text stringlengths 83 79.5k |
|---|
H: Are there any examples of rings $R$ such that $\mathrm{End}(R,+,0)\not\cong R$?
In a handful of examples, I've noticed that the endomorphism ring $\mathrm{End}(R,+,0)$ is isomorphic to the ring $R$ itself. For instance, $\mathrm{End}(\mathbb{Z},+,0)\cong\mathbb{Z}$ and $\mathrm{End}(\mathbb{Z}/n\mathbb{Z},+,0)\cong... |
H: Rational numbers and cardinality of some subset set of them.
Let $G$ be the set of rational numbers of the form $m/n$ , where $m,n$ are positive integers and $n \leq g $ for some possitive integer $g$. Suppose it is bounded by $1/k$ , k is a positive integer less than g.
Prove or disprove that, cardinality of set $... |
H: Proof of Non-Ordering of Complex Field
Let $\mathcal F$ be a field. Suppose that there is a set $P \subset \mathcal F$ which satisfies the following properties:
For each $x \in \mathcal F$, exactly one of the following statements holds: $x \in P$, $-x \in P$, $x =0$.
For $x,y \in P$, $xy \in P$ and $x+y \in P$.
... |
H: Please help me understand conditional probability in relation to dice
If 2 fair dice are rolled, what is the probability that the sum is 6, given both dice show odd numbers?
When the sum is 6, there combinations are: (1,5),(2,4),(3,3),(4,2),(5,1). Probability = 5/36
Probability of 2 odd numbers is 1/2.
What I'm no... |
H: The Lie algebra of a group of matrices
How does one find the Lie Algebra of a Lie group G which is given by matrices
$$\left(\begin{array}{cccc} \cos \theta & -\sin \theta & x & y \\ \sin \theta & \cos \theta & z & w \\ 0 & 0 & \cos \theta & -\sin \theta \\ 0 & 0 & \sin \theta & \cos \theta \end{array} \right)?$$
F... |
H: Convention on comparing cardinality
When showing that two sets $A$ and $B$ have the same (finite or infinite) cardinality, it is usually done by constructing a function and showing that it is a bijection. However, in some cases, constructing such a bijection is not always easy. This can be worked around by construc... |
H: Find perimeter of trapezium
Find perimeter of a trapezium with 3 sides given and distance b/w parallel sides given.
Let the given sides be a,b,c and distance b/w parallel lines be h. How to go about this problem?
AI: There are four possible variatoins for the given situation:
$\hspace{2cm}$
Thus, depending on which... |
H: What does the notation $[0,1)$ mean?
I am studying the procedure for bucket sort from Introduction To Algorithms by Cormen et al, which assumes that the input is generated by a random process that distributes the elements uniformly and independently over the interval $[0,1).$ What does this mean? Why there is no "]... |
H: What is the link (if there is one) between separability and the separation axioms?
I was wondering about this topic. Is there a connection between the $T_n$ separation axioms and separability itself?
AI: No; there is no real connection between the two notions. There are both separable and non-separable spaces with ... |
H: Count permutations.
Hi I have a compinatorial exercise:
Let $s \in S_n$. Count the permutations such that
$$s(1)=1$$ and
$$|s(i+1)-s(i)|\leq 2 \,\, \mathrm{for} \, \, i\in\{1,2, \ldots , n-1 \}$$
Thank you!
AI: This is OEIS A038718 at the On-Line Encyclopedia of Integer Sequences. The entry gives the generating fun... |
H: Intersection of Two Simplices
How to find vertices a the polytope-intersection of two simplices, if I know the vertices of these simplices.
More precisely:
Let $T_1$ and $T_2$ be two regular $n-1$ dimensional simplices with vertices $(t,0,\ldots,0), (0,t,\ldots, 0),\ldots, (0, 0, \ldots, t)$ and $(t-n+1,1,\ldots, 1... |
H: Tensor product algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$
I want to understand the tensor product $\mathbb C$-algebra $\mathbb{C}\otimes_\mathbb{R} \mathbb{C}$. Of course it must be isomorphic to $\mathbb{C}\times\mathbb{C}.$ How can one construct an explicit isomorphism?
AI: An explicit isomorphism of $\mat... |
H: How does one construct a 'sign chart' when solving inequalities?
I'm working on solving inequalities for an assignment. The instructions also request that I draw a 'sign chart' along with each solution. I've never heard of a 'sign chart' before, and the internet also seems to have a limited amount of information. F... |
H: Proving a certain set in a compact metric space is open
The question is:
Let $(X,d)$ be a compact metric space. Let $f : X \rightarrow X$ be continuous. Fix a point $x_0 \in X$, and assume that $d(f(x), x_0)\geq1$ whenever $x\in X$ is such that $d(x,x_0)=1$. Prove that $U\setminus f(U)$ is an open set in $X$, where... |
H: Determine characteristics for the method of characteristics
I'm trying to understand the method of characteristics to solve first-order PDEs. As an example in his course, my professor solve this PDE for $u(x,y)$:
$$x\frac{\partial u}{\partial x}-y\frac{\partial u}{\partial y}=R $$
with $u(s,s)=f(s)$ given along the... |
H: Generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$
I would like to find a generator of the multiplicative groups of units in $\mathbb{Z_5[x]}/(x^{2}+3x+3)$. This is a field since $x^{2}+3x+3$ is irreducible, so every coset with $bx+a\not=0$ as a representative should be a unit...
I d... |
H: Reference request: stability theory in infinite dimensional dynamical systems/ partial differential equations
I am looking for some references (text books, elementary review papers, journal articles etc) regarding the phenomenon of breakdown in stability for (nonlinear) partial differential equations, i.e if we sta... |
H: Solving $f'(x) +f(x)=cf(x-1)$
To show that $f(x) =Ae^{nx}$ for constant $n$ and $A$ starting with this thing:
$$f'(x) +f(x)=cf(x-1)$$
Where $c$ is constant and $c\not= 0$.
If it wasn't for the $f(x-1)$ bit, I would just use the integrating factor where $I=e^x$ and plug it into the equation. But the $f(x)$ throws me... |
H: How low can the approval rating of a majority candidate be?
“Ostrogorski's paradox” describes a strange situation in which voters decide on candidates based on issues in platforms, but on each issue of the platform, the majority of voters disapprove of the majority winner.
What is the lowest possible approval rati... |
H: What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension?
What is the tensor product $M_n(L)\otimes_K L$, where $L/K$ is a quadratic extension?
Let $K$ be a field of characteristic $0$, $L/K$ a quadratic extension.
Let $\rho\in \operatorname{Gal}(L/K)$ denote the nontrivial element of th... |
H: Proving that $\lim\limits_{n\to\infty}n\left( \int_0^1 f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\left(\frac k n\right) \right)=\frac{f(1)-f(0)}{2}$
Let $f\in C^2([0,1])$. Prove that $$ \lim_{n\to+\infty}n\left( \int_0^1
f(t)\, dt -\frac1n\sum_{k=0}^{n-1}f\Big(\frac k n\Big)
\right)=\frac{f(1)-f(0)}{2}. $$
The second... |
H: (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
Since the specific space $\mathbb{R}^k$ is given, this might be provable in ZF.
Let $\{F_n\}_{n\in \omega}$ be a family of closed subset of $\mathbb{R}^k$, of which the union is $\mathbb{R}^k$.
Suppose $\forall n\i... |
H: Every Ring is Isomorphic to a Subring of an Endomorphism Ring of an Abelian Group
Show that for every ring $(R,+,\cdot)$, there is an abelian group, $(A,+)$, such that $R$ is isomorphic to a subring of $(\operatorname{End}(A),+,\circ)$.
$(\operatorname{End}(A),+,\circ)$ is the set of homomorphisms of $A$ that form ... |
H: Any infinite set of a compact set $K$ has a limit point in $K$?
I'm reading principles of mathematical analysis and have a question about a theorem 2.37.
Theorem 2.37
If $E$ is an infinite subset of a compact set $K$, then $E$ has a limit point in $K$.
The proof is
If no point of $K$ were a limit point of $E$, e... |
H: Proof that $1$ is the least natural number using the Principle of Well Ordering
I'm doing a question from a PDF on Abstract Algebra.(The pdf can be found here http://abstract.pugetsound.edu/ 2012 edition). I have to show that the Principle of Well-Ordering implies that $1$ is the smallest natural number; this is on... |
H: A specific example of the GNS construction
In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. Can someone demonstrate that, with the appropriate notions of equivalenc... |
H: Is there a "simple" proof of the isoperimetric theorem for squares?
"Simple" means that it doesn't use any integral or multivariable calculus concepts.
A friend of mine who's taking a differential calculus course came up with the problem
Prove that among all the quadrilaterals with a given perimeter, the one with ... |
H: Why can't $\int_0^1\sin(x^2) dx$ be equal to $2$?
Why can't $\int_0^1\sin(x^2) dx$ be equal to $2$?
What makes this true? Intuitively, it makes sense. But why?
AI: While Peter's answer is right, I think there's a more intuitive estimate. We know that $|\sin(y)| \leq 1$, and thus $\displaystyle \left|\int_0^1 \sin... |
H: How to evaluate the integral $\iint_C \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$
I would like to evaluate
$$\iint\limits_C \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where $C$ is the first quadrant, i.e.
$$\int\limits_0^\infty\int\limits_0^\infty \sin (x^2+y^2)\,\mathrm dx\,\mathrm dy$$
where
$$\int\limits_0^\infty \s... |
H: What's the difference between a set and a language?
I'm very familiar with languages such as $L$={$a^n$ where $n$ is a prime number}. I know this language is decidable. However, I ran across a similar problem that asked if the set $X$ = {$p$ where $p$ is a prime number} is decidable or not. It probably is, but what... |
H: Does $x^2 \equiv 211\pmod{ 159}$ have a solution?
Note that 159=3*53. The answer to this question is yes. I managed to find two of the solutions. They are $x=23,136$ but there are two more. The main question that I have is whether there is an easier way to find all solutions of a quadratic congruence when the modu... |
H: Cardinal exponentiation problem from Halmos' Naive Set Theory
In chapter 24 of Halmos' Naive Set Theory the following problem is posed as an exercise (page 96):
Prove that if $a, b$ and $c$ are cardinal numbers such that ${a}\le{b}$, then $a^c\le{b^c}$. Prove that if $a$ and $b$ are finite, greater than $1,$ and i... |
H: $H$ is normal whenever $Ha\not = Hb \implies aH\not =bH$
Topics in Algebra- Hernstein pg-47,Q.9
If $H$ be a subgroup of a group $G$ such that $Ha \not=Hb$ implies that $aH\not=bH$. Then how can I show that $gHg^{-1}\subset H$ $\forall$ $g\in G$?
This is what I have done:
For any $h\in H$,$(ghg^{-1})(gh^{-1}g^{-1... |
H: Proving $(n+1)^n > 2^n n!$
If $n>1$ is a natural number, how to prove that
$$(n+1)^n > 2^n n!$$
AI: By the AM GM inequality, $$\eqalign{
& \root n \of {n!} = \root n \of {1 \cdot 2 \cdot 3 \cdot 4 \cdots n} \leq \frac{{1 + 2 + 3 + \cdots + n}}{n} \cr
& \frac{{\left( {n + 1} \right)n}}{{2n}} = \frac{{n + 1... |
H: Recursively enumerable set? Any hints?
I don't mean to be pulling answers out of you, but I'm stuck. Any advice on the right direction would be appreciated.
I have the following set
$X$ ={$n$ where $n$ is a number of a turing machine $M$ that does not halt when given $n$ as input}
My gut instinct is that it's not. ... |
H: Artinian local algebras over a complete local noetherian ring.
So let $\Lambda$ be a complete local noetherian ring.
The author claims that $\Lambda[t]/(t^i)$ is an Artinian local $\Lambda$-algebra with the same residue field as $\Lambda$.
I don't see this. For example, of $\Lambda$ is a complete DVR with uniformiz... |
H: How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?
From Berkeley Problems in Mathematics, Spring 1999, Problem 17.
Let $f(x)\in \mathbb{Q}[x]$ be an irreducible polynomial of degree $n\ge 3$. Let $L$ be the splitting field of $f$, and let $\alpha\in L$ be a zero of $f$. Give that $[L:\mathbb{Q}]=n!$, prove th... |
H: Prove $\sum_{n=2}^{k} \frac{1}{n}= \sum_{n=2}^{\infty}\left(\frac{1}{n}-\frac{1}{n+k-1}\right)$
The question is how to prove this equality
$$\sum_{n=2}^{k} \frac{1}{n}= \sum_{n=2}^{\infty}\left(\frac{1}{n}-\frac{1}{n+k-1}\right)$$
I wasn't sure how to start proving this.
AI: Write out some terms of the righthand s... |
H: What does the $A(m)$ denote in Engelking's book?
Recently, I'm reading Engelking's book. In this book, it always uses as the example the space $A(m)$, see for instance the example 1.1.8, page 15.
For this space, I'm not very clear, for the description is complex to me. Sometimes I view it as a sequence with a li... |
H: When collapsing a cardinal, what ordinal does it become?
Work in $V$. Let $P = \text{Col}(\omega, \omega_1)$ and suppose that $G$ is generic for $P$ over $V$. Then $V[G]\models |\omega_1^V|=\aleph_0$ and $\omega_2^V=\aleph_1$. In particular, $V[G]\models\omega_1^V$ is an ordinal between $\omega$ and $\omega_1^{V... |
H: A tricky but silly doubt regarding the solutions of $x^2/(y-1)^2=1$
Motivation :
I have been confused with some degree 2 equation. I suddenly came across a simple equation and couldn't get the quintessence behind that.
I have an equation $$\dfrac{x^2}{(y-1)^2}=1 \tag{1}$$ and I was looking for its solutions. It w... |
H: Finding the minimum number of selections .
Lets assume I have three boxes that contains only apples, only oranges and a mix of apples and oranges respectively and all of them are mislabelled . What is the minimum number of selections that I need to make so that I can be certain which box contains what fruits assum... |
H: Computable function and decidable sets: For a computable $g$ and decidable set $A$ , Does $g(A), g^{-1}(A)$ necessarily decidable?
I'm trying to solve the following exercise from an old exam:
For a computable function $G: \mathbb{N} \to \mathbb{N} $, and a set of numbers $A$, we define $g^{-1}(A)=\{x|g(x) \in A\} $... |
H: Is this a Polygon?
According to the definition of Polygon, If a Poly-line's first and last points are connected then it is called Polygon. See the image below. I have P1, .... P5 Polyline. If I draw a line from P5 to P1 then it will be called polygon or not? I am confused here.
AI: I think conventionally you would... |
H: Lower bound of $J=\frac{x^TAx}{x^TBx}$
Consider two symmetric positive semi-definite matrices $A, B \in \mathbb{R}^{n\times n}$.
Suppose that $A$ and $B$ have the same null space $\mathcal{N}\subset \mathbb{R}^n$.
Now consider the objective function
$$J=\frac{x^TAx}{x^TBx}$$
where $x\in\mathbb{R}^n$ is an arbitra... |
H: Evaluating $\lim\limits_{n\to\infty} \frac1{n^3}\sum\limits_{\ell=1}^{n-1}\sqrt{(n^2-\ell^2)(n^2-(\ell-1)^2)}$
Along the way to proving a solution for this stubborn question of mine, I've come upon this expression which I would like to evaluate:
$$
\lim_{n\to\infty} \frac{1}{n^3}\sum_{\ell=1}^{n-1}\sqrt{(n^2-\ell^2... |
H: Want to learn differential geometry and Riemannian geometry by myself
I want to follow a Ph.D. programme on geometric analysis. The main focus are Monge-Ampere equations and convex level set of $p$-harmonic equations. The theory of PDEs is very familar to me. However, I have little knowledge of the geometry.
There... |
H: Where is the symmetric group hidden in the Yoneda lemma?
In extension to the question
Yoneda-Lemma as generalization of Cayley`s theorem?,
can someone point out to me where, in the categorical notation and analyzation of the Cayley's theorem, the symmetrc group of transformations of the group elements is?
Put anot... |
H: Relationship between cross product and outer product
Since inner products $(V)$ are generalisations of dot products $(\mathbb{R}^n),$ then are outer products $(V)$ also related to cross products $(\mathbb{R}^3)$ in some way?
A quick search reveals that they are—yet the outer product of two column vectors in $ \math... |
H: Calculating the norms of a triangle based pyramid
Hi I have the following co-ordinates, which make up my triangle based pyramid. I need to calculate the normals of each face. However Im struggling to find the best simplest way to do this?
-0.5, 0, 0.5,
0, 0, -0.5,
0.5, 0, 0.5,
0, 0, -0.5,
0.5, 0, 0.5,
0, 1, 0... |
H: Wedge Sum of Axes Not a Manifold
Note: for this question a topological manifold is defined to be a locally Euclidean, second countable, Hausdorff space. Also, I am using the subscript $x$ and $y$ just to keep track of which copy of $\mathbb{R}$ I am calling the $x$ and $y$ axis, respectively.
Intuitively I can see ... |
H: Checking Graph Isomorphism via Generator of Automorphism Group
Let $G$ and $H$ be two connected graphs and $K$ be the disjoint union of them. Also assume that we can find generating permutation for $Aut(X)$ or any graph $X$. $G$ and $H$ are isomorphic if and only if some generator of $Aut(K)$ interchanges the two c... |
H: How to find matrix form of an operator on a vector space V which is direct sum of its two subspaces?
I am studying a lecture notes where I found this result:
Let $A$ be an operator in vector space $V$ . If $V$ is equal to direct sum of its two subspace $U$ and $W$ ie $V = U\bigoplus W$, and if the basis $\{e_1,. . ... |
H: Relation between Interior Product, Inner Product, Exterior Product, Outer Product
Following my previous question Relationship between cross product and outer product where I learnt that the Exterior Product generalises the Cross Product whereas the Inner Product generalises the Dot Product, I was wondering if the s... |
H: How I Prove The Cardinality of the Symmetric difference of $n$ sets
I try to prove that the cardinality of the the symmetric difference of $n$ sets is
$$
\sum\limits_{i}|A_i|-2\sum\limits_{i<j}|A_i\cap A_j|+4\sum\limits_{i<j<k}|A_i\cap A_j\cap A_k|-\ldots
$$
I try to use the Characteristic Function $\chi(x)$.
I kn... |
H: The emptiness problem for "lunatic" and "crazy" Turing machines
Crazy Turing Machine is the same as Turing machine with one stripe , except of the fact that after each ten steps the head jumps back to the beginning of the stripe.
Lunatic Turing Machine is the same as Turing machine with one stripe , except of the f... |
H: Is this a norm? (triangle inequality for weighted maximum norm)
I've been trying to prove that the following is a norm, but wasn't successful. I also cannot find a counterexample. So help is greatly appreciated.
Let $x \in \mathbb{R}^N, \ w_i \in \mathbb{R}_+,\ i=1,\ldots,N$.
$$\lVert x \rVert_w := \max \lvert w_i ... |
H: Simple Limit Divergence
I am working with a definition of a divergent limit as follows:
A sequence $\{a_n\}$ diverges to $-\infty$ if, given any number $M$, there is an $N$ so that $n \ge N$ implies that $a_n \le M$.
The sequence I am considering is $a_n = -n^2$, which I thought would be pretty simple, but I kee... |
H: What is unitary space
In https://www.encyclopediaofmath.org/index.php/Unitary_space, unitary space seems to be Hilbert space. But in http://www.answers.com/topic/unitary-space, "finite dimensional" is required. My question is, which definition of unitary space is commonly used?
AI: Answers.com is wrong. Unitary spa... |
H: An entire function as the projection in a fibre bundle
Let $f$ be a non-constant entire function. I'd like to view $f$ as the projection mapping in a fiber bundle, where the base space is $X =f(\mathbb{C})$ (which according to Picard is either $\mathbb{C}$ or $\mathbb{C} \setminus \{z_0\}$), and the bundle space is... |
H: Vector analysis can anyone clarify whether my assumptions make sense?
In this problem there are three particles, with velocities $\vec{v_1}, \vec{v_2}$ and $\vec{u}$.
Relative to the particle moving at $u$, the velocities $v_1$ and $v_2$ are of equal magnitude and are perpendicular. Accordingly, show:
$$\left|\vec... |
H: How do you calculate the derivative of an integral
How can you calculate $dy/dx$ here?
$$y=\int_{2^x}^{1}t^{1/3}dt$$
I get that the anti derivative is $3/4t^{4/3}$, but I don't understand what I'm supposed to do next.
The answer is
$$\int_x^1\sqrt{1-t^2}dt + 2$$
I have no idea how to get there
AI: What about using... |
H: Complete course of self-study
I am about $16$ years old and I have just started studying some college mathematics. I may never manage to get into a proper or good university (I do not trust fate) but I want to really study mathematics.
I request people to tell me what topics an undergraduate may/must study and the ... |
H: Motion profile, the maximum velocity allowed from stop
I´m trying to calculate what the maximum allowed velocity for a mechanical axis traveling towards a stop. The maximum velocity must be a secure speed so the axis have is able to stop in time.
I have:
current position
current distance to stop
current velocity
a... |
H: An element conjugate to an odd number of elements
I have the following problem I've encountered while studying for quals that I just can't seem to tackle:
(revised, apologies - left out an important detail)
An element g of order greater than 2 such that the conjugacy class of G has an odd number of elements. Prove ... |
H: Is similarity surjection?
How to prove that similarity $f:\mathbb R^n \to \mathbb R^n$, i.e there is $0<r<1$ such that $d(f(x),f(y))=r \space d(x,y)$ for any $x,y\in\mathbb R^n$, is surjection? Intuitively this is the case, but i can't prove it rigorously or give any counterexample. Thanks.
AI: You can compose a si... |
H: Conjugate elements of G that are in the center of a Sylow p-subgroup
I've had a lot of trouble with this question, as well as my advisor.
Let x and y be two elements of $Z(P)$, where $P$ is a Sylow $p$-subgroup of $G$. If $x$ and $y$ are conjugate in $G$, prove that $x$ is conjugate to $y$ in $N_G(P)$, the normali... |
H: Question about continuous functions and indefinite integrals
My math books, in the introductory chapter of indefinite integrals (they call them primitives, and a primitive of a function is any function who's derivative is the original function) concludes the following thing:
Any continuous function on the reals adm... |
H: If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field
Let $R$ be a commutative unitary ring and suppose that the abelian
group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal
of $R$.
Then $R/P$ is a finite field.
Well, the fact that the q... |
H: Appropriate Notation: $\equiv$ versus $:=$
With respect to assignments/definitions, when is it appropriate to use $\equiv$ as in
$$M \equiv \max\{b_1, b_2, \dots, b_n\}$$
which I encountered in my analysis textbook as opposed to the "colon equals" sign, where this example is taken from Terence Tao's blog :
$$S(... |
H: Euler characteristic of a quotient space
I have a question relating to an answer on MathOverflow.net. The cited answer says:
Let $X$ be a topological space for which [the Euler characteristic] $\chi(X)$ is defined and behaves in the expected way for unions, Cartesian products, and quotients by a finite free actio... |
H: How to maintain equal proportion of investment after buying out a partner
Let's say I have five investors, each owning 20% of a company that is worth $1,000,000. One investor wants to be bought out. So the other investors would, to maintain their equal proportion after the buyout, each pay $50,000 and hence would n... |
H: Characteristic function of a sum of Uniform random variables
Suppose I have $S=U_1+U_2+\dots+U_n$ where $U_i$ are distributed Uniform$[-1,1]$.
I am trying to show a couple of things. First, what is the characteristic function. I can show this easily enough for a single uniform RV, but for some reason the sum is not... |
H: Image of complex circle under polynomial
The question is, how many times does $f(C_r)$ go around $0$ for $C_r = \{x \in \mathbb{C}: |x|=r\}$ and a polynomial with complex coefficients $f$.
The answer is clear to me when $f(x) = ax^n + b$ but I don't see how exactly middle terms affect the result. How to see this ... |
H: Complement of co-dense set.
Asaf's argument : (ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
Let $X$ be a separable complete metric space.
Let $D$ be a countable debse subset.
Let $F$ be a closed subset with empty interior.
Here, how do i show that $D\setminus ... |
H: Trigonometry word problem?
Two ships have radio equipment with a range of 200 km.One is 155km North42 degrees 40 minutes east and the other is 165 km north 45 degrees 10 minutes west of a shore station. Can the two ships communicate directly?
How would I solve this problem I know I have to make a triangle but I am... |
H: Questions about adjointness of quantifiers in first-order logic
I have a category theory homework problem which asks: "In first-order logic, why does $\forall$ not have a right adjoint?" The typical argument is that:
If for some operator $\cdot$ it could be asserted $\forall \dashv \cdot$, then $\cdot$ would be a... |
H: Using the invariance principle: how to solve $n+d(n)+d(d(n))=m$?
Let $d(n)$ be the digital sum of $n$. How to solve $n+d(n)+d(d(n))=m$, where $n$ and $m$ are natural?
AI: Not a solution, simply a large size observation. May be deleted once a full solution is posted.
Notice that the left-hand-side is always a multi... |
H: Multiplying exponents, solving for n
When solving for n in this equation I get stuck.
Question: What is the smallest value of n such that an algorithm with running time of $\ 100n^2 $ runs faster than an algorithm whose running time is $\ 2^n $ on the same machine?
Straight out of CLRS chapter 1. Class starts in 2 ... |
H: Integration substitution problem
$$\int_{-1}^1 x^2 ~dx = \frac{2}{3}.$$
Now let us substitute $u=x^2$. Then $du=2x ~dx$ then the definite integration becomes from $1$ to $1$, i.e. $0$. I know I need to partition this from $-1$ to $0$ and $0$ to $1$. But my question is when we substitute anything we do not concern o... |
H: Reasonable strategy for simple game
I have a very simple probabilistic process which I have to deal with in the software project I'm involved in, yet I can't figure how to do it. I can describe the situation as a money game (in reality it is not, but this is the most simple way to go):
At each moment $t$ my program... |
H: Are all infinities equal?
A friend of mine was trying to explain to me how all infinities are equal. For example, they were saying that there are the same amount of numbers between $0$–$1$ as there are between $0$–$2$.
The way they explained it, you could prove that there are the same amount of numbers by getting a... |
H: Baire Category Theorem
This is Asaf's argument;
(ZF) If $\mathbb{R}^k$ is a countable union of closed sets, then at least one has a nonempty interior
Suppose that $(X,d)$ is a separable complete metric space, and $\{a_k\mid k\in\omega\}=D\subseteq X$ is a countable dense subset.
By contradiction assume that we can ... |
H: Sylow 7-subgroups in a group of order 168
Another question from my qual studying that's been stumping me. I'm still a little lost on normalizers. The question is:
Let G be a group of order 168, and let P be a Sylow 7-subgroup of G. Show that either P is a normal subgroup of G or else the normalizer of P is a max... |
H: How to prove this ln inequality?
I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$:
$$
1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}}
$$
I've been struggling to find some known inequalities involving logarithms that could be applied here (and lack of condition for $x$ to be... |
H: Is the identity matrix an elementary matrix?
My reasoning is yes, as you can switch row i with row i in the matrix... But I'm not sure if it's a "legal" elementary operation to switch a row with itself.
AI: This is a question of convention, but I would certainly consider the identity an elementary matrix, as I thin... |
H: If $\lim_n f_n(x_n)=f(x)$ for every $x_n \to x$ then $f_n \to f$ uniformly on $[0,1]$?
This is a self-posed question, so I do not know the answer and I would like to know what you think about.
Let $f,f_n:[0,1]\to \mathbb R$ be continuous functions. Assume that for every sequence $(x_n)_{n\in \mathbb N}$ converging... |
H: Difference of two Cauchy Sequences
Suppose that $\{a_n\}, \{b_n\}$ are both Cauchy sequences of real numbers, and that $a_n \le b_n \ \forall n$. Prove that $\lim_{n \to \infty} a_n \le \lim_{n \to \infty} b_n$.
The definition of a Cauchy sequence I am using is
A sequence $\{a_n\}$ is Cauchy if, given $\varepsi... |
H: How do I find the derivative of a definite integral in which the variable of differentiation is a limit in the integral?
The derivative I would like to find is
$$\frac{\partial}{\partial z} \int_{-z_0}^z (z - z^{\prime}) \; f(z^{\prime}) \; dz^{\prime}$$
where $f(z^{\prime})$ is some arbitrary function of $z^{\pri... |
H: Why do the rationals, integers and naturals all have the same cardinality?
So I answered this question: Are all infinities equal? I believe my answer is correct, however one thing I couldn't explain fully, and which is bugging me, is why the rationals $\mathbb Q$, integers $\mathbb Z$ and naturals $\mathbb N$ all h... |
H: Calculate closest point without calculating hypotenuse or linear length in 2D space.
Given $n$ number of $(x, y)$ points, I want to find out which is the closest (linear distance) to the point of origin $(1, 1)$. Both $x$ and $y$ will always be positive integers.
Due to the magnitude of numbers I am dealing with, m... |
H: Visualizing quotient groups: $\mathbb{R/Q}$
I was wondering about this. I know it is possible to visualize the quotient group $\mathbb{R}/\mathbb{Z}$ as a circle, and if you consider these as "topological groups", then this group (not topological) quotient is topologically equivalent to a circle.
But then, what doe... |
H: A qualitative, yet precise statement of Godel's incompleteness theorem?
I read online a statement to the effect that (I'm paraphrasing):
Goedel's incompleteness theorem shows that we cannot even have a complete and consistent theory for the natural numbers.
I am under the (qualitative) impression that this statem... |
H: Definition of the rank of the isometry group of a manifold.
Let $M$ be a manifold (which you can assume compact if it helps) and consider the natural action of the isometry group of $M$, Iso($M$) on $M$. The we can define the symmetry rank of $M$ as the rank of Iso($M$). Here is were I have trouble: I think I'm mis... |
H: Showing that a function is not differentiable
I want to show that $f(x,y) = \sqrt{|xy|}$ is not differentiable at $0$.
So my idea is to show that $g(x,y) = |xy|$ is not differentiable, and then argue that if $f$ were differentiable, then so would $g$ which is the composition of differentiable functions $\cdot^2$ an... |
H: Homotopy equivalence involving cell - complexes
I have two small questions related to some point-set topology involved in the following. My question is inspired after reading the proof of Proposition 1.26 of Hatcher.
Suppose I have a space $X$ that is built out of a subspace $A$ that is path-connected by attaching... |
H: Help with a problem with matrices
Hi everyone I've been messing with a problem with matrices and I can't get it to work. I think I am not seeing it the right way because else I get an 8 variable quadratic system of equations.
If $A$ and $B$ are $2\times 2$ matrices and $A^2 = B^2 = I$ and $$AB =
\begin{pmatrix}
0 ... |
H: Congruence inequality
Given $n>2$, by calculation or otherwise deduce that $5^{2^{n-3}} \neq -1 \pmod {2^n}$
Note:The problem arose when I tried to deduce $\langle5\rangle \cap \langle2^n-1\rangle=\{1\}$ in the group $\mathbb{Z}^{*}_{2^n}$, I have showed $\operatorname{ord}(5)=2^{n-2}$
AI: We show by induction that... |
H: Apostol's Calculus - Prove the set is open (inequality)
This is from Apostol's Calculus, Vol. II Section 8.3 #3(a).
Prove that $S=\{(x,y,z) \mid z^2 - x^2 - y^2 -1>0\}$ is open.
The only way to prove that a set is open which has been covered so far is to prove that for an arbitrary point $\mathop a\limits^{\small \... |
H: Why does input size matter in NP theory?
When my prof introduced us to the N/NP topics, the first thing he mentioned is input size, which he defines as the number of bytes needed to describe and write a problem's input into a file. Could someone please explain to my why the input size matters to these topics? Thank... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.