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H: Calculating closest and furthest possible diagonal intersections.
Calculating closest and furthest possible diagonal intersections.
Please refer to the image attached. It represents a $2D$ grid with the following properties:
The grid origin is $(1,1)$ at the top-left.
The x coordinates are $1, 2, 3... infinity$.
T... |
H: Munkres Order Topology example question (Section 14 Example 4)
in section 14, Munkres introduces the order topology, and gives this example:
The set $X$ = {1,2} $\times \mathbb{Z}_{+}$ in the dictionary order is an example of an ordered set with a smallest element. Denoting 1 $\times$ $n$ by $a_{n}$ and 2 $\times$ ... |
H: $C^\infty$ dense in Sobolev spaces
I'm reading a theorem which says that $C^\infty$ intersection with $W^{k,p}$ is dense in $W^{k,p}$. I don't understand why they take the intersection. Isn't it $C^\infty$ a subspace of $W^{k,p}$? Thanks.
AI: You don't say what space the functions are on, but here let's consider ... |
H: Trigonometry- why we need to relate to circles
I'm a trigonometry teaching assistant this semester and have a perhaps basic question about the motivation for using the circle in the study of trigonometry. I certainly understand Pythagorean Theorem and all that (I would hope so if I'm a teaching assistant!) but am l... |
H: Continuity of powers in a Banach algebra.
There are some theorems that say in a unital C* algebra $A$ when one can deduce that the functional calculus of a continuous function f is continuous as map from some subset of $A$ to $A$. In the proof of one of these theorems it assumes that $A \mapsto A^n$ is continuous,... |
H: Why is gradient noise better quality than value noise?
I have been reading about the mathematics behind Perlin noise, a gradient noise function often used in computer graphics, from Ken Perlin's presentation and Matt Zucker's FAQ.
I understand that each grid point, $X$, has a pseudo-random gradient associated with ... |
H: meaning of $Hx \ /xH$
$H$ is not normal group, take $x \in G$ such that $Hx \ /xH$. That is, there is $h ∈ H$ such that $hx \not \in xH$. Then $hxH \not = H$, and, if the same definition were to work, supposedly $hH ∗ xH = (hx)H \not = xH$
But, on the other hand, since $hH = eH$,
$hH ∗ xH = eH ∗ xH = (ex)H = x... |
H: For what values of $x$ would $nx \equiv 0\pmod{(x-n)}$?
Given the positive integer $n$, for which integers values of $x$ will the equation below be satisfied?
$$nx \equiv 0\pmod{(x-n)}$$
Also, how many positive integers $x$ exist for a given $n$?
AI: Since $x\equiv n$ modulo $x-n$ we have the logical equivalence (s... |
H: What type of object is a differential form?
This is a naive question; apologies in advance.
For a point $p \in M$ on a smooth manifold $M$, a differential form can be viewed as
a map
$$T_p M\times \cdots \times T_p M \to \mathbb{R} \;.$$
What puzzles me about this object is that it is not "differential."
Yes, I kno... |
H: Is the following a ring?
Is $\{a+bi: a,b \in \mathbb{Z} \} $ and $i^2=-1$, a ring under the usual operations of addition and multiplication?
It needs to be closed under addition and multiplication which it seems like
Addition needs to be commutative and associative which is quite straightforward.
It must have an i... |
H: Differential of Det Map
I want to prove differential of det map at some matrix $A$ is given by $f(A)\operatorname{tr}(X)$
let $f:GL_n(\mathbb{R})\rightarrow\mathbb{R}$ is det map i.e $f(A)=\det A$,
claim: $f_{*}(AX)=(f(A))\operatorname{tr}(X)$
proof of claim:
Let a curve $c(t)=Ae^{tX}$ such that $c(0)=A,c'(0)=AX$. ... |
H: Can't understand the solution of this combination
It's a journey with 9 people using two boats.
A boat can have maximum 7 people
B boat can have maximum 4 people
How many ways journeys can be done?
My solution:
A B
7 2
6 3
5 4
So, number of ways= ${^7C_7}*{^4C_2} + {^7C_6}*{^4C_3} + {^7C_5}*{^4C_4}$ = 12+28+7... |
H: Proving there are infinitely many primes of the form $a2^k+1.$
Fix $k \in \mathbb{Z}_+$. Prove that we can find infinitely many primes of the form $a2^k +1,$ where $a$ is a positive integer.
We can use the result that:
If $p \ne 2$ is a prime, and if $p$ divides $s^{2^t}+1$, for $s > 1$ and $t \ge 1$, then $p \equ... |
H: Question from Rudin: Jensen?
I came across this question from Rudin's Real and Complex Analysis, 3rd Edition (p.75 # 25)
"Suppose that $\mu$ is a positive measure on $X$ and $f:X\rightarrow (0,\infty)$ satisfies $\int_X f d\mu = 1$. Prove for every $E \subset X$ with $0 < \mu(E) < \infty$ that $\int_E(\log{f})d\m... |
H: Convergence of an ugly series
I am wondering for $n > 1$ if the following series converges and if possible what it equates to.
$$\sum _{m=0}^{\infty }\dfrac {\log\left( \dfrac {\left( n+m-1\right) !} {\left( n-1\right) !}\right) -m} {\prod _{t=n}^{t=n+m-1}\log t}$$
I suspect it should sum to 1 as they are mutually ... |
H: Finite summation
Possible Duplicate:
why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$
What is the proof without induction for :
$(1)$ $\sum_{i=1}^n\ i^2= \frac{n(n+1)(2n+1)}{6}$
$(2)$ $\sum_{i=1}^n\ i^3=\frac{n^2(n+1)^2}{4}$
AI: One can give a proof that has a combinatorial flavour. We wa... |
H: Eigenvectors of matrices formed from self-outer products of non-orthogonal vectors
Let $v_1$ and $v_2$, with $v_1\neq v_2$ and $\langle v_1,v_2\rangle\neq 0$, be two non zero vectors in an $N$ dimensional Hilbert space. Let $\mathbf{S}=v_1\otimes v_1 +v_2\otimes v_2$ be an $N\times N$ Hermitian matrix ($\otimes$ de... |
H: Finding all roots of polynomial system (numerically)
I want to numerically find all the roots of a system of polynomials (n equations in n variables). Since I can compute the Jacobian for the system (analytically or otherwise), I can use the Newton Raphson method to find a single root (e.g. as described in the Nu... |
H: Borel linear order cannot have uncountable increasing chain
I am trying to make sense of what this theorem from C.I. Steinhorn, Borel Structures and Measure and Category Logics, says.
Theorem 1.3.3. A Borel linear order cannot have an uncountable increasing or decreasing chain.
I didn't find definitions in the pape... |
H: Calculating Laplace's law for bigrams
Reading this PDF, I encountered a very simple simplification that I can't obtain. Basically, it asks what is the probability of the occurrence of a word $w_{n}$ given that we know another word $w_{n-1}$ already appeared before. Usually, this is calculated using the Maximum Like... |
H: Integrating $\int \frac{dx}{(x+\sqrt{x^2+1})^{99}}$
I am bugged by this problem: how do I evaluate this?
$$\int \frac{dx}{(x+\sqrt{x^2+1})^{99}}.$$
A closed form will be convenient and fine. Thanks (it does not seem particularly inpiring).
AI: If you sub $u=x+\sqrt{x^2+1}$, that alone will make the integral a simpl... |
H: About finiteness of trees
I am reading a book of Michael Sipser "Introduction to the theory of computation", and there is a theorem, which he gives without a proof:
"If every node of a tree has only finitely many children and every branch of the tree has finitely many nodes, the tree itself has only finitely many n... |
H: Is there a good description of the series $\sum_{j=0}^{n}a^j b^{n-j}$?
To what does the serie $\sum_{j=0}^{n}a^j b^{n-j}$ converge? Does this serie have a name?
AI: I'm assuming you are asking for the convergence for $n\to\infty$.
We have (assuming $a\ne b$)
$$\sum_{j=0}^na^jb^{n-j} = b^n\sum_{j=0}^n\left(\frac{a}{... |
H: A module with non-zero dual that does not canonically embed in its bidual
I'd like to find a ring $A$ and an $A$-module $E$ such that $E^*\neq\{0\}$ and the canonical mapping of $E$ into $E^{**}$ is not injective.
As a hint (it's an exercise in Bourbaki's Algebra) I have "consider a module containing an element wh... |
H: Probability - multinomial partitioning
Question:
Joe is in his hunting blind when he locates $20$ geese, $25$ ducks, $40$ eagles, $10$ cranes, and $5$ flamingos. Joe randomly selects six birds to target, what is the probability that at least one of each species is targeted?
The correct answer is apparently $0.0398... |
H: Gram-Schmidt Orthogonalization - does it distort?
I am writing a 3D solar panel positioning programme and have a section of code where I use the Gram-Schmidt Orthogonalization process to go from 3D to 2D for easier calculations.
(For reference, here is the process in more detail: 3D to 2D rotation matrix)
My simple... |
H: Is there any way to determine the first $3$ digits of $2^m-2^n$ ($n\leq m\leq 10^{100}$)
It's a problem in my ACM training.
Since $n,m$ are really huge, I don't think the algo $2^n=10^{n\log2}$ will work. Also, it's not really wise to calculate the value of $2^n$, I think. So I stack.
Can any one come up with a way... |
H: $|P(x)|$ differentiable at a root $x_0$
Let $p(x)$ be a polynomial and suppose that $x_0 \in \bf R$ is a real root i.e. $p(x_0) = 0$. When will $|p(x)|$ be differentiable at $x_0$?
My Thoughts
For polynomials such as $f(x) = x$, we run into trouble at roots of odd multiplicity i.e. $|x|$ is not differentiable at $x... |
H: If $(f'_n)$ converges uniformly, does $(f_n)$ necessarily converge uniformly?
I've been studying complex analysis problems, and get stuck on the following:
Let $D \subseteq \mathbb{C}$ be a domain (open connected set) and $z_0 \in D$. Assume that $(f_n)$ is a sequence of analytic functions in $D$ such that $\lim_{... |
H: Need refresher on z-transforms and difference equations
I recently tried showing someone else how to solve a difference equation using z-transforms, but it's been a long time and what I was getting didn't look right. I was trying to solve the recurrence we all know and love
$$x[n]=x[n-1]+x[n-2],x[0]=0,x[1]=1$$
usi... |
H: $I^2$ does not retract into comb space
In Hatcher's book on Algebraic Topology in Chapter 0 (where the Homotopy Extension Property is discussed) the claim is being made that there is no retraction of $I^2$ into $I \times \{0\} \cup A \times I$ where $A = \{0, 1, \frac 12 , \frac 13, \ldots \}$. However I couldn't c... |
H: Show angle between $P-P_0$ and plane tangent to surface at $P_0$ approaches 0
Let $f(x,y) \rightarrow R$ be a function with continuous partial derivatives, and let $S$ be the surface $z=f(x,y)$ in $R^2$. Let $P_{0}=(x_0,y_0,z_0 )$ be a point in S and $P=(x,y,z)$ be some other point in S. We're asked to show that $a... |
H: Is there a non-contradictory non-trivial axiomatic system in which Gödel's theorem is undecidable?
Gödel's Incompleteness Theorem states that in any non-contradictory non-trivial axiomatic system there are certain statements or theorems whom cannot be proved in that system,
For example in ZFC, assuming it is non-co... |
H: Application of Radon Nikodym Theorem on Absolutely Continuous Measures
I have the following problem:
Show $\beta \ll \eta$ if and only if for every $\epsilon > 0 $ there exists a $\delta>0$ such that $\eta(E)<\delta$ implies $\beta(E)<\epsilon$.
For the forward direction I had a proof, but it relied on the use of ... |
H: Showing that $a+b\sqrt{2}$ is associative over multiplication
I need to show that $a+b\sqrt{2}$ is associative over multiplication. This is what I have so far. I may be taking a wrong route so just please let me know.
$$(a+b\sqrt{2})*((c+d \sqrt{2})*(e+f \sqrt{2} )) = \\
(a+b\sqrt{2})*(ce+2df+(cf+de)\sqrt{2}) $$
a... |
H: Create a C++ program to evaluate the following series: $\sin x \approx x - \frac{x^3}{3! }+\frac{x^5}{5!}-\frac{x^7}{7!}\cdots\pm\frac{x^n}{n!}$
This is a problem that my friend has. He says he has to create a C++ program to evaluate the series (He says we must use functions):
$$\displaystyle\large\sin x\approx x- ... |
H: Derivation of formula for estimating error in bulk-volume
In my textbook, a formula for estimating error in bulk-volume measurements is derived, but I don't quite follow one step in the derivation. The book writes the following:
The bulk volume of a porous-rock core sample can be measured in two steps, by weighing... |
H: book with lot of examples on abstract algebra and topology
I have read a lot of blogs and forums to find out the best of the books on Abstract algebra and Topology but on going through the books I realized that they are full of proofs and all kind of theorems and corollaries.
But the disappointing point is that the... |
H: Limit of Integral of square of function
Let $f: [0,1)\rightarrow \mathbb{R}$ be continuous and nonnegative on $[0,1)$. Prove that if $\lim_{a\rightarrow 1^{-}}{\int_{0}^{a}{f(x)^{2}dx}}$ exists, then $\lim_{a\rightarrow 1^{-}}{\int_{0}^{a}{f(x)dx}}$ exists too.
AI: The Cauchy-Schwarz inequality (taking the inner pr... |
H: Group of order $1225$ is abelian
A problem asked me to show that a group, $G$, of order $1225= 5^{2}\times 7^{2}$ is abelian.
I did this by using Sylow's Theorem to show that we have two normal subgroups: one of order $49$ and the other of order $25$. Since the intersection of these subgroups must be $1$, it follow... |
H: I want to find out the angle for the expression $a^3 + b^3 = c^3$.
like in pythagorean theorem angle comes 90 degree for the expression $a^2 + b^2 = c^2$, however I know that no integer solution is possible.
AI: There is no single angle corresponding to the relationship $a^3+b^3=c^3$.
Suppose that a triangle has si... |
H: Problem with ratios of integrals
I have the following integral from a paper I'm reading:
$$f(z)=\frac{\displaystyle\int_0^{\pi/2}\,\tan \alpha\, J_0(z \sin\alpha)\, d\alpha}{\displaystyle \int_0^{\pi/2}\tan\alpha\,d\alpha}$$
where $J_0$ is the Bessel function of first kind and zeroth order. The authors claim that... |
H: Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $
I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:
DEFINITION Given $x\in \Bbb R$, the integer part... |
H: Map $\{ z=x+iy : |x|\geq y \}$ with a branch cut on the negative imaginary axis from $[-i,0]$ to the unit disc
I have a conformal map that I've been having problems with.
Map the set $\{ z=x+iy : |x|\geq y \}$ with a branch cut on the negative imaginary axis from $[-i,0]$ to the unit disc.
I've tried a few thin... |
H: What is the integral of $\int\frac{(x-1)e^{1/x}dx}{x}$?
I have been trying to solve this integral $\int\frac{(x-1)e^{1/x}dx}{x}$
I used WolframAlpha to solve it but it doesn't show the process.
The solution is $e^{1/x}{x} + constant$
AI: We can integrate this by separating the integrand and integrating by parts:
$... |
H: Fourier transform of even/odd function
How can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is odd and purely imaginary function?
AI: Let $f: ... |
H: Is the subgroup of quadratic residues modulo $N = pq$ a cyclic group if $p$ and $q$ are primes?
I saw this question When is the group of quadratic residues cyclic? and the answers to that.
I have a similar question. Assume $N=pq$ where $p$ and $q$ are primes. We know that $\mathbb{Z}^*_N$ is not cyclic. So my ques... |
H: If $\exp(itH) A \exp(-itH) = A$ for all $t$, do $A$ and $H$ commute?
Let $H$ be a self-adjoint $n \times n$ matrix with complex entries. $H$ gives rise to a continuous 1-parameter group of unitaries $t \mapsto U_t = \exp(itH) : \mathbb{R} \to U(n)$.
Let $A$ be some other $n \times n$ matrix. If $A$ commutes with $H... |
H: Relationship of Fourier series and Hilbert spaces?
I just read in a textbook that a Hilbert space can be defined or represented by an appropriate Fourier series. How might that be? Is it because a Fourier series is an infinite series that adequately "covers" a Hilbert space?
Apart from this I (a mathematical novice... |
H: Finding $F(x)$ so that $F'(x) = e^{-x^2}$
Question: Let $f(x) = e^{-x^2}$. Find a formula for a function $F$ so that $F'(x) = f(x)$.
My Thoughts:
We have just proven the fundamental theorem of calculus, part II of which states:
Let $f$ be a continuous function on a finite interval $[a,b]$. Define $$F(x) := \int_a^... |
H: Prove that Brownian Motion absorbed at the origin is Markov
I'm trying to prove that Brownian motion absorbed at the origin is a Markov process with respect to the original filtration $\{\mathcal{F}_{t}\}$. To be more specific, let $(B_{t},\mathcal{F}_{t})_{t \geq 0}$ be a 1-d brownian motion such that $B_{0}=x,\ a... |
H: Implicit function theorem and PDE; do we get uniqueness?
Please see this page: The implicit function theorem: A PDE example.
In the implicit function theorem they quote, uniqueness is not mentioned. But the inverse function theorem (which is equivalent to the IMFT), we do get uniqueness. So am I right that the solu... |
H: 10 most significant digits of the sum of a 100 50-digit numbers
This is about Project Euler #13. You are given a 100 50-digit numbers and are asked to calculate the 10 most significant digits of the sum of the numbers.
The solution threads stated that we are only required to sum, the 11 most significant digits of e... |
H: Can $\mathbb CP^n$ be the boundary of a compact manifold?
Can $\mathbb CP^n$ be the boundary of a compact manifold?
For example, when $n=1$, $\mathbb CP^n=S^2$, therefore it is the boundary of $B^3$.
AI: Short answer: For $n$ odd, yes. For $n$ even, no.
The explanation for each case:
$n$ odd: There is a fibration
... |
H: A polynomial can be considered a number?
According to wikipedia the Euler's number is:
$$e = 1 + \frac{1}{1} + \frac{1}{1\times 2} + \frac{1}{1\times 2\times 3} + \frac{1}{1\times 2\times 3\times 4}+\cdots$$
And I see it's structure is quite similar to the structure of a polynomial:
$$a_nt^n+a_{n-1}t^{n-1}+\cdots+a... |
H: Galois extension and Tensor product
The following theorem is proved in Bourbaki's algebra.
They use the technique of Galois descent.
I'd like to know the proof without using it if any.
Theorem
Let $K$ be a field.
Let $\Omega/K$ be an extension.
Let $N/K$ and $L/K$ be subextensions of $\Omega/K$.
Suppose $N/K$ is a ... |
H: How was the Monster's existence originally suspected?
I've read in many places that the Monster group was suspected to exist before it was actually proven to exist, and further that many of its properties were deduced contingent upon existence.
For example, in ncatlab's article,
The Monster group was predicted to ... |
H: What is the meaning of the term "inductively P map"?
In this page is the definition of an inductively open map. But in this pdf is the definition of a inductively P map, where P is a property of maps.
But there is a difference in the definitions. In the former it's said that "the image of the subset is the same as ... |
H: Using the sum of squares formula to solve more complex sums.
I'm studying integration and trying to figure out how to use the sum of squares formula to solve more complicated sums.
For example: knowing that $$\sum_{i=1}^n i^2 = \frac{n (n+1) (2 n+1)}{6}$$
how can we simplify
$$\sum_{i=1}^n (i/n)^2$$
AI: The key h... |
H: Uniform integrability question
I know I keep asking similar sorts of questions, but I want to understand exactly what is going on here. So, I will list a full definition and then the problem. (Basically, I want to prove something using Vitali's theorem)
A sequence of functions $\{f_n\} \in L_1(\mu)$ is called un... |
H: Finding a coordinate system for the kernel and nullity of a linear map
Let $T : C^{\infty}(\mathbb{R}) \rightarrow C^{\infty}(\mathbb{R})$ be a linear map, which is defined by $T = \frac{d^2}{d x^2} - 3\frac{d}{dx} + 2 \text{id}$, that is, a map which takes a function $f(x)$ and maps it to
$$f''(x) - 3f'(x) + 2f(x)... |
H: Need help to understand a proof using functional calculus.
I am having trouble with understanding part of a proof, I wrote the entire proof out, and I marked the part which I do not understand. I am hoping that someone could explain it to me. I guess I don't understand functional calculus that well :( .
Suppose tha... |
H: What is the lower bound and upper bound on time for inserting n nodes into a binary search tree?
So given a $n$ array of few numbers(say $n$) we can sort them using the binary search tree (BST) as a black box . In order to that we first build a BST out of the array taking all the elements in order and then do an i... |
H: showing that $X$ is measurable iff $X^{-1}(C')\subset\beta$
suppose $X:\Omega\rightarrow\Omega'$ where $(\Omega,\beta)$ and $(\Omega',\beta')$ are two measurable space. suppose $C'$ generate $\beta'$;how can show $X$ is measurable iff $X^{-1}(C')\subset\beta$
AI: Let $C'$ be a system of subsets of $\Omega'$ such th... |
H: Match covered graph is 2-connected
Seems to be an easy question, but I can't find the right direction.
Let $G$ connected graph on at least 4 vertices, such that every edge in it, participates in a perfect matching. Prove that $G$ is 2-connected.
Any help will be appreciated.
Thanks in advance.
AI: Gerry Myerson has... |
H: How to use Fermat's theorem in congruence problems
Two days ago I asked about how to solve questions of the type:
Find last digit of $27^{27^{26}}$.
or
Find the remainder when $27^{45}$ is divided by $7$, using congruences.
I have now got this method completely, but I am still not able to solve these ... |
H: Why isn't math on the sine of angles the same as math on the angles in degrees?
I noticed something just now. This is probably a stupid question, but I'm going to ask it anyway. Because when I discover that my understanding of a topic is fundamentally flawed, I get nervous.
Basically I'm suppose to show that the... |
H: Covariance matrix of a complex random variable
If one considers a complex random variable as the joint random variable of the real and complex part, the covariance matrix of two complex random variables $Z_{1}=X_{1}+iY_{1}$ and $Z_{2}=X_{2}+iY_{2}$ becomes a $4\times 4$ matrix
$$\begin{pmatrix}C^{(rr)}&C^{(ri)}\\C^... |
H: Noetherian domains with finitely many primes
For any domain $A$ let $A^\times$ be its group of units.
Let $A$ be a noetherian domain with only finitely many prime ideals, and field of fractions $K$.
Is the group $K^\times/A^\times$ finitely generated?
AI: The answer is no in general.
First by an answer to the que... |
H: Limit of a sequence of matrices
I'm preparing or my exam in linear algebra and I'm stuck with a question. I've tried to find some help in my textbook (Linear Algebra and its applications, 4th Edition, By David C. Lay). I can't find anything about it (maybe because the question is written in danish and I'm having t... |
H: Complex valued function $ \cos\sqrt z$
I'm looking for the right argument why the function $ \cos\sqrt z$ is analytic on the whole complex plane. As far as I understand, a holomorphic branch of $\sqrt z$ can only be found on the cut plane (without negative numbers) since the Argument function isn't continuous every... |
H: find the range for the expression, $f(n)=\frac{n^2+2\sqrt{n}(n+4)+4^2}{n+4\sqrt{n}+4}$
find the range for the expression, $f(n)=\frac{n^2+2\sqrt{n}(n+4)+4^2}{n+4\sqrt{n}+4}$ for $36 \le n \lt 72$
$f(n)=\frac{(\sqrt{n}+n+4)^{2}-9n}{(\sqrt{n}+2)^2}$,
$\sqrt{36}=6$
$\sqrt{72}=6\sqrt{2}$
AI: I decided to post this as ... |
H: What do I need to know to simulate many particles, waves, or fluids?
I've never had a numerical analysis course so I don't know what I need to know. I'm just wondering what kind of books I should get to make me able to simulate these things. I'm wanting to simulate in 3 dimensions. I was thinking about getting a... |
H: Matrix Manipulation : trick to sum elements of vector
For example, I have vector A = mx1. I want to sum each elements of vector A by :
1) Define vector B = 1xm.
2) Take : AxB = 1x1. So, this vector will be the value of sum elements of vector A.
But, I don't know how to choose vector B to this trick.
So, my questi... |
H: Why $(\mathbb{N} \rightarrow \mathbb{N}) \subseteq (\mathbb{N} \leadsto \mathbb{N})$?
Written in natural language, the sets of all total functions from naturals to naturals is a subset of the sets of all partial functions of such.
$$(\mathbb{N} \rightarrow \mathbb{N}) \subseteq (\mathbb{N} \leadsto \mathbb{N})$$
We... |
H: Matrix representing operation, with respect to coordinate systems
Let $T : \mathbb{R}[x]_{\leq 1} \rightarrow \mathbb{R}[x]_{\leq 2}$ be the linear map defined by the differential operator $T = (x-2) \frac{d}{d x} - \text{id}$.
Now my exercise asks me to find the matrix representing $T$ with respect to the standard... |
H: Safe use of generalized inverses
Suppose I'm given a linear system $$Ax=b,$$ with unknown $x\in\mathbb{R}^n$, and some symmetric $A\in\mathbb{R}^{n\times n}$ and $b=\in\mathbb{R}^n$. Furthermore, it is known that $A$ is not full-rank matrix, and that its rank is $n-1$; therefore, $A$ is not invertible. However, to ... |
H: Is a countable set $A$ dense in an uncountable set $B$ by analogy of $\mathbb{Q}$ being dense in $\mathbb{R}$?
Let $A$ be any countable set and $B$ any uncountable set with the same infinity as $\mathbb{R}$. Then we must have a one-to-one map $\phi:A\rightarrow\mathbb{Q}$. So, can we say by all this that $A$ is in ... |
H: What is the difference between the supremum (least upper bound) and the minimal upper bound
I understand the difference between the supremum and the greatest element and between the supremum and the maximal elements. But I'm not sure about the difference between the supremum (least upper bound) and the minimal uppe... |
H: Using the (Complex) Identity Theorem with a limit point on the Boundary.
Question: If we have a limit point for the set of zeros which is on the boundary of our domain $D$, does the (complex analysis) identity theorem still hold?
Motivation: The identity theorem states that if two holomorphic functions agree at a l... |
H: What does mantissa mean here?
I was going through this SO post on Math.random() vs Random.nextInt(int) and encountered the following line :
Random.nextDouble() uses Random.next() twice to generate a double that
has approximately uniformly distributed bits in its mantissa, so it is
uniformly distributed in the ... |
H: Simple Function Design
I want to design a function that satisfies the following:
generally speaking, $f(x) = y$
$$f(5.51) = 1$$
$$f(95.31) = 200$$
How can I go about designing the function to satisfy these requirements?
AI: If you want a function of the form $f(x)=a e^{bx}$ then simply solve for $a$ and $b$ using y... |
H: bound on the growth of a function
I was playing around with some functional equations and I came across the following question..
Suppose we have a $C^2(\mathbb R^2, \mathbb R)$ function $f$ with first and second derivatives globally bounded..
Moreover we know that $f(x, 0)=f(0, y)=0$ for any $x, y\in \mathbb R$.
Is... |
H: Why is the last digit of $n^5$ equal to the last digit of $n$?
I was wondering why the last digit of $n^5$ is that of $n$? What's the proof and logic behind the statement? I have no idea where to start. Can someone please provide a simple proof or some general ideas about how I can figure out the proof myself? Than... |
H: Inequality. $\frac{a}{3a-b+c}+\frac{b}{3b-c+a}+\frac{c}{3c-a+b} \geq 1$
Let $a,b,c$ be the side lengths of a triangle. Prove that
$$\frac{a}{3a-b+c}+\frac{b}{3b-c+a}+\frac{c}{3c-a+b} \geq 1 . $$
I found this inequlity in the chapter entitled Cauchy-Schwarz, but I cannot find a proof for this inequality. I used t... |
H: How to make something that will cap on 20?
A user on the chat asked how could he make something that would cap when it gets a specific value like 20. Then the behavior would be as follows:
$f(...)=...$
$f(18)=18$
$f(19)=19$
$f(20)=20$
$f(21)=20$
$f(22)=20$
$f(...)=20$
He said he would like to perform it with a regu... |
H: Proving a function is not integrable
Let $(X, \Sigma,\mu)$ be a $\sigma$-finite measure space, and let $f:X\to\mathbb R^+$ be a measurable function such that $$\mu(\{x\in X\,\colon\,f(x)>t\})>\frac{1}{1+t},\; \forall t>0.$$ Prove then that $f$ is not integrable.
I've tried to derive a contradiction, however my orig... |
H: what's the general form of 3D projective mapping?
I know that the general form of a 2D projective transformation is following rational linear mapping:
$(u, v) \mapsto (x, y)$
$$
x = \frac{\mathit{a}\mathit{u}+\mathit{b}\mathit{v}+c}{\mathit{g}\mathit{u}+\mathit{h}\mathit{v}+i}\\
y = \frac{\mathit{d}\mathit{u}+\math... |
H: How to put eggs in baskets
A farmer has c chickens who have each laid e eggs, which she will put into b baskets. Each basket has a probability p(d) of being dropped, which breaks all the eggs in the basket. How should the farmer distribute the eggs into the baskets in such a way that she minimizes the number of c... |
H: Help proving the primitive roots of unity are dense in the unit circle.
I'm having difficulty understanding how to prove that the primitive roots of unity are in fact dense on the unit circle. I have the following so far:
The unit circle can be written $D=\{x\in\mathbb{C}:|x|=1\}$.
The set of primitive $m$-th roots... |
H: Countability of irrationals
Since the reals are uncountable, rationals countable and rationals + irrationals = reals, it follows that the irrational numbers are uncountable. I think I've found a "disproof" of this fact, and I can't see the error.
Since $Q$ is countable, list $q_1, q_2, \dots$ such that $q_i < q_{i+... |
H: Why is $\pi$ irrational if it is represented as $C/D$?
$\pi$ can be represented as $C/D$, and $C/D$ is a fraction, and the definition of an irrational number is that it cannot be represented as a fraction.
Then why is $\pi$ an irrational number?
AI: A rational number is a number that can be expressed as $p/q$, whe... |
H: Intersection of compact and discrete subsets
I have difficulties with a rather trivial topological question:
A is a discrete subset of $\mathbb{C}$ (complex numbers) and B a compact subset of $\mathbb{C}$. Why is $A \cap B$ finite? I can see that it's true if $A \cap B$ is compact, i.e. closed and bounded, but is ... |
H: We have to select a group of 7 out of a group of 9 men and 11 women,how many seven member teams consist of at least one man?
We have to select a group of 7 out of a group of 9 men and 11 women
Q : How many seven member teams consist of at least one man ?
Now I know that the answer is ${20 \choose 7}-{11 \choose 7... |
H: How to combine ratios? If $a:b$ is $2:5$, and $c:d$ is $5:2$, and $d:b$ is $3:2$, what is the ratio $a:c$?
How would I go about solving this math problem?
if the ratio of $a:b$ is $2:5$ the ratio of $c:d$ is $5:2$ and the ratio of $d:b$ is $3:2$, what is the ratio of $a:c$?
I got $\frac{a}{c} = \frac{2}{5}$ but tha... |
H: When does $n$ divide $a^d+1$?
$\newcommand{\ord}{\operatorname{ord}}$
For what values of $n$ will $n$ divide $a^d+1$ where $n$ and $d$ are positive integers?
Apparently $n$ can not divide $a^d+1$ if $\ord_n a$ is odd.
If $n\mid (a^d+1)\implies a^d\equiv -1\pmod n\implies a^{2d}≡1\pmod n \implies\ord_na\mid 2d$ but... |
H: How do we prove that the Pythagorean theorem holds for a right angled isoceles triangle with sides, $a,b,a$.
Possible Duplicate:
What is the most elegant proof of the Pythagorean theorem?
How do we prove that the Pythagorean theorem holds for a right angled isoceles triangle with sides, $a,b,a$.
For a right angl... |
H: Solving a literal equation containing fractions.
I know this might seem very simple, but I can't seem to isolate x.
$$\frac{1}{x} = \frac{1}{a} + \frac{1}{b} $$
Please show me the steps to solving it.
AI: You should combine $\frac1a$ and $\frac1b$ into a single fraction using a common denominator as usual:
$$\begin... |
H: A little help with algebra (homework)
I was brushing up on my algebra a little, and I've seen that I've become rusty with some of the concepts. Indications for solving these would perhaps be better suited rather than full answers, as the goal is to figure it out myself.
Show that for any $a, b \in \mathbb{R}$ (wit... |
H: Evaluating a simple definite integral
I'm currently teaching myself calculus and am onto the Mean Value Theorem for Integration.
I am finding the value of $f(c)$ on the function $f(x)=x^3-4x^2+3x+4$ on the interval $[1,4]$.
So, with the equation $(b-a)\cdot f(c)=\int_1^4f(x)dx $, you get
$(4-1)\cdot f(c)=\int_1^4(x... |
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