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H: Geometric Series of Matrices when $I-A$ is singular
I need to understand the behavior of the following series as $n$ grows very large, where $A$ is an $n\times n$ matrix, $B$ is a $n\times 1$ vector, and $y_0$ is any vector in $\mathbb{R}^n$
$$
y_n = A^ny_0 + \sum_{k=0}^{n-1}A^kB
$$
Doing a little bit of analysis o... |
H: Is pointwise convergence permutation-invariant?
I am interested in convergence between countable sequences of real numbers. (Perhaps the definitions to follow are nonstandard. Sorry!)
Say that the sequence $\langle \langle x^1_1,x^1_2,x^1_3,...\rangle, \langle x^2_1,x^2_2,x^2_3,...\rangle, ...\rangle$ pointwise con... |
H: How can I justify that the open interval $(0,1)$ is the infinite union of closed intervals?
I have to show that:
$$\bigcup_{i=2}^\infty [\frac{1}{n},\frac{n-1}{n}] = (0,1)$$
The first part:
$$\bigcup_{i=2}^\infty [\frac{1}{n},\frac{n-1}{n}] \subset (0,1)$$
is easy to show, but for the second part:
$$(0,1) \subset \... |
H: sum of pairs and sum of squares $\left(\sum x_i\right)^2-2\sum_{cyc}x_ix_j=\sum x_i^2$
I was messing around today and came across the following which I believe to be true:
$$\left(\sum x_i\right)^2-2\sum_{cyc}x_ix_j=\sum x_i^2$$
for a set of $x=(x_1,x_2,...x_n)$ where $n\ge2$ and it is easy to prove for small value... |
H: If $\int_E f\,d\mu=\int_E g\,d\mu$ then $f=g$ a.e?
Let $(E,\mathcal{A},\mu)$ be a finite measure space.
Let $f$ and $g$ two real-valued measurable functions such that $$\int_E f\,d\mu=\int_E g\,d\mu$$
Can we say that $f=g$ a.e $(*)$? Or it is necessary that: $\forall A\in \mathcal{A}$: such that $$\int_A f\,d\mu=\... |
H: $I=\langle \{2,3,4,...\}\rangle $ is maximal ideal of $(P(\mathbb{N}),\Delta,\cap)$
This was given to me by my friend.
Prove that $I=\langle \{2,3,4,...\}\rangle $ is maximal ideal of $(P(\mathbb{N}),\Delta,\cap)$
My thoughts:-
Let $A\in P(\mathbb{N})$ Then $A\cap \mathbb{N}=\mathbb{N}\cap A=A$ ,so $\mathbb{N}$ is ... |
H: sequence of function, what is wrong with my solution?
Prove that the following series converges uniformly in $[0,\infty)$
$$\sum\limits_{n=1}^\infty\frac{(-1)^n}{x+n}$$
So, I need to prove that for every $x\geq0$ and every $\epsilon > 0 $ the following is true: $$\lim \sup |S(x)-S_n(x)|=0$$
So, I found:
$$\lim \sup... |
H: Confusion about argument result
As part of a larger problem, I wish to find the arguments $\theta$ for the complex numbers $-\frac{1}{2}\pm\frac{\sqrt{3}}{2}i$. This is $\theta = \arctan(-\sqrt{3})$, which is $-60^{\circ}=300^{\circ}$ for the positive and $\theta = \arctan(\sqrt{3})=60^{\circ}$ for the negative.
H... |
H: Proof that the nonnegativy of squares does not follow from the rest of the prepositive cone definition
A prepositive cone $P$ is a subset of a field $F$ defined as exhibiting the following properties:
Additive closure: $p+q\in P$;
Multiplicative closure: $p\cdot q\in P$;
No negative identity: $-1\notin P$;
Nonnega... |
H: General Solution of IVPs
I was trying to obtain the general solution the IVP(INTIAL VALUE PROBLEM)
$$ y'=y^2-\frac{y}{x}-\frac{1}{x^2}$$ ;${x}>0$ ; $y_1$($x)=\frac{1}{x}$ ; $y(1)=2$.
Where $y_1(x)$ denotes a solution to the differential equation.
But was clealry stuck since I couldn't apply bernoulli or ... |
H: Do there exist $2k+1$ irrational numbers such that their product and sum are both rational?
I recently found a problem saying "Find 2 irrational numbers such that their sum and product is both rational."
After a while I noticed any pair like $(a+\sqrt{b},a-\sqrt{b})$ work .From this I could easily say this statemen... |
H: Unboundedness of ODE with trigonometric coefficients
The task I am working on involves proving that the solutions to the system
$$\dot x = \begin{pmatrix}\cos{t} & \sin{t} \newline \sin{t} & -\cos{t}\end{pmatrix}x$$
are unbounded. To start I introduced the new variable $z = x_1 + ix_2$ and reformulated the system ... |
H: Squeeze Theorem and Second Partial Derivative of a Piecewise function
$$f(x,y)=\arctan((xy^3)/(x^2+y^2))$$ for $(x,y) \ne(0,0)$ and $$0$$ for $(x,y)=(0,0)$.
I have tried to prove its continuity at the point $(0,0)$ by using the Squeeze Theorem for $$y=x$$ and $$y=mx$$ such that $$\lim_{x\to 0}\arctan(x^2/2)\le\arct... |
H: Are projections always bounded?
Let $X$ be a Banach space and $P : \mathcal{D}(P) \subset X \to X$ such that $P\mathcal{D}(P)\subset \mathcal{D}(P)$ and that $P^2x = Px,~\forall x \in \mathcal{D}(P).$
Does this imply that $P$ is bounded?
I really think that the answer is no. But could not handle an example so far. ... |
H: How to Prove a Special Case of Stokes' Theorem?
I am currently in Calculus 3, or Multivariable Calculus and need to prove this special case of Stokes' theorem. Please forgive me as I do need this simplified to the bones to understand the explanations.
This version is below.
$$ \int_{\partial S}\mathbf{F}(x,y,z)\cdo... |
H: How would one find the value of this summation?
How would one go about finding the value of this summation? ${\sum_{n=1}^{\infty}}\frac{(-1)^{n+1}}{n^2+1}$
AI: You can verify that if $s\notin \mathbb Z$,
$$\cos{st} = \frac{\sin{\pi s}}{\pi s} \left [1+2 s^2 \sum_{n=1}^{\infty} \frac{(-1)^{n+1} \cos{n t}}{n^2-s^2} \... |
H: Show then that the inequality $(z-x)\int_{y}^zf(u)du≥(z-y)\int_{x}^zf(u)du$ holds for any $0 ≤ x < y < z.$
QUESTION: Let $f : [0,∞) → \mathbb{R}$ be a non-decreasing continuous function. Show then that the inequality $$(z-x)\int_{y}^zf(u)du≥(z-y)\int_{x}^zf(u)du$$ holds for any $0 ≤ x < y < z.$
MY APPROACH: We ob... |
H: Is there a special way to solve a problem like this with the indicated setup?
My Answer, and the Question
My question is if the problem was converted to the following:\begin{align}\cos(x)y'+\sin(x)y&=2\sin(x)\\ u'y'+uy&=2u \end{align} How can I advance from here to solve the problem if substitution were my aim in a... |
H: Showing a set is closed in the product topology
We have some topological space $X$ with continuous functions $f,g:X \rightarrow \mathbb{R}$ equipped with the usual topology I want to show that then set:
$$E = \{(x,y):f(x)=g(y)\} \subset X \times X $$ is closed in $X \times X$ with the product topology.
So I need to... |
H: Understanding how $E(X)=\frac{91}{36}$ was calculated
I've found an older question here and I just can't understand how they got these values. The question says:
"We a roll a fair dice 2 times. ( independent ):
a) X denotes the number of the first throw; Y denotes the sum of the two throws. Calculate Cov(X,Y). Calc... |
H: Check if the last digit in a 4 digit number is a 1
Is there a way to check if the last digit of a 4-digit (or any digit) number is a 1?
I am trying to make something that checks if the first and last digit of a number are both 1. Suppose our number is 1581. To check if the first digit is 1 is trivial. I can go..... |
H: Doubt about chromatic number proof
I'm in a discrete math course and was trying to prove the following theorem:
A graph G with $\Delta(G) = k$ ($\Delta(G)$ is the max vertex degree) is $(k+1)-$coloreable.
I've tried my own, and I've read the answers in here, here and this but I still have a big doubt using an in... |
H: Is there a proper ideal of $B(H)$ that contains a proper projection
Let $H$ be a infinite-dimensional separable Hilbert space and $\mathcal{I}$ be a proper closed two-sided ideal of $B(H)$. Can $\mathcal{I}$ contain a projection for a infinite dimensioal proper closed subspace? If $H$ is not separable (or $\mathcal... |
H: Field norm well-behaved with respect to minimal polynomial
I'm not sure if this property is standard but this is what some examples suggested:
Let $\alpha$ and $\beta$ be algebraic numbers. Is it true that
$$|N_{\mathbb Q(\alpha) / \mathbb Q}(min_{\beta / \mathbb Q}(\alpha))| = |N_{\mathbb Q(\beta) / \mathbb Q}(min... |
H: Application of calculus in kinematics Newton 2nd law
As we all know it the Newton's second law of motion describes the relationship between force and acceleration of a body as diectly proportional. Now lets say a body in motion and is of mass of 9kg and its acceleration is $a($t)=-9$t$. we are to find its displace... |
H: Why does the number of possible probability distributions have the cardinality of the continuum?
Wikipedia's article on parametric statistical models (https://en.wikipedia.org/wiki/Parametric_model) mentions that you could parameterize all probability distributions with a one-dimensional real parameter, since the s... |
H: Mean of a function of Binomial Distribution
Let $X$ be a random variable following the Binomial Distribution with parameters $n$ and $p$. Show that
$$
\mathbb{E}\left[\frac{1}{1+X}\right]=\frac{1-\left(1-p\right)^{n+1}}{p(n+1)},
$$
where $\mathbb{E}[\cdot]$ is the mean value funtion.
My textbook had this exercise a... |
H: $re^{i\omega} \rightarrow re^{2i\phi}$ not holomorphic over $\mathbb{C} \backslash \{0\}$
Try to solve this problem:
Show that function $ f: re^{i\phi} \rightarrow re^{2i\phi}$ not holomorphic over $\mathbb{C} \backslash \{0\}$
My solution:
We have
$$f(z) =f(x+yi) = f(re^{i\phi}) = re^{2i\phi} = rcos(2\phi) + i s... |
H: Does the property hold almost everywhere?
Suppose a property holds in every compact subset of $I$. Does it follow that this property holds almost everywhere on $I$?
I want to use the dominated convergence theorem but do not know almost anything about Lebesgue Measure. I know that the function series converges on ev... |
H: Difficulty in understanding the proof of infinitude of primes in a certain arithmetic progression
Let $m$ as a fixed odd prime. How to show there are infinitely many primes of the form $2km+1$ (for some positive integer $k$).
Can someone please help? Any help would be appreciated. Thanks in advance.
AI: The only o... |
H: $x \in \operatorname{Int}(A) \iff d(x, A^c) > 0$.
In $\Bbb R$ with the usual metric $|\cdot|$. I have to show that
$$x\in \text{Int}(A) \iff d(x,A^{c})>0$$
where $\text{Int}(A)$ is the interior of $A$. When i suppose that $d(x,A^{c})>0$ I have to show that $x\in \text{Int}(A)$. For this, let $x\in A$ and I consi... |
H: Priority of properties in Laplace Transform.
Supose I have a function f(t) which corresponding Laplace Transform is F(s)
Now I want to calculate the Laplace transform of $f(a (t-b))$ using Laplace Transform properties.
If $L(g(t))= G(s)$
Property A: Time shift $L( g(t-a)) = e^{-a s} G(s)$
Property B. Time scaling ... |
H: Can we always construct a matrix using its eigenvectors?
In physics, a Hermitian matrix represents an observable and can be constructed using its eigenvalues and eigenvectors in the following way:
$$ A = \sum_i \lambda_i v_iv_i^\dagger \qquad \qquad (1)$$
where $\lambda_i$ and $v_i$ are the $i^{th}$ eigenvalue and ... |
H: The representation theory of a group
Let $V$ be the $\mathbb KG$-module. Denote by $V^G$ the subspace of $V$ consisting of all the invariant elements under the action of $G,$ i.e., $V^G = \{v\in V| g\cdot v = v\}.$
Consider $W = V/V^G.$ Prove that $W = W^G.$
Clearly, $W^G \subseteq W.$ How to prove $W \subseteq W^G... |
H: Finding irreducible polynomial in finite field
I would like to find an irreducible polynomial of degree $3$ in $\mathbb{F}_4$, where
$$\mathbb{F}_4 = \{a+b\alpha| \ a, b\in \mathbb{F}_2, \alpha^2 = \alpha + 1\}.$$
I first tried to find an irreducible polynomial of degree 2. Since $\mathbb{F}_4 = \mathbb{F_2[\alph... |
H: Complicated infinite sum convergence
Do the following infinite sums
$$
\sum_{n=0}^{\infty}\binom{m+n}{n}b^{n}\text{ and }\sum_{m=0}^{\infty}\binom{m+n}{n}a^{m}
$$
converge? If so how to calculate their limit (the $m+n$ in the binomial coefficient confuses me)?
AI: Depends on $b$ and $a$, obviously... and they are r... |
H: $x$ is a member of $X$ if it is a member of $Y$--what is proof that $X=Y$?
$x$ is a member of $X$ if it is a member of $Y$. From this fact can we get to the statement $X=Y$?
AI: You can’t make this conclusion unless the converse is true as well that if y is a member of Y then it is also a member of X |
H: A boundary point is a limit point of $M$ and $M^{c}$?
I have been studying the basic concepts in topology and i'm wondering if i have a boundary point i can see it like a limit point of $M$ and $M^{c}$ where $M\subset X$ and $(X,d)$ is a metric space.
AI: That’s almost right: it’s a point $x$ that is in both the cl... |
H: Using the sum disturbance method, find a compact form of the following sums:
I tried to solve these two examples, but without success, could someone help me solve it because I got stuck on them and don't understand how to solve them.
Using the sum disturbance method, find a compact form of the following sums:
\begi... |
H: If $2a^2+8b^2+5c^2=2c(a+6b)$ then find $a:b:c$
It's a question involving three variables $a, b$ and $c$. One just has to find the ratio of the three variables.
AI: $$ P^T H P = D $$
$$\left(
\begin{array}{rrr}
1 & 0 & 0 \\
0 & 1 & 0 \\
\frac{ 1 }{ 2 } & \frac{ 3 }{ 4 } & 1 \\
\end{array}
\right)
\left(
... |
H: Extending of a sheaf by zero (exercise II.1.19 (b) from Hartshorne)
I'll reproduce part of Hartshorne's exercise II.1.19 (b) (the rest is not important to the question):
Let $Z$ be a closed subset of a topological space $X$ an $i:Z\to X$ the inclusion. If $\mathcal{F}$ is a sheaf on $U:=X\setminus Z$, let $j_!\mat... |
H: Does polynomial equality hold for multi-variables polynomials?
Let
$$f(x)=a_0x^n+a_1x^{(n-1)}y+a_2x^{(n-2)}y^2...+a_{(n-1)}xy^{(n-1)}+a_ny^n$$
$$g(x)=b_0x^n+b_1x^{(n-1)}y+b_2x^{(n-2)}y^2...+b_{(n-1)}xy^{(n-1)}+b_ny^n$$
If $$f(x)=g(x)$$
for all x and y. Does $$a_i=b_i$$
And, what if I include terms with power less t... |
H: Let $a$ & $b$ be non-zero vectors such that $a · b = 0$. Use geometric description of scalar product to show that
Let $a$ & $b$ be non-zero vectors such that $a \cdot b = 0$. Use geometric description of scalar product to show that $a$ & $b$ are perpendicular vectors.
What I've done so far is to state that $\cos 90... |
H: Proof the the field of rational numbers has the Archimedean property
I was tasked by the question to prove that the field of rational numbers has the Archimedean property
Proof:
Let $x$ and $y$ $\in$$ Q$ and $n$ be a positive integer. If $Q$ does not have the Archimedean property, then $nx\leq y$
For the case whe... |
H: Establishing a bijection
Let $A,B$ be finite sets. Prove that $|A\cup B|=|A|+|B|-|A\cap B|$ by establishing a bijection from $A\cup B$ to $\{1,2,\ldots,|A|+|B|-|A\cap B|\}$.
These are some hints that I got but I'm still confused on how to come up with a bijection.
Prove for disjoint finite sets $A,B$ ( i.e. $A\ca... |
H: Is there a notion of maps that can "expand" spaces "linearly"?
For linear transformations, the dimension of the image is at most the dimension of the domain. More generally, given vectors $v_1, ..., v_n$ in the domain, the vectors $Tv_1, ..., Tv_n$ span the image. So, intuitively, we can only either preserve the di... |
H: How many length-$k$ ternary strings have evenly many of a given symbol?
I write down a string of $k$ letters, where each letter is $X, Y, \text{or } Z.$ The letter $X$ appears an even number of times. How many different sequences of letters could I have written down?
I think I need to start by setting up some cases... |
H: Equivalence relation proof, problem $x∼y \iff x^2 − 2x + 1 = y^2 + 4y + 4$
Determine if the following relationships are equivalent. If they are, determine
the equivalence classes, give a set of indices and the quotient set. Sketch out
the graph of each relationship (whether or not it is an equivalence one):
In $\ma... |
H: Functional equation involving three different functions: $ f ( x + y ) = g ( x ) + h ( y ) $
If $ f , g , h : \mathbb R \to \mathbb R $ all are continuous functions such that
$$ f ( x + y ) = g ( x ) + h ( y ) \text , \quad \forall x , y \in \mathbb R \text , $$
find $ f $, $ g $ and $ h $.
I have literally no id... |
H: Proving $\operatorname{cos}(x+y)=\operatorname{cos}(x)\operatorname{cos}(y)-\operatorname{sin}(x)\operatorname{sin}(y)$ using differentiation
While proving $\operatorname{cos}(x+y)=\operatorname{cos}(x)\operatorname{cos}(y)-\operatorname{sin}(x)\operatorname{sin}(y)$
by this $$\operatorname{sin}(x+y)=\operatornam... |
H: Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.
420 satisfies the condition since $7<$ $\sqrt[3]{420}<8$ and $420=\operatorname{lcm}\{1,2,3,4,5,6,7\}$
... |
H: Need to simplify $(A ∩ \varnothing)' \cap (A \cup B)'$
I have the following expression I need to simplify:
$$(A \cap \varnothing)' \cap (A \cup B)'$$
So, far my solution is to use DeMorgan's Law to simplify it as follows:
$$(A' \cup \varnothing') \cap (A' \cap B')$$
But I'm not sure where to go from here. I was per... |
H: If $R \cup S$ is an equivalence relation on $A$, then $R$ and $S$ are equivalence relations on $A$
Let $A$ be a non-empty set, and let $R$ and $S$ be relations on A.
If $R \cup S$ is an equivalence relation on $A$, then $R$ and $S$ are
equivalence relations on $A$.
How can i prove it? I think it is not necessaril... |
H: do all matrices with $\det(A)=\pm 1$ form a group under multiplication?
all matrices with determinant one form the special linear group.
it is explained that because $\det(A) \det(B)=\det(AB)$ it is closed as $1*1=1$
and because the general linear group is a group, and special linear group is a part of the general ... |
H: About minimum and maximum inequality.
Let $a_i , b_i \geqslant 0$ $\forall i \in \{1 , 2 , 3 , .... , n\}$
Prove that $$min\{\frac{a_i}{b_i}\} \leqslant \frac{\sum_{i=1}^n a_i}{\sum_{i=1}^n b_i} \leqslant max\{\frac{a_i}{b_i}\}$$
Given that $min\{...\}$ and $max\{...\}$ equals to the minimum and the maximum elemen... |
H: Can we deduce that $\mu(X)\leq \mu(Y)$?
Given a measurable function $f: X \to Y$, if $f$ is injective and measure-preserving that $\mu(f(A))=\mu(A)$ for all subsets $A$ and $\mu$ is a probability measure, can we deduce that $\mu(X)\leq \mu(Y)$?
Do we have $\mu(f(X))=\mu(X)\leq \mu(Y)$ since $f(X)\subset Y$?
AI: T... |
H: Example of the non-commutative ring with the set of units are commutative
I was looking for an Example of the non-commutative ring with the set of units are commutative. it will be a great help. Thanks in advance.
AI: Take the ring of noncommutative polynomials $R\langle X_1, X_2\rangle$ over any commutative ring $... |
H: Why is the expected value, $E(X)$, after $5$ games $0$?
In a game the probability that you win is $1/2$. If you win the game, you get $1\$$ and if you loose the game you loose $1\$$. Let $X$ denote the total amount of money after $5$ games. The expected value $E(X)=0$. Why is that?
AI: The chance you win is $50$ pe... |
H: $f''(x) = g(x)$ and $g''(x) = f(x).$ Suppose also that $f(x)g(x)$ is linear in $x$ on $(a,b).$ Show that $f(x) = g(x) = 0$ for all $x ∈ (a,b).$
QUESTION: Let $f$ and $g$ be two non-decreasing twice differentiable functions defined on an interval $(a,b)$ such that for each $x ∈ (a,b), f''(x) = g(x)$ and $g''(x) = f... |
H: Is the sets in density topology Euclidean $G_\delta$?
It has been shown that every Borel subset of density topology X is d-$G_\delta$. I'm curious about its connection to the euclidean topology. For example, is the close/open set in the density topology a $G_\delta$ set in Euclidean topology? It seems false to me; ... |
H: Residue at essential singularity of $\frac{\mathrm{e}^{\frac{1}{z}}}{z^2-2z+2}$
Find the residue at $z=0$ of the function $$f(z)=\frac{\mathrm{e}^{\frac{1}{z}}}{z^2-2z+2}.$$
This is a very small part of a previous exam question of complex analysis. One way of doing it is decomposing $\frac{1}{z^2-2z+2}$ into parti... |
H: Proof of closedness of discrete subgroup of a topological group
Here is a lemma of John Ratcliffe's book:
Lemma: If $G$ is a topological group with a metric topology, then every
discrete subgroup of $G$ is closed in $G$.
Proof. Let $\Gamma$ be a discrete subgroup of $G$ and suppose that $G − \Gamma$ is not
open... |
H: ${\displaystyle \int_{0}^{7}f^{(7)}(x)(x-1)^6dx}$ where $f(x)=e^{-\frac{1}{x^2}}\sin(x)$ for $x>0$ and $f(0)=0$.
How to find the below integral: $$\int_{0}^{1}f^{(7)}(x)(x-1)^6dx$$ where $f(x)=e^{-\frac{1}{x^2}}\sin(x)$ for $x>0$ and $f(0)=0$.
My attempt:
I tried to find the derivative $f^{(7)}(x)$, but at $f^{(3... |
H: PROPOSED PROOF for: If $p$ is a prime number such that $p|ab$, then $p|a$ or $p|b$.
Since $p|ab$, then $\exists \alpha \in\ \mathbb{Z}$ such that:
$$\alpha p=ab$$
By dividing both sides by $a$, we obtain:
$$(\alpha \div b)p=a \Rightarrow\ p|a \ \ ( \text{Similarly to prove that} \ \ p|b)$$
Is this proof comprehens... |
H: How to prove: $\:a\:
if
$0<\lambda <1$ and $\:a\:<\:x_1<b,a<x_2<b$ and $ \:x=\lambda x_1+\left(1-\lambda \right)x_2$
how to prove :
$\:a\:<x<b$
AI: Notice that
$$ x < \lambda b + (1-\lambda) b = b $$
and
$$ x > a(1-\lambda) + a \lambda = a $$ |
H: For periodic $f,g$ (and continuous), does $\lim (f-g) = 0 $ imply $f=g$?
Take $f, g: \mathbb{R} \to \mathbb{R} $ continuous and periodic and assume $\lim_{x \to
\infty} (f(x)-g(x)) = 0$. Does it follow $f=g$? Prove or give counter
example.
I cant find any counterexample, so maybe we should prove it:
we given t... |
H: If $a_1a_2 = 1, a_2a_3 = 2, a_3a_4 = 3 \cdots$ and $ \lim_{n \to \infty} \frac{a_n}{a_{n+1}} = 1$, find $|a_1|$
If $a_1a_2 = 1, a_2a_3 = 2, a_3a_4 = 3 \cdots$ and $\displaystyle \lim_{n \to \infty} \frac{a_n}{a_{n+1}} = 1$. Find $|a_1|$
I could conclude that $\displaystyle \lim_{n \to \infty}a_n$ must be $\infty$... |
H: Examples where $1+I$ is inverted but $I$ is not mapped into the Jacobson radical
Let $f:A\to B$ be a commutative ring morphism. Let $I\vartriangleleft A$ be an ideal. If $f(I)\subset \mathrm J(B)$ is contained in the Jacobson radical then $f(1+i)=1+f(i)\in B^\times$ is a unit so $1+I$ is inverted.
On the other hand... |
H: Why can WLOG be used in binary integer representation theorem?
I was trying to understand the uniqueness portion of the proof for integer representation theorem.
Then I saw this:https://math.stackexchange.com/a/607774/789305.
He made an assumption that $r>s$, reached a contradiction and then claimed $r$ is not $>$ ... |
H: Find $c$ such that $P(Z^2 > c) = 0.95$
I was wondering if any of you could help me with this statistics problem.
AI: If $Z\sim N(0,\,1)$, you can proceed in one of two ways:
Since $Z^2\sim\chi_1^2$, a $\chi_\nu^2$ table will tell you that for $\nu=1$ the value with CDF $1-0.95=0.05$ is $0.00393$.
Let $\Phi$ denote... |
H: Almost sure convergence of random variables with same mean and the difference goes to zero on the product
Let $X_n$ be a sequence of independent real valued random variables on the same event space, with the same (finite) mean $\mu$.
Suppose that for almost every couple of points $(\omega,\omega')$ in the squared ... |
H: Matrix of the differentiation operation
Exercise: Find the matrix of the derivative operation $D$ related to the base $\{1, t, t^2,..., t^n\}$ $$D: \mathcal P_{n} \to \mathcal P_{n}$$
I found a possible solution to this exercise, given that $D(t^k)=kt^{k-1}$ $$ \begin{equation*}
D_{n+1,n+1} =
\begin{pmatrix}
0 & 1... |
H: How does error analysis and significant figures actually work?
I could never quite grasp the precise concept. For starters, some books seem to use absolute error to find the number of correct digits the aproximation shares with the exact value whereas others use relative error. To me, it makes more intuitive sense ... |
H: All prime divisors of $\frac{x^m+1}{x+1}$ are of the form $2km+1$.
Let $m$ be an odd prime and $x$ be the product of all primes of the form $2km+1$. Then all prime divisors of $\frac{x^m+1}{x+1}$ are of the form $2km+1$.
What I know is that $\frac{x^m+1}{x+1}$ is an integer.
Here is the link to the answer which pro... |
H: Prove sum of $k^2$ using $k^3$
So the title may be a little bit vague, but I am quite stuck with the following problem.
Asked is to first prove that $(k + 1)^3 - k^3 = 3k^2 + 3k + 1$. This is not the problem however. The question now asks to prove that
$$ \sum_{k=1}^{n} k^2 = \frac{1}{6}n(n+1)(2n+1)$$ using the fac... |
H: If a commutative ring with $1$ does not have a unit of order $2$, then $R$ does not have an element of order $2$.
Let $R$ be a commutative ring with $1$. This is a simple question, but I can't see whether it is true or not.
Suppose $R$ does not have a "unit" of (additive) order $2$, i.e. , a unit $u$ such that $u=-... |
H: Show that there are no integer solutions to $2x^{11}+3y^{11}=6z^{11}$
I have managed to show that $x$ must be a multiple of $3$ and $y$ must be even, which produces the equation
$$3^{10}s^{11}+2^{10}t^{11}=z^{11},$$
with $s=x/3$ and $t=y/2$. I have tried to approach this by applying Fermat's Little Theorem mod $1... |
H: Unique differentiable linear operator mapping $\mathbb{R} \rightarrow \mathbb{R}^N$?
I am trying to find a mapping $\phi$ such that:
$\phi$ uniquely maps $\mathbb{R}$ to the subspace of $\mathbb{R}^N$ (for bounded $N$), where every dimension of the vector is bounded between $-1$ and $1$.
$\phi$ is differentiable
$... |
H: How to find the degree of the extension $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$?
How to find the degree of extension for $[\mathbb{Q}(\sqrt[4]{3+2\sqrt{5}}):\mathbb{Q}]$? I believe that the minimal polynomial of $\sqrt[4]{3+2\sqrt{5}}$ is $x^8-6x^4-11$, but I don't know how to show that it is irreducible o... |
H: 'Irregular' conformal mapping of square onto circle?
Assume that I want to find the conformal mapping from a square onto the unit disk. In the regular case, where the four edge points are positioned along cardinal directions, this mapping seems to have a relatively simple solution. I have plotted this in the figure... |
H: Existence of a strange Group
It seems to me that if $G$ is a finite group and $|G| \geq 3$ and odd then there exists an element $a \in G$ such that $a \neq a^{-1}$. I can easily show this for order $3$ and $5$ but for higher orders the computations become unwieldy.
My attempt:
I want to prove this by contradictio... |
H: How do I solve this problem using binomial theorem with square root in it?
The question is: Find the coefficient of $\frac{1}{x\sqrt x}$ in the expansion of $(x^2 - \frac{1}{2\sqrt x})^{18}$.
I have included a photo to make it easier to read because I do not know how to format the question.
AI: Hint:
Any term in t... |
H: $A=\{(x,y) : x = u+v,y = v,u^2+v^2 ≤ 1\}$ Derive the length of the longest line segment that can be enclosed inside the region $A$.
QUESTION: Let $A$ be the region in the $xy$-plane given by $$A=\{(x,y) : x = u+v,y = v,u^2+v^2 ≤ 1\}$$Derive the length of the longest line segment that can be enclosed inside the reg... |
H: Verifying that the identity functor is indeed a functor
The identity functor $1_{\mathscr C}:\mathscr C \to \mathscr C$ has $1_{\mathscr C}(a)=a$, $1_{\mathscr C}(f)=f$. Verify that it is indeed a functor, specifically that it is a function that assigns:
To each $\mathscr C$-object $a$, a $\mathscr C$-object $1_{... |
H: How to calculate $\int_0^\frac{\pi}{2} \sum_{k=0}^\infty \frac{1-k\sin{kx}}{2^k} dx $?
I have to find $\int_0^\frac{\pi}{2} \sum_{k=0}^\infty \frac{1-k\sin{kx}}{2^k} dx$, but can't figure out how to do it.
If $\frac{1-k\sin{kx}}{2^k}$ converges it would be easy to calculate $\int_0^\frac{\pi}{2} \frac{1-k\sin{kx}}... |
H: Result of number with decimal point modulo $10$.
This is a very simple question, very trivial to many, but I have to resolve this doubt!
I have done the division $2.2/10$ by hand, and the result is $0.22$, without any remainder:
But why if I do with calculator, I obtain this:
$$2.2 \mod 10 = 2.2$$
why it is not e... |
H: Prove that if $|f(z)| \geq |f(z_{0})|$ then $f(z_{0})=0$
Let $U \subseteq \mathbb{C}$ be an open connected subset and $f: U \rightarrow \mathbb{C}$ an holomorphic function.
Let $z_{0} \in U$ and $r > 0$ with $B_{r}(z_{0}) \subseteq U$.
If $\forall z \in B_{r}(z_{0})$ $|f(z)| \geq |f(z_{0})|$ then $f(z_{0})=0$.
As $... |
H: An application of Fubini’s theorem on Fourier transform
Given $f,g\in L^1(\mathbb{R}^n)$ and we denote the Fourier transform of $f$ by $\widehat{f}$.
I want to prove that
$$\int_{\mathbb{R}^n}\widehat{f}(x)g(x)~dx= \int_{\mathbb{R}^n}f(x)\widehat{g}(x)~dx.$$
Here’s my attempt:
\begin{align}
\int_{\mathbb{R}^n}\wid... |
H: Bias of uniform maximum
Let $X_i$ be i.i.d. uniform random variables in $[0,\theta]$, for some $\theta>0$ and $M_n = max(X_i)$
I am trying to find the bias of $M_n$ as an estimator of $\theta:$
$$E[M_n] - \theta = $$
Computing $E[M_n] = \frac{n}{n+1}$ I guessed the answer would be $\frac{n}{n+1} - \theta$
But this ... |
H: Upper semicontinuous decomposition
I'm reading a paper from Y. Ünlü called Lattices of compactifications of Tychonoff spaces. I've bumped into some definitions that I've never seen; while I find most of them understandable, there's one that I can't break down.
Definition 1. A decomposition $\alpha$ of a space $X$ i... |
H: Show that every contractible space is connected.
So I'm trying to show every contractible topological space is connected. At the moment I've established that $X$ is contractible iff there is a homotopy equivalence $f:\{x_0\}\to X$ for some $x_0\in X$, but I'm having trouble showing that if $X$ and $Y$ are homotopy ... |
H: Statement true for all prime numbers -- can this be done by Math Induction?
Let's say that you want to prove a statement is true for all prime numbers. Can this be done by Math Induction?
AI: Let's say that you want to prove a statement is true for all prime numbers.
Such as Fermat's Last Theorem:
$$ X^p+Y^p=Z^p$$... |
H: Calculating Distance From a Line To Point
Hello everyone I have a point $P$ and a line $l$.
And I need to find all the $X$ points that for them
the distance from $X$ to $P$ is the half distance from $X$ to $l$ in $2d$.
I tried to use the distance formula but I didn't success.
AI: You can very well use the distance ... |
H: summation of random variable into harmonic number
In our textbook "Algorithm Design" we are given an example of a deck of $n$ cards and we have to guess the correct card, and every time a card is drawn, we remember that cards so we are uniformally guessing only among the cards not yet seen.
I don't understand the ... |
H: What do first derivatives, factorials, and alternating signs have to do with explicit and recursive forms of sequences?
I'm a math teacher now, although a few years ago I was finishing up my M.Ed. As part of my studies, I was tasked with conducting my own study of high school level topics and finding unique results... |
H: Variance of the difference of Brownian Motions
I have a question about the variance of the following formula:
$Var(W(t) - \frac{t}{T}W(T-t))$. Where $W(t)$ is a Brownian motion. I tried the following:
$Var(W(t) - \frac{t}{T}W(T-t)) = E[W(t) - \frac{t}{T}W(T-t)]^2 - (E[W(t) - \frac{t}{T}W(T-t)])^2$. Now
$E[W(t) - \f... |
H: Computing a trace containing $\gamma$-matrices
I want to compute the following trace
$$Tr \Big( Y(\not{\!p_1'}+m) \Big) \ \ (1)$$
Where
$$\not{\!A} := \gamma^{\alpha} A_{\alpha} \ \ (2)$$
$$Y:= 4 \not{\!f_1} \not{\!p} \not{\!f_1} + m[-16(pf_1)+16 f_1^2] + m^2 ( 4 \not{\!p} - 16 \not{\!f_1})+16 m^3 \ \ (3)$$
The a... |
H: Show that not exists any polynomial function such that $f(x) = \log (1+x)$.
Does anyone have any idea on that problem?
Let $f : \mathbb{R} \to \mathbb{R}$ be a polynomial function. Show that not exists any $f$ such that $f(x) = \log (1+x)$.
It's easy to show that $a_0 = 0$ and $a_1 = 1$. But after i don't have an... |
H: When two Algebraic vector bundles on a Noetherian quasi-affine scheme are equal in $K_0$ of the scheme
Let $X$ be a (connected) Noetherian scheme and $K_0(X)$ denote the Grothendieck group of the category of Algebraic vector bundles (coherent sheaves that are locally free and of constant rank ( as $X$ is connected)... |
H: Relation between squared norms and sets of orthonormal vectors having the same span
I was asked to show the following claim but I'm stuck. It seems I can't find the right reasoning path.
Given a matrix $M\in\mathbb{R}^{n\times d}$ and two sets of pairwise orthogonal unit vectors {$u_1, ..., u_k$} and {$v_1,..., v_k... |
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