instruction stringlengths 12 30k |
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Let $C_i\sim C$ and $R_i \sim R$ denote the durations (in ticks) of the $i$th coupled periods of clear and rainy weathers, respectively. We want to find
$$p(t)=\mathbb P \left (A_t:=\text{at time $t$ the weather is $\bf{rainy}$} \right ).$$
Let us define
$$q(t)=1- p(t)=\mathbb P \left (A'_t:=\text{at time $t$... |
If $A\to B$ is an integral ring morphism, so is $A[[x]]\to B[[x]]$? |
As I understand it, infinities typically have a one-to-one mapping with known sets. The denumerable infinity has a one-to-one mapping with integers, rational numbers, and natural numbers. The nondenumerable infinity has a one-to-one mapping to the set of real numbers and the power set of the integers.
Is there any ... |
In wheel theory, does $1/0$ have a one-to-one mapping to any set? Or is the concept of a one-to-one mapping outside of its definition? |
How to prove that: A rotation in complex plane corresponding to opposite rotation in Riemann's sphere (Complex Analysis by Theodore W. Gamelin 1.3.3)? |
Two variables change every year, e.g. $a_1=500$, $a_2=600$ and $b_1=1000$ and $b_2=800$. Their ratio change is
$$YoY=a_2/b_2-a_1/b_1=600/800-500/1000=0.25$$
The negative change in the denominator is a negative event and I want to reflect this, so e.g. in the example above $YoY$ is smaller if the change in the denomin... |
Problem. *Let $C$ be a $\mathrm{C}^1$ Jordan curve, and $ \gamma: [-1,1]\to\mathbb R^2,$ a segment crossing $C$ perpendicularly at $x_0\in C$, with $\gamma(0)=x_0$. Show that there exists an $\varepsilon>0,\,$ such that the segments
$$
\ell_-=\big\{\gamma(t): t\in (-\varepsilon,0)\big\}\quad\text{and}\quad
\ell_+=\b... |
$\mathscr{L}(\mathcal{H})=$ Set of linear operators from $\mathcal{H}\to \mathcal{H}$. For $T\in \mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B)$ specified through $T=\sum\limits_{i,j}\gamma_{i,j}A_i\otimes B_j$, where $A_i\in \mathscr{L}(\mathcal{H}_A)$, $B_i\in \mathscr{L}(\mathcal{H}_B)$, $\gamma_{i,j}\in \mathbb{C}... |
I attempted to separate the real and imaginary parts of the function $Tr(R_T(i\lambda)[A] B)$, where $R_T$ represents the resolvent and $T$ is a super-operator. The parameter $\lambda$ is a real number, and both $A$ and $B$ are Hermitian operators. However, I encountered difficulties in achieving this separation using ... |
As I understand, Cantor’s diagonal set can be empty, that is, there could be a mapping from the the Natural Numbers to the Power Set of the Natural Numbers in which the empty set is not mapped. The function is still not a bijection but there is only one element not mapped (one set that is there only because it is vacuo... |
I understand that torsion is a concept specific to three-dimensional spaces. Despite searching on Google, I've struggled to find how to extend the concept of torsion to an n-dimensional space.
Is it not feasible to adapt torsion for higher dimensions, or is there a formula in the academic literature that addresses t... |
Is torsion defined in n-dimensional space? |
Problem. *Let $C$ be a smooth Jordan curve, and $ \gamma: [-1,1]\to\mathbb R^2,$ a segment crossing $C$ perpendicularly at $x_0\in C$, with $\gamma(0)=x_0$. Show that there exists an $\varepsilon>0,\,$ such that the segments
$$
\ell_-=\big\{\gamma(t): t\in (-\varepsilon,0)\big\}\quad\text{and}\quad
\ell_+=\big\{\gam... |
I think the step you suspect is illegal is illegal indeed.
Note that the strategy $X^*$ is optimal, when we are sure that the other player plays $Y^*$, but we don't know yet, if it is optimal, when the other player plays any other strategy.
I'd rather approach the problem in the following way. Since $E(X, Y^*) \l... |
They represent different Kinds of Quantities. Trying to analyse equations to infer whether you're dealing with a torque or a work value would be rather difficult in general, especially within large programmes. A simpler way would be to use a symbolic checker that allows for naming, and performs dimensional analysis. Se... |
$\mathscr{L}(\mathcal{H})=$ Set of linear operators from $\mathcal{H}\to \mathcal{H}$. For $T\in \mathscr{L}(\mathcal{H}_A\otimes \mathcal{H}_B)$ specified through $T=\sum\limits_{i,j}\gamma_{i,j}A_i\otimes B_j$, where $A_i\in \mathscr{L}(\mathcal{H}_A)$, $B_i\in \mathscr{L}(\mathcal{H}_B)$, $\gamma_{i,j}\in \mathbb{C}... |
Given a family of curves, an envelope is defined as a curve that it is tangent to every curve in the family of curves at some point on it.
To derive the equation for it, the first step is to parameterize the envelope with parameter $t$, which also indexes which curve in the family.
But what if there is a curve ta... |
According to its truth table, $P \lor \neg P$ is a tautology, i.e. it is true for all truth values of its constituent propositions. But how comes that is true in *classical* logic? $\lor$ is the *inclusive* 'or', so $P \lor \neg P$ means 'either $P$, or $\neg P$, or both of them'. But in classical logic it is never tru... |
"But [. . . ] these extensions of definitions no longer allow scope for arbitrariness; on the contrary, they follow with compelling necessity from the earlier restricted definitions, provided one applies the following principle: Laws which emerge from the initial definitions and which are characteristic for the concept... |
What did Richard Dedekind mean exactly by his statement about generality? |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\quad\quad\quad (1)$$
Under which assumptions is i... |
As I understand, Cantor’s diagonal set can be empty, that is, there could be a mapping from the the Natural Numbers to the Power Set of the Natural Numbers in which the empty set is not mapped. The function is still not a bijection but there is only one element not mapped (one set that is there only because it is vacuo... |
Does Cantor’s theorem rely on the Empty Set being in the power set of a set? |
I'm looking for a simple proof of the identity
$$ \sum_{k=0}^n \frac{(-1)^k}{\binom{n}{k}} = \frac{n+1}{n+2} (1+(-1)^n) $$
relying only on elementary properties of binomial coefficients. I obtained this by starting with the integral representation
$$ \frac{(a-1)!(b-1)!}{(a+b-1)!} = \int_0^{\infty} \frac{t^{b-1}}{(1+... |
I'm currently trying to work out the following.
Given two uniformly distributed random variables on the unit flat torus (i.e. $[0,1]^2$ which wraps around), what is the probability that their distance is less than some real number $r$? Further I will need the p.d.f, but this is just differentiating. Also if there's ... |
Let $X_t$ be a Markovian Itô diffusion process, defined by an SDE
\begin{equation}
dX_t = \mu(X_t)dt + \sigma(X_t)dW_t\,.
\end{equation}
Let $f(x,t|x_0,0)$ denote its transition density function. Under suitable conditions, as is my case, we know that $f$ is Lipschitz on every relatively compact set $S$ with assoc... |
I am reading the John Lee's Introduction to Riemannian manifolds, 2nd Edition, proof of Theorem 12.28 ( Cheng's Maximal Diameter Theorem ) and stuck at final statement.
First, I will extract and ask main point. It seems easy ( works by unwinding definitions ? ) but can't prove rigorously until now.
Let $(M,... |
I am looking for an expression of the asymptotic behaviour for $|z|\rightarrow\infty$ of the Hypergeometric function ${}_2F_1(a,b,c,z)$ when the condition $a-b \not\in {\mathbb Z}$ is *not* satisfied, in particular when $a=b$.
Previous questions and answer, like [here][1] using the reciprocation formula, have not cons... |
If $\lim_{x\to a}f(x)=L$, then for all $\epsilon\gt 0$ there exists a $\delta$ such that if $|x-a|\lt \delta$, $|f(x)-L|\lt \epsilon$. If $f(x)=x^2$ and $a=2$ and $L=4$, this implies that for all $\epsilon\gt 0$ there exists a $\delta$ such that if $|x-2|\lt \delta$, $|x^2-4|\lt \epsilon$
Working backwards, $|x^2-4|... |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with... |
The **independence number** $\alpha(G)$ of a graph $G$ is the cardinality of the largest independent vertex set.
My question is as stated in the title. We can observe from a path with $n$ vertices that its independence number is $\lceil \frac{n}{2} \rceil $ and its complement has independence number 2. So $\alpha(... |
What is the upper bound for the sum of the independence numbers of a graph and its complement? |
The **independence number** $\alpha(G)$ of a graph $G$ is the cardinality of the largest independent vertex set.
My question is as stated in the title. We can observe from a path with $n$ vertices that its independence number is $\lceil \frac{n}{2} \rceil $ and its complement has independence number 2. So $\alpha(... |
What is the lower bound for the sum of the independence numbers of a graph and its complement? |
The **independence number** $\alpha(G)$ of a graph $G$ is the cardinality of the largest independent vertex set.
My question is as stated in the title. We can observe from a path with $n$ vertices that its independence number is $\lceil \frac{n}{2} \rceil $ and its complement has independence number 2. So $\alpha(... |
One [stack**overflow** question](https://stackoverflow.com/q/76064709/3789665) asks how to check divisibility of a natural number $n$ by 3 with a processor using binary arithmetic mod $2^8$, no divide/modulus instruction (some models not supporting multiplication).
Revisiting computation of a number $s<2^4, s... |
Computing remainder mod 3: Why/how does below procedure work? |
This problem is in my book:
>Show that as $z$ traverses a small circle in the complex plane in the
positive (counterclockwise) direction, the corresponding point $P$ on
the sphere traverses a small circle in the negative (clockwise) direction with respect to someone standing at the center of the circle and with... |
This is more of a comment, but too long for a comment.
We have operations that you mention ($+,*,-,/$), and you've proposed a way to apply these to functions (pointwise) and a way to define the natural domain of a combination (say $f+g$) in terms of the domains of $f,g$. I like this approach, but I would make the f... |
One [stack**overflow** question](https://stackoverflow.com/q/76064709/3789665) asks how to check divisibility of a natural number $n$ by 3 with a processor using binary arithmetic mod $2^8$, no divide/modulus instruction (some models not supporting multiplication).
Revisiting computation of a number $s<2^4, s... |
Started from
\begin{align}
\int{\frac{1}{\cos x}dx}&=\int{\frac{\cos x}{1-{{\sin }^{2}}x}dx}\\
&=\tanh^{-1}(\sin x)+C\quad\ldots\ldots\ldots\ldots\text{ (*)}\\
&=\ln\sqrt{\frac{\sin x+1}{\sin x-1}}+C
\end{align}
(*) can be solved by setting $\tanh Y=\sin x=\frac{e^{2Y}-1}{e^{2Y}+1}$.
On the other side, if Eu... |
Are the two functions different by a constant? |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
According to its truth table, $P \lor \neg P$ is a tautology, i.e. it is true for all truth values of its constituent propositions. But how come that is true in *classical* logic? $\lor$ is the *inclusive* 'or', so $P \lor \neg P$ means 'either $P$, or $\neg P$, or both of them'. But in classical logic it is never true... |
What does this sentence mean in polynomial rings? |
I have encountered some note on complex analysis:
**In particular, given a polynomial in the real variables x and y , with
complex coefficients, these properties tell us that the polynomial
is analytic on C precisely when, in its representation as a linear
combination of terms , terms involving z do not occur.** ... |
a note on complex analysis that I dont understand?Could you help me formulate it? |
Out of the different [divisibility tricks][1] there's a really simple rule that works for more than one divisor: The digit-sum. Specifically, if the sum of the digits of a number is a multiple of $1,3$ or $9$ then so was the original number (although the $1$ case is trivial it'll make sense to include it for later).
... |
If $V$ is a vector space over the field $F$, the general linear group of $V$, written $GL(V)$, is the set of all bijective linear transformations $V\to V$, together with functional composition as group operation.
My question: Let $V$ and $W$ be infinite dimensional vector spaces over a field $F$. Is the following s... |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
I have encountered some note on complex analysis:
**In particular, given a polynomial in the real variables x and y , with
complex coefficients, these properties tell us that the polynomial
is analytic on C precisely when, in its representation as a linear
combination of terms $z^{j} \overline{z}^{j}$ , terms inv... |
Why is a polynomial in $z,\bar z$ analytic iff it does not involve monomials $z^i\bar z^j$ with $j\ge1$? |
Solve the following equation in real numbers:
$\log_2(\sin x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)$
My approach:
$\log_2(\sin x) + \log_3\left(\frac{\sin x}{\cos x}\right) = -2\log_2(\cos x) + \log_5\left(\frac{\cos x}{\sin x}\right)$
Now the equation can be written as:
$\log_2(\sin x) + \lo... |
First question here. Apologies for any rookie mistakes, gladly take comments on how to improve phrasing etc.
Conventions: I define $[k]$ to be the one-dimensional vector space concentrated in degree $-k$. Define $A[k]:= [k]\otimes A$. I also work with the Koszul sign convention.
Let $f: A\to B$ be a map of cochai... |
How to define the induced map of shifted complexes? |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
Let $N \geq \lfloor 1/\alpha \rfloor > 0$ and consider a weakly geometric $\alpha$-Hölder rough path $\textbf{x}$ that preserves the origin, i.e. an element $\textbf{x} \in C^{\alpha\text{-Höl}}_o([0,T],G^{\lfloor 1/\alpha \rfloor}(\mathbb{R}^d))$. This is standard notation, and is the one in the book "Multidimensional... |
I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry plane, computed using numerical integration, assuming a "wetting" surface, i.e. a contact angle of zero:
[![menisci prof... |
Given $\mathcal{G}$ a sigma-field and $X$ a $\mathcal{G}$-measurable random variable, on an intuitive level, why do we need $Y$ to be independent of $\mathcal{G}$ to pull out $X$ from the conditional expectation? $$E[f(X,Y)|\mathcal{G}]=E[f(x,Y)|\mathcal{G}]|_{x=X}$$ I understand that if $X$ is $\mathcal{G}$-measurable... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
While reading Paul J. Nahin's "Hot molecules, cold electrons", while solving the heat equation for a semi-infinite mass with infinite thickness, when comparing the result with the initial conditions, he writes $1=\int_{0}^{\infty}B(\lambda)sin(\lambda x)d\lambda$, then, recalling the Fourier sine transform $f(x)=\int_{... |
One [stack**overflow** question](https://stackoverflow.com/q/76064709/3789665) asks how to check divisibility of a natural number $n$ by 3 with a processor using binary arithmetic mod $2^8$, no divide/modulus instruction (some models not supporting multiplication).
(More fluent with the two's complement, shift&... |
Take two Banach spaces $X,Y$ and consider their tensor product $W=X\otimes Y$.
Any $w\in W$ can be written, generally not in a unique way, as a linear combination of simple tensors: $$w=\sum_{i=1}^n (x_i\otimes y_i)\ \ \ \ \text{ with }x_i\in X \text{ and } y_i\in Y\ \forall i\quad\quad\quad (1)$$
Under which assu... |
One [stack**overflow** question](https://stackoverflow.com/q/76064709/3789665) asks how to check divisibility of a natural number $n$ by 3 with a processor using binary arithmetic mod $2^8$, no divide/modulus instruction (some models not supporting multiplication).
(More fluent with the two's complement, shift&... |
While reading Paul J. Nahin's "Hot molecules, cold electrons", while solving the heat equation for a semi-infinite mass with infinite thickness, when comparing the result with the initial conditions, he writes $1=\int_{0}^{\infty}B(\lambda)\sin(\lambda x)d\lambda$, then, recalling the Fourier sine transform $f(x)=\int_... |
Started from
\begin{align}
\int{\frac{1}{\cos x}dx}&=\int{\frac{\cos x}{1-{{\sin }^{2}}x}dx}\\
&=\tanh^{-1}(\sin x)+C\quad\ldots\ldots\ldots\ldots\text{ (*)}\\
&=\ln\sqrt{\left|\frac{\sin x+1}{\sin x-1}\right|}+C
\end{align}
(*) can be solved by setting $\tanh Y=\sin x=\frac{e^{2Y}-1}{e^{2Y}+1}$.
On the othe... |
I also want to try to answer this question, even though it may be late. My style is that I first give the intuition and then the calculations. Thus, try to imagine whatever I say in $\mathbb{R}^3$.
Assume two Identical spheres centered respectively around points x and y. Imagine I can change some parameter for the r... |
I am currently studying "A Classical Introduction to Modern Number Theory", a book by Kenneth Ireland and Michael Rosen.
In that, we have the following two theorems:
$(1)$ $\textit{The Law of Cubic Reciprocity}:$ Let $\pi_1$ and $\pi_2$ be primary, $N\pi_1,N\pi_2 \neq 3,$ and $N\pi_1 \neq N\pi_2$. Then $\chi_{\p... |
You can think of it like this:
If $P$ is false, then $P\to Q$ is true but $P$ is false, so $(P\to Q)\to P$ is false.
Therefore, if $(P\to Q)\to P$ is true, it must be that $P$ is true.
----------
You asked for a "real world" example. One issue with this is that we need to treat $\to$ as the material condit... |
I think it's better to rearrange like below $$\log_2(\sin x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)\\\log_4(\sin^2 x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)\\\log_4(\sin^2 x) -\log_4(\cos^2 x) = \log_5(\cot x)- \log_3(\tan x)\\\log_4(\frac {\sin^2 x}{\cos^2 x})=\log_5(\tan x)^{-1}-\log_3(\... |
I am interested in a particular 16-dimensional representation of $A6$, the alternating group on 6 things. I first construct an amalgam, gamma, of two copies of SymmetricGroup(4):
F:=FreeGroup(["s1","s2","s3","t1","t2","t3"]); \\
AssignGeneratorVariables(F); \\
rel:=Union(
[s1^2,s2^2,s3^2,s1*s3*... |
Can GAP compute this 16-dimensional representation of AlternatingGroup(6)? |
Started from
\begin{align}
\int{\frac{1}{\cos x}dx}&=\int{\frac{\cos x}{1-{{\sin }^{2}}x}dx}\\
&=\tanh^{-1}(\sin x)+C\quad\ldots\ldots\ldots\ldots\text{ (*)}\\
&=\ln\sqrt{\left|\frac{\sin x+1}{\sin x-1}\right|}+C
\end{align}
(*) can be solved by setting $\tanh Y=\sin x=\frac{e^{2Y}-1}{e^{2Y}+1}$.
On the othe... |
Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such that $$ \vert f(x)-f_n(x) \vert \leq \frac{C(x)}{\sqrt{n}}, \quad \vert g(x)-g_n(x) \vert \leq \frac{C'(x)}{\sqrt{n}}$$ wh... |
I am currently studying "A Classical Introduction to Modern Number Theory", a book by Kenneth Ireland and Michael Rosen.
In that, we have the following two theorems:
$(1)$ The Law of Cubic Reciprocity: Let $\pi_1$ and $\pi_2$ be primary, $N\pi_1,N\pi_2 \neq 3,$ and $N\pi_1 \neq N\pi_2$. Then $\chi_{\pi_1}(\pi_2)... |
I have an homogeneous equation $Lf=0$ where $L$ is an operator and $f$ a function, with boundary conditions $f(\partial\Omega)=g(\partial\Omega)$ where $g$ is a known function.
In the case when the Green function for the operator $L$ is known, how is it possible to use it to find $f$ ? |
I am interested in a particular 16-dimensional representation of $A6$, the alternating group on 6 things. I first construct an amalgam, gamma, of two copies of SymmetricGroup(4):
F:=FreeGroup(["s1","s2","s3","t1","t2","t3"]);
AssignGeneratorVariables(F);
rel:=Union(
[s1^2,s2^2,s3^2,s1*s3*s1^-... |
Started from
\begin{align}
\int{\frac{1}{\cos x}dx}&=\int{\frac{\cos x}{1-{{\sin }^{2}}x}dx}\\
&=\tanh^{-1}(\sin x)+C\quad\ldots\ldots\ldots\ldots\text{ (*)}\\
&=\ln\sqrt{\left|\frac{\sin x+1}{\sin x-1}\right|}+C
\end{align}
(*) can be solved by setting $\tanh Y=\sin x=\frac{e^{2Y}-1}{e^{2Y}+1}$.
On the othe... |
We are interested in calculating the spectral moments of random matrices sampled from [the Wishart distribution][1].
Let $N,T$ be positive integers with $T> N$ .
Then the quantities in question are defined as $m_p:= 1/N \cdot Tr \left[ c^p \right]$ where $p \in {\mathbb Z}$ and where $c:= 1/T \cdot {\bf X} \cdot {\... |
I also want to try to answer this question, even though it may be late. My style is that I first give the intuition and then the calculations. Thus, try to imagine whatever I say in $\mathbb{R}^3$.
Assume two Identical spheres centered respectively around points x and y. Imagine I can change some parameter for the r... |
Say there is a competition with 3 participants A, B, and C. There are no ties.
We know that A does not come 3rd. So we can say that the probability of A coming 1st/2nd is 0.5
So will it be correct to say that the probability of the 1st/2nd position being occupied by A is also 0.5 with the probability of it being... |
One [stack**overflow** question](https://stackoverflow.com/q/76064709/3789665) asks how to check divisibility of a natural number $n$ by 3 with a processor using binary arithmetic mod $2^8$, no divide/modulus instruction (some models not supporting multiplication).
(More fluent with the two's complement, shift&... |
Below is a more complete and rigorous proof than the other existing answers.
## Abstract
We will prove the formula for the number of $k$-combinations $_nC_k$ by showing that $_nP_k = k! \cdot {_nC_k}$. We show the latter by first proving $_nP_k \ge k! \cdot {_nC_k}$, and then showing that $_nP_k \not> k! \cdot {_... |
I also want to try to answer this question, even though it may be late. My style is that I first give the intuition and then the calculations. Thus, try to imagine whatever I say in $\mathbb{R}^3$.
Assume two Identical spheres centered respectively around points x and y. Imagine I can change some parameter for the r... |
Let $X \subset \mathbb{R}^n$ be a compact, connected non-empty subset. Does there exist an analogue to a tubular neighborhood for $X$, i.e. something like a smooth manifold with boundary that retracts to $X$? One can assume that $X$ has the homotopy type of a finite CW-complex.
I thought about using regular neighbor... |
Let $f$ be a real decreasing function (resp. $g$ a real increasing function), and $(f_n)$ be a sequence of real decreasing functions (resp. $(g_n)$ be a sequence of real increasing functions) such that $$ \vert f(x)-f_n(x) \vert \leq \frac{C(x)}{\sqrt{n}}, \quad \vert g(x)-g_n(x) \vert \leq \frac{C'(x)}{\sqrt{n}}$$ wh... |
I have a continuous and nondecreasing function $F:\mathbb{R}\to\mathbb{R}$ satisfying following conditions:
1. $F(x) = 0$ for any $x\in (-\infty, 1]$,
2. $\lim\limits_{x\to +\infty} F(x) = 1$,
2. for any $x\in \mathbb{R}$ we have
$$F(x) \;=\; 0.8 F(2x - 1) + 0.2 F(x/2 - 1).$$
Is there any solution to this "fun... |
Working in the first order language of set theory.
By $R$-bounded quantifiers its meant those of the form $\forall x \ R \ a \, ( \cdots) $ , or $ \exists x \ R \ a \, ( \cdots) $, and these are defined as $\forall x \, ( x \ R \ a \to \cdots)$ and $\exists x \, (x \ R \ a \land \cdots) $ respectively. Here $R$ is ... |
I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry plane, computed using numerical integration, assuming a "wetting" surface, i.e. a contact angle of zero:
[![menisci prof... |
So I'm currently doing an advanced linear algebra module and I'm confused on two statements from two different lectures. I'll attach a section of two lectures and highlight the statements that confuses me. (I apologize for poorly typing the equations, I don't know yet how to type formulas on this site)
My understand... |
Solve the following equation in real numbers:
$\log_2(\sin x) + \log_3(\tan x) = \log_4(\cos^2 x) + \log_5(\cot x)$
My approach:
$\log_2(\sin x) + \log_3\left(\frac{\sin x}{\cos x}\right) = \log_2(\cos x) + \log_5\left(\frac{\cos x}{\sin x}\right)$
Now the equation can be written as:
$\log_2(\sin x) + \log_... |
From https://math.stackexchange.com/questions/2350219/how-to-show-orthogonality-of-associated-laguerre-polynomials or https://math.stackexchange.com/questions/92612/finding-a-generating-function-for-the-laguerre-polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defi... |
Let $\nabla$ be an affine connection on a smooth manifold $\mathcal M$. We say $\nabla$ is torsion-free if for any vector fields $X, Y$ on $\mathcal M$, we have
\begin{equation}
\nabla_X Y - \nabla_Y X = [X, Y],
\end{equation}
where $[X, Y] = X \circ Y - Y \circ X$ is the Lie bracket of $X$ and $Y$. Is it equivalen... |
Let $(X,\mathcal{X},\lambda)$ be a measure space and $(\Theta,d)$ a metric space.
Assume a function $f:X\times\Theta\to \mathbb{R}$ be uniformly bounded, meaning $\sup_{x\in X,\theta\in\Theta}|f(x,\theta)| < \infty$. Assume that
\begin{equation}
\Theta\ni\theta\mapsto\int_X f(x,\theta) \mathrm{d}\lambda(x) \in \mat... |
From https://math.stackexchange.com/questions/2350219/how-to-show-orthogonality-of-associated-laguerre-polynomials or https://math.stackexchange.com/questions/92612/finding-a-generating-function-for-the-laguerre-polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defi... |
I copied my answer from Quora which I wrote on the same day as I saw this.
Let’s take the numbers **1 2 3 4 5 6 7 8 9 10 11** and call it set **A**
Ask yourself: how many ways can I make triplets from **A** (e.g. **1 2 3** or **10 11 6**), where triplets with exactly the same numbers count as one (e.g. **1 2 3**... |
From https://math.stackexchange.com/questions/2350219/how-to-show-orthogonality-of-associated-laguerre-polynomials or https://math.stackexchange.com/questions/92612/finding-a-generating-function-for-the-laguerre-polynomials it appears that the expression, $L^q_p(x)$ is that of an associated Laguerre polynomial and defi... |
I came across a somewhat interesting differential equation while studying the shape of a meniscus ring formed at the bottom of a cylinder. Here are some 2D cross sections through a cylinder symmetry plane, computed using numerical integration, assuming a "wetting" surface, i.e. a contact angle of zero:
[![menisci prof... |
In the past, I came across a very elegant definition (below) of a function, which is based on the fundamental concepts of triples, pairs, and sets. However, I find it difficult to search the internet for any citations that I could refer to. One professor once told me that this definition could be over 200 years old, ... |
Oscar Lanzi has already given quite a nice overview over a general method to attack a certain set of polynomials trying to disprove eventual divisibility. Let's explore this idea a bit further and give a nice summary of the method:
We start with defining $q(x) := x^2+ax+b$ and examine solutions to $q(x) = x$ and $q... |
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