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I notice that your use of the **closure** terminology is careful: you say "the closure of $A$ in $N$", as you should.
But your use of the **closed** technology is not similarly careful. It should be!
You must not ever ask "Is $A$ closed?" in some kind of absolute sense. It's not like asking "Is $A$ finite?" which... |
I am trying to get some practice in computing tensors in Riemannian and Pseudo-Riemannian manifolds. Is there somewhere that I can check my results? For example right now I am computing the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, Einstein and Weyl tensor and the expressions that solve the Killing ... |
In Velleman's "Calculus: a Rigorous Course," Example 9 from Section 1.3 tasks us with computing $ h(h(x)) $, where $ h(x) = \lfloor 5x - 2 \rfloor $.
My initial solution:
\begin{align*}
h(\lfloor 5x - 2 \rfloor) &= \lfloor 5(\lfloor 5x \rfloor - 2) - 2 \rfloor \\
&= \lfloor 5\lfloor 5x \rfloor - 10 \rfloor - ... |
I am physicist, so I am sorry if I am not too rigorous in the following. I have two (closely related I guess) questions:
1. Let me consider a triangulated manifold $M$ and its simplicial cohomology. Here the Steenrod square is an operation $Sq^q: H^p (M,Z_2) \to H^{p+q}(M,Z_2)$. My manifold is smooth and I have als... |
Steenrod squares and higher cup products for differential forms? |
I am physicist, so I am sorry if I am not too rigorous in the following. I have two (closely related I guess) questions:
1. Let me consider a triangulated manifold $M$ and its simplicial cohomology. Here the Steenrod square is an operation $Sq^q: H^p (M,Z_2) \to H^{p+q}(M,Z_2)$. My manifold is smooth and I have als... |
In a algorithm book I'm reading there's this paragraph (it concerns a recursive algorithm for integer multiplication):
*In general, a number $x$ with an even number $n$ of digits can be expressed in terms of two $n/2$-digit numbers, its first half $a$ and second half $b$:*
$x=10^{n/2} * a + b$
I'm not sure how... |
In Algebraic Topology by Hatcher, p. 199,
> ... we first dualize the short exact sequence $$0 \to C_n(A) \xrightarrow{i} C_n(X) \xrightarrow{j} C_n(X,A) \to 0$$ by applying $\operatorname{Hom}(-,G)$ to get $$0 \to C^n(X,A;G) \xrightarrow{j^*} C^n(X;G) \xrightarrow{i^*} C^n(A;G) \to 0$$
Now, Hatcher shows this b... |
I am trying to make the dynamic system associated with this differential equation by its phase space.It is a simple differential equation since it has separate variables.In class we have seen how to build a dynamic system from an autonomous differential equation And it is clearly seen that this equation is not autonomo... |
I have a parameterized optimization problem
\begin{eqnarray}
\max_{x\in D(\theta)} f(x,\theta)
\end{eqnarray}
Here state space $\Theta=(0,1)$; constraint correspondence $D:\Theta\to (0,+\infty)$ is a non-empty, continuous, compact-valued correspondence; the objective function $f(x,\theta)$ is continuous on $(0,1)... |
Let $\Gamma$ be a congruence subgroup of $\text{SL}_2(\mathbb{Z})$ of level $N$ and let $\pi:\text{SL}_2(\mathbb{Z})\to\text{SL}_2(\mathbb{Z}/N\mathbb{Z})$ be the reduction map, which is surjective. Let $$
C_N=\bigg\{\begin{pmatrix}x\\y
\end{pmatrix}\in(\mathbb{Z}/N\mathbb{Z})^2\colon\text{$x$ and $y$ generate $\math... |
I got stuck on the following problem:
> Which of the following sets are domains in the complex plane:
$$D_1=\{z\in \mathbb{C}:|z^2-3|<1\}$$
$$D_2=\{z\in \mathbb{C}:|z^2-1|<3\}$$
These sets are domains if they are arcwise connected open non-empty subsets of the complex plane. I first tried to sketch these re... |
Are $\{z\in \mathbb{C}:|z^2-3|<1\}$ and $\{z\in \mathbb{C}:|z^2-1|<3\}$ complex domains? |
This specific question has arisen out of my attempts to do a wider exercise ([link][1]).
---
**Context**
We say two functions, $f$ and $g$, are equal if all three of the following conditions are met:
- (i) they have the same **domain**
- (ii) they have the same **codomain**
- (iii) they map the same el... |
That is indeed quite a confusing way to put it. The idea is that for any $y \in Y$, you can find an $x$, written as $g(y) = x$, such that $f(x) = y$. Below I have written a few things out, also for injective maps (because the idea is similar) and I hope this makes it more clear.
Suppose $f : X \to Y$ is a function. ... |
Why is the $\nabla g\neq0$ condition needed for the method of Lagrange multipliers? |
Does this matrix exist? A commonly encountered puzzle |
I have a the following likelihood function:
$L(w)=\frac{1}{n}\sum_{t}log(p(y_{t}/x_{t};\omega))$
and the following probability density:
$p(y_{t} = 1/x_{t};\omega) = \sigma(w^{T}x_{t})$
$p(y_{t} = 0/x_{t};\omega) = (1 - \sigma(w^{T}x_{t}))$
so, $p(y_{t}/x_{t};\omega)$ is binary.
From what I saw in text... |
I have a the following likelihood function:
$L(w)=\frac{1}{n}\sum_{t}\log(p(y_{t}/x_{t};\omega))$
and the following probability density:
$p(y_{t} = 1/x_{t};\omega) = \sigma(w^{T}x_{t})$
$p(y_{t} = 0/x_{t};\omega) = (1 - \sigma(w^{T}x_{t}))$
so, $p(y_{t}/x_{t};\omega)$ is binary.
From what I saw in tex... |
I have a parameterized optimization problem
\begin{eqnarray}
\max_{x\in D(\theta)} f(x,\theta)
\end{eqnarray}
Here state space $\Theta=(0,1)$; constraint correspondence $D:\Theta\to (0,+\infty)$ is a non-empty, continuous, compact-valued correspondence; the objective function $f(x,\theta)$ is continuous on $(0,1)... |
Let's say a particle moves in plane according to $\tag{1} r = f(t), \theta=g(t)$
We want to prove that $\tag{2}A'(t) = \frac{1}{2}r^2\frac{d\theta}{dt}$
Tom Apostol proceeds by assuming that it is possible to eliminate $t$ from equations in $1$, and express $r$ as a function of $\theta$, i.e. $\tag{3} r = R(\thet... |
Is this Proof Correct: If $C$ is a Convex Set with Finite Volume, then $C$ is Unbounded? |
Theorem: If $E$ is a separable metric space and $K$ is continuous on $E \times$ $E, \mathbb{H}(K)$ is separable and the functions in $\mathbb{H}(K)$ are continuous on $E$.
The notation $\mathbb{H}(K)$ denotes the RKHS with reproducing kernel K.
Proof: By assumption, there is a countable set $\{t_1, t_2, \ldots\}... |
The functions in RKHS are not only continuous on $E$, but they are also uniformly continuous on $E$? |
Context and level: Freshman undergraduate just out of calculus 1 doing intro to proofs course
Questions:
- Is my proof correct?
- Is it too verbose, too terse or just right?
- Any other feedback about style or anything?
Theorem. Suppose $F$ is a family of sets. Prove that there is a unique set $A$ that ... |
Consider the dynamical system
\begin{align}
x[k+1]=f [x[k], u[k]]
\end{align}
where $x[k]\in \mathbb R^n, u: \mathbb N\bigcup\{0\}\to \mathbb R^p$
and
\begin{align}
x[k+1]=f [x[k], u[x[k]]]
\end{align}
where $x[k]\in \mathbb R^n, u: \mathbb R^n\to [0,\infty)$
Could anyone explain to me what is the dif... |
Theorem: If $E$ is a separable metric space and $K$ is continuous on $E \times$ $E, \mathbb{H}(K)$ is separable and the functions in $\mathbb{H}(K)$ are continuous on $E$.
The notation $\mathbb{H}(K)$ denotes the RKHS with reproducing kernel K.
Proof: By assumption, there is a countable set $\{t_1, t_2, \ldots\}... |
Jay Cummings real analysis, question 4.14 - can be proved without Cauchy-Shwartz? |
Consider the dynamical system
\begin{align}
x[k+1]=f [x[k], u[k]]
\end{align}
where $x[k]\in \mathbb R^n, u: \mathbb N\bigcup\{0\}\to \mathbb R^p$
and
\begin{align}
x[k+1]=f [x[k], u[x[k]]]
\end{align}
where $x[k]\in \mathbb R^n, u: \mathbb R^n\to [0,\infty)$
Could anyone explain to me what is the dif... |
My question is the following: An integer partition $\lambda$ can be represented as an integer sequence $(f_1,f_2,f_3, \cdots)$ where $f_i$ is the number of parts used in $\lambda$. For instance, $4 + 2 + 2 + 1 + 1 +1$ can be represented as $(3 , 2 , 0 , 1, 0 , 0, \cdots)$. Let $C$ be the set of all sequences $(f_1,f_2,... |
Proving that $D_1$ and $D_2$ are open is not so hard, for instance by noting that for any $a \in \mathbb{C}$ the map $\mathbb{C} \to \mathbb{R} : z \mapsto |z^2 - a|$ is continuous. But I don't think you struggle with this?
For $D_1$ your intuition is correct!To prove it is not connected it suffices of course to fi... |
In order to resolve the definitional issues, I think it is the best way to consider the general form of the notions and then, compare the specific exposition to them.
Remarkably, this is the only textbook I am aware of that treats propositional calculus in a formal language setting; probably because propositional c... |
Theorem: If $E$ is a separable metric space and $K$ is continuous on $E \times E$, $\mathbb{H}(K)$ is separable and the functions in $\mathbb{H}(K)$ are continuous on $E$.
The notation $\mathbb{H}(K)$ denotes the RKHS with reproducing kernel $K$.
Proof: By assumption, there is a countable set $\{t_1, t_2, \ldots... |
Is this Proof Correct: If $C$ is a Convex Set with Finite Volume, then $C$ is Bounded? |
I am trying to get some practice in computing tensors in Riemannian and Pseudo-Riemannian manifolds. Is there somewhere that I can check my results? For example right now I am computing the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, Einstein and Weyl tensor and the expressions that solve the Killing ... |
Determine the general term of the sequence $(a_n)_{n>1}$, strictly decreasing, of strictly positive numbers, which satisfies the properties:
(a) $n \cdot a_n \in \mathbb{N} \setminus \{0\}$ for every $n \in \mathbb{N} \setminus \{0\}$.
(b) $\sum_{k=1}^{n} (-1)^{k-1} \cdot a_k \cdot \binom{n}{k} = \sum_{k=1}^{n... |
Remarkably, this is the only textbook I am aware of that treats propositional calculus in a formal language setting; probably because propositional calculus is often the very initial subject with which teaching of logic begins. Let us review the relevant definition from that part:
> **Definition 3.1.1** For each s... |
Have a look at Fig. 1 modelizing the position of the extremity of a string rolled/unrolled along a convex polygon (here the particular case of a regular octagon) generating a kind of spiral (modelizing rather well a half-egg-shape) ?
[![enter image description here][1]][1]
Fig. 1.
But one can obtain much g... |
the way Fourier transform$$f(ξ) = \int_{-∞}^{∞}
f(x)
e^{-2πixξ}dx$$ describe the frequency extent of the function is derived from the Fourier series by taking that $c_{n}$
of the Fourier series and make the period goes to infinity, so the ${c_n}$
is equivalent to $f(ξ)$
but $c_{n}$
for discrete frequencies a... |
Let's say a particle moves in plane according to $\tag{1} r = f(t), \theta=g(t)$
Let $A(t)$ be the area of the region swept out by the position vector from a fixed time point (e.g. 0), to a later time $t$. We want to prove that $\tag{2}A'(t) = \frac{1}{2}r^2\frac{d\theta}{dt}$
Tom Apostol proceeds by assuming tha... |
Tutte proved the famous result: **Every planar 4-connected graph has a hamiltonian cycle**. But I read in Section 111.6.5 on book *Eulerian Graphs and Related Topics* that the author Herbert Fleischner places **great importance on the impact of simplicity on this theorem**. Even its proof in the end is based on refut... |
Are Tutte's results still valid for planar graphs with multiple edges? |
Let $p\in [1,\infty)$. On $X=L^p(\mathbb{R}; \mathbb{C}^2)$ we consider the operator $D=A+B$, where
$A=\begin{pmatrix}-\partial_x & 0 \\ 0 & \partial_x \end{pmatrix}\quad{and}\quad B=c(x)\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$.
Here, $c\colon \mathbb{R}\to \mathbb{R}$ is a real-valued $L^\infty$-function. ... |
> **Problem:** We have a stick of length $1$ that we are cutting into $6$ parts with $5$ uniform and random cuts. What is the probability that no piece is greater than $\frac13$ in length?
The original problem asked was to find the probability that no piece is greater $\frac12$. That seems quite simpler than my foll... |
Is this claim true? The following is my attempt at the proof. I am unsure about the proof because I did not have to use the fact that $X_n\geq 0, \forall n\in \mathbb{N}$. Any feedback for confirmation will be greatly appreciated.
**Claim:** Suppose the sequence of nonnegative random variables, $X_n\geq 0$ satisfies... |
Decaying inequality in expectation implies almost sure convergence to zero? |
I have the matrix A= \begin{bmatrix}
1 & 4 & 1\\
0 & 2 & 5\\
0 & 0 & 5
\end{bmatrix}. I have found the T=\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 & 1 & 3/5
\end{bmatrix}. How can I find the T^(-1)? I already know the 1/|T| part but I am confused with the adjugate matrix. |
I'm having a hard time understanding principal bundles.
- I understand that a fiber bundle can be defined $(E, B, \pi, F)$ where $E, B, F$ are topological spaces and $\pi: E\to B$ is a surjection. For a fiber bundle, for each $b\in B$ there is a neighborhood $U\supset b$ such that $\pi^{-1}(U)\subset E \simeq U\ti... |
Difference between $G$-principal bundle and fiber bundle with fiber $G$? |
Let $p\in [1,\infty)$. On $X=L^p(\mathbb{R}; \mathbb{C}^2)$ we consider the operator $D=A+B$, where
$A=\begin{pmatrix}-\partial_x & 0 \\ 0 & \partial_x \end{pmatrix}\quad{and}\quad B=c(x)\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}$
(with domain $\mathsf{Dom}(D)=W^{1,p}(\mathbb{R};\mathbb{C}^2)$). Here, $c\colon... |
Let $\Omega$ be an open set with boundary such that $\partial\Omega = \Gamma_1 \cup \Gamma_2$ where $\Gamma_1$ and $\Gamma_2$ are closed and smooth but disconnected, in particular think of $\Omega$ to be an annulus.
I'm looking for existence and regularity results (in Sobolev or Sobolev-Bochner spaces) for paraboli... |
While trying to get an expression for [the number of deals from a generalised bridge deck with nobody being void in any suit](https://math.stackexchange.com/q/3443423/357390) I encountered the following subproblem.
From a generalised bridge deck with $r$ ranks instead of just $13$ deal four hands of $r$ cards each. ... |
The function $g(x)$ is a function series, so it is differentiable when $g'(x)$ converges uniformly. So I should just check uniform convergence of $g'(x)$ by using Weierstrass M-test:
$$g'(x) = (\sum_{n=0}^{+\infty}\frac{1}{2^n+x^2})' = \sum_{n=0}^{+\infty}(\frac{1}{2^n+x^2})'$$
then
$$|-\frac{2x}{(2^n+x^2)^2}| = \f... |
Compute $\int_C (y+z,z+x,x+y) dr$ (r should have an arrow above it)
where $C$ is the intersection of the cylinder $x
^2 +y
^2 = 2y$ and the plane $y = z$.
Am I correct that all I can do is apply Stokes Theorem since $C$ is a closed curve and $S$ is a surface bounded by $C$ such that orientations of $C$ and $S$ ar... |
I have the matrix
$$ A=
\begin{bmatrix}
1 & 4 & 1\\
0 & 2 & 5\\
0 & 0 & 5
\end{bmatrix}.
$$
I have found the
$$ T=
\begin{bmatrix}
1 & 4 & 1\\
0 & 0 & 1\\
0 & 1 & 3/5
\end{bmatrix}.
$$
How can I find the $T^{-1}$? I already know the $\frac{1}{|T|}$ part but I am confused with the adjugate matrix. |
The function $g(x)$ is a function series, so it is differentiable when $g'(x)$ converges uniformly. So I should just check uniform convergence of $g'(x)$ by using the Weierstrass M-test:
$$g'(x) = \left(\sum_{n=0}^{+\infty}\frac{1}{2^n+x^2}\right)' = \sum_{n=0}^{+\infty}\left(\frac{1}{2^n+x^2}\right)',$$
then
$$\le... |
The problem: unit square, ABCD. Extend AB as a Ray and trace a Ray from D, crossing the side BC at E and the Ray AB at F.
EF = 1. What is the length of BE?
(Image attached to scale)
[Contructed problem to scale][1]
This problem is "easy" to solve:
Similarity of BEF and CED. The problem is: you get a quartic ... |
r0 = 1 # radius of circle
w0 = ? # angle of interpolation
x0 = 0 # x-coordinate from center of circle
y0 = 0 # y-coordinate from center of circle
x1 = 3 # x-coordinate from reference point
y1 = 5 # y-coordinate from reference point
a0 = x0 + r0 * cos(t) # parametric equation for ... |
How to control a circular interpolation using a reference point? |
I've a question about the following. Take an affine space $(E,V)$ and consider the tangent space $T_pE$ at a point $p$.
From John Lee book "Introduction to smooth manifolds" section 3 there exist a canonical isomorphism between the "geometric tangent space" $\mathbb R_a^n$ at point $a$ and the vector space of deriva... |
Let A be a $n\times n$ matrix, $s$ be real number and $I$ be the $n\times n$ identity matrix. Why does $A$ have a worse condition number than $A-sI$?
I suspect it is something to do with the singular values being closer together for the matrix $A-sI$ than for $A$. I was wondering how I could mathematically prove th... |
I have a polynomial over a specific the range which has two imaginary roots. The first derivative is positive in the lower bound and upper bound. Does that mean that the function is monotone (increasing) over the range? |
Is a polynomial with two imaginary roots monotone? |
I've a question about the following. Take an affine space $(E,V)$ and consider the tangent space $T_aE$ at a point $a$.
From John Lee book "Introduction to smooth manifolds" chapter 3 there exist a canonical isomorphism between the "geometric tangent space" $\mathbb R_a^n$ at point $a$ and the vector space of deriva... |
Compute $\int_C (y+z,z+x,x+y) d\vec{r}$, where $C$ is the intersection of the cylinder $x^2 +y^2 = 2y$ and the plane $y = z$.
Is is true that all I can do is apply Stokes Theorem: since $C$ is a closed curve, I need to choose a surface $S$ bounded by $C$ such that orientations of $C$ and $S$ are compatible? Since th... |
I came across this question,
Compute $$L=\lim\limits_{x\to 0}{\frac{\sqrt[3]{1+\sin^2 x} \hspace{2mm}-\sqrt[4]{1-2 \tan x}}{\sin x + \tan^2 x}}$$
Can I use **Binomial Approximations** here?
As $x\to 0 \implies \sin^2 x\to 0$
$\therefore \left(1+\sin^2 x \right)^\tfrac{1}{3} \approx 1+\frac{sin^2 x}{3}$
... |
Can we use Binomial Approximations when evaluating limits? |
Compute $\int_C (y+z,z+x,x+y) d\vec{r}$, where $C$ is the intersection of the cylinder $x^2 +y^2 = 2y$ and the plane $y = z$.
Is is true that all I can do is apply Stokes Theorem: since $C$ is a closed curve, I need to choose a surface $S$ bounded by $C$ such that orientations of $C$ and $S$ are compatible? Since th... |
Let $R= k[x,y,z]$ and $\mathfrak p_1 = (x,y), \mathfrak p_2 = (x,z), \mathfrak m = (x,y,z) \triangleleft R$. Then
$$\mathfrak p_1 \mathfrak p_2 = \mathfrak p_1 \cap \mathfrak p_2 \cap \mathfrak m^2$$
is a minimal primary decomposition of $\mathfrak p_1 \mathfrak p_2$. Which components are embedded and which are isola... |
Is there any specific reason for which integrals start at zero? Logically when we think of an integral as a summation of area, it must start from negative infinity right? Why do indefinite integrals start at zero? Like when I plug in a value in the indefinite integral, we get the area from 0 to the plugged in value. Th... |
I am trying to get acquainted with the [arithmetical hierarchy][1], and as I wrote down some examples, I got a bit confused.
Consider the language $L=\{+\cdot,<,=,0,1\}$ of $\mathsf{PA}$.
For example, the sentence $\phi:=\forall x\,\exists y\,(y<x\wedge x=y+1)$ is a $\Pi_{1}$-sentence because it is of the form $\... |
Let $\langle x_n\rangle$ be a sequence of real numbers. Let $\langle \phi_n\rangle$ be an enumeration of the Turing machines (in ZFC?). Consider the following two "definitions" of a computable sequence:
> **Correct definition**: We call $\langle x_n\rangle$ computable if there exists a Turing machine that given $n$ ... |
When asserting that a Turing machine exists, how do you know you can find it? |
I'm largely a self-taught highschooler in basic Calculus and I'm utterly confused regarding what Indefinite integrals (or antiderivatives) do mean geometrically (if they really do), physically or mathematically at all (in the intuitive level). My exact confusion is what relation do they have with $x$? For example, the ... |
This specific question has arisen out of my attempts to do a wider exercise ([link][1]).
---
**Context**
We say two functions, $f$ and $g$, are equal if all three of the following conditions are met:
- (i) they have the same **domain**
- (ii) they have the same **codomain**
- (iii) they map the same el... |
Let $V$ be a complex inner product space. Suppose $V$ is the orthogonal direct sum $V = U\oplus W$ of two of vector subspaces $U$ and $W$. Suppose further that $\text{dim}U = \text{dim}V = n$, $\{e_{1},...,e_{n}\}$ is a basis for $U$ and $\{f_{1},...,f_{n}\}$ is a basis for $W$.
Note that $V$ is isomorphic to $\mat... |
I am trying to get acquainted with the [arithmetical hierarchy][1], and as I wrote down some examples, I got a bit confused.
Consider the language $L=\{+\cdot,<,=,0,1\}$ of $\mathsf{PA}$.
For example, the sentence $\phi:=\forall x\,\exists y\,(y<x\wedge x=y+1)$ is a $\Pi_{1}$-sentence because it is of the form $\... |
I'm writing a research paper and considering the following settings:
There are three discrete variables which construct a simple causal graph: Treatment **T**, outcome **Y**, and confounders **U**. Here we have the classical causal path {T, U}->Y, U-> T.
We consider the following setting: We only have **the marg... |
Bessel functions have nice properties, so it is usually not hard to find closed-form solutions of integrals that contain them. Let:
$$I=\int{J_2(x)\over x^2}dx \tag 1$$
We use the property that can be found [here][1]:
$$ xJ_2(x)=2J_1(x)-xJ_0(x) \tag 2$$
$$ J_2(x)={2J_1(x)-xJ_0(x) \over x} \tag 3$$
Insertin... |
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are
used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values of $n \ge 33 $ so that anyway we color the painting... |
My question comes from section 1.6 of *Complex Analysis* (4th edition) by Serge Lang: <br/>
Let $U$ be an open subset of $\mathbb{C}$ and let
\begin{align*}
f(x + iy) = u(x,y) + i v(x,y), \qquad x + iy \in U
\end{align*}
be a complex-valued function on $U$. Lang proceeds to argue that if $f$ is holomorphic... |
r0 = 1 # radius of circle
w0 = ? # angle of interpolation
x0 = 0 # x-coordinate from center of circle
y0 = 0 # y-coordinate from center of circle
x1 = 3 # x-coordinate from reference point
y1 = 5 # y-coordinate from reference point
a0 = x0 + r0 * cos(t) # parametric equation for ... |
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are
used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values of $n \ge 33 $ so that anyway we color the painting... |
I am trying to get acquainted with the [arithmetical hierarchy][1], and as I wrote down some examples, I got a bit confused.
Consider the language $L=\{+\cdot,<,=,0,1\}$ of $\mathsf{PA}$.
For example, the sentence $\phi:=\forall x\,\exists y\,(y<x\wedge x=y+1)$ is a $\Pi_{1}$-sentence because it is of the form $\... |
Let $\langle x_n\rangle$ be a sequence of real numbers. Let $\langle \phi_n\rangle$ be an enumeration of the Turing machines (in ZFC?). Consider the following two "definitions" of a computable sequence:
> **Correct definition**: We call $\langle x_n\rangle$ computable if there exists a Turing machine that given $n$ ... |
>where is $Q$ in that case?
When you move $P$ to $T$, then $R$ moves to $R'$ where $\vec{RR'}=\vec{PT}$ implies $R'_x=R_x+T_x-P_x=-9.9$ and $R'_y=R_y+T_y-P_y=-233.5$.
The equation of the circle whose center is $R'$ with radius $250$ is given by
$$(x+9.9)^2+(y+233.5)^2=250^2$$
So, the slope of the tangent line... |
The only purpose of this answer is to exhibit a less well known universal coefficient theorem and specialise it to give a(n) (alternative?) verification of the facts offered in Qiaochu's answer.
---
By one of the many variants of the universal coefficient theorem, namely Spanier's Algebraic Topology Theorem $6.5.... |
In the coupon collector problem, I open a cereal box every morning and get a random coupon from a set of $n$ different coupons randomly.
It is known that on average, it will take me $n H_n$ days to get all coupons (at least once).
My question is whether it is known how long it will take me to get $n$ coupons cond... |
How long does it take to get n coupons with exactly k coupons being drawn at least twice? |
In an art museum, there are $n$ paintings, $n \ge 33$, for which there are
used a total of $15$ different colors so that any two paintings have at least one common color and there are no two paintings that have exactly the same colors. Determine all possible values of $n \ge 33 $ so that anyway we color the painting... |
I am trying to get acquainted with the [arithmetical hierarchy][1], and as I wrote down some examples, I got a bit confused.
Consider the language $L=\{+\cdot,<,=,0,1\}$ of $\mathsf{PA}$.
For example, the sentence $\phi:=\forall x\,\exists y\,(y<x\wedge x=y+1)$ is a $\Pi_{1}$-sentence because it is of the form $\... |
I see that for $n^2 = 0, 4, 144, 4900, 166464$, etc. we get a perfect square, but is it possible to obtain an analytical solution? |
For which integers $n$ will $2n^2+1$ be a perfect square? |
Can we exactly determine a class of non-decreasing functions defined on the set of non-negative real numbers satisfying:
$$f(1+ax)\leq a f(1+x)$$ and $$f(xy)\leq f(x)+f(y)$$
for any $x,y\in [0,\infty)$, assuming that $a\in[0,1)$ can vary?
Par example, log fulfills second inequality, but not the first one...
Lookin... |
I recently came across [this][1] paper by Friedgut that shows low sensitivity boolean functions are close to juntas. This was an unintuitive result to me, as I thought I could easily imagine a function on the boolean hypercube that is a linear combination of many low-degree monomials, and therefore has low sensitivity,... |
Here are my questions:
Simplify the following $O$-notation statements as much as possible, e.g. $O(n + 25 + \log(2n)) = O(n)$.
(i) $O(n^2 + n)$
(ii) $O(\log(2n)) + O(n) + O(n^2)$
(iii) $O(n) * O(\log(2n)) $
(iv) $O(2^n) * O(n^2 + 2^n + \log(2n))$
I know the first three answers are:
(i) $O(n^2)$
... |
How to simplify Big $O$ notation? |
I am studying Algorithmic Game Theory and am totally new to the topic, but wanted to clarify some items (apologies for the length beforehand). My book of reference is [link](https://www.cs.cmu.edu/~sandholm/cs15-892F13/algorithmic-game-theory.pdf).
On the topic of External Regret Minimization, in section 4.3.1, it ... |
Consider the following question
>Consider the Poisson structure
$$
\{F, G\}=\int_{\mathbb{R}} \frac{\delta F}{\delta u(x)} \frac{\partial}{\partial x} \frac{\delta G}{\delta u(x)} d x,
$$
where $F, G$ are polynomial functionals of $u, u_x, u_{x x}, \ldots$ Assume that $u, u_x, u_{x x}, \ldots$ tend to zero as $|... |
Let $(M,g)$ be a Riemannian manifold. Let $p\in M$. Is it possible to find a chart $\varphi: p\in U \to \mathbb R^d$ such that the geodesic distance $d_M$ on $U$ is close to the euclidean distance $d_E$ on $\phi_i(U_i)$?
More precisely, if I take $\phi_i = \exp^{-1}_p: \{x\in M: d_M(x,p)\le \delta\} \to \{v\in \math... |
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