Algorithm to solve equation
I'm trying to solve the following function via the Extended Euclidean Algorithm, but I'm stuck at the last step where I need to sub in sub 2.
d * 7 = 1 (mod 180)
d = 1 / 7 (mod 180)
d = 7-1 (mod 180)
180 = 7 * 25 + 5
7 = 5 * 1 + 2
5 = 2 * 2 + 1
1 = 5 – 2 * 2
s... |
H: Prove: If $|a_n|$ doesn't converge to $\infty$ then $a_n$ must have a finite partial limit.
Prove: If $|a_n|$ doesn't converges to $\infty$ then $a_n$ must have a finite partial limit.
My thoughts:
if $|a_n|$ doesn't converges to $\infty$ there must be two other posibilities:
$|a_n|$ converges to a finite num... |
H: Just need my math checked! Integration!
Let $b \in \mathbb{R}$, and define $f:[0,2]\to\mathbb{R}$ by $$f(t)=\begin{cases}t&\text{if $0 \leq t < 1$}\\b-t^2&\text{if $1 \leq t \leq 2$}\end{cases}$$ and let $$F(x)=\int_0^xf(t)\,dt.$$ $(\text{a})$ Find a formula for $F(x)$.
$(\text{b})$ For what value(s) of $b$ is $F$ ... |
H: Cool simple solution to: $-A^2$ is not the identity matrix
this is not a question per se, just a simple cool solution to a potentially difficult question, that I want to share. I liked it.
The question is:
Let $A$ be a $3\times3$ matrix with real values.
Show that $A^2 \neq -I_3$
There are probably many solutions f... |
H: Is every complex (smooth) manifold a scheme?
The question in the title doesn't quite make sense. I was always wondering if the scheme is the generalization of manifold. The precise statement should be like following:
If $X$ is a complex (smooth)manifold, is there a scheme $Y$ over $\mathbb{C}$ such that the corresp... |
H: Inverses in the homotopy classes of maps into $RP^{\infty}$
One can define bilinear maps $\mathbb{R}^n \times \mathbb{R}^n \rightarrow \mathbb{R}^{2n-1}$ by considering the elements in $\mathbb{R}^n$ as polynomials and doing multiplication. This defines an $H$-space structure on $RP^{\infty}$ since elements of this... |
H: Is the following expression a tautology?
$\forall x\,(P(x)\rightarrow Q(x))\rightarrow (\exists y\,P(y)\rightarrow\exists z\,Q(z))$
I believe the sentece is a tautoloogy. Can someone confirm?
AI: $\forall x\,(P(x)\rightarrow Q(x))\rightarrow (\exists y\,P(y)\rightarrow\exists z\,P(z))$ is trivially a logical truth,... |
H: determine basis for topology on $\mathbb{R}^2$
Determine whether the collection of subsets below form a basis for a topology on $\mathbb{R}^2$.
All subsets of the form $T_{\epsilon}(x)=\lbrace (y_1,y_2) : |x_1+x_2-y_1-y_2| <\epsilon \rbrace$ for all $x \in \mathbb{R}^2$ and all $\epsilon>0$
By doing some algebra, w... |
H: Calculus series homework with variable a
I have a series:
$$\sum_{n=1}^{\infty}(-1)^n\sin\frac{a}{n}$$
I'm supposed to check how it behaves with different $a$ values. First I would check if it converges absolutely:
$$\lim_{n\to \infty}\left| (-1)^n\sin\frac{a}{n}\right| \sim \lim_{n\to \infty} \left| (-1)^n\frac{a}... |
H: Give an example of relation $R$ and $S$ on $A$ such that $R$ and $S$ are nonempty, and $R \circ S$ and $S \circ R$ are empty
Let $A = \left \{a, b, c, d\right \}$, give an example of relation $R$ and $S$ on $A$ such that $R$ and $S$ are nonempty, and $R \circ S$ and $S \circ R$ are empty
I'm thinking of ways that a... |
H: Several part sum of power series question
(a) Prove that $\sum_{j=0}^{\infty}x^j$ is differentiable on $(-1,1)$ and
$\frac {d}{dx}\sum_{j=0}^{\infty}x^j = \sum_{j=0}^{\infty}(j+1)x^j$.
(b) Use the fact that $\sum_{j=0}^{\infty}x^j = \frac {1}{1-x}$ on $(-1,1)$ to find a formula for $\sum_{j=0}^{\infty}(j+1)x^j$
(... |
H: Number of equivalence relations on a set with fixed class
For A={a,b,c,d,e,f}, how much equivalence relations can we get if a,b and c are in relation?
The total is: $\sum_{k=1}^6 S(6,k)$. But since a,b and c are already in the same class, i would say the answer is $\sum_{k=1}^3 S(3,k)$.
I know that answer is not o... |
H: How to Prove Below Inverse Sin
How to prove below equation .
$$ \sin^{-1}\frac{3}{5}+\sin^{-1}\frac{8}{17} = \sin^{-1}\frac{77}{85}.$$
I am not able to prove the above equation how can we prove it.
As we know
$ \sin^{-1} = y$ if $\sin y = x$ where $-1\leq x\leq 1, -\pi/2 \leq y \leq \pi/2.$
AI: Remark: The qu... |
H: Degrees of interpolating polynomials
Given a collection of $m+1$ points $\{(x_0,y_0), (x_1,y_1), ..., (x_m,y_m)\}$, we can form the interpolating Lagrange polynomial $L(x)$: $$ L(x) = \sum_{i = 0}^{m} y_i l_i(x) \\ l_i(x) = \prod_{\substack{0 \le k \le m\\ k \ne i}} \frac{x - x_k}{x_i - x_k} $$ and it will be the u... |
H: Prove that for any square matrix, an invertible matrix B exists, so that BA is triangular
I'm given a matrix A, its dimensions are n x n.
I am required to prove that an invertible matrix B exists, such that the product of the matrices BA is triangular.
Any help?
AI: If $A=QR$ is a QR factorisation of $A$, then $Q^*... |
H: Remainder Question
What process do I use to show what is the remainder when 14 × 7^36 + 92 when divided by 8?
Is it the same to show the remainder of 5^2003 when divided by 7?
I tried out the problem using congruent modulo but cannot help
AI: HINT:
For the first $7^2=49\equiv1\pmod8\implies 7^{2n}\equiv1$ where $n$... |
H: What is the hitting time distribution for white noise?
What is the distribution of the hitting time for a stochastic process
$(W_t)_{t\in [0,T]}$,
where $W_t$ are i.i.d. Gaussian random variables?
How about in cases, in which $W_t$ are i.i.d. with a common distribution other than Gaussian?
AI: If $H=\inf\{t\in\ma... |
H: Proving inequalities by induction
I'm having trouble understand the inductive when proving inequalities; Here's an example:
Show that $2^n \gt n^2 $ for any integer $n \gt 4 $.
Well for the basis $n=5$, it shows: $32>25$
Now, assume: $2^{n+1} \gt (n+1)^2 $ for some integer $n$.
Well, RHS:
$$(n+1)^2 = n^2 + 2n +1$$... |
H: equivalent functions?
I have this two functions in ($1<x<50$)
$y = -1/x$
and
$ y = \frac{x - \sqrt{x^2+4}}{2} $
why this are very similar ?
AI: The reason is that $$\sqrt{4+x^2}=x+\frac{2}{x}-\frac{2}{x^3}+O((\frac{1}{x})^5)$$
Plugging this into your equation $y=-\frac{1}{x}+\frac{1}{x^3}+O((\frac{1}{x})^5)$. |
H: $k$ colorings of the non empty subsets of $[n]$ gives the same color to two disjoint sets and their union.
This question was already asked but I didn't get enough information from the answer. Here is a link to the question.
Here is the question restated. Show that for $n$ large enough, every $k$ coloring of the no... |
H: Proofs from the Book - need quick explanation
I've been recently reading this amazing book, namely the chapter on Bertrand's postulate - that for every $n\geq1$ there is a prime $p$ such that $n<p\leq2n$.
As an intermediate result, they prove that $\prod_{p\leq x}p \le 4^{x-1}$ for any real $x\geq2$, where the prod... |
H: Showing that $\int_0^{\pi/4} \frac{1-\cos{16x}}{\sin{2x}}\,\mathrm{d}x=\frac{176}{105}$
Wolfram Alpha tells me that $$\int_0^{\pi/4} \frac{1-\cos{16x}}{\sin{2x}}\,\mathrm{d}x=\frac{176}{105}$$
What are some quick/elegant ways of proving this?
AI: $$1-\cos(16x) = 2\sin^2(8x) = 8 \sin^2(4x) \cos^2(4x) = 32 \sin^2(2x)... |
H: Is there a power series which converges to $f(x) =| x|$ for all $x$?
I'm confused how to solve the following problem:
"Is there a power series which converges to $f(x)$ = $\left| x\right|$ for all $x$?"
Your help is greatly appreciated. Thanks a lot!
AI: No. Any power series defines an everywhere-differentiable fun... |
H: Maths-Physics question, can I solve this situation for $x$?
So Let's say I have an object going at velocity $V$, initially. Each second, the current velocity $v$ is reduced by $v/x$ . After $250$ (arbitrary) seconds the velocity has been reduced to below/equal $0.01$ (arbitrary small amount). What is $x$?
The only ... |
H: what is so great about having an invariant measure?
I am a student who just started to learn basic concepts of ergodic theory.
It seems like that given a dynamical system, people are very excited to find various invariant measures of the system. But the books I am reading doesn't really convince me why it is good ... |
H: Finding a limit , dyadic pavings
I need to show that the following limit equals $\pi/4$ :
$$\lim_{k \to \infty}\sum_{n=1}^{2^k-1}\frac{ \left\lfloor\sqrt{4^k-n^2}\right\rfloor\ }
{2^{2k}}$$ I don't know if it is even possible to do so. I was trying to prove that it is possible to pave a unit disk with dyadic squar... |
H: Equal balls in metric space
Let $x$ and $y$ be points in a metric space and let $B(x,r)$ and $B(y,s)$ be usual open balls. Suppose $B(x,r)=B(y,s)$. Must $x=y$? Must $s=r$?
What I got so far is that: $$r \neq s \implies x \neq y$$ but that's it.
AI: No, it’s not necessary that $x=y$ or that $s=r$. Consider the disc... |
H: $\int_{|z|=1} \frac{f(z) }{z-a} \, dz = 0$ for $f(z)=\sin \pi/z$.
Let $f$ be analytic for all $z$ where $0 < |z| < 2$ and $a \in \mathbb{C}$ is in this domain as well. I wish to prove that $$\int_{|z|=1} \frac{f(z) }{z-a} \, dz = 0$$ for $f(z)=\sin \pi/z$. We can rewrite the integral as a line integral
$$\int_{|z|=... |
H: How to prove $\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\pi$?
How can I prove the following identity?
$$\sum_{n=1}^\infty\operatorname{arccot}\frac{\sqrt[2^n]2+\cos\frac\pi{2^n}}{\sin\frac\pi{2^n}}=\operatorname{arccot}\frac{\ln2}\p... |
H: Proving a Sequence's Uniform Convergence
Another homework problem that's been giving me headaches for about a week now.
Prove that the following sequence of functions $(f_n)$ converges uniformly on the interval $[1,2]$:
$$f_n(x) = \frac {nx^2 - 2}{x^4 + nx}.$$
Then, find the following integral on the same interval ... |
H: Prove $x + y$ is divisible by $11$. Is my solution correct?
If $x$ & $y$ are natural numbers, and $56 x = 65 y$, prove that $x + y$ is divisible by $11$.
Solution)
$56$ and $65$ are relatively prime
So, $65∣x$ and $56∣y$
Let $x = 65m$ and $y = 56n$
Then,
$56x = 65y$
$56.65m = 65.56n$,
$m = n$
Thus, the solutions ... |
H: A sphere is a surface.
How to show a sphere is a surface.
$x^2+y^2+z^2=R^2$
Note that I need to find a homeopmhism $\phi : U \to S$ for $U$ in $\Bbb R^2$ and $S$ in $\Bbb R^3$
Let $U=[0,2\pi]\times [0,\pi]$
Then I defined a surface patch $\phi (u,v)= (R\sin u \cos u, R\sin v \sin u, R\cos v)$
After there, what ... |
H: Fibonacci induction stuck in adding functions together
Using Fibonacci...
I am Proving: $$f_3 + f_6 + \cdots + f_{3n} = \frac12(f_{3n+2}-1) $$
I did the assumption of $f_1$ which gave $\mathrm{LHS}=2=\mathrm{RHS}$.
For the second part where it is $n+1$ I am having problem adding the RHS:
$$f_3 + f_6 + \cdots + f_{3... |
H: How to find the ring of regular function on $\mathbb{P}^2\backslash\mathbb{V}(x_0^2+x_1^2+x_2^2)$
Let $X=\mathbb{P}^2$ and $U=X\backslash\mathbb{V}(x_0^2+x_1^2+x_2^2)$, could anyone show me how to find $\mathcal{O}_X(U)$?
I see examples in affine case, but have no idea how to calculate the ring in the projective ca... |
H: Find length of triangle side
There is a triangle $ABC$ where $|CB|=a$, $|AC|=b$ and medians of these sides intersect at a right angle. Find |AB|.
I don't know how to use a right angle in this problem. I have a idea to link a middle of $|CB|$ with $|AC|$, let $K,L$ be a centre of these sides and we have $2|KL|=|AB|... |
H: $\frac{1}{\infty}$ - is this equal $0$?
I've seen that wolfram alpha says:
$$\frac{1}{\infty} = 0$$
Well, I'm sure that:
$$\lim_{x\to \infty}\frac{1}{x} = 0$$
But does $\frac{1}{\infty}$ only make sense when we calculate it's limit? Because for me, $1$ divided by any large amount of number will be always almost zer... |
H: Linear algebra: Matrix multiplication problem
I need to prove something in my homework I just don't know how to approach it and need some guidance.
"Show that for a matrix $A$ ($n \times m$) and a vector $\vec{x}$ ($m \times 1$) it applies that:
$A\vec{x} = \sum_{j=1}^{j=m} x_jA_j$
s.t the multiplication of $x_jA_j... |
H: Substitution To Find Most General Unifier
Could someone give me some advice on how to do this problem?
For the the following pair of expressions, find the substitution that is
the most general unifier (mgu) or explain why the two expressions cannot be unified. b and c are constants, f and g are functions, and w, x... |
H: A weird idea on definition of completing a metric space
Please note the definition below, captured from Page 102, Real Analysis, Carothers, 1ed:
Completions
Completeness is a central theme in this book; it will return frequently. It may comfort you to know that every metric space can be "completed." In effect, thi... |
H: Laguerre polynomials and least squares polynomial
A similar question from Orthogonal polynomials and Gram Schmidt says:
Use the Laguerre polynomials, i.e, $L_1(x)=x-1$,$L_2(x)=x^2-4x+2$, and $L_3(x)=x^3-9x^2+18x-6$, to compute the least squares polynomials of degree one, two, and three on the interval $(0,\infty )$... |
H: Proving $(2n-1)^n + (2n)^n ≈ (2n+1)^n$
As I do, I was messing around and I thought to myself this simple thing:
$3^2 + 4^2 = 5^2$
I just thought that this is only Pythagorean triplet with sequential integers. I know that there are no others and there are no others to higher powers due to Fermat's Last Theorem. Howe... |
H: Real Numbers as Well Defined Sets
For every construction of the reals, we define a real number to be some kind of set of rational numbers (such as cuts or sequences).
However, the number of symbols we have to formulate the description of a set is finite (analytic functions, infinite sums, etc), and I assume sets ca... |
H: Binary relations, closures and equivalences
Let $R$ be the relation on $Z$ such that $xRy \iff x-y=c$.
Well, what I have so far is $R=\{ 0,-1,1,0,-1,1,0 \cdots\}$
Is $R^* $ and equivalence relation? Why not?
This is where problems start: I don't know what the definition of $R^* $. In fact, I cannot seem to find a... |
H: Why $\lim_{n\to \infty} ({3^n+2^n\over 3^n-2^ni^n}) = \lim_{n\to\infty} ({3^n+2^n\over 3^n-2^n}) \ $?
This week I was introduced to the limits of complex sequences. It is actually pretty simple because it's mostly the same compared to real sequences. However, there is one thing - Why is:
$\lim_{n\to \infty} ({3^n+... |
H: Solve, $\cos(x)=\frac 25, \frac{3\pi}{2}
A. Draw and label two triangles, one containing angles $x$ and one
containing angle $y$. (can use the $x-y$ axis version of triangles)
B. List $\sin(x), \cos(x), \sin(y)$, and $\cos(y)$
c. Find $\sin(x+y)$
d. Find $\sin(2x)$
e. Find $\cos(\frac y2)$
AI: HINT:
To obtain $\cos... |
H: Volume of $n$-dimensional parallelepiped as determinant
Let $V$ be a vector space of dimension $n$ and $B:V\times V\rightarrow\mathbb{R}$ be an inner product. Let $\sigma_B:V^n\rightarrow\mathbb{R}$ be the map $$ \sigma_B(v_1,\ldots,v_n)=(\det[b_{i,j}])^{\frac12},$$ where $b_{i,j}=B(v_i,v_j)$. Show that $\sigma_B(... |
H: What does $\Bbb N \to \Bbb R^{\ge 0}$ mean?
What is the interpretation of this:
Functions $$a,b,c\colon\mathbb{N} \rightarrow \mathbb{R} ^{\ge 0}?$$
AI: Standard notation for functions, a function $f$ has domain $X$ and range $Y$. We write $f:X \longrightarrow Y$. In this case we have functions from $\mathbf{N}$ to... |
H: Finding the smallest relation that is reflexive, transitive, and symmetric
Find the smallest relation containing the relation $\{ (1,2),(2,1),(2,3),(3,4),(4,1) \}$ that is:
Reflexive and transitive
Reflexive, symmetric and transitive
Well my first attempt:
Reflexive: $ S_1 = \{ (1,1),(2,2),(3,3),(4,4) \}$
Symme... |
H: Determine whether series is convergent or divergent $\sum_{n=1}^{\infty}\frac{1}{n^2+4}$
I still haven't gotten the hang of how to solve these problems, but when I first saw this one I thought partial fraction or limit. So I went with taking the limit but the solution manual shows them using the integral test.
W... |
H: Precalculus in a Nutshell, Geometry, Appendix B, Section 5, Question 14.
A sphere is circumscribed about a cube. Find the ratio of the volume of the cube to the volume of the sphere.
So I drew this diagram:
Next, I want to relate s and r.
I apply Pythagorean theory to my diagram.
$$\left(\frac12 s\right)^2+\left(\... |
H: How to understand the solution?
Two days ago,I have a problem about $\sum _{n=2}^{\infty } \frac{1}{n \log (n)}$,and the @Julien Clancy give me a solution:
To see whether $\sum_2^\infty 1/(n \log n)$ converges, we can use the integral test. This series converges if and only if this integral does:
$$
\int_2^\infty \... |
H: Formula for $n$-dimensional parallelepiped
What is the formula for the volume of a parallelepiped in $n$ dimensions with $v_1,v_2,\ldots,v_n$ as edges?
In an exercise it's given as $(\det[b_{i,j}])^{\frac12}$, where $b_{i,j}=v_i\cdot v_j$. But I also remember seeing somewhere that the volume is given by $|\det A|$,... |
H: Construction of a continuous function which is not bounded on given interval.
Actual Question is :
On Which of the following spaces is every continuous (real valued) function Bounded?
$X_1=(0,1)$
$X_2=[0,1]$
$X_3=[0,1)$
$X_4 =\{t\in [0,1] : t \text { is irrational}\}$.
I could see that $1,3$ are spaces in which n... |
H: Abstract Algebra: Homomorphism, Kernel, Image
To prove:
Let G be the group of affine functions from R into R, as defined in A. (A = {f_m,b: R -> R | m is not equal to 0 and f_m,b(x) = mx + b}. Define phi: G -> R^x as follows: for any function f_m,b in G, let phi(f_m,b) = m. Prove that phi is a group homomorphism a... |
H: non integer p adic expansion (special case)
I need to calculate the 5 adic expansion of $\frac{1}{45}$. Since i cannot compute it normally, i expand $\frac{1}{45}$ into $\frac{1}{5}*\frac{1}{9}$.
I calculated the 5 adic expansion of $\frac{1}{9}$, but i still cannot calculate the expansion of $\frac{1}{5}$
Please ... |
H: Determinant of the Sum in an Inequality
Given that: $detA > 0$ and $detB > 0$, is it the case that $det(A+B) \ge 0$?
AI: $\det\left(\begin{bmatrix}2&1\\ 1&2\end{bmatrix}+\begin{bmatrix}-1&1\\ 0&-1\end{bmatrix}\right)=\det\begin{bmatrix}1&2\\ 1&1\end{bmatrix}=-1$. |
H: If every continuous function $f$ in $X \subset \mathbb{R}^2$ is bounded then $X$ is compact.
To prove:
"If every continuous function $f$ in $X \subset \mathbb{R}^2$ is bounded then $X$ is compact."
My attempt :
In $\mathbb{R}^n$ a set $X$ is compact iff it is closed and bounded. I can show $X$ is bounded, but can n... |
H: Problem of Harmonic function.
If H is a harmonic function on an unit disk; And $H=0$ on $R_1\cup R_2$, here $R_1, R_2$ are radius of $D(0,1)$. The angle between $R_1$ and $ R_2$ is $r\pi$; here $r\in (0,1]$. If $r$ is an irrational number then is $H$ identically zero on $D(0,1)$?
I think if $r$ is irrational then... |
H: $2\times2$ matrices are not big enough
Olga Tausky-Todd had once said that
"If an assertion about matrices is false, there is usually a 2x2 matrix that reveals this."
There are, however, assertions about matrices that are true for $2\times2$ matrices but not for the larger ones. I came across one nice little exam... |
H: Logical Equivalence Of Quantified Implications
$\forall x (P(x)\rightarrow Q(x))$ is logically equivalent to $(\exists x P(x)\rightarrow\exists y Q(y))$
The proofs I've seen use logical reasoning to prove the equivalence.
Can someone supply a proof using the laws of boolean algebra?
AI: The two sentences are not... |
H: Calculating $\det(A+I)$ for matrix $A$ defined by products
Let $b_1,\ldots,b_n\in\mathbb{R}$. I have an $n\times n$ matrix $A$ whose entry is given by $a_{ij}=b_ib_j$, and I'd like to show that $\det(A+I)=\sum_{i=1}^nb_i^2+1$.
Define $b=(b_1,\ldots,b_n)$. I know that $Ab=\left(\sum_{i=1}^nb_i^2\right)b$, and $Ac=0$... |
H: Find four groups of order 20 not isomorphic to each other.
Find four groups of order 20 not isomorphic to each other and prove why they aren't isomorphic.
So far I thought of $\mathbb Z_{20}$, $\mathbb Z_2 \oplus\mathbb Z_{10}$, and $D_{10}$ (dihedral group), but I can't find another one. Would $U(50)$ work? I kn... |
H: Where to go from here...graduate school
I am an undergrad math major (minor in applied stats) set to graduate next month. I have been considering graduate school for a long time, and I know I want to pursue at least a Master's in the near future.
For a while, I wanted to pursue a Master's in Applied Statistics and... |
H: $\int_{y=-1}^{y=1}\int_{x=y^{2/3}}^{x=(2-y)^2}f(x,y)\ \mathrm dx \mathrm dy$ what does the region look like?
More specifically, I have a double integral
$$\int_{-1}^{1}\int_{y^{2/3}}^{(2-y)^2}f(x,y) \ \mathrm dx \mathrm dy$$
It is mostly the $y^{2/3}$ that is confusing me.
AI: I try to draw your region by sage. Tha... |
H: probability and combinations with the word REGULATIONS
If the letters of the word REGULATIONS are arranged at random,what is the probability that there will be exactly 4 letters between R and E?
The answer in my book is given as 11!/(9C4 x 4! x6!x2!) .Shouldn't the answer be upside down because 11!=total number of ... |
H: Show that X $\times$ Y is compact if X and Y are compact.
Show that X $\times$ Y is compact if X and Y are compact.
I know there are different solutions in this site to this question, but I want to use this exact statement:
Suppose X and Y are two topological spaces with Y compact, and $x_0$ a point in X. If M i... |
H: How can I solve $y-xy'-\sin(y')=0$
How can I solve $y-xy'-\sin(y')=0$? Are there any general techniques for solving ODE of the form $y=f(y')$, where $f$ is a trigonometric function?
AI: Assuming you mean $y(x)$, what you have is what is known as a Clairaut's equation.
Separate your $y$ and $y'$ and then differenti... |
H: What is the difference between a communicating class and a closed communicating class?
What is the difference between a communicating class and a closed communicating class?
I checked the definition on Wikipedia
http://en.wikipedia.org/wiki/Markov_chain#Properties
but I couldn't see any difference.
AI: Assume that ... |
H: Confusion on a problem on limit
Is it true that $$lim_{n->\infty}(1-(0.75)^n)^{2^n} = 0$$ and
$$lim_{n->\infty}(1-(0.25)^n)^{2^n} = 1$$. Why ?
AI: Let $\displaystyle y=\lim_{n\to\infty}(1-x^n)^{2^n}$ where $|x|<1$
$$\ln y=\lim_{n\to\infty}2^n\ln(1-x^n)=-\lim_{n\to\infty}\frac{\ln(1-x^n)}{-x^n}\cdot\lim_{n\to\infty}... |
H: How to prove that if $\sum_{n=0}^{\infty}a_n$ absolutely convergent $\Rightarrow \sum_{n=0}^{\infty}(a_n)^2$ convergent
I need to prove that if $\sum_{n=0}^{\infty}a_n$ absolutely convergent $\Rightarrow \sum_{n=0}^{\infty}(a_n)^2$ convergent.
Do you have any idea haw can I prove it?
Thank you!
AI: $\sum_{n = 1}^{... |
H: Show that a paraboloid is asurface .
That I know about paraboloid is all in the picture.
I wrote its surface patch. (Hopefully, it is correct)
From there, what do I need to do in order show that a paraboloid is a surface.
Definition of a surface: S in $\Bbb R^3$ isa surface if every point $p \in S$ admits an o... |
H: Find the horizontal tangent line
Fine the $x$-coordinate of all points on the curve $y=\sin(2x)+2\sin(x)$ at which the tangent line is horizontal. Consider the domain $x=[0,2\pi)$. There are three $x$ values.
Can you explain how to find the horizontal tangent line and vertical tangent line.
Also, $x=[0,2\pi)$ means... |
H: Reasoning why the implication $t - \epsilon \le x \le t + \epsilon$ for $\epsilon \ge 0 \Rightarrow x = t$ holds using sequences.
In texts I've seen the following reasoning used several times:
Suppose $t - \epsilon \le x \le t + \epsilon$ holds for $\epsilon \ge 0$. Then it in particular holds for $t - \frac 1 n \l... |
H: Algebraic Fractions, no I'm not kidding..
I know most of the people on this forum/site are very advanced, and I'm just sitting here wondering how in the world you do this equation, or "simplify" it.
I know how to do equations like these when there are only two terms, as in (a/b + c/d), but I can't figure out why th... |
H: Prove that $Re+Rf=Re\oplus R(f-fe)$
Let $e,f$ be idempotent elements of a ring $R$. Prove that $Re+Rf=Re\oplus R(f-fe)$.
This post solves the first part of my question. How should we prove the second part, that $e,f$ are orthogonal?
Help me. Thanks a lot.
AI: First of all, it is a good idea to say what you have t... |
H: Limits - Direct Substitution - with 1/3 power functions
First image is the question from my textbook in Latex form.
Second image shows the actual question from my book along with the solution.
What i am actually confused about is that how did they open the power 1/3 and got the following result on each step.
Is t... |
H: $\mathbb{Z}_{p}\left[x\right]/\left\langle f\right\rangle $ is a semilocal ring.
Let $p$ be a prime, $f$ be a nonconstant polynomial that is contained in $\mathbb{Z}_{p}\left[x\right]$. Prove that $\mathbb{Z}_{p}\left[x\right]/\left\langle f\right\rangle $ is a semilocal ring.
AI: Hint: If $f$ is a polynomial of de... |
H: Can we make rectangle from this parts?
I have next problem:
Can we using all parts from picture (every part exactly one time) to make rectangle?
I was thinking like: we have $20$ small square, so we have three possibility: $1 \times 20$, $2 \times 10$ and $4 \times 5$. I can see clearly that $1 \times 20$ and $2 \... |
H: Diffie hellman and the discrete algorithm problem
Suppose Alice and Bob are exchanging keys using Diffie-Hellman Key-Exchange Algorithm.
a - Alice secret key
g - generator
p - prime
x - the public key passed from Alice to Bob.
Eve is listening to the communication and she is exposed to the three parameters g,p,x.
S... |
H: Given T is normal trasformation, and $T^2 = \frac12 (T+T^*)$, prove that $T^2=T$.
if T is normal, than there exists a unitary matrix Q such that:
$Q^*TQ=diag(\lambda_1, \lambda_2, ... , \lambda_k)$
where $\lambda_1, \lambda_2, ... , \lambda_k$ are the Eigenvalues of T.
$Q^*TQ Q^*TQ= Q^*T^2Q = diag(\lambda_1, \lambd... |
H: What is the minimal polynomial of $x$ over $k(x^p - x)$?
Let $k=\Bbb F_p$, and let $k(x)$ be the rational function field in one variable over $k$. Define $\phi:k(x)\to k(x)$ by $\phi(x)=x+1$. I know that $\phi$ has order $p$ in $\operatorname{Gal}(k(x)/k)$ and $k(x^p-x)$ is the fixed field of $\phi$, but I'm confus... |
H: Prove/refute: Every tautology is contingent
I'm asking to prove/refute the following statement:
Every tautology is contingent.
According to definition of contingent:
A statement that is neither self-contradictory nor tautological is called a contingent statement. A contingent statement is true for some truth-va... |
H: Kernel of surjective ring homomorphism and induced isomorphism
Problem: Let $\phi: R\to S$ be a surjective ring homomorphism (the rings are not necessarily commutative) such that for every $a\in \ker\phi$ there is a $n\in\mathbb{N},n\ge 1$ s.t. $a^n=0$.
i) The set $N:=1+\ker\phi$ is a normal subgroup of $R^*$, the... |
H: Some questions about the function $xe^{-x}$
I want to prove that $xe^{-x}$ is bounded by some constant for all the $x \in [0, \infty)$. So I take derivative of this guy, and found that on $x \in [0, \infty)$, its derivative is positive from $[0, 1)$, $0$ at $x=1$, and negative from (1, in infinity). So $xe^{-x}$ a... |
H: The inverse of $f(x)=\sqrt[3]{1-x^3}$ is itself
As I was studying this function, $f(x)=\sqrt[3]{1-x^3}$, I checked that the function is one-to-one, and so is invertible.
Then: $$y=\sqrt[3]{1-x^3}$$
$$y^3=1-x^3$$
$$y^3-1=-x^3$$
$$1-y^3=x^3$$
$$x=\sqrt[3]{1-y^3}$$
$$f^{-1}(y)=\sqrt[3]{1-y^3}$$
What kind of functions ... |
H: The double cone is not a surface.
My question is that
A double cone ( also named as "circular cone") is not a surface.
I know its reason. But I cannot show this mathematically.
Suppose $\sigma : U \to S\cap W$ Is a surface patch.
Because the vertex $(0,0,0,)$ is a problem, S is not a surface.
I can see it. B... |
H: Find $E[(2+X)^2]$
Given $E(X)=1, Var(X)=5$. Find $E[(2+X)^2]$.
I have two approaches to solve the problem, but they're not giving the same result.
First:
$E[(2+X)^2]=E(4+4X+X^2)=E(4)+4E(X)+E(X^2)=4+4+[Var(X)+E^2(X)]=4+4+(5+1)=14$
Second:
$E[(2+X)^2]=E[(3+X-1)^2]=Var(3+X)=Var(X)=5$
Which is wrong and what mistake di... |
H: Does a clique always have a Hamiltonian path?
This isn't homework I'm just preparing for an exam and I came up with this question while I was reviewing the lecture notes.
AI: This is trivially true: the path $1-2-3-\dots-n$ always exists in the $n$-clique, and is Hamiltonian. You can even complete it in a Hamiltoni... |
H: Group theory notation
What does the notation $(G,.)$ mean in group theory? I have seen in places that $.$ implies the binary operation multiplication on group $G$. But then, why do we show an abelian group as $(G, +)$? And what is additive and multiplicative notation?
AI: $(G,\cdot)$ denotes the ordered pair with f... |
H: Given $T^2=\frac12(T+T^∗)$, prove that T is normal trasformation.
using the fact that:
$T = T_1 + T_2$
$T^* = T_1^* + T_2^*$
when:
$T_1 = \frac12 (T + T^*), T_2 = \frac12 (T - T^*)$ and $T_1^* = T_1, T_2^*=-T_2$.
then:
$T^*T = (T_1^* + T_2^*)(T_1 + T_2)= T_1^*T_1 + T_1^*T_2+T_2^*T_1+T_2^*T_2 = T_1^2 + T_1T_2-T_2T_... |
H: Cayley tables and cyclic subgroups
Given a Cayley table, how can I spot the subgroups, without missing any of them? I know which orders of subgroups that I should look for because of Lagrange's theorem, but how can I create an exhaustive list of subgroups without skipping any?
AI: Subsets of the group are created b... |
H: Zeros of a Polynomial and maximum principle
Let $P: \mathbb C \to \mathbb C$ be a non-constant polynomial and $c>0$. Let $\Omega =\{z\in\mathbb C : |P(z)|<c\}$.
I can't understand how does the maximum principle implies that every connected component of $\Omega$ contains at least one zero of $P$.
AI: Let $U$ be a c... |
H: Computing the integration $\int_0^{+\infty}{\frac{\sin(x)}{\sqrt{x}}}dx$.
For any $A>0$,$|\int_0^A{\sin(x)}|\le2$ and ${\frac{1}{\sqrt{x}}}$ is decreasing to $0$ as $x$ tend to $+\infty$.By Dirichlet's test,the integration $\int_0^{+\infty}{\frac{\sin(x)}{\sqrt{x}}}dx$ makes sense.
Moreover,I know the method of com... |
H: Proof of an inequality in arithmetic, geometric and harmonic means
Prove that $$\left[\frac{x^2+y^2+z^2}{x+y+z}\right]^{x+y+z}\gt x^xy^yz^z\gt \left[\frac{x+y+z}{3}\right]^{x+y+z}$$
I could prove the inequality between the first two terms using the fact that $A.M\gt G.M$ in the following way. $$\left[\frac{x(\frac{... |
H: Definite Integral $\int_{\frac{-\pi}{2}}^{\frac{\pi}{2}}\frac{(\cos(x))^{\arcsin(x)+1}}{(\cos(x))^{\arctan(x)}+(\cos(x))^{\arcsin(x)}}dx$
How can I prove that
$${\large\int_{-\pi/2}^{\pi/2}}\frac{(\cos(x))^{\arcsin(x)+1}}{(\cos(x))^{\arctan(x)}+(\cos(x))^{\arcsin(x)}}dx=1$$
AI: HINT:
Let $\displaystyle \arcsin x=... |
H: Distribution of the indicator function of a Gaussian variable.
Let $\xi$ be a random variable distributed according to a Normal distribution with given mean $\mu$ and standard deviation $\sigma$. Find the probability density function of
$$
\psi = c\,\mathbb{1}_{\left\{\xi\leq 0\right\}},
$$
where $\mathbb{1}_{\le... |
H: Prove, in a $\Delta ABC$ with medians $BE, CF$, $BE + CF > BC\cdot\frac32$
Consider a $\Delta ABC$ with medians $BE$ and $CF$. Prove that
$$BE + CF > BC\cdot\frac32$$
Consider the following inequalities, given by the triangle inequality:
$$BE + CE > BC \implies BE + \frac{AC}{2} > BC$$
$$\implies BE > BC - \frac{A... |
H: Finding the values of a
What are the values of $a$ for which all the roots of the equation $x^4-4x^3-8x^2+a=0$ are real?
My approach:
On differentiating the polynomial and equating it to $0$, we get $(x)(x+1)(x-4)=0$. So, the derivative of the given equation has all the three roots real. Now, what should be the bin... |
H: Finding the probability of winning the tennis game
Mark plays best when it's sunny. Mark wins a set with probability P(s) when it's sunny, and with probability P(r) when it's raining. The chance that there will be sun for the first set is P(i). Luckily for mark, whenever he wins a set, the probability that there wi... |
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