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H: Definition of ordinal exponentiation
I found that the usual ordinal exponentiation $\alpha^{\beta}$ is the set of functions from $\beta$ to $\alpha$ with finite support, ordered by antilexicographic order. (least significant position first) It cannot be defined on whole functions, since $f,g:\omega \to 2$ given by ... |
H: Smooth homotopic maps and closed forms..
Does anyone have any idea for showing the following: Let $f_0, f_1:M\rightarrow N$ smooth homotopic maps between the manifolds $M$ and $N$. Suppose $M$ is compact with no boundary. Show that for every closed form $\omega\in \Omega^m(N)$ (where $m=\dim M$), $$\int_{M}f_0^*\om... |
H: $\sqrt 2$ is even?
Is it mathematically acceptable to use Prove if $n^2$ is even, then $n$ is even. to conclude since 2 is even then $\sqrt 2$ is even? Further more using that result to also conclude that $\sqrt [n]{2}$ is even for all n?
Similar argument for odd numbers should give $\sqrt[n]{k}$ is even or odd whe... |
H: Optimum fitting for flanges in a rectangular plate
I have a $2500~\text{mm}\times6300~\text{mm}\times25~\text{mm}$ (width $\times$ length $\times$ thickness) steel plate I want to cut flanges of diameter $235~\text{mm}$ can anyone please suggest
$1)$ How many flanges would fit in this plate?
$2)$ A method of cuttin... |
H: Identifying traitors
Logic is one of the facets of math that is more 'fun', but this one is beyond me. Consider using logic statements and/or truth tables. 'Or' here is inclusive, that is 'A or B' means 'A or B or both, but not neither'.
King Warren suspects the Earls of Akaroa, Bairnsdale, Claremont, Darlinghurst,... |
H: How to show that whether $[ \frac{-p}{q}] =-[ \frac{p}{q}] $ or not?
How to show that whether $[ \frac{-p}{q}] =-[ \frac{p}{q}] $(1) or not?
I think it does not hold, because if p=3 and q=5, then by Euler criterion you get that $3^{\frac{5-1}{2}}= 3^2=9 \equiv 4 \equiv -1 \pmod 5$. So $-[ \frac{p}{q}]=-(-1)=1$. No... |
H: Find the area of a circle that is NOT covered by the rectangle
Using the following image for a visual:
Is there a formula or equation I can use to find the area of the circle NOT overlapped with the rectangle (i.e. the filled in orange part)? I know all of the coordinates & sizes of both the rectangle and the circ... |
H: Does a continuous embedding preserve gaps between subspaces?
I have a separable, reflexive Banach space $(V,\|\cdot\|)$ that is continuously and densely embedded in a Hilbert space $(H,|\cdot|)$. This means, there is a bounded linear injection map $j\colon V \to H$, i.e. $|j(v)|\leq C_j\|v\|$, and $j(V)$ is dense i... |
H: $K[x_1, x_2,\dots ]$ is a UFD
I wonder about how to conclude that $R=K[x_1, x_2,\dots ]$ is a UFD for $K$ a field.
If $f\in R$ then $f$ is a polynomial in only finitely many variables, how do I prove that any factorization of $f$ in $R$ only have factors in these indeterminates, i.e. takes place in the UFD $K[x_... |
H: Solving the next Integral (by parts?)
Can anyone give me a hint / help with the next integral? Thanks!
$$\displaystyle\int_{0}^{t}{x^{a-1}(t-x)^{b-1}dx}$$
AI: This integral equals $t^{(a+b-1)}\frac{\Gamma(a)\Gamma(b)}{\Gamma (a+b)}$
Take $y=xt$
Then we have,
$\displaystyle\int_{0}^{t}{x^{a-1}(t-x)^{b-1}dx}=t^{(a+b-... |
H: Definition by Recursion and a Question about Induction
I have some questions to ask.
Suppose I want to define some sequence of propositional formulas $\{\varphi_{j}\}_{j\in\mathbb{N}}$. First, I define it this way. Fix an enumeration $p_{1},p_{2},\ldots$ of propositional variables. For any $j\in\mathbb{N}$, define... |
H: Is there some nomenclature to get the integer value of a fraction?
There is some math nomenclature to represent the integer value of a fraction?
Say,
$$x \in \mathbb{R},\, \textbf{foo}(x) = \text{integer part of }x$$
Then
$$x = 1.823,\, \textbf{foo}(x) = 1$$
AI: The integer part of $x$ is given by the somewhat ugly... |
H: Simmetric complex matrices
I must show a counter example to the sentence "every simmetric complex matrix is diagonalizable." But Im having issues on guessing one of them. Can someone help me?
AI: As you have found a counterexample, I can give you a full answer now. Just try this:
$$
A=\pmatrix{i&1\\ 1&-i}.
$$
Since... |
H: Why are only fractions with denominator 2 and 5 non-repeating?
Given a rational number $\frac{n}{d}$, I understand that in the base $10$ number system, the number can be represented as a non-repeating decimal number if and only if $d$ has only prime factors of $2$ and $5$.
I have a hypothesis that the reason for th... |
H: What (filtered) (homotopy) (co) limits does $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$ preserve?
Consider the functor $\pi_0:\mathbf{sSets}\to\mathbf{Sets}$.
$\pi_0$ does not preserve arbitrary limits
$\pi_0$ does not send homotopy limits to limits
$\pi_0$ does preserve filtered colimits
$\pi_0$ does preserve arbitrar... |
H: Unbiasedness of product/quotient of two unbiased estimators
An answer to this question might just be "it depends", however I am wondering:
Given unbiased estimators $\hat{\mu}_X$ and $\hat{\mu}_Y$ for the means $\mu_X$ of $X$ and $\mu_Y$ of $Y$ respectively. Under what conditions are the following true: $$E[\hat{\... |
H: Please help me to prove this inequality: $|x|+|y|≥|x+y|$
Please help me to prove the following inequality:
$|x|+|y|\geq|x+y|$
in which $x$ and $y$ are real numbers.
Any help or hint would be appreciated. Thanks :)
AI: Hint: If $w$ and $z$ are real numbers, $w,z \in \mathbb{R}$, (and not sets, as implied by your... |
H: Solve a system of linear congruences
I have this system:
$$ \begin{align}
a_{11} x_1 + a_{12} x_2 + \ldots + a_{1n} x_n &= b_1 \mod p \\
a_{21} x_1 + a_{22} x_2 + \ldots + a_{2n} x_n &= b_2 \mod p \\
\vdots \\
a_{n1} x_1 + a_{n2} x_2 + \ldots + a_{nn} x_n &= b_n \mod p \\
\end{align} $$
Can I solve ... |
H: little problem about open set in the definition of topology
Definition 1
Let $X$ be a set of points. A collection of subsets $U = \left\{U_{\alpha }\right\}$ forms a topology on $X$ if
Any arbitrary union of the $U_{\alpha }$ is another set in the collection $U$.
The intersection of any finite number of sets $U_{... |
H: Is there a symbol for matrix multiplication operator?
Title says it all.
Is there any specific operator symbol for matrix multiplication?
Not just write down side by side but symbols like cross ($\times$).
AI: Juxtaposition is the standard notational convention (to "write side by side") without an intermediary oper... |
H: Are there any distinct finite simple groups with the same order?
In finite group theory, it's often quite easy to show that there are no simple groups of a given order $n$. My question is different: is there some natural number $n$ such that there are two non-isomorphic simple groups of order $n$? The two easiest... |
H: How do you invert a characteristic function, when integral does not converge?
I need to find the probability density of some distribution with characteristic function given by:
$$\frac{1}{9} + \frac{4}{9} e^{iw} + \frac{4}{9} e^{2iw}$$
I know the formula for inverting a characteristic function is:
$$f_X(x) = \frac{... |
H: How to solve this polynomial?
How to get the solution for this polynomial?
If $x/y + y/x = -1$ where $x$ and $y$ are not equal to zero, then what would be the value of $x^3 -y^3$
AI: Hint:
$x^3 - y^3 = (x - y)(x^2 + y^2 + xy)$. Factor out $xy$ from the second term. |
H: Understanding Formulas in First Order Logic
I'm reading a text on Set Theory which states that any formula, say $\phi$, is ultimately built up from atomic sentences of form $x \in y$ and $x = y$ via the logical connectives.
So then my question is as follows: suppose in reading this text I come upon a formula of for... |
H: what is the meaning of "vector" in the context of algorithms?
I'm familiar with the concept of vectors in math and physics, but this wording in a computer science textbook is unfamiliar to me.
A configuration in the shared memory model is a vector
$$C=(q_0,\dots, q_{n-1}, r_0,\dots, r_{m-1}) $$
AI: This use of "ve... |
H: What subsets of a covering space cover their image?
Say I have a covering map $p \colon E \to B$. Then for which subsets $F$ of $E$, is $p|_F \colon F \to p(F)$ a covering map?
If it makes things easier, assume $E$ is simply connected, that is, the universal cover of $B$.
I was looking at the group of deck transfor... |
H: Can it happen that the image of a functor is not a category?
On Hilton and Stammbach's homological algebra book, end of chap. 2, they wrote in general $F(\mathfrak{C})$ is not a category at all in general. But I don't quite get it. I checked the axioms of a category for the image, and I think they are all satisfied... |
H: A group that has a finite number of subgroups is finite
I have to show that a group that has a finite number of subgroups is finite. For starters, i'm not sure why this is true, i was thinking what if i have 2 subgroups, one that is infinite and the other one that might or not be finite, that means that the group i... |
H: Chinese Remainder Theorem and matrix
In $\operatorname{SL}_2\left(\Bbb Z\right)$, theorem 3.2 in p.5, it states that (for an integer $N$ such that $(a,b,N) = 1$) there exists a $b' \equiv b \pmod{N}$ such that $(a,b')=1$ and can be done by using CNT. But I can't do this. I write $N=p_1^{e_1}\cdots p_k^{e_k}$ and us... |
H: Understanding the Definition of the Axiom Schema of Specification
Consider the Axiom Schema of Separation:
If $P$ is a property (with paramter $p$), then for any $X$ and $p$
there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those
$u \in X$ that have property $P$.
Now consider two potential inter... |
H: Show that V is a subspace of M2x2 Matrices and Determine a basis
A bit of information to start us off: Let V denote the set of all 2x2 matrices with equal column sums.
Show $V$ is a subspace of $M_{2\times 2}$ matrices:
and....
Determine a basis for $V$:
So for the first bit... $M$ is the set of all $2\times 2$ mat... |
H: Estimation of the number of prime numbers in a $b^x$ to $b^{x + 1}$ interval
This is a question I have put to myself a long time ago, although only now am I posting it. The thing is, though there is an infinity of prime numbers, they become more and more scarce the further you go.
So back then, I decided to make an... |
H: Difference between covariant and contraviant tensor
Let we write a tensor,
$$P_\mu A^\mu = - P_ \mu A_\mu $$
Where, P= momentum and A is vector potential.
My query is, when we interchange the covariant and contraviant tensor, we get a negative sign, why is that? Is there any physical or geometrical significance... |
H: derivative of parameter integral in $\mathbb C$
Let $f:\mathbb R\rightarrow\mathbb R$ be continuous and let $g(x):=xf(x)$ be absolutely integrable. Then $\widehat f'=-i\widehat g$.
I know this would be true if I can differentiate in the integral of $\widehat f(w)=\int_{-\infty}^\infty f(x)\exp(-iwx)dx$.
But when ca... |
H: Is there always a prime number between $p_n^2$ and $p_{n+1}^2$?
The following table indicates that there is a prime number p between the square of two consecutive primes.
$$
\displaystyle
\begin{array}{rrrr}
\text{n} & p_n^2 & p_{n+1}^2 & \text{p} \\
\hline
1 & 4 & 9 & 7 \\
2 & 9 & 25 & 23 \\
3 & 25 & 49 & 47 \... |
H: How can i compute this probablity?
If we have:
P(C) = 0.01 , P(!C) = 0.99
P(+|C) = 0.9 , P(-|C) = 0.1
P(-|!C) = 0.8, P(+|!C) = 0.2
How can i compute this probablity?
P (C | (T1 and T2)).
T1 and T2 are independent and T1 = +, T2 = +.
This is where i've reached:
P(+ and C) = P(C).P(+|C) = 0.01 x 0.9 = 0.00... |
H: Rolle's theorem for limits
Given a function defined by $a_n(x) = \frac{d^n}{dx^n} e^{-x^2}$. Every function can be written as $a_n(x) = h_n(x) e^{-x^2}$ where $h_n(x)$ is a polynomial (the Hermite-Chebyshev polynomial of degree $n$ to be precise). I want to prove that every polynomial $h_n(x)$ has $n$ real valued r... |
H: How to determine $\lim\limits_{x\to \pi/2 }\frac{\tan 2x}{x - \pi/2 }$?
$$\lim_{x \to \pi/2 } \frac{\tan 2x}{x-\pi/2}$$
Could anyone help me with this trigonometric limit? I'm trying to evaluate it without L'Hôpital's rule and derivation.
AI: Putting $x-\frac\pi2=y,$ as $x\to\frac\pi2,y\to0$
$$\lim_{x\to\frac\pi2}\... |
H: Determining bounds for the sum $\sum\limits_{n=1}^\infty \frac{1}{2^n - 3^n }$
I have to give low and high bounds for the following:
$$
\sum_{n=1}^\infty \frac{1}{2^n - 3^n }
$$
How do I determine an upper bound? How can I show this sum exists?
edit: removed erroneous conclusion.
AI: I strongly suggest not working ... |
H: How to plot biconditional graphs?
I'm using biconditional graphs for the lack of a better name, I don't know the name of it, if you know the real name, feel free to edit it.
Draw the set of points given by: $|x-3|\leq1$ and $|y-2|\leq5$
I have evaluated the values for which the conditions are true and I obtained ... |
H: Question Involving Transitive Sets
In Jech's Set Theory, we are asked to show the following two statements:
1.3 If $X$ is inductive, then the set $\{x \in X : x \subset X\}$ is inductive. Hence $N$ is transitive, and for each $n$, $n = \{n \in N
> : m < n\}$.
1.4 If $X$ is inductive, then the set $\{x \in X : $x ... |
H: How to compute Legendre symbol $\Bigl(\frac{234987}{9086}\Bigr)$?
How to compute $(\frac{234987}{9086})$? I know that Legendre symbol is $(\frac{p}{q})$ where $p \in \mathbb{Z}$ and $q$ is odd prime and Jacobi symbol is $(\frac{p}{n})$ where $p \in \mathbb{Z}$ and $n$ is odd integer. But in this case $n=9086$ is ev... |
H: Finding the x-coordinate of the max point of $y = x\sqrt {\sin x} $ so that it satisfies the equation $2\tan x + x = 0$
The maximum point on the curve with equation $y = x\sqrt {\sin x} $, $0 < x < \pi $, is the point A, Show that the x-coordinate of point A satisfies the equation $2\tan x + x = 0$
I understand t... |
H: A functional equation with inequality
Find all (at least one) functions $f\colon \mathbb{R}\to \mathbb{R}$ (or show there is none), such that
$$
f(x^3+x)≤x≤f(x^3)+f(x), \quad \text{for all $x\in \mathbb{R}$}.
$$
This is a problem asked by a friend of mine, and I do not know how to proceed.
Known easy facts:
$f(0... |
H: On the magnitude of vectors
Imagine a vector a in three dimensional Cartesian coordinates, the vector's endpoint coordinates are ($X_a,Y_a,Z_a$). Now let's assume this vector is in standard form (the vector "begins" at the origin).
The formula for the distance between the endpoint and the origin in Cartesian three-... |
H: Graph with no even cycles has each edge in at most one cycle?
As the title says, I am trying to show that if $G$ is a finite simple graph with no even cycles, then each edge is in at most one cycle.
I'm trying to do this by contradiction: let $e$ be an edge of $G$, and for contradiction suppose that $e$ was in two ... |
H: Prove $S\cap S ^\bot=\{0\}$
Let $S$ be a nonempty subset of the inner product space $V$ and $S^\bot$ be the orthogonal complement of $S$.
Prove $S\cap S ^\bot=\{0\}$
AI: If $s\in S$ is also in $S^{\perp}$ then by definition $\langle s, r\rangle=0$ for all $r\in S.$ Then taking $r=s$ gives $s=0.$ |
H: when changing order of integration
Let $f,g$ and $f\cdot g$ be a functions with $\int_{-\infty}^\infty|h(x)|dx<\infty$ for $h=f,g,fg$.
I have now given in a proof $$\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(u)g(w)\exp(-iwt)du\right)dw=\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(u)g(w)\exp(-iwt)dw\ri... |
H: Linear Programming - values of x and y yielding maximum values for the following problem:
I found the following linear programming question in a past exam paper while preparing for an upcoming exam:
"The daily production of a sweet factory consists of at most 100 kg chocolate covered nuts and at most 125 kg chocola... |
H: How does the divisibility graphs work?
I came across this graphic method for checking divisibility by $7$.
$\hskip1.5in$
Write down a number $n$.
Start at the small white node at the bottom
of the graph. For each digit $d$ in $n$, follow $d$ black arrows in a
succession, and as you move from one digit to... |
H: Proving tautologies using semantic definitions
I know how to prove tautologies with truth tables, but not with semantic definitions. How can I prove, using the semantic definitions, that the following are tautologies?
\begin{gather}
\bigl(\exists x\,P(x) \lor \exists x\,Q(x)\bigr) \to \exists x\,\bigl(P(x) \lor Q(... |
H: A very strange deck - probability and expected number of draws
Say we have a virtual deck of 70 cards of four suits and each player has access to his/her own unique independent deck (one players' actions do not affect another player's):
$$
\begin{array}{r|rr}
&\text{probability}& \text{quantity of card}\\ \hline
\t... |
H: Can you use combinatorics rather than a tree for a best of five match?
The Chicago Cubs are playing a best-of-five-game series (the first team to win 3 games win the series and no other games are played) against the St. Louis Cardinals. Let X denotes the total number of games played in the series. Assume that the... |
H: Irreducibility of locally closed set
We use the Zariskii topology. Let $\phi:\mathbb C^n\rightarrow \mathbb C^m$ be a polynomial map and
let $S\subset \mathbb C^n$ be a locally closed set (that is, $S$ is the intersection of an open set and a closed set).
Assume that $\phi$ induces an isomorphism of $S$ onto a non... |
H: Rectify the formula
Trying to rectify this formula. I'm having problems eliminating the correct symbols.
$$F = ∀x[∀y(P(x,y)∪Q(x,z)) ∩∃z∃x∀y(¬P(x,y) ∪ ∀z¬Q(x,z))]$$
AI: $$F = ∀x[∀y(P(x,y)∪Q(x,z))∩∃z∃x∀y(¬P(x,y)∪∀z¬Q(x,z))]$$
Notice that $∃z$ is not needed.
$$F = ∀x[∀y(P(x,y)∪Q(x,z))∩∃x1∀y1(¬P(x1,y1) ∪ ∀z1¬Q(x1,z1))... |
H: How to show that $x^2 \equiv 23 \equiv 5 \pmod 9$ is not solvable?
How to show that $x^2 \equiv 23 \equiv 5 \pmod 9$(2) is not solvable?
I got that $x^2≡23≡2 (mod 3)$(1) solution would be $x \equiv \pm 2^{\frac{3+1}{4}}\equiv\pm2(mod 3)$ because $3\equiv3 (mod 4)$(3), but because by Euler's criterion $(2/3)\equiv2... |
H: Closed form of $\sum_{j=0}^k {k \choose j} (-1)^{k-j} j^b$ where $k,b$ are positive integers.
I am wondering about the closed form (if it exists) of $$\sum_{j=0}^k {k \choose j} (-1)^{k-j} j^b$$ where $k,b$ are positive integers. I know it is equal to $0$ when $b<k$ and the sum is equal to $k!$ when $b=k$. However,... |
H: Hilbert's 17th Problem - Artin's proof
In this expository article, it is mentioned that Emil Artin proved Hilbert's 17th problem in his paper:
E. Artin, Uber die Zerlegung definiter Funktionen in Quadrate, Abh.
math. Sem. Hamburg 5(1927), 110–115.
Not being able to speak German, my question is
Does anyone know... |
H: Given basis spanning the vector space
I am learning Linear algebra nowadays. I had a small doubt and I know it's an easy one. But still I am not able to get it. Recently I came across a statement saying "((1,2),(3,5)) is a basis of $ F^2 $ ".
According to the statement a linear combination of the vectors in the l... |
H: Why is the operator $2$-norm of a diagonal matrix its largest value?
I read this in my textbook have tried working through it - I keep getting max 2-norm(Ax), which is just the magnitude of Ax.
How should I do this proof? (note, this is not for homework, I'm just trying to understand why as no proof is provided).
... |
H: Why is a set a subspace only when its determinant is equal to zero?
I am reading in my book that a given set ${(1,2,-1),(0,3,4),(0,2,1)} $ is a subspace of $R^n$ when the determinant does not equal zero of the coresponding table that set creates, like this :
\begin{array}{c c c}
1& 2 & -1 \\
0& 3 & 4 \\
0&... |
H: How To Set The Area Between Two Functions Equal To A Constant?
Please pardon the broad nature of this question. Suppose two functions encompass an area between them. What approach might be taken to adjust either function through adding constants to set the area between them equal to a constant? Part of the problem ... |
H: Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ preserve (homotopy)colimits?
Does the functor $S:\mathbf{Top}\to \mathbf{sSets}$ given by $S(X)_m=Hom_{\mathbf{Top}}(\Delta^m,X)$ preserve colimits? If not, what is a counterexample?
The only things I can say are that it preserves limits because it is a right... |
H: Simple question about circumference of circle
Q: The physical education teacher asked to one classroom, by vote, choose a sport between volleyball, basketball and football, to practice in class the following week.
pie chart:
The segment AB, which is the diameter of the pie chart, measuring 12cm, and the length of ... |
H: Derivative of absolute value over the complex numbers
Given the function $f: \mathbb{C} \rightarrow \mathbb{C}, z \mapsto z\bar{z}$. I am supposed to determine where f is differentiable and where it is holomorphic. So I tested the Cauchy Riemann Differential equations and found out that for $u(x,y)=x^2+y^2,v(x,y)=0... |
H: Can you solve current version of Quadratic reciprocity from Gauss version?
Can you solve current version( after: "He notes that these can be combined") of Quadratic reciprocity from Gauss version?
I have tried it but I have problem to understand especially if $ q \equiv 3 \pmod 4 \Rightarrow (\frac{p}{q}) =1 \Left... |
H: Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is similar to $B$. Prove trace($A$) = trace($B$).
Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is similar to $B$. Prove $\operatorname{trace}(A) = \operatorname{trace}(B)$.
I'm not sure where to go on this. So far I have this:
If $A$ is similar to $B$, th... |
H: Partial Sums of partial sums of a converging series.
I've been studying for an analysis qual and this problem was on one of the past exams.
Let $\{a_{j}\}$ $\subseteq$ $\mathbb{R}$ such that $\sum_{j=1}^{\infty}a_{j}=\frac{3\pi}{4}$. For every $n\in \mathbb{N}$, define
$T_{n}=\frac{1}{n}\sum_{j=1}^{n}S_{j}$ where ... |
H: Show that $N_n \mid N_m$ if and only if $n \mid m$
Let $N_n$ be an integer formed of $n$ consecutive $1$s. For example $N_3 = 111,$ $N_7 = 1 111 111.$ Show that $N_n \mid N_m$ if and only if $n \mid m.$
AI: Note that $$N_n=\frac{10^n-1}9$$
Bill Dubuque once proved the following
Let $\{f_n\}$ be a sequence of integ... |
H: Let A and B be n by n matrices. Suppose A is invertible. Show that AB is similar to BA.
Let $A$ and $B$ be $n\times n$ matrices. Suppose $A$ is invertible. Show that $AB$ is similar to BA.
I started with $BA = P^{-1}(AB)P$, but I'm drawing a blank now.
AI: Since $AB=BA$, and A is invertible, then we have ${A^{-1}}... |
H: Laplace-Beltrami on a sphere
I'm trying to compute the Laplace-Beltrami of the function $u(r,\varphi,\theta) = 12\sin(3\varphi)\sin^3(\theta)$ on a unit sphere. Note that $\varphi$ is the azimuth, i.e. $\varphi \in [0,2\pi]$ and $\theta$ the inclination, i.e. $\theta \in [-\frac{\pi}{2},\frac{\pi}{2}]$. For instruc... |
H: Proof of Zassenhaus Lemma
Lemma 4.52 from Advanced Modern Algebra by Rotman:
Given four subgroups $A \triangleleft A^*$ and $B \triangleleft B^*$ of a group G, then $A(A^* \cap B) \triangleleft A(A^* \cap B^*)$, $B(B^* \cap A) \triangleleft B(B^* \cap A*)$, and there is an isomorphism
$$\frac{A(A^* \cap B^*)}{A(A^... |
H: Subspaces and Dimension of sum of two subspaces.
A simple question.i am in the initial phase of learning linear algebra. Need your help.
I have made it through my own understanding. Wanted to know if i am thinking right.
Lets take the subspace {(a,b,c,d,e,0,0,0,0,0,0) $ \forall $ a,b,c,d,e $ \in $ R} . What i want... |
H: help understanding step in derivation of correlation coefficient
I'm looking to understand the starred step in the derivation below (also, if someone could help with the LaTex alignment, I'd appreciate it).
The regression line is $y= b_0 + b_1 x$, where $b_0$ and $b_1$ can be found by:
1) taking the difference betw... |
H: If $a, b$ are positive integers, does $\;b\mid(a^2 + 1)\implies b\mid (a^4 + 1)\quad?$
If $a, b$ are positive integers, does $\;b\mid(a^2 + 1)\implies b\mid (a^4 + 1)\;$?
Explain if this is true or not. If no, give a counterexample.
AI: Suppose $b = 5, a = 3$:
$$5\mid (3^2 + 1)\;\;\text{but}\;\;5 \not\mid (3^4 + 1)... |
H: Why does this summation of ones give this answer?
I saw this in a book and I don't understand it.
Suppose we have nonnegative integers $0 = k_0<k_1<...<k_m$ - why is it that
$$\sum\limits_{j=k_i+1}^{k_{i+1}}1=k_{i+1}-k_i?$$
AI: In general, $$\sum_{k=m+1}^n 1=n-m$$
This is because $$\sum_{k=1}^m1=m$$ $$\sum_{k=1}^n ... |
H: find the complex number that satisfies the following conditions
Find all values of $z \in \Bbb C$ such that: $z + \bar{z} = 18$ and $z.\bar{z} = 84$.
I don't know how to get that values, someone can help me to solve this?
AI: Hint: (Please check first that you have written the question correctly, because $z+z$ look... |
H: Angle preserving linear maps
In Spivak's Calculus On Manifolds, in part (c) of question 1-8, he asks the following question: What are all angle preserving $T:\mathbf{R}^n \to \mathbf{R}^n$?
I already showed that if $T$ is diagonalizable with a basis $\{x_1,\ldots,x_n\}$ where $Tx_i = \lambda_i$, then $T$ is angle p... |
H: Basics of Infinitary Formal Languages
Reading through Hodges' "A Shorter Model Theory", he gives the following symbolism (pgs. 23-25) for the first-order language constructed in the normal way with only finitely many formulas conjoined/disjoined together and only finitely many quantifiers in a row: $L_{\omega \omeg... |
H: The series expansion of $\frac{1}{\sqrt{e^{x}-1}}$ at $x=0$
The function $ \displaystyle\frac{1}{\sqrt{e^{x}-1}}$ doesn't have a Laurent series expansion at $x=0$.
But according to Wolfram Alpha, it does have a series expansion with terms raised to non-integer powers:
$$ \frac{1}{\sqrt{e^{x}-1}} = \frac{1}{\sqrt{x}... |
H: Angle between different rays (3d line segments) and computing their angular relationships
I have several positions (say A,B,C,..) and I know their coordinates
(3d). From each point, if a certain ray is passing in a way to
converge them at a given (known) point (say O). This point O is lie on
a known planar s... |
H: Proving $q\Rightarrow r \models (p\land q) \Rightarrow (p \land r)$ using only natural deduction.
I'm trying to prove
$$q\Rightarrow r \models (p\land q) \Rightarrow (p \land r)$$
using only the natural deduction rules in this handout.
Any hints? I am not allowed to do transformational stuff, such as converting eve... |
H: Prove $a-b$ is prime
Let $n\in\Bbb{N}^{>1}$,$P_n$# is the product of the first $n$ primes,$P_n$ is $nth$ prime number.Suppose $a$ and $b$ are coprime, and $a*b$ is multiple of $P_n$# and $\sqrt{a-b}<P_n$ ,then $a-b$ is prime.
A Formula for Prime Numbers?
AI: $\sqrt{a-b} \le p_n\ $ so $\,a\!-\!b,\,$ if composite, ... |
H: What can we say about the order of a group given the order of two elements?
If I know that a group of finite order has two elements $a$ and $b$ s.t. their orders are $6$ and $10$, respectively. What statements can be made regarding the order of the group?
I know by Lagrange's that the elements should divide the ord... |
H: Which book to read on quantum-related mathematics
Recently I watched the "Big Bang Theory" and decided to google about quantum mechanics. It really intrigued me. But I also understood that I am too stupid to understand even the basic mathematics in there. So I decided first to improve my knowledge of mathematics. I... |
H: Do the tangents of two circles define concentric circles?
Given two non-overlapping circles, $R_1$ and $R_2$. The radii of $R_1$ and $R_2$ may be different. The distance between the centers of $R_1$ and $R_2$ is defined as $x$.
Draw the four tangents between $R_1$ and $R_2$. There will be two tangents that cross be... |
H: Uniform Continuity of $1/x^3$ on compact intervals.
So, I have the function $f : (0,\infty) \to \Bbb R$. With $f(x) = \frac{1}{x^2}$ and I have to show that $f(x)$ is uniformly continuous on the set $[1,\infty)$. Is the following proof correct?
$[1,\infty) = \cup_{n=1}^{\infty} [n,n+1]$. And each of those sets is c... |
H: Find the equation of a circle given two points and a line that passes through its center
Find the equation of the circle that passes through the points $(0,2)$ and $(6,6)$. Its center is on the line $x-y =1$.
AI: Hint: Find the equation of the line passing through the two given points. This is a chord of the circle... |
H: Proving that $\frac{n!-1}{2n+7}$ is not an integer when $n>8$
How can I prove that
If $n$ is a positive integer such that
$$n>8$$
then
$$\frac{n!-1}{2n+7}$$
is never an integer?
Some of the first things that came to my mind is that $n!-1$ is not divisible by all numbers from $2$ to $n$, so if
$$2n+7<n^2$$
That is, ... |
H: basic doubt about topological manifold
In his book "Introduction to Smooth Manifolds", J.M. Lee defines a topological manifold to be a second countable, Hausdorff space with every point having a neighbourhood homeomorphic to an open subset of $\mathbb{R}^{n}$ for some $n$. I was wondering should it not say that it ... |
H: number of subgroups index $p$ equals number of subgroups order $p$
I'm doing an exercise in Dummit's book "Abstract Algebra" and stuck for a long time. I think I'm doing in the right way but I can't finish it. Hope someone can help me. I really appreciate it.
Let $A$ be a finite abelian group and let $p$ be a prim... |
H: Subgroup of elements of order at most $2^{m}$
The problem A5 in Putnam 2009 reads as follows:
Is there a finite abelian group $G$ such that the product of the
orders of all its elements is $2^{2009}$?
The answer is No. I am reading the official solution here. The solution starts by observing that if such group ... |
H: Computing taylor series, getting all 0's
I started out by finding the first and second derivative.
For $f'(x)$ I got $\;\;\dfrac{(12x^2-x^4)}{(4-x^2)^2}$
For $f''(x)$ I got $\;\;\dfrac{(4-x^2)(24x-4x^3)-(12x^2-x^4)(-4x) }{ (4-x^2)^3}$
After evaluating $f'(0)$ and $f''(0)$ I got $0$ for both of those ($f'(0)=0$ and... |
H: Pointwise convergence not enough to show incompleteness of continous functions from $[0,1]$ under the $L^2$ norm.
If we want to show the space of continuous functions from $[0,1]$ under the $L^2$
norm $(f,g) = \int_0^1 f \bar g$ is not complete, then we have to find a Cauchy sequence in this space which does not c... |
H: Adjust a range of given values.
If I have a number anywhere on the range 140 - 350 and I want to match it to the correlated range "0 - 360" what function can I run it through?
i.e.:140 would go through the function and return 0.350 would go through the function and return 360.245 (midpoint of Range 1) would go thr... |
H: Solving $0.0004<\frac{4,000,000}{d^2}<0.01$
From Stewart Precalculus, P86.
This question is from the chapter about inequalities and we are supposed to set up a model using inequalities to solve this problem.
my working out was
$$0.0004<\dfrac{4,000,000}{d^2}<0.01$$
In this chapter, the author did not show us how ... |
H: Strong equidistribution of points on the n-sphere
The vertices of a Platonic solid are equally distributed on its circumscribing sphere in a very strong sense: each of them has the same number of nearest neighbours and all distances between nearest neighbours are the same.
It seems clear to me that the Platonic sol... |
H: why is this Markov Chain aperiodic
I have this Matrix:
$$P=\begin{pmatrix} 0 & 1 \\ 0.3 & 0.7 \end{pmatrix}$$
this markov chain is said to be aperiodic, I dont understand how it comes to it. Period $\delta$ is the gcd of the set of all diagonal elements, right? if $\delta>1$, $P$ is periodic, if $\delta=1$, then ... |
H: Sum of distances from triangle vertices to interior point is less than perimeter?
Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$.
The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm not sure what theorems she can rely on in... |
H: Show that the $k$th forward difference of $x^n$ is divisible by $k!$
Define the forward difference operator
$$\Delta f(x) = f(x+1) - f(x)$$
I believe that if $f(x)$ is a polynomial with integer coefficients, $\Delta^k f(x)$ is divisible by k!. By linearity it suffices to consider a single monomial $f(x) = x^n$. I... |
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