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H: Showing the bijection between countable dense subset of a metric space $E×\mathbb{Q}→$neighborhoods of $E$ with rational radius
Let $X$ be a separable metric space.
Let $E$ be a countable dense subset of $X$.
Let $p_i$ enumerate $E$ and $q_j$ enumerate $\mathbb{Q}$.
Let $G=\{N(p_i,q_j) \subset X | i,j\in \omega\}$
... |
H: What is an example of a vector field that is not left-invariant?
Let $G$ be a Lie group, $L_g$ the left-translation on this group with differential $d L_g$. A vector field $X$ on $G$ is called left-invariant if
$$ X \circ L_g = d L_g \circ X \quad \forall g \in G$$
i.e.
$$ X_{gh} = (d L_g)_h (X_h) \quad \forall g,h... |
H: Find the minimum value of $P=x^2+y^2+\frac{x^2y^2}{(4xy-x-y)^2}$
Given that $\frac{1}{3}<x \le \frac{1}{2}$ and $y\ge1$
Find the minimum value of $P=x^2+y^2+\frac{x^2y^2}{(4xy-x-y)^2}$
AI: As it seems to be a homework question and I don't have Latex-MathType in my job's computer (I'll edit my answer late to make i... |
H: Relation between areas of two quadrilaterals
If it is given that:
in a quadrilateral $ABCD$, $X$ is the mid point of the diagonal $BD$, then prove that
area of $AXCB$ = $\frac12$ area of $ABCD$.
I do not know where to start, but I think we can take two cases, something like:
$1$. When $AXCB$ is quadrilateral.
$2$... |
H: Vector and matrix norm definitions?
I've questions on these four norms whose definitions I'm memorizing like this:
Vector euclidean norm: $(x_1^2+x_2^2+\cdots+x_n^2)^{1/2}$
Vector max norm: $\max\{|x_1|, |x_2|, \ldots, |x_n|\}$
Matrix norm: $\max\limits_{x \neq 0} \frac{\|Ax\|}{\|x\|}$
Matrix max norm: $\max\limit... |
H: Existence of non-trivial solution of Sylvester equation.
I'm trying to solve a special case of Sylvester equation
in my case it looks like
$$A*X=X*B$$
so it can be written in form
$$A*X+X*(-B)=C$$ where C consist of all 0 items.
I tried to solve it in Mathematica with LyapunovSolve
but it give me all =0 trivia... |
H: Showing that $\{0w:w \in A\} \cup \{1w: w \notin A \}$ computes $A$
I'm trying to construct a reduction from $A \in RE \setminus R$ under $\sum=\{0,1\}$ to $B$ which defined: $B=\{0w:w \in A\} \cup \{1w: w \notin A \}$.
I need to show that $B\notin RE \cup co-RE$ with the the defining of $A$ and the use of the theo... |
H: Finding $\lim_{E \to U} \frac{1}{4}\left[\frac {U^2}{E(E-U)}\right]\sin^2 k'L$ where $k' = \left[2m(E-U)/\hbar\right]^{1/2}$ and $L$ is a constant
I am trying to find the limit of this equation:
$$\lim_{E \to U} \frac{1}{4}\left[\frac {U^2}{E(E-U)}\right]\sin^2 k'L$$
Isn't there no limit, because $(E-U)$ would be 0... |
H: Does $\int_{0}^{\infty} \cos (x^2) dx$ diverge absolutely?
I believe it does, but i would like some help formulating a proof.
AI: It's equivalent to the convergence of $\int_\pi^{\infty}\frac{|\cos t|}{\sqrt t}dt$, after having used the substitution $x^2=t$.
We have
$$
\int_{\pi}^{N\pi}\frac{|\cos t|}{\sqrt t}dt... |
H: Correct order of books for a beginner
what should be the order of the books in which a beginner should do the following books in algebra:
-1.E.J. Barbeau POLYNOMIALS
-2. Polynomials and Polynomial Inequalities (Graduate Texts in Mathematics) - (Springer) - Peter Borwein -Tamas Erdely.
3.Geometry of Polynomials - (... |
H: How to `bound' $L^\infty$ by the constant function $1$
Let $(S,\mathcal A, \mu)$ be a measure space and consider the Riesz space $L^\infty=L^\infty(S,\mathcal A, \mu)$ (under point-wise ordering). Let $1_X$ denote the indicator function on $S$ (which is contained $L^\infty$).
Given an arbitrary $f\in L^\infty$, is ... |
H: Evaluating the line integral of $F=\frac{-y}{x^2+y^2}i+\frac{x}{x^2+y^2}j$ along $0 \le t \le 2\pi $
Recently, I had an exam and in that I was asked to evaluate the line integral of the function $$F=\frac{-y}{x^2+y^2}i+\frac{x}{x^2+y^2}j$$ alongside the unit circle, $0 \le t \le 2\pi $ . Moreover, it was asked if t... |
H: preorders induced by continuous functions to the reals
Any function to a (total) order induces a (total) preorder on its domain. What can be said about the total preorders induced by a continuous function from a "nice" topological space (for instance, a Euclidean space) to the reals?
AI: It is possible to characte... |
H: A mathematical notation for the mean value theorem?
I'm looking for a stringent definition of the mean value theorem i.e. stated with mathematical symbols. I think it should be something like "there exists..." and then the mean value if there is an integral between the surrounding values which to my knowledge is th... |
H: Finite $G$ with involutory automorphism $\alpha$ with no nontrivial fixed points. Proving properties of $\alpha$.
The question at hand is:
Let G be a finite group and $\alpha$ an involutory automorphism of G, which doesn't fixate any element aside from the trivial one.
1) Prove that $ g \mapsto g^{-1}g^{\alpha}... |
H: What are some examples of vector spaces that aren't graded?
From wikipedia: a vector space $V$ is graded if it decomposes into direct sum $ \oplus_{n \geq 0} V_n$ of vector spaces $V_n$.
So as far as I understand things, any vector space with a countable basis is graded: Let $V$ be a vector space over a field $k$ ... |
H: Proof of a Proposition on Partitions and Equivalence Classes
I stumbled upon a seemingly rudimentary proposition that I am having trouble writing out a proof for. The proposition goes something like,
Proposition: If $\{A_i|i\in I\}$ is a partition of $\mathcal A$, then there is an equivalence relation on $\mathc... |
H: Four-parameter Beta distribution and Wikipedia
Sorry if it is not an appropriate place for such questions, but anyway can anybody please confirm that the formula for the density function of the four-parameter Beta distribution is correct in Wikipedia. It seems $(c - a)$ is missing in the denominator. Thank you.
Bes... |
H: Prerequisites for studying smooth manifold theory?
I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to self study. Besides some basic topological definitions, are there... |
H: The expected payoff of a dice game
There's a question in my Olympiad questions book which I can't seem to solve:
You have the option to throw a die up to three times. You will earn
the face value of the die. You have the option to stop after each
throw and walk away with the money earned. The earnings are not ad... |
H: Properties of homomorphisms of the additive group of rationals
Let $f : (\mathbb{Q},+) \longrightarrow (\mathbb{Q},+)$ be a non-zero homomorphism.
Can we conclude that $f$ is bijective (or, if that fails, that $f$ is injective or surjective)?
Context
The additive group of integers has non-surjective nonzero endo... |
H: Is it possible to 'approximate' compact, convex sets in $\ell^2$ by the Hilbert cube
Define $H=\{(x_n)_n\in\ell^2:|x_n|\le \frac1n, n\in\mathbf N\}\subset\ell^2$. This set is known as the Hilbert cube and it is well-known that $H$ is compact, convex and non-empty. Let $\overline{\mathrm{conv}}(C)$ denote the closur... |
H: The print problem: How to show it is not decidable?
I wonder the following reduction is correct.
I'm trying to show that the following problem "PRINT_BLANK" is not decidable.
Input: (a coding of) Turing machine M.
Question: Does the machine never types "blank" on the stripe when it runs on x?
An attempt for reducti... |
H: some elementary questions about cardinality
I know little about set theory and while reading some Algebra proof I had difficulty on some details. So my questions are :
If $X$ is an infinite set and $Y$ is the set of all finite subsets of $X$, they have the same cardinality. How can I prove it ?
If $X$ is infinite,... |
H: Formation sequence for a logic formula
I will start with some definitions from An Introduction to Mathematical Logic and Type Theory: To Truth through Proof by Peter B. Andrews then give the exercise that I am working along with my attempt at a proof. My actual question will be at the bottom, if you wish to skip ah... |
H: General conditions for $f \in K[\zeta_p] \Rightarrow f(\zeta_p) \in K$
suppose there is a polynomial $f$ in $\zeta_p$ (root of unity) with coefficients in $K$, $\text{char}(K)>0$. Are there conditions when $f(\zeta_p) \in K$?
AI: Suppose that $f(\zeta_p) = k \in K$. Then $f(\zeta_p) - k = 0,$ and
so $f(X) -k $ is... |
H: Relation of compositum of fields
Let $E/k$ be a finite field extension, $\operatorname{char}(k)=p>0$. Suppose that $E^p k = E$. Is it then true that $E^{p^n}k = E$ for any positive integer $n$? If yes, why?
Thanks.
AI: Yes, it is true. I will show that $E=E^{p^2}k$ and leave to you the proof of the general case $E=... |
H: Point set topology
Some time back, I tried reading Rudin's Principles of Mathematical Analysis and I found no trouble with the introductory chapter. In chapter II I encountered point set topology. The number of theorems packed into some pages kind of overwhelmed me. I got stuck there as I felt I did not fully under... |
H: Find the area of the region inside the limaçon
I'm struggling to figure out the answer to this:
Find the area of the region inside the limaçon, $r=3 + \sin(\theta)$
Could someone please help me out?
AI: It would help to plot it first.
Now use this formula $$A = \frac{1}{2}\int_{\alpha}^{\beta}f(\theta)^2\ d\theta... |
H: Combination - How many different ways
I am stomped on the following question
How many different ways are there to draw $6$ cards from a standard deck of cards and obtain $4$ kings and $2$ jacks? (The Answer is $6$)
I believe I am starting the question all wrong since I am doing this for
How many different way... |
H: Graph Homology and Rank-Nullity Theorem
Let $G$ be a connected, directed graph with $v$ vertices and $e$ edges. According to Massey (Ch. VIII, Section 3), the euler characteristic satisfies
\begin{align}
v - e = \chi(G) = \text{rank} \, H_0(G) - \text{rank} \, H_1(G) = 1 - \text{rank} \, H_1(G).
\end{align}
Based... |
H: When is the geometric multiplicity of an eigenvalue smaller than its algebraic multiplicity?
I was kinda crushed to discover that two different matrices with different properties can actually share the same characteristic polynomial ($-\lambda^3-3\lambda^2+4$):
$A=\begin{pmatrix}
1 & 2& 2\\
-3 &-5 &-3 \\
3& 3... |
H: Proving inequality $\prod _{i=1}^n\frac {1-a_i} {a_i}\geqslant \left( n-1\right) ^n$
Let $a_1,a_2,\ldots ,a_n\in \left( 0,1\right)$ be real numbers such that $\sum\limits_{i=1}^n a_i=1$. Prove that
$$\prod _{i=1}^n\dfrac {1-a_i} {a_i}\geqslant \left( n-1\right)^n.$$
AI: By AM-GM
$$\dfrac {1-a_i} {a_i} = \frac{a_1+a... |
H: Surjective and injective functions
So let's say that we have some function $f: \mathbb{A} \rightarrow \mathbb{B} $
Is it possible to have some function such that not all elements of A map to some value in B?
Like for example, in these pictures for various surjective and injective functions:
Would it be possible ... |
H: Calculating: $\lim_{n\to \infty}\int_0^\sqrt{n} {(1-\frac{x^2}{n})^n}dx$
Possible Duplicate:
Prove: $\lim\limits_{n \to \infty} \int_{0}^{\sqrt n}(1-\frac{x^2}{n})^ndx=\int_{0}^{\infty} e^{-x^2}dx$
I need some help calculating the above limit.
What i have observed so far is that:
For all x the limit of the seq... |
H: Should this be $\int u^2 \sqrt{u+1} du$ after substitution of $u=e^x$?
Should this be $\int u^2 \sqrt{u+1} du$ after substitution of $u=e^x$?
http://www.wolframalpha.com/input/?i=integrate+%28exp%28x%29%29^2*sqrt%281%2Bexp%28x%29%29 -> show steps ->3rd row
AI: No, when you do the substitution $u=e^x$ then $du=e^x d... |
H: easy "show this is a subset" question
I am having a hard time showing this simple relation. Here, all the letters below are in $\mathbb{N} \cup \{0\}$.
$$A = \{(a,b) : a + 2b \leq n+2\}$$
$$B = \{(a,c+1) : a + 2c \leq n\}$$
How to show that $B \subset A$?
This is something I got from a sum over the above indices. ... |
H: Proving inequality $\left( \frac {x} {y}\right) ^{x}\left( \frac {1-x} {1-y}\right) ^{1-x}\geqslant 1$ for $x, y\in (0, 1)$
Let $x,y\in \left( 0;1\right).$ I want to prove that
$$\left( \dfrac {x} {y}\right) ^{x}\left( \dfrac {1-x} {1-y}\right) ^{1-x}\geqslant 1$$
AI: You need to show $y^x(1 -y)^{1 - x} \leq x^x(1 ... |
H: Need to clarify the "At-least Concept" in Combination.
I managed to solve this question but I had some inquiries regarding the solution.
If two cards are chosen at random from a standard deck of playing cards, how many different ways are there to draw the two cards if at least one card is a jack, queen or a king?... |
H: Is there notation denoting that one sigma-algebra is sub-sigma-algebra of another?
The question is self-describing.
AI: I have never seen such a notation, per se. The closest I've seen is something like:
Let $\mathcal{F}, \mathcal{G}$ be $\sigma$-algebras, with $\mathcal{F} \subset \mathcal{G}$.
That is, using $... |
H: Projectile Motion
Hello Stack Exchangers!
I'm developing a video game. One feature requires an archer be able to target an enemy and shoot an arrow at it. I've looked around and found plenty of guides on how to do this with the quadratic formula with the appropriate variables and am receiving what I believe to be... |
H: What does "the orthogonal basis vectors spanning the subspace perpendicular to vector $\vec{e}_1$" mean?
I am reading a paper titled "A Robust Real-Coded Genetic Algorithm using Unimodal Normal
Distribution Crossover Augmented by Uniform Crossover : Effects
of Self-Adaptation of Crossover Probabilities" by Ono, Kit... |
H: "L'Hôpital's rule" vs. "L'Hospital's rule"?
I know this is not strictly a mathematical question, and I considered putting it on Linguistics SE, but I decided that seeing as this is most probably a mathematical history question, it would be better placed here on math SE.
My question is:
Why is "L'Hôpital's rule" of... |
H: Set notation: subtracting elements with given cardinality from the powerset
I have a set $S = \{1,2,\ldots,n\}$ of $n$ elements and I denote with $P(S)$ the powerset of $S$.
Which is a correct and accepted notation to say that the set $Z$ is composed by all the elements in $P(S)$ with the exception of all the subs... |
H: Local rings and flatness
Let $A \rightarrow B$ be a flat and local homomorphism of commutative local rings.
Let $M,N$ be two $B$-modules which are free of finite rank as $A$-modules.
Consider the product $M \otimes_B N$ as an $A$-module. Is this $A$-module flat?
AI: In general the tensor product may not be torsion-... |
H: A basic group question
Let $G=\{0, \cdot\}$.
I'm arguing with someone over if $G$ is a group with the regular multiplication since I don't see why it isn't.
Addition:
Now, $G=\{\mathbb{Z},\triangle \}$ with $x \triangle y=x+y+xy$. Is it true that $G$ is not a group and the only subset of $\mathbb{Z}$ to form a gro... |
H: Using the pumping lemma to show that a language is not regular (Computer Science)
Show that $L=\{a^{n^2} | n \ge 0\}$ is not regular
Hey guys. I'm taking a CS class and this stuff is really new to me so bear with me.
I tried to look if I get some contradiction by using the pumping lemma for regular languages and ... |
H: Find the domain of $f(x)=\frac{3x+1}{\sqrt{x^2+x-2}}$
Find the domain of $f(x)=\dfrac{3x+1}{\sqrt{x^2+x-2}}$
This is my work so far:
$$\dfrac{3x+1}{\sqrt{x^2+x-2}}\cdot \sqrt{\dfrac{x^2+x-2}{x^2+x-2}}$$
$$\dfrac{(3x+1)(\sqrt{x^2+x-2})}{x^2+x-2}$$
$(3x+1)(\sqrt{x^2+x-2})$ = $\alpha$ (Just because it's too much to ... |
H: Generalization of metric spaces
The Wikipedia article for metrics mentions several generalizations of metric spaces, but all of them seem to have the property that the metric must be non-negative for all x and y. To me it seems like a space where distances don't have to be non-negative would be an obvious generaliz... |
H: Combination - Inverse Way of solving this problem
Regarding my previous post , I'll repeat the question
A five member committee is to be selected from among four Math teachers and five English teachers. In how many different ways can the committee be formed if the committee must contain at least three Math teach... |
H: A qualifying exam question concerning compactness
Here is a qualifying exam question which I hope someone can help me with. I have done all of it except having problem to "visualize" $\partial S$ and hence have no idea, is the intersection of $K \cap \partial S$ empty or not?
Here is the question
Assume $f: \mathbb... |
H: Basic help with sigma algebras and borel sets
In non-rigorous, intuitive terms, can someone briefly define:
(i) a measurable set
(ii) a borel set
(iii) a sigma algebra
(iv) a borel sigma algebra
Im studying these concepts independently in preparation for a course in the fall and want to make sure I have a functiona... |
H: Prime numbers $(x,c,p)$ such that $x^3-p x^2-cx-5c=0$
How should I proceed to find all prime numbers $x,c,p$ such that
$$x^3-px^2-cx-5c=0$$
AI: If $x^3-px^2-cx-5c=0$, then $x$ divides $5c$. Since $x$ and $c$ are prime, we have the two possibilities $x=c$ and $x=5$.
Suppose $x=c$. Substitute. We get $c^3-(p+1)c^2-... |
H: limit of exponential function and applying l'Hospital's rule
The eqaution goes the following:
$$\lim_{x\to \infty} \left(\frac {x}{e^x-1}\right)^2 e^x = \ ? $$
The first question is I tried to use l'Hopistal's rule, but unsure whether this is the right approach, as the limit goes to the infinity. (and this way did ... |
H: Nonexistence of Pythagorean triple with largest side prime.
The Problem: In the Pythagorean triplets (a,b,c) when a < b then b can't be a prime number.
The Background: While searching the properties of Pythagorean triplets in web I saw quite a few listed, but didn't see the above one which I thought was true, beca... |
H: Pick out the correct statements
Pick out the correct statements from the following list:
a. A homomorphic image of a UFD (unique factorization domain) is again a
UFD.
b. The element $2 ∈ \Bbb{Z}[\sqrt{−5}]$ is irreducible in $\Bbb{Z}[\sqrt{−5}].$
c. Units of the ring $\Bbb{Z}[\sqrt{−5}]$ are the units of $\Bbb{Z... |
H: Nilpotent Element And Jacobson Radical
I am looking for a ring with nilpotent elements such that $J(R)=0$ where $J(R)$ is Jacobson radical.
Any suggestion?
AI: The best examples are the matrix rings over a field. These are simple, so they've got trivial Jacobson radical, and yet already the $2\times 2$ matrices hav... |
H: $im(I)=im(R)$ implies what?
I'm studying an ideal $I \trianglelefteq R$ and noticed that for a certain non-injective, non-zero homomorphism $\varphi: R \rightarrow S$ I can show that $\varphi(I)=\varphi(R)$. I'm wondering if this implies that $I=R$.
It holds for the one little example I could think of. Let $R=\math... |
H: Proving an identity involving the derivative of the Laguerre polynomials with respect to $n$
I've recently come across the following equality in a paper: suppose one defines an analytic function $L(n,x)$ which is equal to the $n$th Laguerre polynomial for $n\in\{0,1,\ldots\}$, and let* $L^{(1,0)}(n,x) = \frac{\part... |
H: Let A be an $n\times n$ matrix with complex entries which is not a diagonal matrix. Pick out the cases when A is diagonalizable
Let A be an $n\times n$ matrix with complex entries which is not a diagonal
matrix. Pick out the cases when A is diagonalizable.
(a) A is idempotent.
(b) A is nilpotent.
(c) A is unitary.... |
H: Is this a legitimate way to show that $[-ze^{-z^2/2}]_{-\infty}^{\infty}=0$? (Proving a statement about a function of a normal random variable)
The problem is, let Z be a standard normal variable and $n\geq1$ be an integer. Show that $E[Z^{n+1}]=nE[Z^{n-1}]$. Here's what I've got so far, miraculously:
$E[Z^{n+1}]=\... |
H: Suppose $G$ is $2$-connected. Show that there exists a path from $x$ to $y$ containing $z$.
I'm studying for a graph theory exam and am stumped on one of the practice questions:
Suppose $G$ is $2$-vertex-connected. Show that for any distinct vertices $x$,
$y$, $z$ of $G$ there exists a path from $x$ to $y$ conta... |
H: Compute the series : $\sum_{n=1}^{\infty} \frac{4^n n!}{(2n)!}$
How would you compute the following series? I'm interested in some easy approaches that would allow me to work it out.
$$\sum_{n=1}^{\infty} \frac{4^n n!}{(2n)!}$$
AI: It suffices to calculate the sum with the summation index running from 0, not from 1... |
H: The greatest possible geometric multiplicity of an eigenvalue
Wikipedia claims that
"Given an n×n matrix A.... both algebraic and geometric multiplicity are integers between (including) 1 and n."
But how can the geometric multiplicity possibly be n? Since $(A-\lambda I)$ is a square matrix (as opposed to a matrix w... |
H: Hatcher problem 1.2.3 - technicality in proof of simply connectedness
I am trying to prove that $\Bbb{R}^n$ minus finitely many points $x_1,\ldots,x_m$ is simply connected, where $n \geq 3$. For days now I have tried many different arguments but I have found flaws in all of them. I have finally come up with one, e... |
H: $C(X)$ with the pointwise convergence topology is not metrizable
I need to show that if $X$ is an uncountable Tychonoff space, then $C(X)$ is not metrizable. All I've been able to show so far is that that $F(X)$, the space of all functions with pointwise topology, is homeomorphic to $\mathbb{R}^X$ (the product) whi... |
H: Show that any open subsets of the real line are $F_\sigma$-sets.
This is an exercise from a topological book. It is this:
Show that any open subsets of the real line are $F_\sigma$-sets.
Could anybody help to solve it?
AI: First note that open subsets of the real line are countable unions of (pairwise disjoint) o... |
H: Gre Question Complex Number (plug and chug)
This seems like it should be easy, but I can't seem to simplify it: If $z=e^{i\frac{2\pi}{5}}$, then what is $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9$. The choices are $0, 4e^{i\frac{3\pi}{5}}, 5e^{i\frac{4\pi}{5}}, -4e^{i\frac{-2\pi}{5}}, -5e^{i\frac{3\pi}{5}},$ with t... |
H: Period of $f'(x)$
It is easy to prove that if $f(x)$ is periodic with period $T$,then $f'(x+T)=f'(x)$. However I don't know how to prove $T$ is the shortest period for $f'(x)$.
Can anyone help me? Or if it is not true, can anyone give me a counterexample?
AI: It seems that the hint I have given before was too obsc... |
H: group of order 30
What are the steps in showing a group of order 30 is solvable/non-solvable?
I don't know how to proceed. All I know is that the group either has a group of order $5$ or $3$. I don't need all the steps for this problem, just an outline what to do.
AI: Some ideas:
1) Show that such a group always ha... |
H: Can an eigenvalue (of an $n$ by $n$ matrix A) with algebraic multiplicity $n$ have an eigenspace with fewer than $n$ dimensions?
Is it possible for a matrix with characteristic polynomial $(λ−a)^3$ to have an eigenline (one-dimensional eigenspace)?
I know that geometric multiplicity can generally be smaller than al... |
H: Ring homomorphism with $\phi(1_R) \neq1_S$
Let $R$ and $S$ be rings with unity $1_R$ and $1_S$ respectively. Let $\phi\colon R\to S$ be a ring homomorphism. Give an example of a non-zero $\phi$ such that $\phi(1_R)\neq 1_S$
In trying to find a non-zero $\phi$ I've done the following observation:
Since for $\forall ... |
H: Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number
Firstly, I give the definition of the epsilon number:
$\alpha$ is called an epsilon number iff $\omega^\alpha=\alpha$.
Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number and there are $\kappa$ e... |
H: Veryify that the union of a co-dense set and a nowhere dense set is a co-dense set
Veryify that the union of a co-dense set and a nowhere dense set is a co-dense set. Give an example to show that the union of two co-dense sets if not necessarily a co-dense set.
Note: A co-dense set $A$ in the topological $X$ denot... |
H: Two sums with Fibonacci numbers
Find closed form formula for sum: $\displaystyle\sum_{n=0}^{+\infty}\sum_{k=0}^{n} \frac{F_{2k}F_{n-k}}{10^n}$
Find closed form formula for sum: $\displaystyle\sum_{k=0}^{n}\frac{F_k}{2^k}$ and its limit with $n\to +\infty$.
First association with both problems: generating functi... |
H: Singular or non-singular matrices
Which of the following matrices are non-singular?
$I + A$ where $A$ not equal to $0$ is a skew-symmetric real $n\times n$ matrix, $n\geq 2$.
Every skew-symmetric non-zero real $5 \times 5$ matrix.
Every skew-symmetric non-zero real $2 \times 2$ matrix.
AI: We will use the properti... |
H: Morphisms between cyclic groups
I'm trying to solve a group theory question involving morphisms:
How many different morphisms do there exist from $ C_n $ to $ C_m $?
Am I correct in saying if $f$ is a morphism, then $f(0) = 0$ and $f(a+b) = (a+b)f(1)$?
If yes, where do I go from here? And if no, how do I start... |
H: Why is the following language decidable? $L_{one\ right\ and\ never\ stops}$
I can't understand how the following language can ever be decidable:
$L= \{ \langle M \rangle | M \ is \ a \ TM \ and\ there\ exists\ an\ input\ that\ in\ the\ computation\ $ $of\ M(w)\ the\ head\ only\ moves\ right\ and\ M\ never\ stops... |
H: Best way to denote some trigonometric functions ("tg" vs "tan", "ctg" vs "cot")
What is the best way to denote tangent and other trigonometric functions: tg or tan, ctg or cot. What notation is commonly used and standardized?
AI: In current US textbooks, $\tan$ and $\cot$ are commonly used and standardized. Also: ... |
H: Duality, Symmetry, Dual Spaces
The following is from the book "Tensor Methods in Statistics", which could be downloaded there
http://www.stat.uchicago.edu/~pmcc/tensorbook/
I have a question regarding section 0.3.1, titled "Duality and dual spaces", there it is said
Let $V$ be a vector space with basis $\{ e_1, \... |
H: What is a PL mapping?
In a proof of the Borsuk-Ulam theorem I've encuntered the notion of a PL mapping between two n-spheres. Does anyone know what it means?
AI: PL stands for "piecewise linear". Roughly, it means a map which can be decomposed as a piecewise linear map with respect to suitable triangulations of so... |
H: Is this a correct way to write the summation?
I have values such as 2,4, 7, 10 , that is not sequential but are stored in array w. Can I use the summation
$$ {\sum_{i = 1}^{n} w[i]} $$
or there is another way to write it down ?
AI: in programming if we have array $w={1,2,3,4....}$,then we write $w[i]$,in mathemati... |
H: How to invert this symmetric tridiagonal Toeplitz matrix?
What's the best way to invert a simple symmetric tridiagonal Toeplitz matrix of the following form?
$$
A = \begin{bmatrix} 1 & a & 0 & \ldots & \ldots & 0 \\\
a & 1 & a & \ddots & & \vdots \\\
0 & a & 1 & \ddots & \ddots& \vdots \\\
\vdots & \ddots & \ddots... |
H: conjecture regarding the height of polynomial's square-free part
About some time I am struggling with the following interesting problem:
There is a well-known theorem of Mignotte which says that for a polynomial $f\in\mathbb{Z}[x]$ of degree $n$ and height (coefficient size) $2^\tau$, the height of its divisors is ... |
H: Global dimension of free algebra.
Is there any easy way to see the global dimension of a free algebra
$$
A=k\langle x_{1},\dots,x_{n} \rangle
$$
is 1?
AI: In "Modules over coproducts of Rings", Bergman proved that the global dimension of a free product is the supremum over the global dimensions of the factors (un... |
H: Goldbach's conjecture and number of ways in which an even number can be expressed as a sum of two primes
Is there a functon that counts the number of ways in which an even number can be expressed as a sum of two primes?
AI: See Goldbach's comet at Wikipedia.
EDIT: To expand on this a little, let $g(n)$ be the numb... |
H: How to show that $\mathbb R^n$ with the $1$-norm is not isometric to $\mathbb R^n$ with the infinity norm for $n>2$?
Could you please give me a hint to prove that
$\mathbb{R}^n$ with the 1-norm $\lvert x\rvert_1=\lvert x_1\rvert+\cdots+\lvert x_n\rvert$ is not isometric to $\mathbb{R}^n$ with the infinity-norm $\lv... |
H: Help me find equation of this graph
I need to find equations of this list: $[1,1,1,0,0,0,1,1,1,0,0,0, ...]$ (it's periodic)
The closest equation I've got is $\left\lceil \sin (\frac{\pi}{3}x)\right\rceil $, which looks like this:
_ _ _
_| |_| |_| |_
But I need it to look like this:
_ _ _
_/ \_/ \_/ \_
... |
H: Puzzle: Guessing the bigger number!
Consider the following interesting puzzle:
"Alice writes two distinct real numbers between 0 and 1 on two sheets of paper. Bob selects one of the sheets randomly to inspect it. He then has to declare whether the number he sees is the bigger or smaller of the two. Is there a... |
H: Drawing a monkey saddle surface in matlab?
I'd like to draw a monkey saddle surface using matlab. But how do I plot a function of several variables in matlab? I never did that before. I can define $x$ and $y$ as two vectors and then according to wikipedia the monkey saddle equation is $x^3-3xy^2$ so all I wanna do ... |
H: Is there a difference between a model and a representation?
I'm thinking of models in logic here, vs. e.g. group representations.
Is there a difference between a model and a representation?
Could one not explain both at the same time?
A model gives an interpretation, but this might be viewed as a side effect to ... |
H: Show that the sequence $\left\{ x^n \right\}$ of functions converges uniformly on $[0,k],k<1$
Show that the sequence of functions $\left\{ x^n \right\}$ converges uniformly on $[0,k],k<1$, but non-uniformly on $[0,1]$.
For $x\in[0,1)$
$$\lim_{n\to \infty}f_n(x)=\lim_{n\to \infty}x^n=0$$
and for $x=1$
$$\lim_{n\to... |
H: What do the $+,-$ mean in limit notation, like$\lim\limits_{t \to 0^+}$ and $\lim\limits_{t \to 0^-}$?
I'm working on Laplace Transforms and have got to a section where they are talking about zero to the power plus or minus and that they are different. I can't remember what this means though.
It's generally used in... |
H: How to integrate $\sec^3 x \, dx$?
Possible Duplicate:
Indefinite integral of secant cubed
How to integrate $\sec^3 x \, dx$?
Can someone please give a method, I tried separating $\sec^3 x$ as $\sec x(\sec^2 x)$ then applying by-parts method but it didn't yield anything useful
AI: $$\sec^3(x)=\frac{1}{\cos^3(x)}... |
H: How to convert any non-negative matrix into a doubly stochastic matrix?
Given a non-negative real matrix $A \in \Bbb R_+^{m \times n}$, how do I convert it to a doubly stochastic matrix (each row and column sums to $1$)
$$\sum_{j=1}^n A_{ij}= 1, \qquad \forall i = 1, \dots, m \tag{row sum}$$
$$\sum_{i=1}^m A_{ij}= ... |
H: Condensation points and derived sets
Let $(X,d)$ be compact metric space and $C$ be the set of condensation points of $X$. Following the notations here, is the equality $\bigcap\limits_{n \geq 1} X^{(n)}=C$ true ?
I succeeded in showing the inclusion $C \subset \bigcap\limits_{n \geq 1} X^{(n)}$ but the other inclu... |
H: a question about a function $g(x) = \sum f(2^nx)$
Let $f(x)$ be non-negative and decreasing for $ x > 0$. Suppose that $\int_0^\infty f(x)dx < \infty$. Let $g(x) = \sum_{n=1}^\infty f(2^nx)$. How do I prove that $\int_0^\infty f(x) dx = \int_0^\infty g(x) dx$?
AI: The conditions imposed on $f$ mean that we can swap... |
H: Answer of $5 - 0 \times 3 + 9 / 3 =$
According to order of operations the answer should be $\mathbf{2}$
But Google and Wolfram calculates as 8
This is last proccess: $5-0+3$
This is how I think: $5-(0+3)$
This is how Google answers: $(5-0)+3$
So, question is which operation is first $+$ or $-$?
AI: You proceed fro... |
H: Same arrow between distinct objects of a category
I'm a bit bothered by something I've come across, and I'd like to know if the misunderstanding is my own (likely) or the author's (unlikely).
Let $\mathcal{C}$ be a category. An arrow $e : A \to A$ in $\mathcal{C}$ is idempotent if $e \circ e=e$. If $\mathcal{E}$ i... |
H: Are there any secure ciphers you can use without a computer?
I have some kids that like encryption schemes such as the Caesar cipher and the Vigenère cipher. I would like to teach them something that's not easily breakable by todays maths and computers, but I want them to be able to use it just using pen and paper.... |
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