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H: 'Shrunken Version' of a convex set is also convex
I'm trying to show that for a convex set $K$ in $\mathbb{R}^n $ (possibly bounded, if that makes things easier), the set $K_{\epsilon}:= \{x\in K: \text{dist}(x,\partial K)>\epsilon\}$ is also convex (I don't really care whether we consider open or closed sets since... |
H: Why is $P(a \text{ and } b)$ maximized when $P(a \text{ or } b)$ is minimized?
I can't seem to wrap my head around why $P(a \text{ and } b)$ is minimized when $P(a \text{ or } b)$ is maximized. This comes from PIE:
$$P(a \text{ or } b) = P(A) + P(B) - P(a \text{ and } b).$$
Can someone please explain the intuition ... |
H: A Map between topological spaces is open iff interior of preimage is a subset of preimage of interior.
Suppose $X$ and $Y$ are topological spaces, and $f: X \to Y$ is any map. Show that $f$ is open iff $\forall B \in Y, int(f^{-1}(B)) \subseteq f^{-1}(int B)$.
AI: Suppose that $f$ is open. Let $B \subseteq Y$ and c... |
H: Probability of a consecutive sequence of 3 balls
You have 6 balls, 3 white, 3 black. What is the probability of a sequence of 3 white balls (the white balls appear next to each other in the sequence) ?
My solution is:
There are $\binom{6}{3}$ ways to arrange the 6 balls on a line, while there are 4 ways you can put... |
H: Could someone suggest a good book for a first course in real analysis for someone who just failed their first course in real analysis?
Okay so basically I've never taken a course that involved any proof writing (other than having done proof by induction in a levels) until my first real analysis course and I did rea... |
H: Stock Technical Analysis: Keltner Channel Calculation
I'm a software engineer and I'm not so good at maths, I am writing some software which performs technical analysis on stocks but it appears my maths is slightly off and I have spent hours and hours trying to figure out which part of my formula is wrong and I can... |
H: Multiplicity of the zero eigenvalue in $A^tA$
If I have a fat matrix $A$ $\in\mathbb{F}^{m\times n}$ (with $m<n$), is it true that $A^tA$ has a zero eigenvalue of multiplicity of $n-m$? I am not sure if it is true but I tried few variation to contradict it without a succes.
AI: This depends on the underlying field ... |
H: Exterior derivative of a vector field
The exterior derivative of a scalar function is
$d f(x,y,z) = (
\frac{\partial f}{\partial x} dx
+ \frac{\partial f}{\partial y} dy
+ \frac{\partial f}{\partial z} dz
)$
Am I correct in assuming then that
$d\left( F_x(x,y,z) e_x + F_y(x,y,z) e_y + F_z(x,y,z) e_z \right)$
woul... |
H: locally free resolution of coherent sheaf on quasi-projective scheme
Sorry for my bad English.
In Hartshorne "Algebraic Geometry " , III. Example 6.5.1
i.e. there is locally free resolution
$\dots \to \mathscr{L}_1\to \mathscr{L}_0\to \mathscr{F}\to 0$.
But I can't understand why $\mathscr{L}_1, \mathscr{L}_2$... ... |
H: Number of words that can be made from the word ALGEBRA
I'm trying to do this combinatorics problems out of KHEE-MENG KOH's "Principles and Techniques of Combinatorics" and am getting overwhelmed.
We want to find the number of distinct 2-letter strings that can be formed from the word ALGEBRA.
My approach:
We have 6... |
H: Find a basis of a subspace
Let $X$ be a set with $X \neq \emptyset$ and $F$ a field. Let $V$ the vector space such $V=\{ f : X \rightarrow F \}$ with the usual operations. Find a basis for the subspace,
$$W= \{ f\in V \mid f(x) =0 \quad \text{for all} \quad x \in X \quad \text{except for a finite number of elements... |
H: Showing the solution of a recurrence relation
I've been working through recurrence relation problems and came across one that I am struggling with
Say we have a relation as follows
$r_k - 7r_{k-1} + 12r_{k-2} = 0$ for all $k \geq 2$ and $r_0 = 1, r_1 = 7$
The problem is essentially asking whether for any $a_n$, doe... |
H: Function field of open set of subvariety
Fulton, in his book Algebraic Curves on classical algebraic geometry, says that if $X$ is an irreducible algebraic set and $V \subset X$ an open subset, then the field of fractions $k(V) = k(X)$; subsequently, he defines the dimension of an abstract variety and says that in ... |
H: If we have a connected graph that remains connected after removing any edge, can we make the following claim
Given we have a connected graph $G$ that remains connected if any edge in $G$ is removed
Say we have $(a, b) \in E(G)$ (where $E(G)$ is the set of all edges in $G$) and $(c, d) \in E(G)$.
Can we assume that ... |
H: Orders in extension of $Q_p$
Suppose we have field extensions $L/K/\mathbb{Q}_p$. If $R$ is a ring such that
$$\mathcal{O}_K \subset R \subset \mathcal{O}_L$$
(where if $M$ is an extension of $\mathbb{Q}_p$ then $\mathcal{O}_M$ is the integral closure of $\mathbb{Z}_p$ in $M$) and $R$ has rank $[L : K]$ as an $\mat... |
H: Volume of the solid bounded by $z = 4-x^2$, $y+z=4$, $y=0$ and $z=0$.
If I am seeing this problem correctly, when $z=0$, $x = \pm 2$, so $-2 \le x \le 2$.
The $y$ coordinate varies from $0$ to $4$, because when $z=0, y=4$ (the plane $y+z=4$ with $z=0$). So $0 \le y \le 4$.
The $z$ coordinate varies from the plane $... |
H: If $R$ is a reduced Noetherian ring, then every prime ideal in the total quotient ring $K(R)$ is maximal.
I know that in $K(R)$, the set of maximal ideals is the set of associated primes of $K(R)$ and that an ideal is maximal if and only if it is the localization of a maximal associated prime of $R$.
So, we know th... |
H: Countability in practice
I am studying the function portion of discrete mathematics and I am wondering to know that how can we practically count the integers?
As they are saying these are countable.
AI: The term "countable" in math just means that we can assign each "object", "label", or "symbol" an integer number ... |
H: Linear Mappings and Ker
Let $\{\vec{v_1},...,\vec{v_k}\}$ be a basis for a subspace $S$ of an $n$-dimensional vector space $V$. Prove that there exists a linear mapping $T:V\rightarrow V$ such that Ker$T=S$.
I reviewed the rank-nullity theorem and I am very lost with this proof.
AI: Two hints:
A basis for a subspa... |
H: Increasing Percentage Question
Question: A population of a colony of bacteria increases by 20% every 3 minutes. If at 9:00am the colony had a population of 144,000, what was the population of the colony at 8:54am?
My solution: If I want to increase something by 20%, then I would multiply it by (.20). So for the pro... |
H: Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $
Prove that $ \sum_{i=1}^{N} a_i \leq \sqrt{N \sum_{i=1}^{N}a_i^2} $. Well i choose $u=(1,\ldots,1)$ and $v=(a_1,\ldots,a_N)$ whit $a_i$ positive and. Apply $u \circ v \leq |u||v|$. With this now i want to prove that $\sum_{i,j}^{N}\frac{\partial u... |
H: limit of a function at infinity from the definition perspective
By definition from Wiki, for $f(x)$ a real function, the limit of $f$ as $x$ approaches infinity is $L$ if for all $\epsilon > 0$, there exists a $M$ such that $|f(x) - L| < \epsilon$ whenever $x > M$.
Now I can prove a function $g(x)$ that for all $\e... |
H: How to prove that the language L={w1#w2#. . .#wk: k ≥ 2, each wi ∈ {0,1}^* , and wi = wj for some i !=j} is not context free using the pumping lemma?
I am having trouble choosing the string to use for the proof. I know that I have to choose a string such that at least two substrings separated by the # are equal to ... |
H: How to integrate :$\int \frac{\sin^4x+\cos^4x}{\sin x \cos x}\:dx$
How to integrate :
$$\int \frac{\sin^4x+\cos^4x}{\sin x \cos x}\:dx$$
$$=\int \:\sin^2x \tan x \: dx+\int \:\cos^2x \cot x \:dx$$
Any suggestion?
AI: Add and subtract in nominator $2 \sin^2x \cos^2x$. Can you continue? |
H: When to use each of the three double-angle identities for cosine?
There are three double angle identities that are all equivalent to each other. The concept of the equations being equivalent sounds fair to me, except I noticed each one has a specific time when to be used precisely with the problem.
How will I know... |
H: Prove that f(m,n)=(m+n−2)(m+n−1)/2+m from $\Bbb Z^+\times \Bbb Z^+$→ $\Bbb Z^+$ is one-to-one.(2)
Since the last post no one gave the solution, so i reopen one and use other approach searched in this forumn.
https://math.stackexchange.com/a/91323/620871
Show that the polynomial function $$f(m,n)=(m+n−2)(m+n−1)/2+m ... |
H: Probability that a book with $200$ pages has $3$ misprints at most using Poisson approximation
A publisher assumes that when printing a typical book, the probability for a misprint on a page is $1.25$%.
I'm trying to find out these two things:
What is the probability that on a book with $200$ pages three misprints... |
H: Why paracompact spaces are required to be Hausdorff
If paracompactness is supposed to be a generalization of compactness. Why is Huasdorfness required in its definition?
It seems like it is more a generalization of compact normal spaces. But the name does not suggest so.
AI: In fact, many texts require compact spac... |
H: Piecewise Function vs Regular Function
I'm into graphing functions and I'm currently working on some project of mine. I'm a little confused, what's the main distinction of a Piecewise Function with just a Regular / $f(x)$ Function? I mean, most Piecewise functions posses the same format of equations with an $f(x)$ ... |
H: A question in proof of Apostol ( Mathematical Analysis) in Theorem 10.27
I am self studying Apostol Mathematical Analysis Chapter->Lebesgue Integration and I was unable to think about an argument used in that proof.
Adding it's image ->
Can someone please tell a rigorous argument which deduces the blue underlined ... |
H: Relationship between induced measure and measure corresponding to a density function.
I am reading these lecture notes, section 2.3 (pg 4), and I've become very confused about relationship between
The induced measure $\mu_X$ -- a random variable $X: \Omega \rightarrow S$ with the original measure $\mu$ induces $\... |
H: An integrable function $f$ on $\Bbb R$ satisfying $\lim_{h\to 0}\int_\Bbb R \frac{|f(x+h)-f(x)|}{h}dx=0$ must be constant
Suppose $f:\Bbb R\to \Bbb C$ is an integrable function, i.e. $\int_\Bbb R|f|~dx<\infty$, satisfying $$ \displaystyle\lim_{h\to 0}\displaystyle\int_\Bbb R \dfrac{|f(x+h)-f(x)|}{h}dx=0.$$
I am try... |
H: If $f(x)=(x^2+6x+9)^{50}-4x+3$ has roots $r_1, r_2, \ldots, r_{100}$, then compute $\sum_i (r_i+3)^{100}$
Let $f(x)=(x^2+6x+9)^{50}-4x+3$, and let $r_1,r_2,\ldots,r_{100}$ be the roots of $f(x)$. Compute $$(r_1+3)^{100}+(r_2+3)^{100}+\cdots+(r_{100}+3)^{100}$$
How should I compute this?
AI: If $r_i$ is root of th... |
H: Which subsets out of S, W and T form subspace of the vector space V?
$V = \mathbb{R}^\mathbb{R}$
$S = \{f : f$ is monotone$\}$
$T = \{f : f(2) = (f(5))^5\}$
$W = \{f : f(2) = f(5)\}$
Note that monotone means either non-decreasing or non-increasing.
My Attempt:
For S:
It's not a subspace of V and it can be show... |
H: Prove the left hand side of the following inequality
$\frac{2ab}{a+b} \le \sqrt{a \cdot b} \le \frac{a+b}{2}$ where $a,b > 0$
The right hand side $\sqrt{a \cdot b} \le \frac{a+b}{2}$ is the AM-GM inequality, it's clear how to solve it. Does the left hand side also a trivial/elementary proof?
I appreciate your answe... |
H: Proof $\mathbb{C}^* \cong \mathbb{C} / \mathbb{Z}$
I am trying to prove the isomorphism between $\mathbb{C}^*$ and $\mathbb{C} / \mathbb{Z}$. I already established the way to do it:
find a surjective homomorphism $f: \mathbb{C} \to \mathbb{C}^*$, such that $Ker(f)=\mathbb{Z}$
take the homomorphism $\phi: \mathbb{C... |
H: A question in Hilbert spaces
Let be $ X=C[-1,1]$ and define $\langle f,g \rangle =
\int^1_{-1} f(t)g(t) dt$.
If $M=\{f \in\ C=[-1,1]\mid f\text{ is odd function}\}$, what is $ M^\perp$?
AI: You can easily show that $$C([-1,1])=O([-1,1])\oplus E([-1,1])$$ where $O([-1,1])$ denotes the continuous odd functions on $[... |
H: Infinite divisibility - two examples on characteristic functions
Are the r.v.s with following characteristic functions infinitely divisible?
$e^{it}e^{-|t|}e^{-t^2}$
$\left(\frac{1}{2}(e^{it}+e^{-it})\right)^n$
The second one is easy because it is just a $\cos(t)^n$ but $\cos(\frac{\pi}{2}) = 0$ and characteristi... |
H: Factoring inequalities using Iverson identity - confused by double summations in Concrete Mathematics book
In chapter 2 section 4 (multiple sums) of Concrete Mathematics(Graham,Knuth,Patashnik) the authors use Iverson Identity to rearrange the variables' bounds.
In particular, they start off with a question like th... |
H: Are All Generating Sets for the Borel Algebra Uncountable?
A related question asks is there a smallest set that generates a given $\sigma$-algebra:
Smallest collection of subsets that generate a sigma algebra
The only existing answer to that (at the time of writing this) says that there must be a smallest cardinali... |
H: System of algebraic equations is solvable iff condition on Gröbner basis
I was reading on Gröbner basis in chapter 9 of Dummit and Foote's Abstract Algebra, and while investigating on my own I stumbled upon an interesting result in Gröbner Bases - Theory and Applications by Franz Winkler which states the following:... |
H: Find $7^{1604} \mod28$
How do I find $7^{1604} \mod28$? 7 and 28 aren't coprime, so I can't use Fermat's little theorem. How do I approach that types of problmes? Do I use Chinese Remainder theorem?
AI: A typical approach to problems with a fixed number to take modulo of is to break the number up to prime powers. I... |
H: how is this a functor?
This is from Page 78 of Rotman's homological algebra book:
Give an (R,S)-bimodule A and left R-module B, then Hom(A, B) is a left S-module, where sf takes a to f(as) and Hom (A, ) is a functor from left R-Mod to left S-Mod.
I've proved everything except the following part:
Suppose f is R-map ... |
H: Order of elliptic curve $y^2 = x^3 + ax^2 + b^2x$ is multiple of $4$.
Let $\mathbb{F}_q$ be a finite field with odd characteristic and let $a,b \in \mathbb{F}_q$ with $a \neq \pm 2b$ and $b \neq 0$. Define the elliptic curve $E: y^2 = x^3 + ax^2 +bx$
The goal is to show that $4 \ | \ E(\mathbb{F}_q)$.
The first ste... |
H: Prove that if $ \lim_{x \to 0}g(x)=0$, then $\lim_{x \to 0} g(x) \cdot \sin(1/x) = 0 $
This is the Problem 21 from Chapter 5 of M. Spivak's "Calculus". It states:
Prove that if $ \lim_{x \to 0}g(x)=0$, then $\lim_{x \to 0} g(x) \cdot \sin(1/x) = 0 $
That's how I approached the problem. First, we know that $\sin(... |
H: proving a set is bounded Metric spaces
I'm trying to show that given the metric space X= $(\mathbb{R}^{2}, d)$ where $d$ is the usual Euclidean metric that the set $D=\{(x,y): \sqrt{x^{2}+y^{2}} <1\}$ is bounded.
I am using Rudin and it gives the following definition of bounded: "$E \subseteq X$ is bounded if there... |
H: Are the following functions positive definite?
Is there a quick way of determining whether the following functions are positive-definite or not?
$f: \mathbb{R}\to\mathbb{C}$
$f(x) = 3$
$f(x) = -3$
$f(x) = x - 3$
$f(x) = x + 3$
So, my attempts:
For $f(x) = x - 3$ if we take $n=1, a_1 = 1$ and $x_1$ arbitrary then ... |
H: $\cos(\alpha-\beta)+\cos(\beta-\gamma)+\cos(\gamma-\alpha)=\frac{-3}{2}$,show that $\cos\alpha+\cos\beta+\cos\gamma=\sin\alpha+\sin\beta+\sin\gamma=0$
I think that I've done a major part of the problem but I'm stuck at a point.
Here's what I've done :
It's given to us that
$$\cos(\alpha-\beta)+\cos(\beta-\gamma)+\... |
H: What is the value of $g'(1)$?
Suppose $f(x,y)$ is real valued function for which $f(1,1)=1$ and its gradient at this point point is given by $\nabla f(1,1)=(-4,5)$.
Define a function $g(t)$ by $g(t)=f(t,f(t^2,t^3))$. Then what is the derivative of $g$ at $t=1$?
I deduce that $g(1)=f(1,1)=1$.
How to find $g'(1)$ ? A... |
H: Convergence in distribution of sum of two random variables, simple case
$X_n, Y_n$ are random variables on the same probability space and it's known that $X_n \rightarrow X$ and $Y_n \rightarrow 0 =: Y$ in distribution. Proof that $X_n + Y_n \rightarrow X$ in distribution.
I have a problem because of the case, when... |
H: Check if a $f \in U$
Let $U$ be a subspace where $U = \{ f \in Abb(\mathbb{R},\mathbb{R}) |\; \forall\; x \in \mathbb{R}: f(x) = -f(-x)\}$ and $f_0: x \mapsto \frac{1}{1+x^2}$.
Check if $b: x \mapsto |x|$ is in $f_0 + U$.
When I write out $f_0 + U = \frac{1}{1+x^2} + f(x) = \frac{1}{1+x^2} -f(-x)$ I can't see a w... |
H: Calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$
I have to calculate the residue of $\exp\left(\frac{z+1}{z-1}\right)$ in every point of $\mathbb{C}$.
So I tried to compute the Laurent Series expansion $\forall z_0 \in \mathbb{C}$.
For $z_0 = 0$ we obtain that $f(z)=\sum_{... |
H: Convergence with probability one of $\sum \frac{1}{n}X_n$ and $\sum \frac{1}{\sqrt n}X_n$ if $X_n$ are i.i.d. $N(0,1)$
Suppose $X_n$ are $N(0,1)$ i.i.d.:
If $Y_n = \frac{1}{n}X_n$ then does $\sum Y_n$ converge with P.1?
If $Y_n = \frac{1}{\sqrt n}X_n$ then does $\sum Y_n$ converge with P.1?
If we have convergenc... |
H: Upper bound "square" of Lebesgue measure of set
Let $\lambda$ be the Lebesgue measure and $A$ a set with $\lambda(A) < \varepsilon$. Consider the set $A^2 = \{a \cdot a \ | \ a \in A\}$. Can we produce an upper bound on $\lambda(A^2)$ in terms of $\varepsilon$?
AI: No. If $A=(n,n+\frac 1 {\sqrt n})$ then $A^{2}=... |
H: Determining $\min\{|z-a|^2+|z-b|^2 \mid z\in \mathbb{C}\}$
I've been doing exercises on understanding the structures of given complex sets, but I'm stuck on this one.
Find $$\min\{|z-a|^2+|z-b|^2 \mid z\in \mathbb{C}\},$$ where $a,b\in \mathbb{C}.$
What's the correct way to tackle this kind of exercises? I've tri... |
H: Is Inclusion–Exclusion Principle still valid if $\cap$ and $\cup$ are exchanged?
The principle for the case of three sets, states:
$$|A\cup B\cup C|=|A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|$$
I wonder whether there exists a similar formula, where $\cap$ and $\cup$ are exchanged; for example:
$$|A\c... |
H: Find a holomorphic function when you know the real part.
I am trying again to learn complex analysys and I have solved an exercise in here but I am not sure if it is correct.
$u(x,y) = x^2-y^2+ln(x^2+y^2)$. I need to find $f$ holomorphic function, such that $Re(f) = u$ This is how I tried to solved it: $f = u+iv$. ... |
H: Prove that any 2 bases of a vector space has the same cardinality
I know this question has been asked before, but I tried to prove it myself and I cant finish my prove because im not sure how to write the contradiction in a foraml and correct way.
Let V be a vector space, and $B_1$, $B_2$ an infinite bases. Assume... |
H: Proving a subset of $H^1(\mathbb{R}^d)$ is compactly embedded in $L^2(\mathbb{R}^d)$.
I was recenty reading about the weighted Lebesgue spaces and came accross an exercise that asks to prove that $H^1(\mathbb{R}^d) \cap L^2(\mathbb{R}^d,|x|^2\,dx)$ is compactly embedded in $L^2(\mathbb{R}^d)$. Where $H^1(\mathbb{... |
H: Corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides trisected?
The corners are cut off from an equilateral triangle to produce a regular hexagon. Are the sides of the triangle trisected?
In the actual question, the first line was exactly the same. It was then asked to find... |
H: Denote that expression is differentiated, without differentiating it
I'm trying to indicate that I'm working with the derivative of an expression, without differentiating it. This is how it would be done, if I differentiate both sides now:
\begin{align}
y &= 5x^2\\
y' &= 10x
\end{align}
However, I want to different... |
H: Showing the Closed unit disk is not open but is closed and perfect - metric spaces
I'm trying to show that the closed unit disk i.e. $D = \{z \in \mathbb{C} : |z| \leq 1\}$ is closed and perfect but it is not open.
I have managed to show that it is closed (I think) but am unsure of my final step.
Showing it is Clos... |
H: if $ i:L^{p}( d\mu )\longrightarrow L^{q}(d\mu) $ is an inclusion map, then $ i $ is bounded.
I want to show that if $ i:L^{p}( d\mu )\longrightarrow L^{q}(d\mu) $ is an inclusion map where $ (X,A, \mu) $ is a measure space and $L^{p}(d\mu)\subseteq L^{q}( d\mu ) $, then $ i $ is bounded.
Firstly, we can use the de... |
H: Proof that $\text{Hom}_R(M, -)$ is left exact in the category of $R$-modules
I'm looking at a proof that $\text{Hom}_R(M, -)$ is left exact for $R$-modules. Specifically at the one that appears in Robert Ash's Abstract Algebra, which you can find here on page 13.
Let $A, B, C$ be $R$-modules for commutative ring $R... |
H: Upper bound Lebesgue measure of "square" of set on closed interval
This is a follow up to this question: Upper bound "square" of Lebesgue measure of set
Let $\lambda$ be the Lebesgue measure, $A$ a set with $\lambda(A) < \varepsilon$ and $M > 0$. Consider the set $A^2 = \{a \cdot a \ | \ a \in A\}$. Can we produce ... |
H: Show if $A$ is open w.r.t to $d_1$ and $d_2$ then $A$ is also open w.r.t $D$.
Let $d_1$, $d_2$ and $D$, given by $D(x,y)=\max\{d_1(x,y),d_2(x,y)\}$, be metrics on $M$. Assume $A\subseteq M$. Show if $A$ is open w.r.t at least one of $d_1$ and $d_2$ then $A$ is also open w.r.t $D$.
CASE 1 Let $x,y \in A$. Assume $A... |
H: Why is $(1 - \frac{\ln n}{n})^n$ approximated by $e^{-\ln n}$ for $n \rightarrow \infty$?
I'm trying to reconstruct the proof for the Erdös-Renyi Theorem from Jackson's "Social and Economic Networks" [1], chapter 4.2.3. This part I don't understand:
Let $p(n)$ be a function s.t. $\lim_{n\to \infty} \frac{p(n)}{\ln ... |
H: Is the successor of a non finite ordinal a non finite ordinal?
Definition:
$$\alpha \text{ is finite iff } \forall\: \beta \: \text{ordinal}, \: \beta \leq \alpha\: \text{and}\: \beta \: \neq \emptyset \: \Rightarrow \exists \gamma( \beta=\gamma\cup\{\gamma\}) $$
(definition corrected thanks to suggestion)
Questio... |
H: example for $\varphi(H \cap K) \subset \varphi(H)\cap \varphi(K)$, $H, K< G$ and $\varphi \in Hom(G,G')$
Let $H$ and $K$ be subgroups of $G$ and $\varphi \in Hom(G,G')$
Give an example of $\varphi(H \cap K) \subseteq \varphi(H)\cap \varphi(K)$ for which the inclusion holds strictly.
I have already proved the rela... |
H: Is the power of set of elements that are less than $x$ little?
Does every set $X$ can be well ordered such that for all element $x$ the power of set of elements that are less than $x$ is less than the power of $X$?
I saw this idea in proof of some problem. Can you show me why it is true?
AI: Yes. Initial ordinals a... |
H: Entire function that is in $\mathcal{L}^1({\mathbb R^2})$
I take the lebesgue measure on $\mathbb R$ .
Take $f:\mathbb R \to \mathbb R$ a continuous function and $\int_{\mathbb R}|f(x)|\mathrm d\mu(x)<\infty$.
Can we say any thing about $\lim_{|x|\to \infty}|f(x)|$ ? or can we say if $f$ is bounded ?
If we can sho... |
H: Prove for $P_n(z)$ if $r<1$ then exist $N$ s.t for every $n>N$ $P_n$ has no roots at $\{z:|z|<1\}$
Prove for $P_n(z)$ if $r<1$ then exist $N$ s.t for every $n>N$ then $P_n$ has no roots at $\{z:|z|<r\}$ where $P_n(z)=1+2z+\ldots +nz^{n-1}$
AI: Hint: Show that $(1-z)P_n(z)=\frac {1-z^{n}} {1-z} -nz^{n}$. Conclude ... |
H: How many trains a train will meet in its way from station $A$ to station $B$?
In each hour, a train starts from station $A$ to station $B$ and another starts from station $B$ to station $A$. Each train has the same speed and takes $5$ hours to reach the other station. How many trains a train will meet in its way fr... |
H: Let $f(x) = x + 2\sin(\ln(\frac{1}{x}))$ for $x \geq 1$. Show that $\lvert f(x) - f(y) \rvert \leq 3|x-y|$ for all $x, y \geq 1$
My attempt:
$$\lvert f(x) - f(y) \rvert = \lvert x-y + 2(\sin(\ln(\frac{1}{x})) - \sin(\ln(\frac{1}{y}))) \rvert$$
Using triangle inequality and $\lvert \sin(x) - \sin(y) \rvert \leq \lve... |
H: How can I understand the definition $(a-ε, a+ε)\thinspace\cap(D\thinspace\backslash{\{a\}})= \emptyset$?
Let D be a subset of $\mathbb {R}$ and $a\in\mathbb {R}$.
The definition for a point of convergence of a set D: $(a-ε, a+ε)\thinspace\cap(D\thinspace\backslash{\{a\}})= \emptyset$
I even drew it for better und... |
H: KL-Divergence of Uniform distributions
Having $P=Unif[0,\theta_1]$ and $Q=Unif[0,\theta_2]$ where $0<\theta_1<\theta_2$
I would like to calculate the KL divergence $KL(P,Q)=?$
I know the uniform pdf: $\frac{1}{b-a}$ and that the distribution is continous, therefore I use the general KL divergence formula:
$$KL(P,Q... |
H: How to prove that this function all over the positive integers gives us this sequence?
On the first hand, I have this sequence : $0,1,1,2,2,2,3,3,3,3,...$ which is the sequence where an $n$ positive integer appears $n+1$ times consecutively.
On the other hand, I have this function : $a_n=\lfloor\frac{\sqrt {1+8n}-1... |
H: Limit of sequence $a_n = (-4)^{\frac{1}{2n+1}}$
Using a calculator, I can see that this sequence $a_n = (-4)^{\frac{1}{2n+1}}$ is convergent and has limit -1. However, I am struggling to prove this in a formal way.
Since the exponent tends to zero when $n$ tends to infinity, I thought that the limit should be 1 (bu... |
H: Calculation with Landau symbol (Big $O$)
I'm not sure about my calculations with the Landau Symbol $O$:
Let $c>0$ and $n\to \infty$. Consider:
$$\frac{1}{c\sqrt{n}+O\left(\frac{\ln(n)}{n}\right)}-\frac{1}{c\sqrt{n}}=
\frac{O\left(\frac{\ln(n)}{n}\right)}{\left (c\sqrt{n}+O\left(\frac{\ln(n)}{n}\right)\right)c\sqrt{... |
H: Prove that G is Hamiltonian
Given $G$ a graph with degrees:$6,6,4,4,4,k,k$ on $7$ vertices and $10$ regions
(and by Euler $n-f+r=2$ I found that $k$=3)
prove $G$ is contains a Hamiltonian cycle
I did find a visual cycle on the actual graph, where in the solutions (by a student) he proved that vertex 3 doesn't have... |
H: How to solve $x^x-x=1$?
I was recently posed the question "solve for $x$ in $x^x-x=1$". The intended answer was $x=0$, assuming that $0^0=1$, but I used brute force and determined another solution, $x\approx1.776775040097$ (which Wolfram Alpha agrees with me on). Is there a closed form or symbolic solution to this ... |
H: Determining an expression for a linear function $f$ such that $f(x_1)=y_1$ and $f(x_2)=y_2$
We say that a function $f:[a,b]\rightarrow\mathbb{R}$ is linear if it is of the form $f(t)=mt+n$ for some $m, n \in\mathbb{R}$. Show that $f$ is determined by its values at two (distinct) points in $[a,b]$. More precisely, ... |
H: Cycle structure in symmetric group
I am studying group representations and to prove that characters from symmetric groups $\chi(g) \in \mathbb{Z}, \forall g \in S_{n}$ I need prove that:
Consider that $\sigma \in S_{n}$, and $\gcd(m, o(\sigma)) = 1$. Then $\sigma$ and $\sigma^{m}$ has the same cycle structure.
So... |
H: Transfinite induction, proving $\operatorname{P}(0)$ although $\alpha = 0$ is out from hypothesis.
In a transfinite induction, if I have to prove that a predicate $\operatorname{P}(\alpha)$ is true $\forall \alpha \gt 0$, can I proceed showing $\operatorname{P}(0), \; \operatorname{P}(\alpha) \rightarrow \operatorn... |
H: Is the orthogonal group convex?
Is the group of real orthogonal matrices convex?
I've read that this space has two connected components, and I don't think that this set is convex, since all convex sets must be path connected. However, I'm just not a specialist in Algebra. Could someone please provide an explanati... |
H: Injection from an open interval into a ball in $\mathbb{R}^{2}$
As an exercise I am trying to show that we can find an injection from some open interval $(0,1)$ say into the open ball $B_{r}(x)$, where $r>0$ and $x \in \mathbb{R}^{2}.$
I'm a bit confused because I've only ever dealt with injective functions where t... |
H: Determine a polynomial function with some information about the function
I am working through some exercises at the end of a textbook chapter on polynomial functions. Till now the questions have been about providing answers based on a given polynomial function. However, with this particular question I am to work ba... |
H: How can I prove $\sum_{n=1}^{\infty}\frac{cos(\frac{nπ}{3})}{n^s}=\frac{1}{2}(6^{1-s}-3^{1-s}-2^{1-s}+1)\zeta(s)$ for $Re(s)>1$
Question:-
Prove that
For $Re(s)>1$
I used this series while evaluating
$\int_{0}^{t} x^2\cot(x)dx$
I got
Evaluating it we get
On letting $ t=\frac{π}{6}$
We have to find $\int_{0} ^{π/... |
H: Why is this theorem about derivatives true? $\frac{dy}{dx}= \frac{1}{dx/dy}$
$$\frac{dy}{dx}= \frac{1}{\;\frac{dx}{dy}\;}$$
Why is the above theorem true as long as $dx/dy$ is not zero? How can you prove it rigorously?
I don’t think it is obvious by the definition of the derivative. I think this says $dx/d(x^2)$ ... |
H: Figure out if the improper Integral exists
please only give a slide hint, not the complete way to the goal, since I want to figure it out myself ;)
So given is the improper integral:
$$\displaystyle\int_{0+0}^{+\infty} x^{\frac{\alpha}{5}}(1+x^2)^{\alpha-3}dx$$
So I want to check for which $\alpha \in \mathbb{R}$ ... |
H: Uniqueness proof: smallest element of an integers set
How do you prove that the smallest element of a nonempty set of positive integers is unique? This is straightforward but how do you show it formally speaking?
AI: I think you can do something like this:
Take a set of positive integers, ${S \subseteq \mathbb{N}}$... |
H: Prove the following hope equality
$X$ is a non negative ramdom variable, prove that $EX = \int_{0}^{\infty} P(X>t)dt$.
Well I don't know how to start to prove this, I have a clue that says to me that is: write $P(X>t)$ as the integral of the index function and use the Fubini's theorem.
AI: As you said, you can use ... |
H: find the hermitian operator of an operator which attaches a matrix and its iverted
$V=M_n\left(\mathbb{C}\right)$ with the standard inner product $<A,B>=tr\left(B^{*}A\right)$ and let $P\in V$ be an invertible matrix. We define an operator $T_p:V->V$ such that $$T_p\left(A\right)=P^{-1}AP$$
I need to find $adj(T_p)... |
H: Show the set $S=\{(x_1, x_2, x_3, x_4, x_5)\in \mathbb{R}^5 \vert x_3^2e^{x_1+x_2^{100}}>2\}$ and another set is open.
Can I please get feedback on proof below? Thank you!!
$\def\R{{\mathbb R}}$
Show the set $$S=\{(x_1, x_2, x_3, x_4, x_5)\in \R^5 \vert x_3^2e^{x_1+x_2^{100}}>2 \text{ and } x_3x_4 - x_5^2<-1\} \sub... |
H: A question of Quantitative aptitude and Logic in School Level
I need help in a question asked by my younger brother in his aptitude exam of Secondary level.
Question's Image ->
I thought answer should be B
Answer is given (A)
I can't reason how it must be independent of both d and n.
Can anyone please explain.
... |
H: I somehow deduced that $\tan x=\iota$ for any real value of $x$ by equating the value of $\tan(\frac{\pi}{2}+x)$ obtained using two identities.
Let's assume that we're familiar with the identity : $\tan \Bigg (\dfrac{\pi}{2} + x \Bigg ) = -\cot x$ which we have derived using the unit circle.
I was trying to equate... |
H: Proving that the statement $f: X \to Y$ is continuous iff $x_n \to x \implies$ $f(x_n) \to f(x)$ maybe false if $X,$ and $Y$ are not metric spaces.
Given that $f: (X, \tau_1) \to (X, \tau_2)$ is a map. Then I want to show that even if $x_n \to x \implies f(x_n) \to f(x),$ f may not be continuous.
I know that f is c... |
H: Interpretation of the word Random
I have previous knowledge of what a random experiment is, but sometimes I get confused by the use of the word Random.
I can express my doubts as the following questions: if something is random them it is aleatory? if something has a defined distribution like in a random experiment ... |
H: Calculating radial derivative from Cartesian derivatives
For a radially symmetric function $f(x, y, z)$, is there a simple method to convert from $\frac{\partial f(x, y, z)}{\partial x}$, $\frac{\partial f(x, y, z)}{\partial y}$, $\frac{\partial f(x, y, z)}{\partial z}$ to $\frac{\partial f(x, y, z)}{\partial r}$?
... |
H: For any pair of sequences $(a_n),(b_n)$ with $a_n
I wrote down a solution, but I'm not sure if it works:
Question: Let $\mu$ be a measure on $\mathscr{B}(\mathbb{R})$ such that $\mu(B)< \infty$ for all bounded $B$. Let $f \geq 0$ be $\mu$-integrable. Show that for any pair of sequences $(a_n),(b_n)$ with $a_n<b_n... |
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