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H: Discrete valuation ring associated with a prime ideal of a Dedekind domain
Let $A$ be a Dedekind domain.
Let $K$ be the field of fractions of $A$.
Let $P$ be a non-zero prime ideal of $A$.
Let $v_P$ be the valuation of $K$ with respect to $P$.
Then the localization $A_P$ of $A$ at $P$ is the valuation ring of $v_P$... |
H: The $100$th derivative of $(x^2 + 1)/(x^3 - x)$
I am reading a collection of problems by the Russian mathematician Vladimir Arnol'd, titled A Mathematical Trivium. I am taking a stab at this one:
Calculate the $100$th derivative of the function $$\frac{x^2 + 1}{x^3 - x}.$$
The derivative is non-trivial (in the se... |
H: Hilbert class field of $\mathbb{Q}(\sqrt{65})$
Let $K = \mathbb{Q}(\sqrt{65})$.
Let $L = \mathbb{Q}(\sqrt{5}, \sqrt{13})$.
Is $L$ the Hilbert class field of $K$?
If yes, how would you prove this?
AI: First, compute the class number of $K$; the answer is $2$.
Now $L$ is a quadratic extension of $K$, which is unramif... |
H: Lifting projective indecomposable modules - Benson's vol. 1
I'm reading Benson's Representations and Cohomology I, Section 1.9.
Could someone please clarify to me the following sentence
at page 18 lines -7,-6,-5:
"So given a simple $\bar\Lambda$-module $S_j$, it has a projective cover $P_j=\bar Q_j$ for some projec... |
H: The probability of bricks arranged randomly.
If someone have $4$ red bricks and $8$ blue bricks and arranges them randomly in a circle,
what is the probability that two red bricks are not side by side?
Is the sample space $\frac{12!}{4!\,8!} \,=\, 495$ ?
and the answer $P(\text{Reds non-adjacent}) \:=\:\frac{70}{4... |
H: Rating system incorporating experience
I am looking to rank players on something and right now I am using a win-loss percentage to "rank" the players.
The issue is that I have some people who just beat out the 5 game threshold and have a win/loss ratio of say 5 wins, 1 loss for 83% and then I have someone who has 5... |
H: Does inversion reverse order for positive elements in a unital C* algebra?
Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related statement be made if we only assume $a \leq b$? (... |
H: What is $\aleph_0$ powered to $\aleph_0$?
By definition $\aleph_1 = 2 ^{\aleph_0}$. And since $2 < \aleph_0$, then $2^{\aleph_0} = {\aleph_1} \le \aleph_0 ^ {\aleph_0}$. However, I do not know what exactly $\aleph_0 ^ {\aleph_0}$ is or how I could compute it.
AI: No. By definition $\aleph_1$ is the least uncountabl... |
H: Puzzle - reordering people in a line
Eighty players numbered I through $80$ are standing in a row, one behind the other, in the
increasing order of their numbers. The physical director of the players performed eight successive
inspections of the players and in each of the respective inspections he sent the first $1... |
H: Spivak Calculus 3rd ed. $|a + b| \leq |a| + |b|$
I'm working through the first chapter of Michael Spivak's Calculus 3rd ed.
Towards the end of the chapter he proves $ |a + b| ≤ |a| + |b| $ using the observation that $|a|= \sqrt{ a^2 }$ when $a$ is $ ≥ 0 $ .
$ |a + b| ≤ |a| + |b| $
$$ (|a + b|)^2 = (a + b)^2 $$ $... |
H: Application of Bezout identity
From Wikipedia: en.wikipedia.org/wiki/Chinese_remainder_theorem
Suppose
$x \equiv a_1 \pmod{n_1} \\
x \equiv a_2 \pmod{n_2}$
where $n_1$ and $n_2$ are coprime to each other.
How does this result in the following:
$n_2 [n_2^{-1}]_{n_1} + n_1 [n_1^{-1}]_{n_2} = 1$
where the notation $... |
H: Finding Satisfiability, Unsatisfiability and Valid well formed formula
I have a confusion regarding how to check whether a wff is satisfiable, unsatisfiable and valid.
As far as I understood, valid means the truth table must be a tautology, otherwise it is not a valid wff. For eg, p or (not p) is valid since it is... |
H: Primitive Element for Field Extension of Rational Functions over Symmetric Rational Functions
A rational function $f$ in $n$ variables is a ratio of $2$ polynomials,
$$f(x_1,...x_n) = \frac{p(x_1,...x_n)}{q(x_1,...x_n)}$$
where $q$ is not identically $0$. The function is called symmetric if
$$f(x_1,...,x_n) = f(x... |
H: Evaluate $\frac{d}{dx}ⁿx$.
Evaluate $\frac{d}{dx}ⁿx$ which $ⁿx=x^{x^{x^{...}}}$, total $n$ $x$'s.
I have tried to observe when $n=1,2,3,4,5$, but it's difficult to see the pattern. Can anyone get it? Thank you.
AI: As I said in my comment above, define $y= ^n\!\!x$. Then
$$\log y=\log ^nx=\log x^{^{n-1}x}=\, ^{n-... |
H: Mathematics base for data mining and artificial intelligence algorithms.
Could you give me some clarification about data mining and artificial intelligence algorithms? What mathematics base they used for? Could you give me starting point, in mathematics, to understand these types of algorithms?
AI: As Paxinum menti... |
H: Compute $\lim\limits_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{{1}/{2}}\cdots\left(1+\frac{n}{n}\right)^{{1}/{n}}$
Compute the limit:
$$\lim_{n\to\infty} \left(1+\frac{1}{n}\right)\left(1+\frac{2}{n}\right)^{\frac{1}{2}}\cdots\left(1+\frac{n}{n}\right)^{\frac{1}{n}}$$
AI: Note at the onset ... |
H: Exact sequence of four sheaves in Beauville: associated l.e.s.?
This question is about an exact sequence of four sheaves on a smooth projective surface $S$ over $k=\mathbb{C}$, to be found in Beauville: complex algebraic surfaces, theorem I.4, page 3 (second edition).
It comes down to this: Denote the exact sequenc... |
H: Calculate width in a grid with lines
This is a web related question but is a mathematical problem.
Look at this:
|O|O|O|O|
|O|O|O|
|O|O|
A vertical line represents a 1 pixel wide horizontal line.
A 'O' represents a space between the lines.
All the three rows should have the same width. The space 'O' is streched ou... |
H: Predicting the number of solutions of a system of equations without solving it explicitely
$$
\left\{
\begin{array}{l}
x + y + az + at = a \\
2x + y - a^2z + 2at = 1 \\
4x + 3y + a^2 z + 4a^2 t = 1
\end{array}
\right.
$$
Is there a way to predict the number of solutions $(x, y, z, t)$, depending on the value of the... |
H: Books on AI, programming, optimization
I'm studying math (just started) and I like programming as well (just started this too), is there a career or a branch of research including deep aspect of this two aspects? Is there someone among you who is doing this perhaps?
The sensation I have is that there's much more "a... |
H: can one derive the $n^{th}$ term for the series, $u_{n+1}=2u_{n}+1$,$u_{0}=0$, $n$ is a non-negative integer
derive the $n^{th}$ term for the series $0,1,3,7,15,31,63,127,255,\ldots$
observation gives, $t_{n}=2^n-1$, where $n$ is a non-negative integer
$t_{0}=0$
AI: You already have an expression for the $n^{\text{... |
H: Sequence of indicator functions is weakly convergent.
Let $E\subseteq [0,1]$ be a measurable set.
Then we recall that
$\chi_E(x)=\begin{cases} 1&\text{ if } x\in E,\\ 0 & \text{ if } x\not\in E.\end{cases}$
We are given a sequence of measurable subsets $\{E_n\}_n$ of $[0,1]$ and a measurable subset $E\subseteq [0... |
H: 2 jars with 50 balls each. Pick any ball with $>50\%$ probability.
You are blindfolded and placed in front a table with two jars. One jar has $50$ red balls and other has $50$ blue balls.
What should be your strategy so that you pick up the red ball with more than $50\%$ probability.
AI: Take one ball from each ja... |
H: Truth Tables - How to identify normals in Knights, Knaves and Normals problems?
To describe my question, I'll illustrate an example of a Knights, Knaves and Normals problem and the way I solve it.
Question
Knights always tell the truth.
Knaves always lie.
Normals sometimes lie and sometimes tell the truth.
Given th... |
H: Clarification of a Proof in Lee's An Introduction to Topological Manifolds
I'm having a bit of difficulty understanding the following proof from Lee's An Introduction to Topological Manifolds:
I don't understand the deduction that since $e_0\cap e \subseteq \overline{e}\setminus e$ and $\overline{e}\setminus e$ i... |
H: Poisson's summation formula
It is said that the Fourier transform $\hat{f}(\omega)$ of a function $f(t)$ and the Fourier transform $\hat{b}(\omega)$ of its samples $b(k)=f(t)|_{t=k}$ are related by Poisson's summation formula and it's given by
$$\hat{b}(\omega)=\displaystyle\sum_{k\in{\mathbb{Z}}}\hat{f}(\omega+2\... |
H: Equivalent metrics determine the same topology
Suppose that there are given two distance functions $d(x,y)$ and $d_1 (x,y)$ on the same space $S$. They are said to be equivalent if they determine the same open sets.
Show that $d$ and $d_1$ are equivalent if to every $\varepsilon$ there exists a $\delta>0$ such tha... |
H: What makes a sufficient proof?
This question is related to the question posted here. Would a shorter proof to those in the answers, such as:
Take the subsequence $\{a_m\}$ of $\{a_n\}$ where $m > 0$. By induction on $m$ $$\forall{m}( a_m < a_{m+1})$$ Therefore $$ \forall{\epsilon} > 0 \ \exists{N} \ \forall{m_0,m_... |
H: Lipschitz Continuous $\Rightarrow$ Uniformly Continuous
The Question: Prove that if a function $f$ defined on $S \subseteq \mathbb R$ is Lipschitz continuous then $f$ is uniformly continuous on $S$.
Definition. A function $f$ defined on a set $S \subseteq \mathbb R$ is said to be Lipschitz continuous on $S$ if th... |
H: Prove that square root of 2 is irrational using the principle of Mathematical Induction
How do I prove that the square root of 2 is irrational using the principle of mathematical induction?
AI: The usual proof starts out something like "suppose for a contradiction $\sqrt 2$ were rational, and write it as $p/q$ in l... |
H: a question about permutation in the digits in the decimal system
This is an exercise from a textbook in Portuguese.
Choose a symbol (number) $abc$ in the decimal system ($a$ being the
hundreds digit, $b$ the tens digit, and $c$ the units digit), in a way
that the hundreds digit $a$ and the units digit $c$ diff... |
H: Possible series of typos in Apostol's Calculus Vol.II involving gradients and directional derivatives
The question as presented, this is from Calculus Vol. II Section 8.14 #4
A differentiable scalar field $f$ has, at the point $(1,2)$, directional derivatives $+2$ in the direction toward $(2,2)$ and $−2$ in the di... |
H: Geometrically interpret the set $\{z \in \Bbb C :$ $|z+a|+|z-a| \leq 2b,$ $b\in\Bbb R^+,$ $|a|
Geometrically interpret and determine the following set of complex numbers:
$$\{z \in \Bbb C : |z+a|+|z-a| \leq 2b , b\in\Bbb R^+,|a|<b\}.$$
I understand it means the sum of distance between $z$ and $a,$ and between $... |
H: How to solve $1/2 \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0$?
How to solve trigonomtry function involving $\sin x \cos x$ and $\sin 2x$:
$$\frac{1}{2} \sin(2x) + \sin(x) + 2 \cos(x) + 2 = 0. $$
AI: Hint:
Using the identity $\sin(2x) = 2 \sin x \cos x$ we have
$$ \sin x \cos x + \sin x + 2\cos x + 2 = 0$$
Factor
$$ (1 ... |
H: Lipschitz Condition on a Set of Measure 0
The question is as follows:
Let $f:[0,1] \rightarrow R$ be a non-decreasing continuous function, and let $A \subset [0,1]$ be a set of Lebesgue measure $0$.
(a)Suppose that $f$ satisfies the following condition:
(*) For some constant $C \geq 0$, $|f(s)-f(t)| \leq C|s-t|$ f... |
H: Modern formula for calculating Riemann Zeta Function
Possible Duplicate:
How to evaluate Riemann Zeta function
I have an amateur interest in the Zeta Function. I have read Edward's book on the topic, which is perhaps a little dated. I would like to know any modern methods for computing estimates of
$\zeta(s)$, s... |
H: Proof of Uniqueness Theorem
I have the following question on the theorem below, which I've been working on while I study for quals.
Suppose $\Omega$ is an open and connected set, with $f$ analytic in $\Omega$, and $a \in \Omega$. Then:
(1) If $f(a)=f'(a)=f''(a)=\ldots=0$ then $f$ is identically zero in $\Omega$.
(... |
H: Understanding how to state the Karush-Kuhn-Tucker Conditions for a given problem
I'm trying to understand an example given by Nocedal & Wright (1999), pg 329, Example 12.4. According to a definition given earlier in this book:
At a feasible point x, the inequality constraint $i\in I$ is said to be active if $c_... |
H: Proof of uniform continuity of $\frac{1}{x}$
Show that the function $f(x) = \frac{1}{x}$ is not uniformly continuous on the interval $(0,\infty)$ but is uniformly continuous on any interval of the form $(\mu, \infty)$ if $\mu > 0$.
My Work
Referring to the definition of uniform continuity, I have that $f$ is unif.... |
H: Getting square root of negative in completing the square problem
I try to solve the equation $f(x) = 7x - 11 - 2x^2 = 0$ for $x$, but run into troubles. I've gone through it over and over again as well as similar problems, but can't find what I'm doing wrong.
$$f(x) = 7x - 11 - 2x^2 = 0$$
$$\iff x^2 - \frac{7}{2}x ... |
H: Question about Cauchy sequence in $C(K)$
In my lecture, I need prove the existence of the anyone Cauchy sequence $(f_n)_{n\in \mathbb{N}}$ belonging $C(K)$ with the norm ${\Vert \cdot \Vert}_{\infty}=\sup_{x \in K}|f(x)|$ where $K$ is a compact space. How I can do it? I'm trying that below
Fixing $\varepsilon>0$, t... |
H: Question about a Proof of Triangularizing a Matrix
I was questioned from my friend about a non-proof in a linear algebra textbook.
The author mentions that the following proof of the theorem, that every square matrix in $\Bbb{C}$ is triangularizable, has a gap and thus is invalid:
First, we have two preliminary le... |
H: Inequality for singular values
I stumbled on the following inequality for singular values (stated without proof), and would like to understand it better:
Let $A,B$ be two $n\times n$ real matrices. Denoting by $\mu_i(C)$ the $i$-th singular value of a matrix $C$ and $by$ $\|\cdot\|$ the operator norm we have for ... |
H: Does this Dirichlet series converge to zero?
Consider the periodic Dirichlet series that has this iterative definition:
$$\text{a1}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2}{\sqrt{3}}-\frac{1}{\sqrt{4}}+\frac{1}{\sqrt{5}}+\frac{2}{\sqrt{6}}+...$$
$$\text{a2}=\frac{1}{\sqrt{1}}-\frac{1}{\sqrt{2}}-\frac{2+a1}{\s... |
H: clarification on binary and decimal representations of integers
This is from the book: Elementary number theory, David M. Burton, page 69.
Given an integer $b>1$, any positive integer $N$ can be written
uniquely in terms of powers of $b$ as
$$N=a_{m}b^{m}+a_{m-1}b^{m-1}+\cdots +a_{2}b^{2}+a_{1}b+a_{0},$$ where... |
H: Standard terminology for the relation between $A$ and $B$ if $B= Q^t A P$?
Let $A,B$ be two rectangular $m\times n$ matrices related by
$$B= Q^t A P$$
with $P$ an $n\times n$ and $Q$ an $m\times m $ matrix.
Is there a standard terminolgy for this relation? If instead of the transposed $Q^t$ one takes the inverse ... |
H: Existence of upper limit and lower limit
Previous Question; Does upper limit and lower limit exist for any sequence in $\mathbb{R}$?
For every sequence $\{s_n\}$ in $\mathbb{R}$, $\{x\in \overline{\mathbb{R}}| s_{n_k} →x\}$ is nonempty.
However, how do i prove the existence of $\sup \{x\in \overline{\mathbb{R}}| s_... |
H: Random Walk Expected Number of Visits
I'm having a bit of trouble with a question involving a random walk with five vertices. The graph is shown below.
The problem I can't figure out reads:
Suppose a walker starts in vertex C. What is the expected number of visits to B before the walker reaches A?
I've done a bi... |
H: What is subgroup $\langle g^d \rangle$?
According to the introductory abstract algebra, it says that $g^n =e$ where $g$ is an element of some finite group. Then, it talks about the subgroup $\langle g^d \rangle$. What is it exactly, and what would be the elements of this group? (where $d$ is some integer.)
AI: The ... |
H: distinct generators of a group $G$
By saying distinct generator of a group $G$, is this saying that elements produced by the generator differ from elements produced by other generators?
The distinct generators of $G$ are the elements $g^r$ where $1 ≤ r < N $ and $gcd(r,N) = 1$. Thus, there are
$ϕ(N)$ of them, wh... |
H: $m\mid rq,sq\Rightarrow m\mid \gcd(r,s)q\,$ and its view via additive order
Let $m,q,r,s$ be integers.
Suppose GCD($r$,$s$) = 1.
Suppose $m|qr$ and $m|qs$.
Show that $m$ divides $q$.
This was a result that was assumed in class. I couldn't see the reasoning behind it though.. please provide some guidance/hints.
... |
H: Is the functional $F(u) = \int_{\Omega} \langle (A_1(x)\chi_{\{u>0\}}+A_2(x)\chi_{\{u\le0\}}) \nabla u, \nabla u \rangle$ convex?
Is the functional
\begin{equation}
F(u) = \int_{\Omega} \langle (A_1(x)\chi_{\{u>0\}}+A_2(x)\chi_{\{u\le0\}}) \nabla u, \nabla u \rangle \, dx
\end{equation}
where $A_i,i=1,2$ is a matri... |
H: A question about a stochastic process being Markov
Let $(X_{s},\mathcal{F}_{s})$ be a stochastic process adapted to a given filtration. I was told that, in order to prove that $X$ is Markov, it suffice to prove that for any nonnegative, Borel-measurable function $f$, $$E[f(X_{t}) \vert \mathcal{F}_{s}]=g(X_{s}) \qu... |
H: Spivak's Calculus (Chapter 2, Exercise 17)
I am having trouble completing exercise 17 of chapter 2 of Spivak's Calculus. In this exercise, the reader is asked to prove that for all natural numbers $n$ and $p$, there exist real numbers $\alpha_1,\ldots,\alpha_p$ such that:
$$ \sum_{k = 1}^n k^p ~=~ \frac{n^{p + 1}... |
H: Help with the integral for the variance of the sample median of Laplace r.v.
When we draw $n$ samples of Laplace-distributed random variable such that $n=2k+1$ and the location parameter is zero, the median $x$ (or the $k$-th order statistic) has the following p.d.f.:
$$f_m(x)=\frac{n!}{(k!)^2}2^{-n}\frac{1}{b}e^{-... |
H: Continuous version of Cauchy-Schwarz
Proving the discrete version of Cauchy-Schwarz is easy:
$$ \left(\sum_{i} a_i^2\right) \left(\sum_i b_i^2\right) \geq
\left(\sum_i a_ib_i\right)^2 $$
can be done via the determinant of the quadratic formula.
Now, however, I want to prove the continuous version, which states... |
H: Find a side of a triangle given other two sides and an angle
I have a really simple-looking question, but I have no clue how I can go about solving it?
The question is
Find the exact value of $x$ in the following diagram:
Sorry for the silly/easy question, but I'm quite stuck! To use the cosine or sine rule, I'... |
H: $f(x) = x^r$ equaling to $r \cdot x$ when $f$ is cyclic group $G \rightarrow G$
Suppose that there is a cyclic group $G$ of order $n$ with a generator $g$. Also suppose that $r$ is a fixed integer. Then define a homomorphism function $f: G \rightarrow G$, $f(x) = x^r$.
How do we get that $f(x) = r \cdot x$?
Quote:... |
H: How to prove Lagrange trigonometric identity
I would to prove that
$$1+\cos \theta+\cos 2\theta+\ldots+\cos n\theta =\displaystyle\frac{1}{2}+
\frac{\sin\left[(2n+1)\frac{\theta}{2}\right]}{2\sin\left(\frac{\theta}{2}\right)}$$
given that
$$1+z+z^2+z^3+\ldots+z^n=\displaystyle\frac {1-z^{n+1}}{1-z}$$
where $z\ne... |
H: Mixture of two mirror-image Gaussians
Suppose we are given a set of points $(x_i, y_i)\in\mathbb{R}^2$ and are told that they are drawn from a normal (Gaussian) distribution. It is a simple matter in that case to find the mean $(\mu_x,\mu_y)$ of the distribution: $(\langle x \rangle, \langle y \rangle)$, where $\l... |
H: Clarification on the proof for the Euclidean algorithm
Lemma: Let $m$ and $n$ be positive integers with $m \leq n$. If $r$ is the remainder of dividing $n$ by $m$, then $(n,m) = (m,r)$.
The proof is given as follows:
We have by the division algorithm that $n = sm + r$ with $0 \leq r < m$. Suppose that $d = (n,m)... |
H: what is the equation to get this type of a curve?
what must be the equation to get this type of a curve??
the curve must increase from -infinity to 0 and then decrease again to +infinity.
Thank you
AI: You can try multiplying a $\sin$ wave with a Gaussian, e.g:
$$\sin(5x)e^{-x^2/2}$$ |
H: The average surface area of a projection of a randomly rotated planar rectangular shape on a two-dimensional surface
I have a planar rectangular shape, of dimensions $N$ by $M$, positioned in 3-space above a two-dimensional surface. Provided a large number of random 3-space rotational orientations of the shape, wh... |
H: Norm of quotients
$\newcommand{\Ker}{\operatorname{Ker}}$
Let $X$ be Hilbert space and let $T:X\to X$ be a bounded operator. Define the operator
$S: X/\Ker T \to X/\Ker T$ via $S(x+\Ker T)=Sx+\Ker T$. I can show that $||S||\leq ||T||$, and I am wondering whether the norms are actually equal. Is this true or false ... |
H: division by integers
let d,k be integers, with k even.
suppose d|2k
suppose d does not divide 2
suppose d does not divide k
show that d equals 2k.
(I'm really just trying to understand the 2nd last line, in this answer to the question: Prove that if $p$ is an odd prime that divides a number of the form $n^4 + 1... |
H: how to solve these questions with congruences
how can I solve these type of questions : using congruences
find the last digit or last two digit of $27^{27^{26}}$ and find remainder when $53^{103}$ is divided by 7.
I can solve 2nd question and a simpler case of first question with binomial theorem but in quest... |
H: Find the constant K in probability distribution?
Find the constant K?
I know how to find K, if there is only one "k" in the question and the rest are given number. I know that all the x value are add upto 1.
But in this question, I don't know how to start.
One thing I know is $\Sigma(x)=1(k)+2(2k)+3(3k)+4(... |
H: A question about applying Arzelà-Ascoli
An example of an application of Arzelà-Ascoli is that we can use it to prove that the following operator is compact:
$$ T: C(X) \to C(Y), f \mapsto \int_X f(x) k(x,y)dx$$
where $f \in C(X), k \in C(X \times Y)$ and $X,Y$ are compact metric spaces.
To prove that $T$ is compact... |
H: Lifting of maps to a covering space
I am reading Algebraic topology by W. Massey and I have a problem with the proof of property 5.1:
Let $(\tilde{X},p)$ be a covering space of $X$, $Y$ a connected and arcwise connected space, $\tilde{x}_0 \in \tilde{X}$, $y_0 \in Y$ and $x_0=p(\tilde{x}_0) \in X$. Given a map $\va... |
H: Why this limit of integration?
I've been solving problems from the book by DeGroot and Schervish and I can't understand why m is the upper limit of integration in the solution to this problem. Why not the lower one?
Here is the problem:
Suppose that a random variable X has a continuous distribution for which the p... |
H: Is $\int_{\mathbb{R}^D}\exp(\mathbf{y}^\top\mathbf{S}\mathbf{y}) \mathbf{y}\,d\mathbf{y}$ equal to $\mathbf{0}$?
I want to show that $\int_{\mathbb{R}^D}\exp(\mathbf{y}^\top\mathbf{S}\mathbf{y})\mathbf{y} \, d\mathbf{y}$, where $\mathbf{S}$ is a symmetric real matrix, is equal to $\mathbf{0}$ .
My intuition (altho... |
H: Calculating roots of $x^3 = 2x^2 - x$, getting wrong result
I am working with a text book problem, trying to find the solutions to $x^3 = 2x^2 - x$, but I keep ending up with a different result than the book gives. I'm thinking the problem lie in getting the wrong square root towards the end, seeing as the text boo... |
H: Question regarding technicalities in the paper Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent
For people on this board I have a probably pretty modest question, but since I'm not a mathematician (just an economist), I'm having trouble.
The full pdf can be found here: http://... |
H: Why does a time-homogeneous Markov process possess the Markov property?
Klenke defines (Definition 17.3, p. 346) a time-homogeneous Markov process independently, rather than as a special case of a stochastic process that possesses the Markov property (Definition 17.1, p. 345) and in addition satisfies certain furth... |
H: How can I figure out the plot of the set $M:=\{(x,y) \in R:\frac{1}{2}\leq x \land 1
I have some trouble to figure out the plot/graph of a set like
$$M:=\{(x,y) \in R:\frac{1}{2}\leq x \land 1<y<\frac{1}{x}\}$$
On wolframalpha I saw the plot.
Is there a trick how to figure out such kind of graph?
Greetings.
AI: I s... |
H: 64 possible values - continuous or discrete?
I have a response variable in a model that is calculated from binary values of an 8x8 grid of pixels, giving a proportion. Technically, there are only 64 possible values the result can take, so would this count as discrete data? There are many values, and there are not i... |
H: is this a subset of the set of all selfadjoint operators?
Is the set $Q$ of all operators that have the property that the image is orthogonal to the kernel, and the kernel isn't the null space, only a subset of the set $T$ of the selfadjoint operators or equal to it ?
(would this hold in an infinite dimensional sp... |
H: Find an equivalent to $(P \lor Q) \land (P \to R) \land (Q \to S)$
I need some help regarding solving a logic. The question is to find an equivalent to the following logic.
$$(P \lor Q) \land (P \to R) \land (Q \to S)$$
The choices are
(a) $S \land R$
(b) $S \to R$
(c) $S \lor R$
(d) none of these
The answe... |
H: Is $z/z$ holomorphic in $z=0$?
To be holomorphic require to have derivative at $z\to0$ but $f(z)$ is undefined.
Does it mean $z/z$ is not holomorphic at $z=0$?
AI: Generally in complex analysis texts, after we are exposed to Riemann's Removable singularity theorem, every function written down that has a removable s... |
H: Follow-up on Sobolev embedding theorem
I asked this question in a comment when I realised that the answerer is away until September so I am posting it here in a new thread.
I've been thinking about the Sobolev embedding theorem, given as follows:
If $k > l + d/2$ then we can continuously extend the inclusion $C^\in... |
H: Neither provable nor disprovable theorem
I wonder about a theorem which can be proven that this theorem is $neither\, provable$ nor $disprovable$ using any kind of mathematical knowledge.
Questions:
1- Is there any such theorem? or is it proposable?
2- If there exists such a theorem about a fact, how should we thin... |
H: direct sum to get a vector space
Concerning this question I asked (specifically my last comment), is it true that the formula $$L\oplus \ker S =H$$
holds in a finite dimensional vector space, where $S:H\rightarrow H$ is linear operator, $(b_1,\ldots,b_k)$ is a basis of the image of $S$ and $L$ consists of the pre... |
H: Counting the genus of a model
I was reading a webpage on Euler-Poincaré Formula and it has a question that I couldn't figure out. It should be simple, but I am not getting the right answer.
The question on the webpage goes like this:
Consider the following model which is obtained by taking out a torus and tube fro... |
H: Proving a Riemann integral theorem
Question: Prove that for $f$ continuous on $[a,b] \in \Bbb R$, for $c \in [a,b]$, then:
$$ \int_a^b f(x)\, dx = \int_a^c f(x)\,dx + \int_c^b f(x)\, dx$$
My Work:
Let $\{P_k\}$ be a sequence of partitions such that the mesh of $P$ goes to $0$ as $k \to \infty$. For each $k$, denote... |
H: Why is a random walk a time-homogeneous Markov process?
Why is a random walk on $\mathbb{R}^d$ (see below) a time-homogeneous Markov process? Specifically, why does it satisfy requirement #2 of definition 17.3 that the map $\left(x,B\right)\mapsto\mathrm{P}_x\left[X\in B\right]$ be a stochastic kernel?
Relevant De... |
H: Solving a system of linear congruences in 2 variables
Given:
$$6x+7y \equiv 17 \pmod{42} \tag1$$
$$21x+5y \equiv 13 \pmod{42} \tag2$$
Here's my initial attempt at solving the above system.
$(2) \times 35$: $$21x+7y \equiv 35 \pmod{42} \tag3$$ $(3)-(1)$: $$15x
\equiv 18 \pmod{42}$$ $$5x \equiv 6 \pmod{14}$$ $$x \eq... |
H: On the solutions of $y''(t)=f(y(t))$.
This topic has been inspired by some time spending on trying to refining my knowledge about PDE's in general. Then everybody knows how to solve $$y''(t)=\pm y(t).$$ Then I tried to slightly modify the question and I focused on $$(\therefore)\;y''(t)=f(y(t)),\; f\in C^1(\mathbb ... |
H: Inequality. $\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab$
I want to prove the following inequality :
$$\frac{1}{16}(a+b+c+d)^3 \geq abc+bcd+cda+dab, $$
$a,b,c,d \in \mathbb{R}_{+} .$
In my book, at the answers chapter the author uses AM $\geq$ GM, but I haven't any idea how I can use that.
Thanks :)
AI: Al... |
H: Historical Development of CW Complexes
I recently started learning about CW complexes. Although my understanding of them is somewhat nascent, I see that one can deduce a number of useful properties of a space if one can show it is a CW complex. For example, if $X$ is a CW complex, then we can immediately conclude t... |
H: How many times does $3$ appear in the $\{3,6,9,\ldots,1002\}$?
A set $S$ is defined as $S = \{3,6,9,\ldots,1002\}.$
How many times does digit three appear in the decimal representations of members of $S$?
I have solved this question by seeing pattern like $3, 6, 9$, but is there a general solution?
AI: Hint: how ma... |
H: A modification of the harmonic series that causes it to converge.
Possible Duplicate:
sum of inverse of numbers with certain terms omitted
We are working in base $b$, a positive integer greater than $1$. Let $S$ be the set of all positive integers that contain a given symbol in their base $b$ expansion. Let's ch... |
H: Advice for study plan.
I'm studying math by myself, and now I'm wondering what is the most appropriate for me to study. I think my interest goes to rather pure mathematics than applied math or math for engineers.
So far I finished studying Thomas' Calculus (12th Edition), and I'm basically done with reading the fir... |
H: software tool for accurate visualization of algebraic curves
First of all, I apologize since this is not strictly speaking a "mathematical" question but I could not find a better place for it.
For a work presentation I need a tool for accurate visualization of implicit
algebraic curves in 2D. I tried ImplicitPlot ... |
H: Manifold/Topology Notation
I have a basic notation related doubt as follows:
Let $M\subset \mathbb{R}^N$ be a manifold. What does $C^\infty(M)$ denote in $f \in C^\infty(M)$?
AI: Each $m \in M$ lies in some coordinate chart $(U, \phi)$, where $\phi: U \to U' \subset \mathbb{R}^k$ is a homeomorphism ($k$ may be les... |
H: How do you pronounce the inverse of the $\in$ relation? How do you say $G\ni x$?
If I am talking about sets $G$ and $H$ and I want to say in words that $G\subset H$, I, like everyone else, will say that $G$ is contained in $H$, or that $H$ contains $G$.
But if I am talking about a set $G$ and a single point $x$, I ... |
H: Proving an Integral Theorem
Theorem: Let $f$ be a continuous function on $[a,b] \in \Bbb R$. Prove that there exists a $c \in [a,b]$ so that $$ f(c) = \frac{1}{b-a}\int_a^b f(x)\,dx $$
I cannot use derivatives or any other integral theorems, I am doing these proofs using Riemann Sums and continuity.
My Thoughts:
... |
H: How to show that e.g. $\cos(z)$ is analytic using Cauchy- Riemann differential equations?
How to show that e.g. $\cos(z)$ is analytic using Cauchy-Riemann differential equations [$u_x(x,y)=v_y(x,y)$ and $u_y(x,y)=-v_x(x,y)$]? Do all analytic functions satisfy Cauchy-Riemann differential equations (CRDE)? What is ... |
H: Closed set and Open set
EXERCISE Show that $F$ is closed set $\iff$ if for all ball centered at $x$ contains points in $F$, then $x\in F$.
Thanks by your replies.
DEFINITION Given a metric space $(X,\rho)$ and a point $x$, an open ball about $x$ with radius $\epsilon$ is the set $B(x,\epsilon)=\{y\in X:\rho(x,y)<\... |
H: In need of tips/suggestions when to add or multiply probabilities
I am having trouble deciding when to add or when to multiply probabilities as in the following example. I know that by constructing Probability tree diagrams we could multiply along branches and add vertically. However I could definitely use more sug... |
H: Suppose that $d$ divides $2^{r+1}$ but does not divide $2^r$ or $2$. Show that $d$ must equal $2^{r+1}$.
Let $r$, $d$ be positive integers, with $d>1$.
Suppose that $d$ divides $2^{r+1}$ but does not divide $2^r$ or $2$. Show that $d$ must equal $2^{r+1}$.
AI: By unique prime factorization, $d = 2^k$ for some $1 \... |
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