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H: Why the digit Sum of a non negative number(n) is always the remainder after division by 9 , until n becomes only one digit?
lets assume for $689$ : ,
$$689 = (6+ 8 + 9 ) = 23$$
$$23 = (2 + 3) = 5$$
We can get this way:
$$689 \!\!\mod \;9 = 5$$
I found a approach like this:
$$10 = (9 * 1) + 1 $$
$$100 = (9 * 11) ... |
H: Convergent Sequences in Extremally Disconnected Hausdorff Spaces
It's written in Willard (15G.3) that the only convergent sequences in a Hausdorff Extremally Disconnected space are the eventually constant sequences. However, it has not provided a proof.
I've tried to derive this myself, but am unable to do so. Ther... |
H: Integral definition of curl
My book, Mathematical Methods for Physics and Engineering - K. F. Riley,
explains that the 'Integral definition of curl' is:-
$$\nabla\times a = \lim_{V\to0}(\frac{1}{V}\int_S dS\times a) $$
and then, after a few pages, it mentions:-
$$\int_S dS \times \nabla\phi = \int_C\phi dr$$
Now, i... |
H: Is it true that $\sum_{i=0}^{n-1} 2^i$ divides $\sum_{i=n}^{2n-1} 2^i$?
Some investigation suggested to me that
$$\sum_{i=0}^{n-1} 2^i \Bigg| \sum_{i=n}^{2n-1} 2^i$$
The point being that binary numbers in the form exclusively of an even number of $1$s appear to be divisible by the number represented by just half th... |
H: What's the name for sequences sitting between geometric and arithmetic, i.e. whose recurrence relation is of the form $ax+b$?
What's the name for sequences sitting between geometric and arithmetic?
E.g. let $x_{n+1}=ax_n+b$
I can't find a general name for these.
These sequences may also be a Lucas Sequence but that... |
H: Easy way to check that real part of $e^{-\frac{1}{z^{4}}}$ is harmonic.
Let $z=x+i y$ and $$ u(x,y) = Re \left(e^{-\frac{1}{z^{4}}}\right) ,~ \text{for}~ (x,y) \ne (0,0)$$ and $0$ otherwise. Then is there any short way to check that $u$ satisfies Laplace equation ?
I can apply brute force to check that given funct... |
H: Why is $\operatorname{Im}{\frac{1-e^{iny}}{-2i\sin{\frac{y}{2}}}}=\frac{1}{2\sin{\frac{y}{2}}}\operatorname{Re}(1-e^{iny})$?
I am studying Fourier analysis and had a question involving the following equality:
$$\operatorname{Im}{\frac{1-e^{iny}}{-2i\sin{\frac{y}{2}}}}=\frac{1}{2\sin{\frac{y}{2}}}\operatorname{Re}(1... |
H: Divergence of $\sum_{n=1}^{\infty}\prod_{k=1}^n q_k$ for some enumeration $(q_n)_{n}$ of $\mathbb{Q}\cap (0,1)$
Given an enumeration $(q_n)_{n}$ of $\mathbb{Q}\cap (0,1)$, let us consider the series
$$\sum_{n=1}^{\infty}\prod_{k=1}^n q_k.$$
Find an enumeration such that the series is convergent.
Find an enumerati... |
H: Prove that set of permutation on $3$ elements is not isomorphic to $(\Bbb Z_6,+)$.
Prove that set of permutation on $3$ elements is not isomorphic to $(\Bbb Z_6,+ )$ (the group with $+$ on $\Bbb Z_6$).
Hello everyone,
I tried to build a function and show that associativity doesnt work because of the permutation b... |
H: System of congurences and the Chinese Remainder Theorem
I have the following system of congruences:
\begin{align*}
x &\equiv 1 \pmod{3} \\
x &\equiv 4 \pmod{5} \\
x &\equiv 6 \pmod{7}
\end{align*}
I tried solving this using the Chinese remainder theorem as follows:
We have that $N = 3 \cdot 5 \cdot 7 = 105$ a... |
H: Physics and Riemann hypothesis
I was reading the article "Quantum physics sheds light on Riemann hypothesis" from Bristol University (http://www.bristol.ac.uk/maths/research/highlights/riemann-hypothesis/) and stopped here:
From a conference in 1996 in Seattle, aimed at fostering collaboration between physicists a... |
H: Why does the irreflexivity of $L^1$ follow from $(L^1)'' \subsetneq L^1$?
My question is about the proof that $l^1$ is irreflexive. I have seen multiple proofs ($L^1$ and $L^{\infty}$ are not reflexive, Dual of $l^\infty$ is not $l^1$ and more), and all proofs stop after showing that there exists $f \in (l^\infty)'... |
H: Finding coefficients in expansions
Show that the coefficient of $x^{−12}$ in the expansion of
$$\left(x^4−\frac{1}{x^2}\right)^5\left(x−\frac{1}{x}\right)^6$$
is $−15$, and calculate the coefficient of $x^2$. Hence, or otherwise, calculate the coefficients of $x^4$ and $x^{38}$ in the expansion of $$(x^2−1)^{11}(x^... |
H: Determine if a statement can haapen
let $ V= M_{10}\left(F\right) $ a vector space over some field $ \mathbb{F} $
I have to determine if its possible that exists $ A\in M_{10}\left(F\right) $
such that $ M_{10}\left(F\right)=span\left\{ A^{i}:0\leq i\leq100\right\} $
I guess it cant be. But I cant prove why. I tri... |
H: Alternate methods to prove $(1+a)(1+b)(1+c)(1+d) \geq 16$ if $abcd =1$.
I found this question some time ago in an Elementary Olympiad book:
If $a, b, c, d$ are positive integers such that $abcd =1$, then prove that $(1+a)(1+b)(1+c)(1+d) \geq 16$.
Evidently this was a direct consequence of Hölder's inequality, so ... |
H: If $\lim_{x\to 0} \frac{x^2 \sin (bx)}{ax-\sin x}=1$, then find $a,b$
$$\lim_{x\to 0} \frac{x^2 \frac{\sin bx}{bx} (bx)}{x(\frac{ax}{x}-\frac{\sin x}{x})}$$
$$=\lim_{x\to 0} \frac{bx^2 \frac{\sin bx}{bx}}{a-\frac{\sin x}{x}}$$
$$=0$$
Which is obviously wrong. I think I am suffering from lack of conceptual clarity i... |
H: Meaning of a phrase from Zorich II
Taken this excerpt directly from Zorich, Mathematical Analysis II:
My question is simply this: I'd like to understand what the author means by the last sentence highlighted in red, which seems to me to seem to be in contrast with the first (always highlighted in red).
AI: It's ju... |
H: When are the zeros of $x^T A x$ equal to kernel of $A$?
I am wondering when the set of solutions to $x^T Ax = 0$ is equal to the kernel of the matrix $A$. Is there a general answer to this?
It seems to be true in the positive-definite case:
If $x^TAx = 0$ then $x$ must be $0$ (otherwise $x^TAx > 0$), so $Ax = 0$.
O... |
H: morphism is epic?
If we have three morphisms f, g and h between objects of a category.
Suppose gf = h. If g and h are epic, can we conclude that so is f ? Any help would be appreciated!
AI: Hint:
In category set let the domain and codomain of $h=g\circ f$ be a singleton.
Further let the codomain of $f$ have more t... |
H: Number of Relations that satisfy a condition
This is a multiple choice question from my Text Book
Let $A=\{1,2,3\}$. The no. of relations containing $(1,2)$ and $(1,3)$ which are reflexive and Symmetric but not transitive is
(A) $1$
(B) $2$
(C) $3$
(D) $4$
My Approach:
$A=\{1,2,3\}$
Relation $R$ must contain $(1,2... |
H: Question about Schur's Lemma
Maybe a very dumb question about Schur's Lemma, but somehow this confuses me.
Let $G, H$ be groups. A function $f: G\rightarrow H$ is called group isomorphism, if $f(g\cdot h) = f(g)*f(h)$, and $f$ is bijective.
Let $(D_1,V_1)$ and $(D_2,V_2)$ be irreducible representations of a group $... |
H: Undertstanding Cup products
I am trying to understand cup products from the following notes and am having some difficulty understanding theorem which in a way defines cup product for cohomology: https://www.math.ucla.edu/~sharifi/groupcoh.pdf#theorem.1.9.5
What does it mean to say that the map 'cup products'
$$H^i(... |
H: Evaluating $\int_0^1dx\int_x^{\frac{1}{x}}\frac{y\,dy}{(1+xy)^2(1+y^2)}$
Evaluate the integral (change of order of integration may be useful):$$\int_0^1dx\int_x^{\frac{1}{x}}\frac{y\,dy}{(1+xy)^2(1+y^2)}$$
I searched the region, I got this graph
But, can't identify the region exactly. In fact, the lower bound of... |
H: Proving uniqueness of antipodes in Hopf algebras
Let $(H,\mu,\nu,\Delta,\epsilon)$ be a Bialgebra where H is the vector space, $\mu, \nu$ are the product and unit whilst $\Delta, \epsilon$ are the coproduct and counit. Now, for $f,g \in end(H)$ define $f@g \in end(H)$ by $f@g=\mu(f \otimes g)\Delta(x)=\Sigma_{(x)}f... |
H: If $\lim_{x\to 0} \frac{1+a\cos 2x + b\cos 4x}{x^4}$ exists for all $x\in\mathbb R$ and is equal to $c$, find $\lfloor a^{-1} +b^{-1} + c^{-1}\rfloor$
$$\lim_{x\to 0} \frac{1+a(1-\frac{4x^2}{2!} + \frac{16x^4}{4!}-\cdots)+b(1-\frac{16x^2}{2!} + \frac{256x^4}{4!}-\cdots)}{x^4}$$
$$=\lim_{x\to 0} \frac{ (1+a+b) -\fra... |
H: Why use limit laws to verify continuity instead of direct substitution?
My textbook (Calculus Early Transcendentals, 8th edition, by James Stewart ) asks to verify a function is continuous at a point using the definition of continuity and the limit laws. However, why would the text explicitly state to use the limit... |
H: How to quantify which expression for a given mathematical quantity converges the fastest?
Which converges faster to $e$: $\lim_{ n \to \infty} \sum_{k=0}^{n} \frac{1}{k!}$ or $\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$?
To check this, I thought of taking ratio of the two and taking limit as n goes to inf... |
H: How to calculate the limit of sequence
The question is calculate limit of sequence
$$
\lim_{n \to \infty}
\frac{\left(2\,\sqrt[\Large n]{\, n\,}\, -
\,\sqrt[\Large n]{\, 2\,}\right)^{n}}{n^2}
$$
I'm trying to simplify the equation, like divide $\,\sqrt[\Large n]{\, n\,}\,$, but can't get more. I drew the continuou... |
H: Are $\forall x\forall y(p(x,y)\leftrightarrow p(y,x))$ and $\forall x\forall y p(x,y) \leftrightarrow \forall y\forall x p(x,y)$ tautologies?
$$\forall x\forall y( p(x,y) \leftrightarrow p(y,x) )$$
I don't know if this formula is a tautology or not.
I think the order inside the predicates is not important and there... |
H: Let $G=\mathbb{Z}\times\mathbb{Z}$ and $H=\{(a,b)\in\mathbb{Z}\times\mathbb{Z}: 8\mid a+b\}$. What is the index $[G:H]$?
Let $G=\mathbb{Z}\times\mathbb{Z}$ and $H=\{(a,b)\in\mathbb{Z}\times\mathbb{Z}: 8\mid a+b\}$. What is the index $[G:H]$?
From a separate exercise part of this problem we are given $H\lhd G$ and... |
H: Construct compact subset of a set
In the context of uniform continuity in our analysis course we often need compact sets or compact subsets to prove certain properties. I am not sure if I have fully understood how to construct a compact subset and why we are allowed to do this. May be someone can give me an explana... |
H: how to find eigenvalues of $T_n:R^n\to R^n$ where $T_n (x)=(0,x_1,\frac{x_2}{2},\ldots,\frac{x_{n-1}}{n-1})$, what happen if $n\to\infty$
how to find eigenvalues of $T_n:R^n\to R^n$ where $T_n (x)=(0,x_1,\frac{x_2}{2},\ldots,\frac{x_{n-1}}{n-1})$ what happen if $n\to\infty$, well by definition i do $T_n(x)=\lambda... |
H: If $M$ is compact subset of a metric space $X$, then it is bounded.
I have proved this theorem using open cover definition. I also read that this result can be proved using sequence definition. That is, if a set $M$ is compact then every sequence in $M$ has a convergent subsequence.
Using this definition, book show... |
H: let $|S| = 3$. Prove $(P(S),\Delta)$ is not isomorphic to $(\mathbb{Z_8},+)$.
Hi everyone I hope somone can help
I have some set $S$ with $3$ elements $P(S)$ is the power set.I need to prove that $(P(S),\Delta)$ is not isomorphic to $(\mathbb{Z_8},+)$. I tried building a function to show that associativity doesn't... |
H: Characterization of noetherian modules via short exact sequences (understanding a step in the proof)
I am currently dealing with this result from William Stein's Algebraic Number Theory notes, and I also underlined the part of which I am not sure yet:
I know that $g(M_0)$ is trivial since we have a short exact seq... |
H: How do you show that $\mathbb{Z}^2$ is a closed set in $\mathbb{R}^2$?
In $(\mathbb{R}, \tau_{st})$, we can write $\mathbb{R} \setminus \mathbb{Z} = \bigcup_{n \in \mathbb{Z}} (n,n+1)$, and hence $\mathbb{R} \setminus \mathbb{Z}$ is an open set. Thus the complement, $\mathbb{Z}$, is closed.
In $(\mathbb{R}^2, \tau... |
H: Suppose $A$, $B$, and $C$ are sets. Prove that $C\subseteq A\Delta B$ iff $C\subseteq A\cup B$ and $A\cap B\cap C=\emptyset$.
Not a duplicate of
Suppose $A$, $B$, and $C$ are sets. Prove that $C ⊆ A △ B$ iff $C ⊆ A ∪ B$ and $A ∩ B ∩ C = ∅$.
Suppose $A, B$, and C are sets. Prove that $C\subset A\Delta B \Leftrightar... |
H: $g_n = m(E_n)^{\frac{-1}{q}}\chi_{E_n}$. Show $\int fg_n \rightarrow 0$
Let $f\in L^p(\mathbb{R})$. Let $E_n$ be measurable sets of finite measure, where $\lim_{n \to \infty}m(E_n)=0$. Let $p,q$ be such that $1<p,q<\infty$ and $\frac{1}{p}+\frac{1}{q} = 1$. And define $g_n = m(E_n)^{\frac{-1}{q}}\chi_{E_n}$. Show $... |
H: The multinomial coefficient and number of unique words
Let's define $C$ as the number of possible sequences of length $N$ we can form using symbols $A_i$ from an alphabet $A = \{A_1,\ldots , A_M\}.$
To my knowledge, we always have $C = M^{N+1}$.
For example, the total count of numbers containing 3 digits that we ca... |
H: Why is $\sum_{i=0}^{t-1}r^i \leq r^t$?
On page 348 of this article by Siegelmann and Sontag, there's a claim that depends on the following inequality:
$$\sum_{i=0}^{t-1}r^i \leq r^t$$
(The claim appears after a line consisting only of "Therefore,". I'm using $r$ for $LW$ in the text. The same claim can also be fo... |
H: In a Hopf algebra $H$, unit $\circ$ counit = Identity?
Let $(H,\mu,\nu,\Delta,\epsilon,S)$ be a Hopf algebra where the convolution is denoted by $*$.
Then $S=S*\nu\epsilon$ and thus $\nu\epsilon = 1_{End(H)}$. This would imply that $\epsilon$ is injective and $\nu$ is surjective. I am surprised that this mysteriou... |
H: Asymptotics of Recursive Bound
Suppose that we knew that for $n>N$, we have $$F(n) \le F(n-g(n)) +h(n)$$ for some well behaved functions $g,h$. (for a concrete example, let’s say $N=1,g(n)=n^\alpha, h(n)=n^\beta$ where $\alpha,\beta \in (0,1))$)
I was wondering then how we can get an asymptotic upper bound of $F$. ... |
H: Basic Geometry: Partitions and Intersections
Once more unto the breach, dear friends, once more!
So I'm currently working on a problem which I have somehow been able to simplify to the point where if I can simply prove that if three lines each partition a compact subset of $\mathbb{R}^2$ into two pairs of equal par... |
H: Order of statements in the Delta-Epsilon Limit definition
So the definition of Limit I see is
$$\lim_{x \to a}f(x) = L$$ means:
for all $\epsilon >0$, there exists a $\delta >0$ such that $$0<|x - a| < \delta \Rightarrow |f(x) - L| < \epsilon $$
I was wondering if modifying the definition to: Limit exists when for ... |
H: When does $[a][b]=[ab]$ hold, where [] is an equivalence class?
Let $A$ and $B$ be subsets of the integers. Define $A \boxtimes B $ = $\{ ab : a \in A,\ b \in B\}$. I want to know what properties an equivalence relation ~ must have such that $[a]\boxtimes[b] = [ab]$. I have already proved that this works for a con... |
H: Are all vector fields gradients of functions?
Just what the title says. I wanted to know if all vector fields are gradients of functions?
AI: If the function is one dimensional then yes. Take partial derivative of each variable to get the gradient.
If the function has higher dimensions with more than 1 variable the... |
H: Seeming contradiction in switching order of limits problem
I had to solve the following problem:
Find $0\neq f \in \mathcal{C}[-1,2]$ such that $\int_{-1}^2x^{2n}f(x) \,dx = 0$ for all $0\leq n \in \mathbb{Z}$.
Now, I have indeed managed to find such a function, and verified my answer with the solutions. But then... |
H: What is E[X-Y]?
I was given X~N(1,3) and Y~N(5,7) and for E[X+Y]= E[X]+E[Y], I just added the 1 and 5 and got the answer correct, but when I subtracted it for E[X-Y] I didn't get the correct answer.
I was lost thinking if E[X-Y] = E[X]-E[Y] does not exist. Can I get help on what E[X-Y] means?
AI: Expectation is a l... |
H: Connection between words and sets of solutions to integral equations
Consider any nonnegative integral solution $x_1, x_2,\ldots, x_n$ of the equation $x_1 + x_2 + \ldots + x_n = m.$ For each $i = 1,2,\ldots,n$, let $y_i = x_1 + x_2 + \ldots + x_i$. Then $0 \le y_1 \le y_2 \le \ldots \le y_n = m$. Conversely, sup... |
H: Hilbert space of multivariate random variable
From this lecture notes (Page 7, Example 4), the space of univariate random variables with finite variance is a Hilbert space, when the inner product is chosen as $\langle X, Y \rangle = \mathrm{Cov} (X, Y)$. I'm asking for answers or references for the following two qu... |
H: Question about ring isomorphism out of a quotient ring
Suppose $f:R\rightarrow S$ is a map of commutative rings with $1$. Further suppose that $I$ is an ideal of $R$ contained in the kernel of $f$. This induces a map on the quotient $f':R/I\rightarrow S$. Suppose further that $f'$ is an isomorphism.
Is it true then... |
H: How to calculate series sum $\sum_{n=1}^{\infty}\frac{\sin(kn)}{n}=\frac{\pi-k}{2}$
According to Wikipedia: $$\sum_{n=1}^{\infty}\frac{\sin(kn)}{n}=\frac{\pi-k}{2}, 0<k<2\pi$$
How do I prove this? What if $k \ge 2\pi$?
AI: Take the function $\;f(x):=\cfrac{\pi-x}2\;$ on $\;[0,\pi]\;$ and extend it to an odd periodi... |
H: Showing that the field of fractions of $\mathbb{Z}[\sqrt{d}]$ is $\mathbb{Q}[\sqrt{d}].$
Let $d\in\mathbb{Z}$ be an integer that it not a square. Let $\sqrt{d}\in \mathbb{C}$ be a square root of $d$. Let $\mathbb{Z}[\sqrt{d}]:=\left\{a+b\sqrt{d}:a,b\in\mathbb{Z}\right\}$. Let $F$ be its field of fractions. Show t... |
H: When Should I Use Symbols in a Proof?
The question I'm about to ask might sound weird, I hope i can deliver the idea.
I have noticed that in some mathematics books (especially English ones) the proofs are written in words and symbols are used only when necessary, example: for all, implies, there exists, if and only... |
H: Difference between "measure" and "metric"
I was reading about mathematical structure and came across the distinction of metric and measure as follows:
A measure: intervals along the real line have a specific length, which can be extended to the Lebesgue measure on many of its subsets.
A metric: there is a notion of... |
H: Expected value of dice game
You throw a die. Each time you get a 4,5, or 6, you get the value on the face of the die. When you get a 1, 2, or 3, you quit the game (but keep any winnings). What's the expected value of this game?
Here's how I'm approaching this. I tried to first write an equation for the EV, as so:
$... |
H: Finite index subgroups of $SL(2,\mathbb Z)$
A complete classification of genus $0$ congruence subgroups of $SL(2,\mathbb Z)$ has been carried out by A. Sebbar [1]. They fall into 33 conjugacy classes with index divisible by $6$. I was wondering if dropping the requirement of a congruence subgroup has also been stud... |
H: How may I teach myself pure mathematics "from scratch"?
I'm in my 19's and I keen on becoming a mathematician yet currently I can't fund going to university. I am to work as a customer service representative in order to linger. I harness mornings to study maths.
How may I teach myself pure mathematics "from scratch... |
H: How do I know if I should divide or subtract cases to remove over counting in combinatorics?
Sometimes I do questions where I divide my answer by a factor to remove excess counting and sometimes do subtraction. Now, I wonder, would both methods be equivalent? Because I definitely think we can just pluck out unfavou... |
H: Cauchy problem and global solution
Let $y(x)$ the unique solution of the Cauchy problem
\begin{cases}
y'(x)=\sqrt{\ln(1+y(x)^2)}
\\
y(0)=y_0
\end{cases}
i) Show that, for $y_0>0$, $y(x)$ is strictly increasing and convex.
ii) Show that $y(x)$ is globally defined.
For i), I note that $$\ln(1+y(x)^2) = 0 \text{ iff... |
H: $\rho(f,g)=\int_E \min(1,|f-g|)dm$. Prove that $f_n$ converges to $f$ in measure if and only if $\rho(f_n,f)\rightarrow 0$ as $n\rightarrow\infty$
Question: Suppose $m$ is a finitemeasure on a measurable space $E$. Define $\rho(f,g)=\int_E \min(1,|f-g|)dm$. Prove that $f_n$ converges to $f$ in measure if and onl... |
H: Trace Inequality for difference of positive definite matrices
Prove that for positive definite matrices $A$ and $B$ where $A - B$ is also positive definite, show
$$2Tr((A-B)^{1/2}) + Tr(A^{-1/2}B) \leq 2Tr(A^{1/2})$$
My attempt so far:
We know that $A - B$ positive definite $\implies Tr(A - B) \geq 0 \implies Tr((... |
H: Is a set of full measure also dense?
So I found this post about subsets of measure $1$ in $[0,1]$ being necessarily dense. I was wondering if this holds true for more general measure spaces, i.e. are sets of full measure necessarily dense?
I asked a friend this question and we think we have a counterexample, althou... |
H: How to compute the solutions of $d\alpha = \omega$ for a given exact form $\omega$?
Let $\omega \in \Omega^k(\mathbb R^n)$ be an exact form. How can I compute all the forms $\alpha \in \Omega^{k-1}(\mathbb R^n)$ such that $\omega = d\alpha$? I am mostly interested in the case $k = n$, but an answer for general $k$ ... |
H: Fractional part of a real number: questions
I was reading this question Evaluate the following integral $ \int_1^{\infty} \frac{\lbrace x\rbrace-\frac{1}2}{x} dx$ and I have seen that the user have used the mantissa of a real number or the fractional part.
I know that the mantissa of $x\in\Bbb R$ is defined by $\te... |
H: Show that $M:= \{x\in \mathbb{R}^n~|~ \Vert x-a\Vert \leq \delta\}$ is closed
Let's consider the following problem:
We want to show that $M:= \{x\in \mathbb{R}^n~|~ \Vert x-a\Vert \leq \delta\}\neq\emptyset$ is closed. According to our definition:
A set $M$ is closed iff $M$ contains all its limit points.
Let be ... |
H: What is the pdf for a jointly uniform distribution inside a triangle?
I have a triangle bounded by $0 \leq x, y \leq 1$ and $x + y \geq 1.5$. I'm told that points are uniformly distributed within this triangle.
I am wondering how I can find the pdf? Is it simply solving for the following?
$$
\int_{0.5}^1 \int_{0.5}... |
H: Existence of two sequences
Does there exists two nonnegative sequences
$\{a_{n, m} \} $ and $\{r_m\} $
such that
(1). $a_{n, m}\in (\frac{1}{N_o},1)$ for some integer $N_o$ for all $n, m$
(2). $\sum_{n=1}^{\infty}a^2_{n, m}=r_m$
and
$\sum_{m=1}^{\infty}r_{ m} $ converges.
If yes, can we find a family of such sequen... |
H: Can basis vectors contain more components than the dimension of the spanned vector space?
For example, can I say $\left(1,0,1\right)$ and $\left(0,1,1\right)$ form the basis for a plane in $\mathbb{R}^{3}$?
AI: The answer to your question is yes. The dimension of the subspace (the plane) is $2$, but since it is a ... |
H: Asking for a hint: Convex sets and inner product spaces
I'm trying to solve the following question:
Suppose $V$ is an inner product space and $B$ is the open unit ball in $V$ (thus
$B = \{ f \in V : \Vert f \Vert < 1 \}$ ). Prove that if $U$ is a subset of $V$ such that
$B \subset U \subset \bar{B}$, then $U$ is co... |
H: Show that $\int_0^1\frac{1}{|f(x)-x_0|}dx$ is unbounded
Let $f$ be a Lebesgue measurable functinon on such that $f:[0,1]\rightarrow [0,1]$. Prove that for any $M$ there exists $x_0\in[0,1]$ such that
$$
\int_0^1\frac{1}{|f(x)-x_0|}dx\geq M.
$$
My attempt:
There exists a sequence of step functions $f_n$ such that $f... |
H: Why is $\det \Phi =\det \Psi =1 $.
I found this in Partial Differential Equations, Evans, page 627.
I would to like understand why Evans claims that $\det \Phi =\det \Psi=1$. Here, $y_i=x_i:= \Phi^{i}$, $i=1,\ldots,n-1$ and $y_n=x_n -\gamma(x_1, \ldots,x_{n-1})$ ($y=\Phi(x)$).
The inverse transform is similarly def... |
H: Generated $\sigma$-algebras identity (unions)
I'm trying to prove
$$\sigma (\mathcal{C}_1 \cup \mathcal{C}_2) = \sigma (\sigma( \mathcal{C}_1 ) \cup \sigma (\mathcal{C}_2))$$
for non-empty $\mathcal{C}_1,\mathcal{C}_2 \in \mathscr{P}(\Omega)$.
So I know that $\mathcal{C}_1 \cup \mathcal{C}_2 \subseteq \sigma( \math... |
H: Prove that $ba^k + ab^k \leq a^{k+1} + b^{k + 1}$ for $a, b \geq 0$, $k \in \mathbb N$
I'm in the middle of a proof by induction that for all $n \in \mathbb N$, $\left(\frac{a + b}{2}\right)^n \leq \frac{a^n + b^n}{2}$ for all nonnegative real numbers $a, b$. I've reached a point in my proof where I want to show th... |
H: Finding Work done by the Force Field $\vec{F}$
I need to find the work required to move a certain block from point A to point B.
$\vec{F}(x,y)=2y^{3/2}\ \textbf{i}+3x\sqrt{y} \ \textbf{j}$, where point A is $(1,1)$ and point B is $(2,9)$. I do know how to solve this problem, but I thought of another way to solve it... |
H: Let $D$ be a dense subset of a banach space $X$. Show that any $x$ can be written as a sum of elements of $D$ with a certain conditon.
Let $D$ be an everywhere dense subset of a Banach space $B$ with norm $\|\cdot\|$. Show that any $x\in B$ can be written as the sum of series $$x=\sum_{k=1}^\infty x_k\,,$$ where $... |
H: propositions of Interior and closure
Let $(X,{\tau})$ a topological space, i have to show:
$A{\bigcup}B={X} \Rightarrow \text{cl } A{\bigcup}\text{int } B={X}$
$A{\bigcap}B=\varnothing \Rightarrow \text{cl }A{\bigcap}\text{int }B={\varnothing}$
Well, my proof of 2) is this:
$x\in (\text{cl }A{\bigcap}\text{int }B... |
H: 2010 USAMO #5:Prove that if $\frac{1}{p}-2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m - n$ is divisible by $p$.
Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let $S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}
$
Prove that if $\frac{1}{p}-2S_q = \f... |
H: $K$-topology of the real line and quotient topology.
Let $K$ be the set $\{1,1/2,1/3,\cdots,1/n,\cdots\}$, which is the set of reciprocal of all positive integers. The $K$-topology on $\mathbb R$ is defined as generated by the usual open intervals $(a,b)$ and also $(a,b)-K$. For the text below allow me to write $\m... |
H: Let $A$ and $B$ be $n\times n$ matrices. Let $\operatorname{rank}(A)=s$ and $\operatorname{rank}(B)=t$. Then rank of $A+B\ge\cdots$
$\newcommand{\rank}{\operatorname{rank}}$Suppose $A$ and $B$ are $n\times n$ matrices. Let $\rank(A)=s$ and $\rank(B)=t$. Then rank of $A+B$ is at least ..............
My attempt
$\ra... |
H: Arnold on Homogeneous Linear Equations
In this section, Arnold formulates the equation that we know as the homogeneous linear equation. My question is why the term $\partial a/\partial X$ vanishes when linearisation is done?
Completing the steps for linearisation around the point $(p, 0)$, we get
$$
\begin{align}
... |
H: Isomorphism in UFD
Is this statement true:
Let $R$ be a UFD and $p,q\in R$ are irreducible which are not associated. Prove that for all $n,m\in\mathbb N$:
$$R/(p^mq^n)\cong R/(p^m)\times R/(q^n).$$
I don't think that the chinese remainder theorem works here, however, it is easy to construct a map $f$ from $R$ to ... |
H: Finding the positive integers that can be written in the form $x^2+xy+5y^2$
I'm working on a homework problem that asks which positive integers can be written in the form $x^2+xy+5y^2$. An example was given on how to find all positive integers that can be written as the difference of two squares (i.e. $x^2-y^2$), b... |
H: Probability that a King of hearts will be chosen, given there is a king
I am currently stuck on this question and how it would be worked out.
I currently have the probaility that at least one king will be 0.341
And that the probability of drawing a King of hearts is 51 c 1 / 52 c 5 [editted! I typed it wrong 51 c ... |
H: The solutions of a linear differential equation!
I can't see how if a differential equation is linear then if it attains a complex solution, then the complex conjugate of the solution is a solution also?
Thanks is advance.
Modification: the linear differential equation has to be real too (from the comments).
AI: As... |
H: Book recommendations for Euclidean/Non-Euclidean Geometry
Request for Book Recommendations:
Background introduction
Disclaimers: If the tone below is a little arrogant I apologize beforehand, but I'm being very specific here because I want to make
sure that I will be learning and accessing the right material, and
... |
H: If $(x^2 - x - 1)^n = a_{2n}x^{2n} + a_{2n - 1}x^{2n - 1} + \cdots + a_2x^2 + a_1x + a_0$, then find the value of $a_0 + a_2 + a_4 + \cdots + a_{2n}$
So here is the Problem :-
If $(x^2 - x - 1)^n = a_{2n}x^{2n} + a_{2n - 1}x^{2n - 1} + \cdots + a_2x^2 + a_1x + a_0$, then find the value of $a_0 + a_2 + a_4 + \cdots... |
H: What is the value of $\int_{0}^{2\pi}f(e^{it}) \cos t \, dt$ if $f$ is analytic?
If $f(z)$ is an analytic function, then find the value of the integration
$$\int_{0}^{2\pi}f(e^{it}) \cos t \, dt.$$
My Work:
Taking $e^{it} = z$ the integrand becomes of the form
$$i\frac{f(z)\operatorname{Re(z)}}{z}$$
on the simple... |
H: Continuity of a function $f: X\to \mathbb Z$
If you have a set $X$ which is equipped with a topology: what is required for the function $f: X\to \mathbb Z$ to be continuous?
The fact that $X$ has a topology should certainly be helpful. But $\mathbb Z$ is a discrete space, so I am not totally sure how to establish c... |
H: When is a closed ball inside another closed ball?
I was solving a question and I came upon a statement which I can't really prove. I know that this is indeed true when our metric space is $\mathbb{R}^2$ with the euclidean metric.
Let $(X,d),$ be a metric space. Take $\overline{B}_r(x)$ be the closed ball around a ... |
H: Does there exist a closed set that is not semi-open?
A subset $A$ of a topological space $(X,T)$ is said to be semi-open if there exist an open set $B \in ( X,T)$ such that $B \subseteq A \subseteq \overline B$.
Now my question is that
Give an example of a closed set that is not necessarily a semi-open set.
My at... |
H: Solving a system of coupled recurrence relations
I am required to solve the below system of recurrence relations:$$\begin{cases}a_n-a_{n-4}=t_n-t_{n-3}\\a_n-a_{n-1}=25t_{n-1}-t_{n-3}\end{cases}$$As you can see, I can't isolate $a_i$ or $t_i$. I would like to know if a non-constant closed-form solution exists and if... |
H: Solving definite integrals with periodic integrand
So, the question is:
If $f(2-x) = f(2+x)$ and $f(4-x) = f(4+x)$ where $f(x)$ is a function for which
$$\int_{0}^{2}f(x)dx =5$$ ,
then prove that :
$$\int_{0}^{50}f(x)dx =125$$
$$\int_{-4}^{46}f(x)dx =125$$
$$\int_{2}^{52}f(x)dx =125$$
also comment whether $\int_{1... |
H: Are $\operatorname{Spec}\overline{\mathbb{Q}}[x]$ and $\operatorname{Spec}\mathbb{Z}$ homeomorphic?
Let $\overline{\mathbb{Q}}$ be the algebraic closure of the field $\mathbb{Q}$.
I know that $\operatorname{Spec}\mathbb{Z}=\{0\}\cup\{(p): p\text{ is prime}\}$ and the closed points of $\operatorname{Spec}\mathbb{Z}... |
H: Binomial exponent to be a Martingale process
Let $0 < p < 1$. Let $\{X_n\}_{n\in\mathbb N}$ be a sequence of independent random variables such that
$$P (X_n = 1) = p, P (X_n = −1) = 1 − p$$
for all $n \in \mathbb N$. For each $n\in\mathbb N$, set
$$S_n := X_1 + X_2 + · · · + X_n$$
for the filtration $F_n := \sigma(... |
H: evaluate the limit of $\frac {x}{|x|^s}$ as $s<1$ and $x$ goes to $0$
Evaluate $lim_{x\to 0} \frac {x}{|x|^s}$ for $s<1$ and $x\in \mathbb R^n$. For $n = 1$, I think this limit is equal to $0$. However I am trying to evaluate it when $x$ is an vector in arbitrary dimension. Does this limit exist?
AI: Hint: $\displa... |
H: determining the quotient group in Mayer-Vietoris sequence
I am having trouble to determine the quotient group in the following Mayer-Vietoris sequence.
I know this problem in Hatcher exists here but my question is not to have a solution (because I do have one). I want to understand how we come up with it.
X is the... |
H: Is $f(x)=\left.\begin{cases}x\,\text{sgn}(\sin\frac{1}{x})&\text{if $x\neq0$}\\0&\text{if $x=0$}\end{cases}\right\}$ Riemann integrable?
For $x\in[-1,1]$, let
$$
f(x)=
\begin{cases}
x\,\operatorname{sgn}(\sin\frac{1}{x}), &\text{if $x\neq0$} \\
0, &\text{if $x=0$}
\end{cases}
$$
where $\text{sgn}$ denotes the s... |
H: counterexample completeness of metric spaces
I'm looking for a counterexample. I already proved that if $(X,d_X)$ and $(Y,d_Y)$ are metric spaces and $X$ is complete and $f:X\to Y$ is a bijection that $Y$ is complete if $f^{-1}$ is uniform continuous. I know that if $f^{-1}$ isn't uniform continuous $Y$ isn't comp... |
H: Problem with sequences and cofinite topology
Let $X$ be an infinite Topological Space with Cofinite Topology.
Let $\lbrace x_{n}\rbrace\subseteq X$ be a sequence and let $a$, $b\in X$ be two points such that $x_{n}=a$ and $x_{n}=b$ for endless values of $n$.
Prove that $\lbrace x_{n}\rbrace$ does not converge.
Hone... |
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