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H: Permutations and symmetric groups
Suppose that a permutation $f$ is the product of disjoint cycles $f_1,f_2,\dots, f_m$. Show that $o(f)$ is the least common multiple of $o(f_1), o(f_2),\dots, o(f_m)$.
Really lost with the question..
Say if there are 3 disjoint cycles, then $f=f_1*f_2*f_3$ say $f=(a,b)(c,d)(e,f)$ w... |
H: Is there a closed form for...
I was working on an analysis question, and was wondering if there's a closed form for $\sum_{i=0}^{log(n)}(1/2^i)log(i)$? Unless I have made a mistake, I am trying to show that $n\sum_{i=0}^{log(n)}(1/2^i)log(i) \in \Theta(n)$.
Thanks!
AI: You are interested in the sum
$$S(n) = \sum_{i... |
H: What is the proof to show that $3^n$ divides $(3n)!$?
Working on a number theory question and I am quite stuck. How would you prove that $(3n)!/3^n \in \mathbb{N}$ with $\mathbb{N}$ being the natural numbers?
AI: HINT: How many multiples of $3$ are there in the set $\{1,2,\ldots,3n\}$? |
H: What does multiplying an inverse of a quotient ring by a ring mean?
I am attempting to prove that a ring $R$ is the intersection of all $R_p$, where $R_p=S^{-1}R$ for S=R\P and the intersection is over all prime ideals P of R. The trouble right now is that I don't quite understand the statement. I understand that S... |
H: Metric Space- open sets
$\qquad\mathit{(i)}\,$ We know that $\sin:\Bbb R\to\Bbb R$ is continuous. Show that, if $\,U=\Bbb R$, then $U$ is open, but $\sin U$ is not.
$\qquad\mathit{(ii)}\,$ We define a function $f:\Bbb R\to\Bbb R$ as follows. If $x\in\Bbb R$, set $\langle x\rangle=x-[x]$ and write $$\langle x\ran... |
H: Is $n \choose k$ defined when $k < 0$? What about $n < k$?
I know that ${n \choose 0} = 1$, and this makes sense to me based on my understanding of combinatorics.
But what about ${n \choose -1}$? My instinct is that this is undefined, since it is equivalent to $\frac{n!}{k!(n - k)!}$, which when $k = -1$ equals $\f... |
H: Find $\lim_{(x,y) \to (0,0)}\frac{\sin(x - y)}{\sqrt{x} + \sqrt{y}}$
I'm trying to find the $\lim_{(x,y) \to (0,0)}$ $\dfrac{\sin(x - y)}{\sqrt x + \sqrt y}$ and I know the answer is $0$ (used WolframAlpha for that), but I need to understand why.
AI: Note that for $x,y>0$, we have
$$\dfrac{\sin(x-y)}{\sqrt{x} + \sq... |
H: Real Analysis: Continuity of a Composition Function
Suppose $f$ and $g$ are functions such that $g$ is continuous at $a$, and $f$ is continuous at $g(a)$. Show the composition $f(g(x))$ is continuous at $a$.
My idea: Can I go straight from definition and take $\delta=\min\{\delta_1,\delta_2\}$, where $\delta_1$ is ... |
H: Find the triple integral of $f(x,y,z)=(x^8+y^6+z^4)$sin$(z^3)+2y-3$.
I am trying to find the triple integral of $f(x,y,z)=(x^8+y^6+z^4)$sin$(z^3)+2y-3$ over the unit ball. I am completely stuck. I have tried converting to cylindrical and spherical coordinates and neither seems to help. Does anyone have any ideas?
A... |
H: Solving logarithms with different bases?
How would I go about getting an exact value for a question like: $\log_8 4$
I know that $8^{2/3} = 4$ but how would I get that from the question?
AI: The logarithm can be rewritten as
$$\log_8(4) = x \iff 8^x = 4$$
Now note that both $8$ and $4$ are powers of $2$ to get
$$(2... |
H: Finding $H'(1)$
Given $H(x)=F(x)G(x)$ Find $H'(1)$
Suppose:
$F(1)=2$
$F'(1)=3$
$G(1)=5$
$G'(1)=-2$
Then using the product rule I assumed $H(1)=11$ because:
$H(1)=((2)(-2))+((3)(5))=11$ using the product rule. However I am unsure as to how to derrive $H'(1)$ from $H(1)$ Would anyone mind giving me some insight on th... |
H: Discarding lower cardinality subset doesn't change infinite cardinality?
This is a basic question about set theory.
I am of the belief that if $A$ is a set of some infinite cardinality and $B$ is a subset with lower cardinality, $A\setminus B$ has the same cardinality as $A$.
This is true, right? How is it proven?
... |
H: Structure of $(\mathbb{Z}/p^k\mathbb{Z})^\times$ other than via $p$-adics?
I think I heard somewhere long ago that $\mathbb{Z}/p^k\mathbb{Z}$'s unit group is cyclic if $p$ is odd and $C_2\times C_{2^{k-2}}$ if $p=2$. I remember trying to prove it and finding it surprisingly difficult.
This morning I think I realize... |
H: A non-UFD where we have different lengths of irreducible factorizations?
A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then they are of the same length (the sum of the... |
H: Question About Lipschitz Maps and Measure Zero
I'm having difficulty with a problem involving measure.
Suppose $F : B^k \rightarrow B^n$ is a Lipschitz map from the unit ball in $\mathbb{R}^k$ to the unit ball in $\mathbb{R}^n$. If $k \lt n$, show that the image of $F$ has measure zero in $B^n$.
Starting from the... |
H: Find the probability that the largest number shown by any throw is r
So I have a pretty hard dice question on a past exam that has thrown me a bit:
A dice is rolled a random number $n$ of times. Let $A_i$ be the event that $n=i$ and suppose that $P(A_i)=1/2^i$
a.) How would i find the probability that the largest n... |
H: Prove that $\phi(n) \geq \sqrt{n}/2$
So I'm trying to prove the following two inequalities: $$\frac{\sqrt{n}}{2} \leq \phi(n) \leq n.$$ The upper bound we get from simply noting that $\phi(n) = n \prod_{p | n}\left( 1 - \frac{1}{p}\right)$ and the fact that $(1 - \frac{1}{p}) \leq 1$. But how can we get the lower b... |
H: Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.
Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$
A start: If $\lim\limits_{n\to\infty}\int_0^{n}xf(x)dx$ is finite, then it's obvious. Otherwise, ... |
H: Finding Binet's formula using generating functions
$\newcommand{\fib}{\operatorname{fib}}$
I am trying to solve the Fibonacci recurrence using generating functions, but I've run into a bit of a snag. Here's what I've done so far:
$$\begin{align}
&\fib(0) = 0\\
&\fib(1) = 2\\
&\fib(n) = \fib(n-1) + \fib(n-2)
\end{al... |
H: Find $a_{2012}-3a_{2010}/3 a_{2011}$ where the sequence $a_n$ is determined by roots of a quadratic equation
If $\alpha$ and $\beta$ are the roots of $x^2-9x-3=0$, $a_n=\alpha^n-\beta^n$ and $b_n=\alpha^n+\beta^n$, then find the value of $\dfrac{a_{2012}-3a_{2010}}{3 a_{2011}}$ and $\dfrac{b_{2012}-3b_{2010}}{3b_{2... |
H: $\log(n)$ is what power of $n$?
Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small $\epsilon$, but what value is $\epsilon$ exactly?
AI: No: If $\epsilon$ is any positive n... |
H: How to prove some statements about divisibility and the $\gcd$ function
Struggling with some number theory homework. Could use a helping hand. The two statements are as follows
$\gcd(c, ab) \mid \gcd(c,a)\gcd(c,b)$
If $c \mid ab$ and $\gcd(a,b)=1$, then $c=\gcd(a,c)\gcd(b,c)$
I'm supposed to figure out a proof fo... |
H: Expected Value and Summation
The question is suppose $X$ ~ $Geometric(\frac{1}{2})$ and I have computed $P(X > 1) = \frac{1}{2}$ and $Var(X) = 2$ where $E[X] = 2$ and $E[X^2] = 6$. How can I verify that $E[X] = \sum_{n=0}^{\infty}P(X > n)$. I know that this formula holds in general but which theorem relates all of ... |
H: Why rows should be proportional for independence?
Suppose that $X$ and $Y$ have discrete distributions; that $X$ can take the values $1,2,\dots,r$; that $Y$ can take the values $1,2,\dots,s$; and that $$Pr(X=i \text{ and } Y=j)=p_{ij}\text{ for }i=1,2,\dots,r \text{ and }j=1,2,\dots,s.$$Then for $1,2,\dots,r$, let... |
H: Fixed field of automorphism $t\mapsto t+1$ of $k(t)$
I'm working on the following problem:
Determine the fixed field of the automorphism $t\mapsto t+1$ of $k(t)$. (Ex 7, section 14.1, Abstract Algebra by Dummit & Foote).
Here is my attempt of a solution:
Let be $\sigma$ the automorphism determined by $t\mapsto t... |
H: Question about distinctness of coset given $H \leq K \leq G$
This question is regarding a detail I am unsure about in a problem in the Dummit and Foote abstract algebra textbook that I am currently working on. The problem itself is that if $H \leq K \leq G$ I need to show $|G:H| = |G:K|\cdot|K:H|$. Within this prob... |
H: Complex Analysis Problem
Find the image under the map f(z) = $ e^{iz} $ of the square
S = {z $\in$ $C$ | 0 $\leq$ Re(z) $ \leq $ 1, 1 $\leq$ Im(z) $\leq$ 2}
and label the corresponding portions of the boundary.
How would I go about doing this?
AI: The paths are relatively straightforward ; the bottom of the square ... |
H: Problem on Lagrange multipliers
This problem has two parts:
$a)$ Let $k>0$, find the minimum of the function $f(x,y)=x+y$ over the set S=$\{(x,y) \in \mathbb R^2_{> 0}:xy=k\}$. $b)$ Prove that for every $(x,y) \in \mathbb R^2_{> 0}$ the inequality $\frac{x+y} {2}\geq \sqrt{xy}$ holds.$$$$
I want to find the minimu... |
H: extending a limit to an integral case
For a finite set of postitive real numbers $\{g_i\}$, it's clear that
$$
\lim_{\lambda->\infty} \frac{1}{\lambda} \ln\left(\sum_i e^{\lambda g_i} \right) = \max \{g_i\},
$$
since the argument of the logarithm becomes dominated by the largest of the $e^{\lambda g_i}$ terms. In t... |
H: Poincaré inequality for $W_0^{1,\infty}$
In the book A first course in Sobolev spaces by Leoni, the following Poincaré inequality for $W_0^{1,p}(\Omega)$ is stated:
Suppose $\Omega\subset \mathbb{R}^n$ has finite width (lies between two parallel hyperplanes) and $p\in [1,\infty)$. Then for all $u\in W^{1,p}_0(\Omeg... |
H: What is the Probability that the Lowest Card out of 4 Cards is X?
If I have four different value cards, what is the probability that the lowest card (ace, lowest -> king, highest) is some value X?
Here is what I have so far:
I know that the lowest value card cannot be a king, queen, or jack, as there must exist a v... |
H: are all dynamical systems described by differential equations?
we defined in lecture a dynamical System as a one-parameter family of maps $\phi^t:M\rightarrow M$ such that $\phi^{t+s}=\phi^t\circ\phi^s$ and $\phi^0=Id$, where $M$ is some (smooth) manifold and $s,t\in (a,b)\subset\mathbb{R}$.
Of course, if we consid... |
H: Boolean Algebra A + AB = A
Hi I have a question about the following algebra rule
A + AB = A
My textbook explains this as follows A + AB = A This rule can be proved as such:
Step 1:
Dustributive Law:
A + AB = A*1 = A(1+B) Huh...? Where do they get the one(1) from?
Step 2:
1 + B = 1 {Question:1 + B = B right? so how ... |
H: Is $(\mathbb R,\tau)$ compact?
Let us consider $\tau=\{G\subset \mathbb R: \mathbb R\setminus G$ is compact set in ($\mathbb R,\tau_u)\}$, where $\tau_u$ denotes the usual topology on $\mathbb R$. Then $\tau$ is a topology coarser than $\tau_u$. Since $(\mathbb R, \tau_u)$ is connected, so $(\mathbb R, \tau)$ is co... |
H: Question related to the spectrum of a bounded operator
If $A$ is a bounded linear operator on a Banach space $X$ and $\lambda\in \sigma(A)$, is it true that for all $\epsilon>0$, there is $ x\in X$ and $||x||=1$ such that
$$ ||(A-\lambda I) x|| <\epsilon \ ?$$
If not, is it true if we consider $X$ a Hilbert space ... |
H: Proof or disproof that $\frac{\pi^{8}}{3150}=\sum\limits_{k=0}^{\infty} \int\limits_{0}^{\infty}\frac{t^{3} e^{-4(k + 1)t}}{1-e^{-4(k + 1)t}}\,dt$
I need a proof or disproof of- $$\frac{\pi^{8}}{3150}=\sum_{k=0}^{\infty} \int_{0}^{\infty}\frac{t^{3} e^{-4(k + 1)t}}{1-e^{-4(k + 1)t}}\,dt$$
AI: For every $\alpha\gt0$... |
H: Symbol for "if any"
I am looking for a symbol if any for the following equation in my algorithm
This is to find closed pattern where $p_i$ is longer than $p$ and $p$ is a sub-pattern of $p_i$ and $support(p) = support(p_i)$
so can I say
if $\forall \; p \subset p_i$ and $support(p) = support(p_i)$ or
if $ \left | ... |
H: how do I solve this seperable equation with so many terms?
Solve given differential equation by separation of variables
$$\frac{dy}{dx}=\frac{xy+3x-y-3}{xy-2x+4y-8}$$
I started by multiplying each side by the denominator to get
$$(xy-2x+4y-8) dy = (xy+3x-y-3) dx$$
Now that there is $xy$ on each side of the equation... |
H: Let $\sum_{n=1}^\infty \frac{a_n}{3^n}.$ Determine (numerically or not) the limit of the infinite series by choosing $a_n=0$ or $2$ randomly.
The problem I have is essentially in the title.
What I'm trying to do in Matlab is to have a set which has two elements $0$ and $2$ and to choose either $0$ or $2$ randomly a... |
H: characterization of weakly convergent to zero sequences on $l^p$ for $1\le p < \infty$
Let $1\le p< \infty$. Show that a sequence $t_k = ({t_{kj}})_{j=1}^{\infty}\in l^p$ converges weakly to 0 iff $||t_k||_p$ is bounded and $\lim_k t_{kj}=0$. I proved that if $t_k$ converges weakly to 0 then we conclude that. I wa... |
H: Show that if $10$ divides into $n^2$ evenly then $10$ divides into $n$ evenly
I'm not sure how to show that if $10$ divides into $n^2$ evenly, then $10$ divides into $n$ evenly.
AI: This answer is an answer to the original form of this question "show that if $20$ divides into $n^2$ evenly, then $20$ divides into $n... |
H: Prove the following identity for Fibonacci numbers
Prove this:
for any positive integer $a,b,c$,
$F_{a+b+c+3}=F_{a+2}(F_{b+2}F_{c+1}+F_{b+1}F_c)+F_{a+1}(F_{b+1}F_{c+1}+F_bF_c)$
Is there any way other than induction to prove this?
AI: Consider matrix formula from here. Let $M=\pmatrix{1 & 1\\ 1 & 0}$. Then $M^{a+1... |
H: If $A \in Mat_{n \times n} (\mathbb{Q}) $ is a matrix, where $n$ is odd show that $A^2 \neq 2I.$
I don't quite see the use of the fact that $n$ is odd.
Anyway, I give a counterexample: Take a matrix $A$ where $a_{ii}$ are the non-zero rational numbers, and all other entries are zero. Clearly, $A^2 \neq 2I.$ If it w... |
H: orthonormality and transpose of vectors
Let $ \{u_1, u_2, \ldots , u_k \} $ be $k$ vectors in $\mathbb{R}^n$. Show that $\{u_1, u_2, \ldots, u_k\}$ is an orthonormal family if and only if the $n \times k$ matrix $U = \begin{bmatrix}
u_1,u_2, \ldots, u_k
\end{bmatrix}$ satisfies the equation $U^{\intercal} U = I_k$... |
H: product of two numbers ending in 6 also ends with 6
We have to prove that
"the product of two numbers ending in 6 also ends with 6" mathematically.
I have no clue how to start. I don't want you to prove it for me! but some hints would be very helpful as I'm totally stuck on how to start this prove.
AI: HINT: If $a$... |
H: Groups with no abelian centralizer
Suppose $G$ is a finite group with no abelian centralizers. Is it true that $G$ must be a 2-group?
Thanks for any help.
AI: No, this is not necessarily the case. If $G$ is a group such that there are no abelian centralizers in $G$, then $G \times H$ also has this property for a... |
H: Does there exist an injection from $P(S)$ to $u(S)$
Let $S$ be an uncountable set , let $u(S)$ denote the set of all uncountable subsets of $S$ and let $P(S)$ denote , as usual , the powerset i.e. the set of all subsets of $S$, then does there exist an injection $f:P(S)→u(S)$ ?
AI: Yes. $|S\times\{0,1\}|=|S|$, so r... |
H: Minimization of Sum of Squares Error Function
Given that $y(x,{\bf w}) = w_0 + w_1x + w_2x^2 + \ldots + w_mx^m = \sum_{j=0}^{m} w_jx^j$ and there exists an error function defined as $E({\bf w})=\frac{1}{2} \sum_{n=1}^{N} \{y(x_n, w)-t_n\}^2$ (where $t_n$ represents the target value). I'm having trouble making sens... |
H: Problem about pointwise and uniformly convergence
We define $f_n$ define the functions with $n \in N$, as follows:
$$f(n) = \begin{cases}
\sin(x) &\mbox{if } 2\pi n \leq x \leq 2\pi(n+1) \\
0 & \mbox{if other way} \end{cases}
$$
The functions $f_n$ are uniformly bounded and equicontinuous and converge
pointwise, ... |
H: number of compositions of [n] that each contain a largest part
I am trying to generalize for any [n] the number of compositions that each contain a largest part. [1] has only one composition with a unique largest part, [2] has 1, [3] has 3 compositions with a unique largest part, [4] has 6 compositions with a uniqu... |
H: Let $\sum_{n=1}^\infty \frac{a_n}{3^n}.$ Determine (numerically or not) the limit of the infinite series by choosing $a_n=0$ or $2$ randomly.
This is a follow up question to Let $\sum_{n=1}^\infty \frac{a_n}{3^n}.$ Determine (numerically or not) the limit of the infinite series by choosing $a_n=0$ or $2$ randomly. ... |
H: Restrict a metric, gives same topology as subspace topology from larger space $X$
Let $A\subseteq X$. If $d$ is a metric for the topology of $X$, show that $d\restriction_{A\times A}$ is a metric for the subspace topology on $A$.
I've shown that $d'=d\restriction_{A\times A}$ is a metric on $A$.
I am letting $\ta... |
H: When is the product of two points on a circle is still on the circle?
Suppose we have a pair $(x,y)\in\Bbb{R}^2$ such that $x^2+y^2=1$. I want to find a pair $(a,b)$ such that $a^2+b^2=1$ and $(xa)^2+(yb)^2=1$. What is $a$ and $b$ in terms of $x$ and $y$?
AI: Given $x^2+y^2=1$ and $a^2+b^2=1$, multiplying, we have ... |
H: Some question about even function
Given that $f$ is an even function, $f''(x)>0$. Which of the following are true?
I. $f(0)<f(1)$
II. $f(4)-f(3)<f(6)-f(5)$
III. $f(-2)<\frac{f(-3)+f(-1)}{2}$
I understand I is true since $f$ is increasing for $x>0$, and II is also true since $f'(x)>0$ for $x>0$. How about III? Any h... |
H: Is this notation correct?
I am writing a paper and I have an expression something like this:
$$\begin{equation}
\notag
x = \text{large_expression} + \begin{cases}
y & \text{if } a<b \\
z & \text{if } a=b \\
\end{cases}
\end{equation}$$
Will this notation be understood just as... |
H: Help with a question regarding basis and dimension?
I am presented with:
Let $a_1$ and $a_2$ be linearly independent vectors in $R^3$ and let x be a vector in $R^2$.
a) Describe geometrically Span($a_1, a_2$)
b) If A=($a_1,a_2$) and b=Ax, then what is the dimension of Span($a_1, a_2$, b)? Explain.
For a), since t... |
H: Hartshorne's proof that $\mathcal{O}_{\operatorname{Spec} A}(D(f)) \cong A_f$, Prop II.2.2(b)
Hartshorne, "Algebraic Geometry," Proposition II.2.2(b) on page 71 reads (roughly):
$\mathcal{O}_{\operatorname{Spec} A}(D(f)) \cong A_f$
The relevant section of the proof reads (after some simplification):
We define $\... |
H: Subset of $\mathbb{Z}$ not a subgroup of $(\mathbb{Z},+)$
I have this question for homework and I can't figure it out.
Find a subset of $\mathbb{Z}$ that is not a subgroup of $(\mathbb{Z},+)$ but adheres to closure.
In other words, a subset whose elements when added are elements of the subset, but it is not a subgr... |
H: Graph Cut Problem
I have a problem as such:
$2n$ players, each of whom has an odd number of friends, are
distributed into two teams. A player is happy if more of his friends
are on the other side than on his own side, and a distribution is
stable if everyone is happy. The cut number of a distribution is the
number... |
H: What use are Fermat’s Little Theorem and Wilson’s theorems in number theory?
Do these theorems have any real life applications? We cannot use them to find primes as both are pretty inefficient for large numbers.
AI: Fermat’s Little Theorem plays a key role in modern life in the proof of correctness of the RSA algor... |
H: Find the radius and centre of the circle $x^2 -6x +y^2 -2y -6=0$
Find the radius and centre of the circle $x^2 -6x +y^2 -2y -6=0$
Can someone please help me with this question? I'm quite lost with what I have to do.
AI: HINT Try to complete the square $x^2-6x$ to get it into the form of $(x-a)^2-c$, do the same for... |
H: Do monotone operators have positive Frechet derivatives?
If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$.
Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces with its dual space $V^*$, via:
The operator $A\colon V \to ... |
H: Generating $A_4$ with two distinct $3$-cycles $x$ and $y$ in $S_4$
Given two distinct $3$-cycles $x$ and $y$ in $S_4$ such that $x\neq y^{-1}$, I would like to show that $A_4$ is generated by any such pair of $x$ and $y$.
The only times where $3$-cycles in $S_4$ can be the inverse of each other is when they can be ... |
H: Order of operations: Matrix product and hadamard product
Is there any convention of the order of operations in a term with both ordinary matrix multiplication and hadamard (elementwise) multiplication?
Obviously,
$$ A ( b \circ c ) \ne (A b) \circ c $$
But how is $ A b \circ c $ (without parentheses) conventiona... |
H: How to solve the inequality $\frac {5x+1}{4x-1}\geq1$
Please help me solve the following inequality.
\begin{eqnarray}
\\\frac {5x+1}{4x-1}\geq1\\
\end{eqnarray}
I have tried the following method but it is wrong. Why?
\begin{eqnarray}
\\\frac {5x+1}{4x-1}&\geq&1\\
\\5x+1&\geq& 4x-1\\
\\x &\geq& -2
\end{eqnarray}
... |
H: Proof that a subset of R has same cardinality as R
I am sitting with a task where I have to prove the following:
Claim:
Every subset of $\mathbb{R}$, that contains an interval $I$ with $a < b$, has the same cardinality as $\mathbb{R}$.
So I think that I should prove that there exist a bijection from $I$ to $\mathbb... |
H: Prove that this is a legitimate PMF
I know that these are two properties of PMF.
Non-negativity
Sum over the support equals 1
However I can't show that this PMF's sum over the support equals 1.
AI: Using the Taylor series for $e^{\lambda}$ you have
$$
\sum_{k=0}^{\infty} p_X(k) =
\sum_{k=0}^{\infty} e^{-\lambda} ... |
H: A point P moves so that AP and BP are perpendicular. Find the equation of the locus of P
A point p(x,y) moves so that AP and BP are perpendicular, given A=(3,2) and B =(-4,1). Find the equation of the locus of P.
Can someone please advise me on what to do for this question. Just need a direction. Thanks.
AI: Direct... |
H: non-transitive, antisymmetric and reflexive binary relation on $\mathbb Z$
Does anybody know about a reflexive, antisymmetric, but not transitive relation on $\mathbb Z$? I really cant figure any out and I am having doubts that something like that exists.
AI: Let $R \subseteq \mathbb Z^2$ denote the relation to fin... |
H: Unknown terms of the proportion
please help me solving this problem. The question is, find the unknown terms of the proportion $$\frac 23 = \frac x{12} = \frac y{15}.$$
AI: $$\dfrac 23 = \cfrac x{12}$$ $$\dfrac 23 = \dfrac y{15}$$
Cross multiplying gives us the following equalities:
$$2\cdot 12 = 3x$$ $$2 \cdot 15... |
H: Upper bound on a sum of complex numbers
Let $A=\{z_1, z_2, z_3,\ldots \} $ be a set of complex numbers with $|z_j|\ge 1$ such that the number of elements of $A$ with modulus $<r$, denoted $N_A(r)$, satisfies
$$
N_A(r) \le C_0r^N
$$
for some positive integer $N$ where $C_0$ is some constant. For any $\delta >0$ I ... |
H: Brownian Motion hitting random point
I got a problem that seems to be quite standard and easy, but I have lots of problems with it. I do already know that $T_a:=\inf\{t\geq 0: B_t=a\}$ is a stopping time for any $a\in\mathbb{R}$ fixed, $B_t$ being a standard BM.
But what about for instance $T_{B^{o}_1}$, where $B... |
H: Plot implicit equation
I'm working with a frequency-response curve of a nonlinear oscillator and came across the following equation (Kovacic & Brennan 2011, p. 179):
$$
A^2 = \frac{f^2}{4 \xi^2 \omega^2 + (\omega^2 - 1 - \frac{3}{4}A^2)^2 }.
$$
Question: what is the strategy for plotting of $A$ vs. $\omega$?
I am... |
H: Isometries of the plane
Let $m$ be an orientation-reversing isometry. Prove algebraically the $m^2$ is a translation.
What I attempted: We know that $m$ is an orientation-reversing isometry i.e. it is either a reflection or a glide-reflection.
I started by consider the case when $m$ is a reflection.
The matrix repr... |
H: Proving ${\sim p}\mid{\sim q}$ implies ${\sim}(p \mathbin\& q)$ using Fitch
I am struggling with proving something in Fitch. How can I prove from the premise ~p | ~q , that ~(p & q). Any ideas on how I should proceed?
AI: I'll have my first stab at an answer here (I've been won over from Stack overflow- first answe... |
H: Opposite function definition
How to define an opposite of a function. If for example I have the function F(x) = y how can use it to define the function f(y) = x
AI: What you are seeking is called the inverse function $f^{-1}(x)$ of a bijective function $f(x)$. A function's inverse exists if and only if the function... |
H: Rolling a die, probability problem
So this is a problem I'm stuck on,
You roll a fair 4-sided dice and with probability 1/3 you get to roll once more, and with probability 2/3 you have to stop. Assume that you get as many coins as the sum of the rolls. What is the probability you will win an even number of coins?
C... |
H: Equivalence of zero divisor in commutative ring
Let $x$ be a nonzero element in a commutative ring, then $\exists y\neq0(xy=0)$ ($x$ is a zero divisor) iff $\exists y\neq 0(x^2y=0)$. $(\rightarrow)$ part is pretty trivial. How to prove the other way?
AI: Hint: $x^2y=x(xy)$. Consider the cases $xy = 0$ and $xy \ne 0... |
H: How to calculate Frenet-Serret equations
How to calculate Frenet-Serret equations of the helix
$$\gamma : \Bbb R \to \ \Bbb R^3$$
$$\gamma (s) =\left(\cos \left(\frac{s}{\sqrt 2}\right), \sin \left(\frac{s}{\sqrt 2}\right), \left(\frac{s}{\sqrt 2}\right)\right)$$
I know the following info about Frenet-Serret equa... |
H: Clarification of L'hospital's rule
I have a question regarding L'hospital's rule.
Why can I apply L'hospital's rule to $$\lim_{x\to 0}\frac{\sin 2x}{ x}$$ and not to $$\lim_{x\to 0} \frac{\sin x}{x}~~?$$
AI: The reason you cannot use L'Hopital on the $\sin(x)/x$ limit has nothing to do with calculus, and more wit... |
H: Prove that $\lfloor0.999\dots\rfloor= ?$ $0$ or $1$?
I think $\lfloor0.999\dots\rfloor= 1$, as $0.999\dots=1$,but I have doubt, as $\lfloor0.9\rfloor=0$,$\lfloor0.99\rfloor=0$,$\lfloor0.9999999\rfloor=0$, etc.
AI: Your first assertion is correct. The other observation just says that the function $x\mapsto\lfloor x\... |
H: Uniform convergence of $\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ on $\mathbb{R}$?
For every $r>0,$ the series $f(x)=\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ converges uniformly on $[-r,r]$. May I know how to prove/disprove that $\sum^{\infty}_{n=0}(-1)^n \frac{x^{2n}}{n!}$ converges uniformly on $\mathbb{R}... |
H: How to find the $f^{-1}(x)$ of $f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}}$
It is a question from a quiz.
The following is the whole question.
Let
\begin{eqnarray}
\\f(x)=x^{3}-12x+\frac{48}{x}-\frac{64}{x^{3}} , \space x\in (-\infty ,0),
\end{eqnarray}
find $f^{-1}(x)$. Hint : $f(x)$ can be written in the f... |
H: Limit induction proof
I need help to verify the following
Prove that if does not equal 0
lim of x approaches a: x^-n = a^-n
I know to prove lim of x approaches a: x^n = a^n requires induction so I believe that this problem requires the same.
lim of x approaches a: [f(x)]^-n ... = k^-n
inductive step
lim of x a... |
H: Prove that $ x_{k}=2^{k} \cdot \sin \frac{\pi}{2^{k}}$ equals $ x_{1}'=2, x'_{2}=2 \sqrt{2}, x_{k+1}=x_{k} \sqrt{\frac{2x_{k}}{x_{k}+x_{k-1}}}$
Prove that $ x_{k}=2^{k} \cdot \sin \frac{\pi}{2^{k}}$ equals $ x_{1}'=2, x'_{2}=2 \sqrt{2}, x_{k+1}=x_{k} \sqrt{\frac{2x_{k}}{x_{k}+x_{k-1}}}$.
This is what I've managed:... |
H: Can we replace the condition $d(E_1,E_2)>0$ with $E_1\cap{E_2}=\emptyset$ in an property of outer measure?
Can we replace the condition $d(E_1,E_2)>0$ with $E_1\cap{E_2}=\emptyset$ in "If $E=E_1\cup{E_2}$,and $d(E_1,E_2)>0$,then $m_\star(E)=m_\star(E_1)+m_\star(E_2)$."? Obviously,$E_1\cap{E_2}=\emptyset$ can not im... |
H: Laurent series of $\frac{1-\cos(z)}{z^2}$
How do I calculate the Laurent series of $\frac{1-\cos(z)}{z^2}$? (I know the general formula as is shown here )
AI: If you don't want to use the general formula, but you know the series for $\cos z$, then you can use them.
I understand that $f(z)=\frac{1-\cos z}{z^2}$ is n... |
H: Meaning of "relative likelihood".
I am quoting this from DeGroot's "Probability and Statistics".
Let $X$ and $Y$ be random variables. The joint probability density function(p.d.f.) $f$ defines a surface over the $xy$-plane for which the height
$f(x, y)$ at each point $(x, y)$ represents the relative likelihood o... |
H: how to determine which cell in a grid a point belongs to
I have a square area which is divided into an N X N grid.I need to insert a point (x,y) into this area.I tried to find out if there is a relation between the value of N and x,y coordinates sothat I can say this particular point belongs in the cell (0,3) or so... |
H: Hints for a complex limit: Prove if $\lim_{z \to \infty} f(z)/z = 0$ then $f(z)$ is constant.
(To clarify, I would just like a hint. Please do not give me the answer to this problem. )
The solution to the following problem has really evaded me here:
Problem:
Assume that $f$ is entire and that $\lim_{z \to \infty} f... |
H: smallest algebra generated by ring
Let $\Omega\neq \emptyset$ and $\mathcal E$ be a ring in $\Omega$ and $\mathcal R_0 := \mathcal E\cup \{E^c:E\in\mathcal E\}$. Show that $\mathcal R_0$ is the smallest algebra in $\Omega$ containing the ring $\mathcal E$, i.e. $\mathcal R_0=\mathcal R_0(\mathcal E)$.
I have been a... |
H: What is the asymptote for the positions of the largest Stirling numbers of the second kind?
The infinite lower triangular array of Stirling numbers of the second kind starts:
$$\begin{array}{llllllll}
1 & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} & \text{} \\
1 & 1 & \text{} & \text{} & \text{} & ... |
H: Chebysev inequality proof
Let $(X,\Omega,\mu)$ be a measure space and let $f$ be an extended real valued measurable function defined on $X$. Proof that
$$
\mu\left(\{x\in X : |f(x)|\geq t\}\right)\leq \frac{1}{t}\int_X |f|~d\mu
$$
for any t>0.
Then, conlude that the measure of the set $$\{x\in X : |f(x)|\geq t\}$... |
H: on two dimensional graded vector spaces
I read the following statement:
Consider a graded vector space $V$ with basis $\{a, b\}$ such that $a \in V^2$
and $b \in V^5$.
Does this mean that $V=\bigoplus_{i\geq 0}V^i$ such that all $V^i$ are $0$ except $V^2$ and $V^5$ hence we can simply write $V=V^2\oplus V^5$ an... |
H: Prove linear operator is a reflection
Prove that a linear operator on $\mathbb{R}^2$ is a reflection if and only if its eigenvalues are $1$ and $-1$ and the eigenvectors with these eigenvalues are orthogonal.
$\Rightarrow$: Let $r: \mathbb{R}^2\ \rightarrow \mathbb{R}^2$ such that $\forall x= \begin{pmatrix} x_1 \\... |
H: Find an Inverse function
I need to find the inverse of those functions:
$x \mapsto \sin e^{x}$
$x \mapsto e^{\sin x}$
I know that the way is to solve the equation $y = f(x)$ for $x$, and I did it with functions like ($x \mapsto x^2$, $2x+3$) but I can't do it with those too.
Thanks for help!
AI: You are looking ... |
H: Calculate the 1 in a value
I am getting different values from a computer program that I designed. I then want to formulate an algorithm to calculate the 1th value below it (and I'm not really sure the terminology for this so bear with me).
Examples if I have:
43535 the result = 10000,
76765 = 10000,
5674 = 1000, ... |
H: A common eigenvector of $A^2$ and $A^{-1}$
Show that the eigenvector of $A^2$ is the same as the one of $A^{-1}$, where
$$A = \begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}.$$
All that I can find is that
$$A^2 = \begin{bmatrix} 4 & 4 \\ 0 & 4 \end{bmatrix}, \quad A^{-1} = \begin{bmatrix} 1/2 & -1/4 \\ 0 & 1/2 \end{b... |
H: Define domain $X,$ predicate $A(X)$ and $B(X)$
I'm having trouble creating a domain $X$ and the predicates $A(X)$ and $B(X)$ to for this set of sentences to be evaluated to be true or false.
$(T)\quad \forall x \in X, (A(x) \rightarrow B(x))$
$(F)\quad \exists x \in X (A(x) \land B(x))$
I've tried letting X be th... |
H: Prove that every integer ending in 3 or 7 has a prime factor that also ends in 3 or 7
Prove that every integer ending in 3 or 7 has a prime factor that also ends in 3 or 7.
I have that such an integer n has n=3 or 7(mod 10) but don't know where to go from there.
Then show that there are infinitely many prime number... |
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