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H: Proof by Induction - Triangles
Given n non-parallel lines such that no three intersect at a point, there are n choose 3 triangles formed.
So far what I come up with is by using proof by induction:
Base Case: For every three lines a triangle is formed.
Inductive Hypothesis: Assume true for k line.
Inductive Step:... |
H: Why is $H^0(C,\mathcal O(D))$ a vector space?
Given a divisor $D$ on a smooth curve, one can define the sheaf $\mathcal O(D)$ by the prescription $\Gamma(U,\mathcal O(D) :=$ $\{$meromorphic functions on $U$ that satisfy $(f) + D \ge 0\}$. Then, one can define a line bundle as $\mathcal L(D) = H^0(C,\mathcal O(D))$.... |
H: Finding minimum value of multi-variable expression without partial derivatives
Minimize where $a$ and $b$ are positive real numbers
$\sqrt{a^{2}\; +\; 4}\; +\; \sqrt{\left( 3-a \right)^{2}\; +\; \left( b-2 \right)^{2}}\; +\; \sqrt{25\; +\; \left( 6-b \right)^{2}}$
I could take the partial derivatives, equate them t... |
H: Posterior distribution as a distribution for a new random variable?
So in Bayesian framework one uses observed data $X=\{x_1,\dots,x_n\}$ to update the prior $p(\theta)$. My question is it justified mathematically to say that $p(\theta\mid x_1,\dots,x_n)$ corresponds to a new random variable itself?
AI: If you beli... |
H: Remainder when $123456789101112\ldots$ is divided by $75$
How would you find the remainder when you divide
$$1234567891011121314151617\ldots201120122013$$
(The number formed by combining the numbers from $1$ to $2013$)
by $75$?
AI: The Chinese Remainder Theorem tells you that to calculate a remainder mod $75$ is th... |
H: Complicated multivariable implicit differentiation problem
Given that the surface $x^{6}y^{5}+y^{4}z^{5}+z^{9}x^{7}+4xyz=7$ has the equation $z=f(x,y)$ in a neighborhood of the point (1, 1, 1) with f(x,y) differentiable, find:
$\displaystyle\frac{\partial^{2} f}{\partial x^2}(1,1)$
I have already found an intermed... |
H: Product of rotation and translation is a rotation
I have a homework question that I'm not sure how to answer.
Given rotation R and translation T (neither of which are the identity), show that T(R) must be a rotation.
My guess is that we can draw a triangle, then rotate and translate it, and then find some sort of i... |
H: If $B \subseteq A$ and $f:A \to B$ is 1-1, it must be onto
Let $B \subseteq A$ and $f: A \to B$ be a 1-1 function, then $f$ must be onto.
I understand that $f$ is onto if and only if every element of $B$ is in the image of $f$...
I believe this statement is true then?
AI: It is true by the Pigeonhole Principle if... |
H: The sum of the discount is the discount of the sum
Suppose I have a till. I have a special that I give a 10 percent discount on all dinner meals. I want to know that if I calculate the discount on each dinner item and round up the discount and apply it to each meal, I will not lose money than if I first added up th... |
H: definition clarification in graph theory
I was studying about Almost Self-Centered Graphs (ASC). ASC graphs are
introduced as the graphs with exactly two non-central vertices. Of course, the remaining two vertices are diametrical. My doubt is that if we have two non-central vertices x and y, does it mean that d(x,y... |
H: On irreducible polynomial over normal extension
Let $L/K$ be a normal extension and a irreducible polynomial $f(X) \in K[X]$. Prove that, if $f$ is reducible over $L$ then $f$ is factored into product of irreducible factors with same degree. Furthermore, if $f$ has roots in $L$ then $f$ splits over $L$.
Help me a h... |
H: The number of ways in which four cards be selected from a pack of 52 cards such that there is exactly one pair ( pair has same number or alphabet)
Problem :
The number of ways in which four cards be selected from a pack of 52 cards such that there is exactly one pair ( pair has same number or alphabet)
What I trie... |
H: integral of $\,x/\ln(x)$
I'm tried to integrate this integral without any success using integration by parts and substitution. $$\int_0^1\int_{e^y}^e \frac{x}{\ln x} dx dy$$
Does this integral require some other technique?
AI: Let $R=\{(x,y):0\le y\le 1,\, e^y\le x\le e\}$. Then you can check that $R=\{(x,y):1\le ... |
H: Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean.
Random variable $V$ has moment generating function $M(t)=e^{3e^{t}-1}$. What is $E(V)$? Note: you do not have to derive the mean.
I have tried deriving the function , which i got... |
H: Trouble with simple consequence of the polarization identity
The polarization identity is
$[u,v] = \frac{1}{4} \sum_{k=0}^3 i^k \|u + i^k v\|^2$,
where $[u,v]$ is any sesquilinear hermitian form. My instructor claims as a simple consequence that:
if $[v,v] = 0$ for all $v\in V$, then $[u,v] = 0$ for all $u,v\in V$.... |
H: Non-trivial finite/infinite subgroups of infinite groups
Does an infinite group whose every non-trivial subgroup is also infinite exist? If yes, what can be an example of such a group?
also,
Does an infinite group whose every non-trivial subgroup is finite exist? If yes, what can be an example of such a group?
AI: ... |
H: If $ p \equiv 3 \mod 4$ and $r$ primitive root, then $\mathrm{ord}_p(-r) = (p-1)/2$
I've been looking at a bunch of number theory problems lately and I need help with a few. One of them is as follows:
Let $p$ be a prime number with $p \equiv 3 \mod 4$ and let $r$ be a primitive root modulo $p$. Prove that $\mathrm{... |
H: Prove that if $x | (3x + 20)$ then the only positive $x$ for which the statement is true are $1,2,4,5,10,20$.
How can I give a valid proof to this problem?
AI: $x$ divides $3x$ since $x$ divides $x$. Now $3x$ and $20$ are in addition.
For $3x+20$ to be divisible by $x$, $(3x+20)/x$ should give an integer value (Pro... |
H: Very elementary set theory inquiry
Suppose $A = \bigcup_{i=1}^{N} E_i$ and $B = \bigcup_{j=1}^{N} F_j $. Does it follow that
$$ A \cap B = \bigcup_{i=1}^{N} \bigcup_{j=1}^{N} (E_i \cap F_j) = \bigcup_{j=1}^{N} \bigcup_{i=1}^{N} (E_i \cap F_j)$$
?????
AI: Do you know the distributive property? Namely, that $(A \cup... |
H: Quadratic residues modulo $p$ are congruent to the even powers of $r$ modulo $p$
This is another number theory problem I've been tackling:
Let $p$ be an odd prime number and let $r$ be a primitive root modulo
$p$. Prove that the quadratic residues modulo $p$ are congruent to the
even powers of $r$ modulo $p$ a... |
H: Find norm of a linear functional
I have a normed space:
$$l_p = \{ (x_n)_{n = 1}^{\infty}: \sum\limits_{n = 1}^{\infty}|x_n|^{p} < \infty \}$$
With norm:
$$||x|| = (\sum\limits_{n = 1}^{\infty} |x_n|^p)^{1/p}$$ where $p = \dfrac{5}{4}$
And I want to find norm of following functional:
$$f(x) = -x_{1} + x_{100} + \su... |
H: Is there an expression using the main constants of mathematics as result of the following infinite sum:
$$\sum_{k=0}^\infty {{\pi^{k\over 2}}\over {\Gamma({k\over 2} +1)}}$$
I've found, that
$\sum_{k=0}^\infty {{\pi^{k\over 2}}\over {\Gamma({k\over 2} +1)}}$ = $e^{\pi} +
2\sum_{k=0}^\infty {{({4\pi})^{k}k!}\over ... |
H: $p=2^n+1$. Prove that every quadratic nonresidue modulo $p$ is a primitive root modulo $p$
This is another one of the number theory problems I've been struggling with as of late (hopefully I'm not posting too many questions at once!).
Let $n$ be a positive integer and let $p=2^n+1$ be a prime number.
Prove that ... |
H: Need help with a certain integrating technique
How do I integrate this?
$$\frac{\text{d}P}{\text{d}t} = \frac{(r(t) - B)}{z} \cdot P(t) + c\cdot w$$
The t is just the top limit in the integral.
Let me be more specific. I have this:
$$\frac{dP}{dt} = g(t)P(t) + k$$ where $k$ is a constant and $g(t)$ is any function ... |
H: Poisson Distribution Lambda, Probability, and Looking for Exactly k
Automobiles arrive at a vehicle equipment inspection station according to a Poisson process with a rate of $ \lambda $ = 10 per hour. Suppose that with probability 0.5 an arriving vehicle will have no equipment violations.
What is the probability t... |
H: Find the sum of $\sum_{n=1}^\infty (-1)^{n+1} (2n-1)x^{2n-1}$
Find the sum of $S(x) = \sum_{n=1}^\infty (-1)^{n+1} (2n-1)x^{2n-1}$.
I know convergence radius is $1$ because $\frac{1}{\sqrt[n] {(2n-1)}} = \frac{1}{1} = 1.$
Then:
$$x^{-1} S(x) = \sum_{n=1}^\infty (-1)^{n+1}(2n-1)x^{2n+2}. $$
$$\int x^{-1}S(x) = \sum_... |
H: How to evaulate $\int \cos x \sqrt{5 + \cos^2 x} dx$?
How do I evaluate
$$
\int \cos x \sqrt{5 + \cos^2 x} dx?
$$
AI: HINT:
$$\int\cos x\sqrt{5+\cos^2x}dx=\int\cos x\sqrt{6-\sin^2x}dx$$
Now put $\sin x=u$ and use Point $\#8$ of this or this |
H: $L^1$ function is bounded almost everywhere
Suppose $f\in L^1(\mathbb{R})$. Is it true that $f$ is bounded (except for a set of measure zero)?
I think it should be true, but can't show it formally. If $f$ is unbounded, why would we have $\int_\mathbb{R}|f|dx=\infty$?
AI: Every measurable function is "approximately ... |
H: Composite of two algebraic extensions is algebraic.
Let $L,F$ be extensions of the field $K$ and are contained in a common field. Prove that, if $L$ and $F$ are algebraic extensions over $K$ then $LF$ is also a algebraic extension over $K$.
Help me a hint.
Thank for any insight.
AI: Suppose $\alpha$ is in $LF$. The... |
H: Validate or invalidate the propositional argument
Validate or invalidate the following arguments
$ p\to t$
$ p \to \lnot r$
$q \to p$
$\lnot t \lor r$
$r \to t$
$\therefore \lnot p \land \lnot q \land (r \iff t) $
I could only see why it is $(r \iff t)$
AI: Hint: You know $r \iff t$. What if $p$ were true? |
H: Solve for: $2\log_3\left(x^2-4\right)+3\sqrt{\log_3\left(x+2\right)^2}-\log_3\left(x-2\right)^2\leq4$
Solve for: $$2\log_3\left(x^2-4\right)+3\sqrt{\log_3\left(x+2\right)^2}-\log_3\left(x-2\right)^2\leq4$$
My try:
$2\log_3\left(x^2-4\right)+3\sqrt{\log_3\left(x+2\right)^2}-\log_3\left(x-2\right)^2\leq4\\\Leftright... |
H: Solve for: $8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$
Solve for: $$8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2$$
My try:
$8\log_4\sqrt{x^2-9}+3\sqrt{2\log_4\left(x+3\right)^2}=10+\log_2\left(x-3\right)^2\\\Leftrightarrow \log_2\left(x^2... |
H: Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field.
Let $L,F$ be extensions over the field $K$ and $L,F$ are contained in a common field. Prove that if $L=K(S)$, with $S$ is a nonempty subset of $L$ then $LF=F(S)$.
Thank for any insight.
AI: Ok,
$S \subset L \subset LF$ and $F \su... |
H: Finding Polar Area Problem
So I've been staring at this problem for about half an hour because I cannot for the life of me understand what exactly I did wrong. I took a picture because with my stress level it would take me about 25 minutes of angry typing to type up everything using the html tags and all of that.
A... |
H: Proving the Frobenius map is an endomorphism
I have prime $p$, and $K$ a field such that $p \cdot 1 = 1+1+\cdots+1 = 0$. I am asked to prove that $F: K \rightarrow K$, $a \mapsto a^p$ is a ring homomorphism.
I can prove this for everything except addition, $F(a+b)=F(a)+F(b)$. For addition, it seems to me that $... |
H: Projection of ellipsoid
Find the projections of the ellipsoid
$$ x^2 + y^2 + z^2 -xy -1 = 0$$
on the cordinates plan
I have no idea how to do this. I couldn't find much on google to help me with it too.
Thanks in advance!
AI: By coordinate planes, I assume you mean the $xy$-plane, $xz$-plane, and $yz$-plane? If so,... |
H: Cardinality of subbasis for topological space
Claim: If $(X,\tau)$ is a topological space, $\mathcal B$ a base for $\tau$ and $\mathcal U$ an open cover of $X$ then there is a subcover $\mathcal V \subset \mathcal U$ whose cardinality is not larger than that of $\mathcal B$.
I appreciate some hints, since since is ... |
H: Indefinite Integral $\int\frac{3\sin(x)+2\cos(x)}{2\sin(x)+3\cos(x)}dx$
How can I evaluate this integral?
$$\int\frac{3\sin(x)+2\cos(x)}{2\sin(x)+3\cos(x)}dx$$
AI: Write $$3\sin x+2\cos x=A(2\sin x+3\cos x)+B\frac{d(2\sin x+3\cos x)}{dx}$$
$$\implies 3\sin x+2\cos x=A(2\sin x+3\cos x)+B(2\cos x-3\sin x)$$
$$\impli... |
H: What is the period
How to find the period of the function \begin{array}{cc} e^{z}\end{array} where "z" is a complex number?? Does this function really have any period??
AI: We say that a function $f(z)$ has period $C$ for some constant $C$ if and only if:
$$ f(z+C) = f(z) $$
for all $z$. In this case the function... |
H: Morse index and degeneracy
The function is as follows: $$f(x,y,z) = e^x(xy-y^2-z^2)$$ I have found the critical points to be $(0,0,0)$ and $(-2,1,0)$. The question asks to determine the morse index of the points and the degeneracy. How am I supposed to determine the morse index? Also, what exactly is a degenerate c... |
H: Determine the interpolating polynomial
Determine the polynomial of $ deg \le 6 $ interpolating function $$ f(x) = x^3 + 2x^2 + x + 1 $$ at the points : $ -3, -2, -1, 0, 1, 2, 3 $.
My first idea it was to use Lagrange's formula, but it's too long.
Maybe you have other, faster method? Because I guess, that I should... |
H: Parametric Curves Existence of Tangent
If $\frac{dy}{dt}$ and $\frac{dx}{dt}$ exist, then does $\frac{dy}{dx}$ always exist when $\frac{dx}{dt} \not=0$?
Indeed, this is a very simple question. Sorry but I'm just a beginner for Calculus so I don't know that much in depth. But here is the reason for asking.
During ... |
H: Inverse of function arcsin
I'm having trouble finding the solution of the inverse of the function
${\rm f}\left(y\right) = \arcsin\left(\,3 - x^2\,\right)$
Isn't $\arcsin$ the inverse of $\sin$ ?. This is what I have now as inverse:
$\sin\left(\,3 - x^2\,\right)$.
AI: $$y(x)=\arcsin(3-x^2)\iff x(y)=\sqrt{3-\sin(y)}... |
H: A Question Regarding Reverse Mathematics and V=L
Can all of "ordinary mathematics" ("ordinary mathematics" as understood by practitioners of reverse mathematics) be formulated in ZF[0]+V=L (ZF[0]is simply ZF without the Power Set Axiom)? It is my understanding that second order arithmetic interprets ZF[0]+V=L (thi... |
H: Why is the expected value (mean) of a variable written using square brackets?
My question is told in a few words: Why do you write $E[X]$ in square brackets instead of something like $E(X)$? Probably it is not a "function". How would you call it then? This question also applies for $Var[X]$.
AI: Mathematicians usua... |
H: Recursive formula for the probability that the starting player wins [DBertsekas & JTsitsiklis P57, 1.21]
Two players take turns removing a ball from a jar that initially contains
$w$ white and $b$ black balls. The first player to remove a white ball wins. Develop a
recursive formula that allows the convenient ... |
H: Characteristic of a field $F$ is prime
If Char$F$ $\neq 0$, then Char$F$ must be prime number.
MY try: If Char$F$$ = nk $ for integers $n$ and $k$, then by definition, $nk = 0 \implies n = 0$ or $k = 0$ which implies Char$F=0$ which is a contradiction.
Is this correct?
AI: We need to claim $F$ is also an integra... |
H: When is a polynomial irreducible over a field with characteristic $\neq2$?
When is a polynomial irreducible over a field with characteristic $\neq2$?
Help me a hint. I have no idea.
Thanks a lot.
AI: We can, in turn, check if this polynomial has roots on this field or not.
If $charF$ is not too large, it is not ter... |
H: Volume 3 of Johnstone's "Sketches of an Elephant"
Recently, I read the Chapter 8 of Johnstone's "Topos theory" and got interested in the homotopy and cohomology theory of Grothendieck toposes. So I'm looking for the textbooks expanding these subjects, and it seems that it will be treated in the Volume 3 of "Sketche... |
H: Defining iteration of function in set theory
I was trying to prove the axiom of dependant choices from AC, and got confronted with the following problem:
Given $x\in X$ and $f : X\rightarrow X$, can we define the sequence $(f^i(x))_{i\in\mathbb N}$?
Let $\mathcal R$ be a relation on $X\times X$ such that for every... |
H: Confused about taking absolute value after integrating reciprocal
I often seem to get caught out when integrating $1/x$ to $\log x$, or similar. Here's an example -- solve
$$ \frac{\mathrm{d}z}{\mathrm{d}x} + \frac{1}{2}z = \frac{1}{2}$$
My first attempt was using separation of variables, which proceeds as follows:... |
H: Is there a means of analytically proving the following identity?
Okay, so before I begin, my background is more in the world of applied, rather than pure, mathematics, so this question is motivated by a physics problem I'm looking at just now. Mathematically, it boils down to looking for a minimum of a real valued ... |
H: Finding the nth derivative of $y=e^{ax+b}$
How to find the $n$-th derivative of $y=e^{ax+b}$
Please provide an explanation of the steps.
Thanks.
AI: Recall that for $y = e^{f(x)},\; y' = f'(x)e^{f(x)}$, where $f(x)$ is a function of $x$. This is making use of the chain rule, as it relates to the exponential functio... |
H: Dividing one equation by another equation
This is from Higher Algebra by Hall and Knight,
$u+v+\sqrt{uv}=39$...(1)
$u^2+v^2+uv=741$...(2)
we obtain by division
$u+v-\sqrt{uv}=19$
I don't know how do you divide one equation by another equation, can someone pls explain.
AI: It's just like any other operation: if y... |
H: Prove that $\mathbb{R}_{\neq0}$ is not a real vector space
Assume a set of $\mathbb{R}_{\neq0}=\{a \in \mathbb{R} \mid a \neq 0\}$, where addition of elements in $\mathbb{R}_{\neq0}$ is the product in scalar $ab$. Prove that this is not a real vector space.
I have made the assumption that the scalar product for the... |
H: Use induction on $n$ to prove that $2n+1<2^n$ for all integers $n≥3$.
Use induction on n to prove that $2n+1<2^n$ for all integers $n\geq 3$.
My attempt: Let $P(n)$ be the statement $2n+1<2^n$.
Base case: Prove that $P(3)$ is true. $LS = 2(3)+1=7$ and $RS=2^3=8$. Since $LS<RS$, $P(3)$ is true.
Inductive Hypothesis... |
H: Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$?
Prove that if $f$ is integrable on $[a,b]$ then so is $|f|$.
I can prove the converse of this is false, I also try using the definition of integrable function $f$, but I don't know what to do after that
AI: Integrability means that certain sums involving ... |
H: Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$
Prove that $SO(3)$ acts transitively on the unit sphere $S^2$ of $\Bbb R^3$.
I think $S^2$ means a 2-D sphere and $SO(3)$ is the usual $SO(3)$ group. I'm unsure how to prove that $SO(3)$ acts transitively.
My guess is to show that like $... |
H: let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$
let $f,g$ be two bounded function on $[a,b]$, let $L_f, L_g, L_{f+g}$ be the lower integral of $f, g, and f+g$. Prove that $L_{f+g}\ge L_f+L_g$
I don't know how to start
AI: Here is how you should attack any problem li... |
H: Projection on a convex set
If I have a convex set $ S$ and if I project an $ x$ onto $S$. Is it true that $x $ would project onto a unique element of $S$. Why? What would be considered different if the set $S$ was non-convex?
AI: First of all, you need $S$ to be closed. For example, $S=(0,1)$ is convex but $2$ has... |
H: Show that $n! \mid (p^n-1)(p^n-p) \cdots (p^n-p^{n-1})$ where $p$ is prime and $n \geq 1$.
So, I'm preparing for an exam and in one of the problems it asks us to find the number of distinct bases that we can have for an $n$ dimensional vector space over a finite field of $p$ elements ($p$ is a prime number, of cour... |
H: Relation between a null space and a line.
I have a matrix $T$ and a line $x=y=z$. Lets say I find the null space of $T$ to be $\{\begin{bmatrix}-1\\ 1 \\ 0\end{bmatrix}, \begin{bmatrix}-1\\0\\1\end{bmatrix}\}$. Is it correct to say that the nullspace of $T$ is the plane of $x=y=z$?
AI: I presume that you mean the n... |
H: Necessary and Sufficient Condition for Vector Space
Problem Assume a finite set $F$, write the necessary and sufficient condition in terms of the number of elements of $F$, such that $F$ is a real vector space. (Assuming that the vector addition and scalar multiplication can be defined)
Attempt at Solution
Since th... |
H: Is there any point in a logician studying $\infty$-categories?
My primary areas of interest lately have been set theory, logic, and category theory, so naturally topos theory has been a large part of what I'm learning (in between getting caught up on some other things I should really know). I've recently been intro... |
H: A relation between sine and cosine
I cannot figure out how this relation:
$$\cos(\omega t)+ \frac{\zeta}{\sqrt{1-\zeta^2}}\sin(\omega t) $$
is equal to:
$$\frac{1}{\sqrt{1-\zeta^2}}\sin\left(\omega t + \tan^{-1}\frac{\sqrt{1-\zeta^2}}{\zeta}\right)$$
I only found that this is not true for $\zeta<0$, and specifical... |
H: Convert time duration a to decimal figure
I am in needs of a formula for converting a duration of a work, already expressed in hour and seconds to a decimal representation. I mean the clock is divided in four parts, of 25 units each.
For example, these are the entries and the desired results:
8h 00m = 8,00
8h 15m ... |
H: $f(x)=x$ if $x$ is rational and $f(x)=1-x$ if $x$ is irrational
Show that $f$ assumes every value between $0$ and $1$ if
$$f(x) =
\begin{cases}
x, & x\ \text{is rational},\\
1-x, & x\ \text{is irrational}
\end{cases}$$
for all $x\in[0,1]$.
Is it enough to say that since the rationals and the irrationals are dense... |
H: Limited function.
Let $f:[0,+\infty) \rightarrow \mathbb{R}$ be a function bounded on each bounded interval. Prove that if $\lim_{x \rightarrow +\infty } [f(x+1)-f(x)] = L$, then $\lim_{x \rightarrow +\infty} \frac{f(x)}{x} = L$.
AI: Assume $|f(x+1)-f(x)-L|<\epsilon$ for $x \ge M$.
Use the triangle inequality as f... |
H: Proving a constant function $f(x) = c$ is Riemann integrable
Prove that a constant function $f(x) = c$, where $c$ is in the Real Numbers, is Riemann integrable on any interval $[a, b]$ and $\int_a^bf(x) dx = c(b-a)$.
By looking at the definition, it looks like I am going to explain that it's bounded (which would b... |
H: Number of elements of group with specific order
Consider the group $(G,\cdot)$ where $$G=\left\{\left(\begin{matrix}1&a\\0&b\end{matrix}\right):a,b\in\mathbb{R}, b\neq0\right\}.$$ How many members of $G$ have order 2?
My Attemt
A member $M$ of $G$ will have order two iff $M^2=I$. I.e. $$\left(\begin{matrix}1&a\\... |
H: In characteristic $p$, the field extension $k(X,Y)$ over $k(X^p,Y^p)$ is not simple
Let $k$ be a field with characteristic $p>0$, $L=k(X,Y)$ be the field of rational fractions of two variables over $k$. Let $K=k(X^p,Y^p)$. Then $[L:K]=p^2$ (for a proof see In characteristic $p$, the field extension $k(X,Y)$ over $k... |
H: $A$ is similar to $B$ if $A\oplus A$ is similar to $B\oplus B$
Question:
If the matrix $\begin{pmatrix} A & 0 \\ 0& A \end{pmatrix}$ is similar to $\begin{pmatrix} B & 0 \\ 0 & B \end{pmatrix}$
show that:
the matrix $A$ is similar the matrix $B$
My try:
since the matrix diag $(A,A)$ is similar matrix diag $(B,B... |
H: Finding the limit of $|x|^{1/x}$ as $x \rightarrow 0$.
I have to find the limit of $|x|^{1/x}$ as $x\rightarrow 0$
I can't use de l'Hopital's method and derivatives.
Now the limit itself does not exist but it have to be divided into two side the left and right. Then it have to be calculated separately. The right s... |
H: Deriving time and distance
The distance an aircraft travels along a runway before takeoff is given by $D=(10/9)t^2$, where $D$ is measured in meters form the starting point and $t$ is measured in seconds from the time the brakes are released.
The aircraft will become airborne when its speed reaches $400\;\text{km/... |
H: Probability Uniform Distribution
If $A$ is uniformly distributed over $[-25, 30]$, what is the probability that the roots of the equation
$$x^2 + Ax + A + 80 = 0$$
are both real?
I kept getting weird answers which was wrong. $$A (1 + x)+ x^2+ 80 = 0$$
$$A (1 + ... |
H: $U^*\otimes V$ versus $L(U,V)$ for infinite dimensional spaces
It's well known that $U^*\otimes V\cong L(U,V)$ for finite dimensional spaces. However, why people say that $U^*\otimes V$ is not isomorphic to $L(U,V)$ for infinite dimensional spaces and many books don't say anything related to this?
AI: Suppose we ar... |
H: Math behind Keynesian Expenditure Multiplier
Take a look at this page: http://wiki.ubc.ca/Keynesian_Multiplier
Why can you find out the sum of the geometric series just by dividing the mps by 1?
AI: Consider the series
$$
1+r+r^2+r^3+\cdots=\sum_{k=0}^\infty r^k
$$
If $|r|<1$, then this is a geometric series whose... |
H: Proof related with prime numbers and congruence
How to (dis)prove this
$ (n-2)! \equiv 1 \mod n$
If n is said to be a prime number. I guess we'll have to use FERMAT’S LITTLE THEOREM, and I just don't know where to start from. Thanks in advance
AI: If $\;n=p\;$ is a prime, then by Wilson's theorem
$$\color{red}{-... |
H: Prime notation in integral?
Recall the definition of potential energy:
$$
U_x-U_{x_0} = -\int^x_{x_0}F_x(x')dx'
$$
I've seen the integral definition of work, but not this - the thing I'm specifically interested in is the notation. I've never seen differential and prime notation intermingled like this. What does... |
H: Multiplication modulo proving a set is a group conform conditions
Given is the following explanation.
A group is a set, together with a binary operation $\odot$, such that the following conditions hold:
For all a, b $\in$ S it holds that a $\odot$ b $\in$ S
Now, i have to show that the Set = {1, 2 ... 16, 17} tog... |
H: Prove a matrix maps to a point that make up the plane perpendicular to a line.
I'm having difficulty understanding what the following question is asking and was hoping someone could explain it to me.
$T = \begin{bmatrix}1/3 & 1/3 & 1/3 \\ 1/3 & 1/3 & 1/3 \\ 1/3 & 1/3 & 1/3 \end{bmatrix}$
Given a point on the line x... |
H: Does there exist a semigroup such that every element factorizes in this way, which nonetheless lacks a left identity?
If a semigroup $S$ has a left identity-element, then for any $y \in S$ we can write $y = xy$ for some $x \in S$. Just take $x$ to be any of the left identities, of which there is at least one, by hy... |
H: If derivative of a function is non-zero then it is monotone. Since function is monotone, variable can be substituted in integration
I came across this in the text Differential Equations-An Introduction With Applications, by Lothar Collatz:
$y'(x)=\frac{dy}{dx}=\frac{f(x)}{g(y)}$
Suppose $f(x)$ is continuous in $[a,... |
H: Tangent line to the curve $x^3+xy^2+x^3y^5=3$
Does the tangent line to the curve $x^3+xy^2+x^3y^5=3$ at the point $(1,1)$ pass through the point $(-2,3)$? (using implicit differentiation)
I got the implicit differentiation as $\frac{dy}{dx}=\frac{-3x^2-y^2-3x^2y^5}{2xy+5x^3y^4}$. Then I got stuck?
Explanation would... |
H: Finding the local minimum of $e^{3x} + e^{-x}$
$${\begin{array}{l} f(x) \; = \; e^{3x} + e^{-x} \\ f'(x) \; = \; 3e^{3x} - e^{-x}\end{array}}$$
The interval on which $f$ is increasing is $\left( \frac{1}{4}\ln \left(\frac{1}{3}\right), \; \infty \right).$
The interval on which $f$ is decreasing is $\left(-\infty, \... |
H: Step in Proof of Cardinality of Product of two Groups
Let $A,B < G$ be two subgroups of some group $G$, then I have a question on the proof of the following:
$$
|AB| = \frac{|A||B|}{|A \cap B|}.
$$
Proof: Let $D = A \cap B$, arrange $A$ and $B$ in left cosets and right cosets regarding $D$ (which is also a subgrou... |
H: Trying to understand matrix image
On my linear-algebra lecture, we were given the definition of the image of a matrix $A \in \mathbb{K}^{m,n}$ as follows:
$$ \mathrm{im} A = \{\vec y \in \mathbb{K} ^{m}: \exists \vec x \in \mathbb{K}^n . \vec y = A\vec x\}$$
I just can't wrap my head around this one. Could someone ... |
H: If $f:[0,1] \to \mathbb{R}$ is continuous and $\int^{x}_{0} f = \int^{1}_xf,$ then $f(x) = 0, \forall x\in [0,1].$
If $f:[0,1] \to \mathbb{R}$ is continuous and $\int^{x}_{0} f = \int^{1}_xf,$ then $f(x) = 0, \forall x\in [0,1].$
May I verify if my proof is valid? Thank you:)
Proof: $\int^{c}_{0} f = \int^{1}_cf \i... |
H: PSD matrices properties
If I have a matrix $X \in R^{n \times n} $ and an index set $ I \subseteq \{1,\dots,n\} $,
Is $X_I$ also positive-semidefinite $\forall \ \ I $? Why ?
$X_I $ is the submatrix that is formed by choosing all rows and columns from index-set $I $
Edit : How would you prove that the determinant i... |
H: Are these isomorphic $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ and $\mathbb{Z}_{9}^{*}$
Is $\mathbb{Z}_{2}\times\mathbb{Z}_{3}$ isomorphic to $\mathbb{Z}_{9}^{*}$
both have orders 6
both have elements with orders 1,2,3,6 (1 element of order 1, 2 elements of order 3, 1 element of order 2 and 2 elements of order 6)
Both... |
H: Let $E \subset R$ and let $f$ be a real-valued function on $E$ that is continuous at $p \in E$
Let $E \subset R$ and let $f$ be a real-valued function on $E$ that is continuous at $p$ in $E$. If $f(p) > 0$, prove that there exists an $\alpha > 0$ and a $\delta > 0$ such that $f(x) \ge \alpha$ for all $x \in N_\delt... |
H: Terminology: Groups, rings, fields, etc.
Groups, semigroups, fields, rings, integral domains, vector spaces, R-modules... these are all approximately the same sort of "stuff", but each one refers to a slightly different combination of required properties. Is there a general term that collectively refers to these ty... |
H: Let $m$ and $n$ be integers in the ring of integers. Show that $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$
I am working on the problem:
Let $m$ and $n$ be integers in the ring of integers. Show that $m\mathbb Z$ contains $n\mathbb Z$ if and only if $m$ divides $n$.
It's an if and only if pr... |
H: Rings whose elements are partitioned between units and zero-divisors.
In $\mathbb{Z}_n$ the elements are fully partitioned between the units and the zero-divisors. I believe this is the case, am I correct?
Now, I take it this does not hold true in general, there may be rings with elements that are neither units nor... |
H: How much Category theory one must learn?
I have learnt very basic category theory (up to Yoneda lemma from Hungerford's Algebra text). My question is how much category theory should every Mathematics student who is not planning to specialize in that area learn ?
I am not sure which area of Mathematics I would like... |
H: approximating a measurable function using simple functions
I would like to understand the proof that shows that for any measurable function, we can approximate it with a sequence of simple functions. The proof is presented in (among others):
https://www.math.ucdavis.edu/~hunter/measure_theory/measure_notes_ch3.pdf
... |
H: linear dependence on $\mathbb R^n$
$S_1$ and $S_2$ finite sets on $\mathbb{R}^n$, $S_1$ is a subset of $S_2$, $(S_1\neq S_2)$.
If $S_2$ is linearly dependent, so:
$S_1$ could be linearly dependent?
$S_1$ Could be linearly independent?
How can i answer this, and show some examples?
AI: You can have both of them. F... |
H: Stuck on equivalence relation question
I have been stuck on this question for a while. I was wondering for a set $A={1,2,3,4,5,6}$, given that its distinct equivalence classes are $\{1,4,5\},\{2,6\},\{3\}$, what is the equivalence relation R on A?
I have tried everything, such as powers, modulo, parity arguments, s... |
H: Prove that the $x$-axis in $\Bbb R^2$ with the Euclidean metric is closed
I want to show that the $x$-axis is closed.
Below is my attempt - I would appreciate any tips on to improve my proof or corrections:
Let $(X,d)$ be a metric space with the usual metric.
Want to Show: $\{(x,y) | x ∈ \Bbb R, y = 0\}$ is clos... |
H: Choosing a congress
Five scientists each have to choose a venue to attend out of 8 possible venues. Their choice is independent and each venue has the same chance of being chosen.
What is the probability that each of the 5 scientists choose a different venue?
What is the probability that 4 out of 8 venues will no... |
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