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Imagine we have N distinct and ordered elements.
(1, 2, 3, 4, 5, 6, 7) # Example with 7 elements.
And we permute them randomly, for example....
(3, 2, 5, 4, 1, 7, 6) # We get one coincidence for the number 4.
What is the probability that at least one element matches its original ... |
What is the probability of at least one coincidence after a permutation? |
Say we have $n$ sets, each set containing $x_k$ elements that can be distinguished. Now, we choose $m$ elements from those sets and number each element sequentially. In each position, an element is placed, but you can't place two elements adjacently if they come from the same set.
\begin{equation*} \text{If }N\text... |
I am currently reading a lecture notes on module theory and more specifically on Morita Theory. Here is a porition of the lecture note that I do not understand.
> Finitely generated projective modules are `rigid' in the following sense. Let $A$ be a ring and $P$ be a finitely generated projective left $A$-module. Th... |
Imagine we have N distinct and ordered elements.
(1, 2, 3, 4, 5, 6, 7) # Example with 7 elements.
And we permute them randomly, for example....
(3, 2, 5, 4, 1, 7, 6) # We get one coincidence for the number 4.
What is the probability that at least one element matches its original ... |
I'm working on a question for a course in mathematical logic.
> Prove that if a sentence $\phi$ expresses a truth function $f$ and every connective in $\phi$ is assigned a self-dual truth function, then $\phi$ expresses $f^*$.
My lecture notes say that $\phi$ expresses $f$ iff for all $\mathcal{A}:$ $|\phi|\_\mat... |
The stability of discrete systems can be understood by partial-fraction expansion of the discrete transfer function and finding the region of conversion for each divided fraction. Yet, there is a frequent argument in a few references about the stability of discrete systems in $z$ domain which is intuitively based on th... |
There are many examples
1. https://math.stackexchange.com/questions/1332652/show-that-w-t-t-geq-0-and-w-t2-t-t-geq-0-are-not-uniformly-integr?rq=1
2. from https://math.stackexchange.com/questions/352570/martingale-not-uniformly-integrable
> Let $\xi_j$ be i.i.d. with $(\forall j \in \mathbb{N}) \mbox{} \mathbb{P... |
There are many examples
1. https://math.stackexchange.com/questions/1332652/show-that-w-t-t-geq-0-and-w-t2-t-t-geq-0-are-not-uniformly-integr?rq=1
2. from https://math.stackexchange.com/questions/352570/martingale-not-uniformly-integrable
> Let $\xi_j$ be i.i.d. with $(\forall j \in \mathbb{N}) \mbox{} \mathbb{P... |
I'm working on a question for a course in mathematical logic.
> Prove that if a sentence $\phi$ expresses a truth function $f$ and every connective in $\phi$ is assigned a self-dual truth function, then $\phi$ expresses $f^*$ (the dual of $f$).
My lecture notes say that $\phi$ expresses $f$ iff for all $\mathcal{... |
This question comes from a previous question on this site. There is the integral $$\int_{-1}^{1}\int_{0}^{\sqrt{1 - x^2}}\int_{0}^{\frac{y}{2}}f(x, y, z) \, dz \, dy \, dx.$$ Our job is to reorder the order of integration so that the differential is $dy \, dx \, dz$.
I thought of $$\int_{0}^{1/2}\int_{-1}^{1}\int_{2... |
I've been studying Von Neumann-Barnays-Gödel axiomatic set theory (NBG) from Dugundji's Topology (1966) presented in chapter 8, and the statement 8.3 in the books still doesn't make sense to me, or at least the logical thinking it takes to get there: "the Russell class $\mathcal{R}(p)$ is not a set".
This is affirme... |
I'm working on a question for a course in mathematical logic.
> Prove that if a sentence $\phi$ expresses a truth function $f$ and every connective in $\phi$ is assigned a self-dual truth function, then $\phi$ expresses $f^*$ (the dual of $f$).
My lecture notes say that $\phi$ expresses $f$ iff for all $\mathcal{... |
https://groupprops.subwiki.org/wiki/Standard_representation_of_symmetric_group:S3
Gives two 2-D irreducible representations for S3. One in the basis (e1-e2) and (e2-e3) and another "viewed as a quotient" in the basis $\overline{e1}$ and $\overline{e2}$. There is enough information given about the basis (e1-e2) and (e... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
What type of mean satisfies "average rate of change is the average of derivatives"? |
I'm completely stumped. I'm trying to figure out a formula for the following...
How much of a solution that is 400 PPM per gallon needs to be added to X amount of gallons of a 0 PPM solution in order for X amount of gallons to be 2 PPM per gallon?
OR
If container A has X amount of gallons of a 0 PPM solution.
... |
Let the vertices of the pyramid be $A,B,C,D,V$, where $ABCD$ is the base and $V$ is the apex. Using coordinate geometry, and letting the given sides of the base be $a = AB, b = BC, c = CD, d = DA $, we can assign $A,B,C,D$ as follows
$ A = (0,0,0) , B= (a, 0, 0), C= (x_1, y_1, 0), D = (x_2, y_2, 0) $
with $y_1, ... |
In Chapter 12 of Jech's Set Theory, the model $L$ is defined as:
$$
\begin{align}
L_0 & = \emptyset, L_{\alpha+1} = \text{def}(L_{\alpha}), \\
L_{\alpha} &= \bigcup_{\beta < \alpha} L_{\beta} \text{ for limit ordinal $\alpha$} \\
L &= \bigcup_{\alpha \in Ord} L_{\alpha}
\end{align}
$$
where $\text{def}(M) =... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
In Chapter 12 of Jech's Set Theory, the model $L$ is defined as:
$$
\begin{align}
L_0 & = \emptyset, L_{\alpha+1} = \text{def}(L_{\alpha}), \\
L_{\alpha} &= \bigcup_{\beta < \alpha} L_{\beta} \text{ for limit ordinal $\alpha$} \\
L &= \bigcup_{\alpha \in Ord} L_{\alpha}
\end{align}
$$
where $\text{def}(M) =... |
Let $V\subset \subset U\subset \mathbb{R}^1$. Can we construct a Lipschitz cut-off function $f$ such that
$$
f=0 \quad \text{on} \quad V ,\quad f=-1 \quad \text{on} \quad \mathbb{R}^1 \setminus U,\quad f^2-|f'|\geq -C \quad \text{a.e.}\quad \text{on} \quad U\setminus V
$$
hold？
Could you please give me some h... |
Let $V\subset \subset U\subset \mathbb{R}^1$. Can we construct a Lipschitz cut-off function $f$ such that
$$
f=0 \quad \text{on} \quad V ,\quad f=-1 \quad \text{on} \quad \mathbb{R}^1 \setminus U,\quad f^2-|f'|\geq -C \quad \text{a.e.}\quad \text{on} \quad U\setminus V
$$
hold？
Is there a concrete example of ... |
Solution A: $400$ ppm/gallon
Solution B: $2$ ppm/gallon
How much solvent ($0$ ppm/gallon) is required? Say $X$
$\frac{1 \text{gallon} \times 400 \text{ppm/gallon}}{(1 + X) \text{gallons}} = 2 \text{ppm/gallon}$
Solve for $X$.
[Answer: $199 \text{gallons}$]
$\frac{a \times b}{c + d} = e$ (formula) |
I would like to know how the interval of definition for an initial value problem be determined without explicitly solving it. For example, the solution to $$y' = e^y + 1, \; y(0) = 0$$ is $$ y(t) = log \left( \frac{e^t}{2 - e^t} \right)$$ hence the interval of definition $$t < log(2)$$
And also in the case of higher o... |
This question comes from a previous question on this site. There is the integral $$\int_{-1}^{1}\int_{0}^{\sqrt{1 - x^2}}\int_{0}^{\frac{y}{2}}f(x, y, z) \, dz \, dy \, dx.$$ Our job is to reorder the order of integration so that the differential is $dy \, dx \, dz$.
I thought of $$\int_{0}^{1/2}\int_{-1}^{1}\int_{2... |
I'm delving into the concept of strict convexity within $L^2$ spaces and its implications for the existence and uniqueness of elements with minimum $L^2$ norms. Given $L^2(\mathbb{R})$'s role as a pivotal space in functional analysis, particularly in optimization and approximation theory, I'm curious about how its st... |
Is the Uniqueness of Minimum Norm Elements in $L^2$ Spaces a Consequence of Strict Convexity? |
Trying to split the logarithm of the sum of two exponential functions ([this question](https://math.stackexchange.com/q/4835130/909869)), I found the following approximation for the [Softplus function](https://en.wikipedia.org/wiki/Rectifier_(neural_networks)#Softplus) $f(x)=\ln(1+e^x)$:
$$\ln(1+e^x) \approx \begin{... |
The permutohedron P4 is the 3d truncated octahedron. Its facets consist of 6 hexagons and 8 squares.
The permutohedron P5 is a 4d polytope with 30 3d facets: there are 10 truncated octahedra, and 20 hexagonal prisms (I believe).
The permutohedron P6 is a 5d polytope with 62 4d facets: there are 12 P5 polytopes, 3... |
What type of mean satisfies "average rate of change is the mean of derivatives"? |
>Do there exist 4 rationals $(a,b,c,d)$ which satisfy the following two relations?,
>
>$a^2+b^2+c^2+d^2=1$ and $2a+b+c+d=0$?
I spent a lot of time with it and tried the criteria of quadratic equations having rational root but it doesn't seem to be working! Any help or comments would be appreciated , Thanks! |
The population OLS coefficient for some $X \in \mathbb{R}^d, Y \in \mathbb{R}$ is defined as
$$
\beta =\mathbb{E}[X_iX_i']^{-1}\mathbb{E}[X_iY_i]
$$
and if $X$ is a scalar RV, the commonly shown formula for simple regression is
$$
\beta = Var(X)^{-1}Cov(X,Y)
$$
I am clearly missing something (maybe some algebra... |
Population OLS coefficient in simple regression? |
I'm self-studying the Quadrics by following https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Projective_geometry/Chapter_2_Quadrics.pdf.
On page 25, there is an Example
> Consider the quadratic form in R3: $B(v, v) = x_1 x_2 + x_2 x_3 + x_3 x_1$.
The notes also define the quadratic form like this:
#... |
Is this quadratic form B(v, v) a symmetric bilinear? |
I would like to know how the interval of definition for an initial value problem be determined without explicitly solving it. For example, the solution to $$y' = e^y + 1, \; y(0) = 0$$ is $$ y(t) = \log \left( \frac{e^t}{2 - e^t} \right)$$ hence the interval of definition $$t < \log(2)$$
And also in the case of higher... |
I was wondering about the Euler's function and I would like to know which are all the natural numbers $m$ such that:
$$\phi (m) \neq \phi (n) \forall n \in \mathbb N$$
After putting some thought into it:
$m$ cannot be odd, because $\phi(m) = \phi(2m)$.
$m$ cannot be twice an odd number for the previous re... |
On which natural numbers do we have injectivity of Euler's totient function? |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
I'm reading a text (https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Projective_geometry/Chapter_2_Quadrics.pdf) to study about Quadrics in the projective space. The following is the example (1) (on Page 26).
## Example
Consider the hyperbola $xy = 1$.
Using the homogeneous coordinate ($x = x_1/x_0, y = x_... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
I was wondering about the Euler's function and I would like to know which are all the natural numbers $m$ such that:
$$\phi (m) \neq \phi (n) \forall n \in \mathbb N$$
After putting some thought into it:
$m$ cannot be odd, because $\phi(m) = \phi(2m)$.
$m$ cannot be twice an odd number for the previous re... |
As pointed out by BigbearZzz in the comments, no, a general subset of $L^2(\Bbb{R})$ (or indeed, any non-trivial normed space) need not have a unique point of minimum norm, as we can take any non-zero point $x$ in the space, and $\{x, -x\}$ has two distinct points of minimum norm.
The failure of this example is that... |
I was wondering about the Euler's function and I would like to know which are all the natural numbers $m$ such that:
$$\phi (m) \neq \phi (n) \forall n \in \mathbb N\setminus \{m\}$$
After putting some thought into it:
$m$ cannot be odd, because $\phi(m) = \phi(2m)$.
$m$ cannot be twice an odd number for ... |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits is easy to settle as the following.
> - The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on followin... |
As pointed out by BigbearZzz in the comments, no, a general subset of $L^2(\Bbb{R})$ (or indeed, any non-trivial normed space) need not have a unique point of minimum norm, as we can take any non-zero point $x$ in the space, and $\{x, -x\}$ has two distinct points of minimum norm.
The failure of this example is that... |
I wonder whether anyone knows a reference for the following result? I can give a shortish proof, but would prefer to cite the literature if possible.
**Theorem** Let $V=F^n$ be an $n$-dimensional $F$-space where $F$ is a field.
The action of ${\rm GL}(V)$ on $\Lambda^2(V)$ induces a homomorphism
$\phi_n\colon{... |
This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. It seems to me that since $\pi_1(GL(N,\mathbb{C}))\cong \pi_1(U(N)) = \mathbb{Z}$, the degree should e... |
You are asking two different questions: is $\mathbf Z[\sqrt{14}]$ not *norm*-Euclidean and is $\mathbf Z[\sqrt{14}]$ not Euclidean? The answers are "yes" and "no": it isn't Euclidean with respect to the absolute value of its norm function, but it is Euclidean with respect to another function.
It is hopeless to appr... |
This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. In particular I am dealing with $f:S_1\to GL(N,\mathbb{C})$ and it seems to me that since $\pi_1(GL(N,... |
I am reading the paper [Minimal surfaces of constant curvature in $S^n$,][1] wrote by Professor Robert Bryant in 1985. In this paper, the author introduced some operators to give a new method in order to classify the minimal surfaces of constant ($>0$, $<0$ or $=0$) curvature in $\mathbb{S}^n$.
In section 1, the aut... |
Let X and Y be CW complexes. Then the inclusion $j:X\vee Y\to X\times Y$ is a cofibration with cofibre $X\wedge Y$. According to Hans J. Baues' book *Principal fibrations and cofibrations, obstructions and differences* Definition 1.2.1, a closed cofibration $X\subset Y$ is a principal cofibration if there is a map
$f:... |
When will a cofibration be a principal cofibration? |
I already read [this post](https://math.stackexchange.com/questions/3397111/why-cant-the-dual-and-primal-linear-program-both-be-unbounded) and its answers and I am still not satisfied.
I want to know how to use weak duality to explain why it is not possible for both primal and dual LP to be unbounded.
Here is one... |
Why it is not possible for both primal and dual LP to be unbounded? |
1. I wholly agree with your interpretation that for real $A$ and nonzero real $B,$ $“A$ is $x$ times more than $B”$ means that $$x=\frac{A-B}{|B|}.$$
So, filling in the blanks with `15`, these assertions are all correct:
- __ is **more** than 3 **by** $\boldsymbol{12}.$
- __ is **more** than 3 **by $\bold... |
It is known that the pdf $f_{ac}$of the product of two independent random variables $a$ and $c$ with standard normal distribution is (see [here][1]):
$$f_{ac}(t)=\frac{1}{\pi}K_0(|t|), \quad t\in \mathbb R$$
where $K_0(t)$ denotes [the modified Bessel function of the second kind][2].
Then, the probability is ... |
It is known that the pdf $f_{ac}$of the product of two independent random variables $a$ and $c$ with standard normal distribution is (see [here][1]):
$$f_{ac}(t)=\frac{1}{\pi}K_0(|t|), \quad t\in \mathbb R$$
where $K_0(t)$ denotes [the modified Bessel function of the second kind][2].
Then, the probability can... |
It is known that the pdf $f_{ac}$of the product of two independent random variables $a$ and $c$ with standard normal distribution is (see [here][1]):
$$f_{ac}(t)=\frac{1}{\pi}K_0(|t|), \quad t\in \mathbb R$$
where $K_0(t)$ denotes [the modified Bessel function of the second kind][2].
Then, the probability can... |
How is, for example, Goedel's Completeness theorem, proved if it requires a formula to be true in *every* model, given that we may not be able to exhaust all possible models? |
How is completeness proved? |
This is probably trivial but I am trying to find whether or not the degree of a map $f:X\to Y$, with $X$ and $Y$ compact oriented manifolds can be extended to the case where $Y$ is non compact like $GL(N,\mathbb{C})$. In particular I am dealing with $f:S^1\to GL(N,\mathbb{C})$ and it seems to me that since $\pi_1(GL(N,... |
Is anything known about the function $P(n)$ where,
$$P(n)=|\{ m\leq n :\text{There is a perfect group of order } m\}|.$$
Like asymptotics or a good upper bound? |
I'm puzzled with how we're supposed to define $L^2(S^{n-1})$ where $S^{n-1}$ is the unit sphere in $\mathbb{R}^n$. How do we even define the inner product there? the only way that comes to mind currently is $(f,g) = \int_{S^{n-1}}fg\,d\mathcal{H}^{n-1}$. But this is not well defined since we need $f, g $ to be function... |
We were asked to solve the following equation in $\mathbb{Z}_8$: $[2]x^2+[7]x+[1]=0.\,$ I know how to do this by brute-force checking all eight possible residues $\!\bmod 8,\,$ but is there a better way? (I mean, using ring properties or something like that). Because if the problem involved a much larger modulus then b... |
I would like to determine the probability that a random quadratic polynomial has positive discriminant, where the 3 coefficients $a, b, c$ are normally distributed and independent:
- That is, given $a,b,c \sim \operatorname{N}\left(0,1\right)$, what is ( a numerical approximation to 5 digits of ) $\operatorname{Pr}\l... |
Is an ideal generated by irreductible elements of a ring a radical ideal? |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta = b^2 - 4 a c$ of such a polynomial is positive, by definition... |
Suppose we knew the expected number $\Bbb E$ of real roots of a random quadratic polynomial, where all $3$ coefficients have standard normal distribution, $\mathcal{N}(0, 1)$. Then, if $\Bbb P(\Delta \geq 0)$ is the probability that the discriminant $\Delta = b^2 - 4 a c$ of such a polynomial is positive, by definition... |
>Suppose that $\rho$ and $\sigma$ are degree n irreducible representations of a group $G$ over $\mathbb{C}$ and that for every $g\in G,$ there is a matrix $T_h\in GL(n,\mathbb{C})$ depending on $g$ so that $\sigma(g) = T_g^{-1} \rho(g) T_g$. Does there exist $T\in GL(n,\mathbb{C})$ so that $\sigma(g) = T^{-1} \rho(g)T$... |
does there exist $T\in GL(n,\mathbb{C})$ so that $\sigma(g) = T^{-1} \rho(g)T$ for all $g\in G$? |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits is easy to settle as the following.
> - The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on followin... |
I just started studying real analysis and am having some trouble with this question:
**Show that the sequence given by x<sub>n+1</sub>=$\frac{1}{5-x_n}$, $x_0$=4 converges and find its limit.**
I can observe that the sequence is decreasing and bounded below and that $\lim_{x\rightarrow\infty}x_{n+1}=0$.
Sinc... |
How to show a decreasing sequence is convergent and how to find its limit? |
I understand why on a set S with n elements there are $n^{n^2}$ binary operations, but how exactly do you count the number of commutative binary operations? I saw this post: https://math.stackexchange.com/questions/375529/counting-binary-operations-on-a-set-with-n-elements but I do not really understand their answer. F... |
I just started studying real analysis and am having some trouble with this question:
**Show that the sequence given by x<sub>n+1</sub>=$\frac{1}{5-x_n}$, $x_0$=4 converges and find its limit.**
I can observe that the sequence is decreasing and bounded below and that $\lim_{x\rightarrow\infty}x_{n+1}=0$.
Sinc... |
A cardinal $\kappa$ is Reinhardt if there is a nontrivial elementary embedding $j:V\to V$ such that $\kappa$ is the critical point of $j$. Kunen proved (using choice) that this large cardinal axiom is inconsistent.
For ordinal $\lambda$, a cardinal $\kappa$ is called $\lambda$-strong if there is a nontrivial element... |
Why is strongness of a cardinal a different notion than Reinhardtness? |
For example you have the cartesian equation:
(x-2)/-2 = y/3 = (z-1)/3
One possible choice for a is the vector <2, 0, 1>. How can you find a different possible vector of a. I understand that another one for v is any scalar multiple of it as the pvf is a span right?
Also: Sorry for the maths not being typed ... |
How can you find multiple answers for vectors a and v in parametric vector form given by the Cartesian equations? |
For example you have the cartesian equation:
(x-2)/-2 = y/3 = (z-1)/3
One possible choice for a is the vector <2, 0, 1>. How can you find a different possible vector of a. I understand that another one for v is any scalar multiple of it as the pvf is a span right?
Note: Sorry for the maths not being typed ... |
If $E \to B$ is a smooth vector bundle, then the tangent bundle $T E$ is a vector bundle over both $E$ and $T B$. (If you like, the latter structure is the derivative of the vector bundle structure on $E$.) If $F \to B$ is another smooth vector bundle and $\pi : T B \to B$ is the obvious projection, I have seen it writ... |
Because the line doesn't go through the origin, you can't just scale up the vectors.
Instead, notice that you can freely set any one of the variables, e.g. $x$, and that will restrict the other values.
For example, set $x = p$, then you get $\frac{p - 2}{-2} = \frac{y}{3} \implies y = 3(1 - \frac{p}{2})$. Similar... |
Problem 4, Chapter 9 in "Special Functions and Their Applications" (N.N. Lebedev)is to show that
$$w(z,t)=(1-t)^{\beta -\gamma }
(1-t+z t)^{-\beta }=\sum
_{n=0}^{\infty } \frac{\gamma
_n \, _2F_1(-n,\beta ;\gamma
;z) t^n}{n!}$$ $| t| <\min \left\{1,\frac{1}{|
z-1| }\right\}$
Following the ... |
Solution A: $400$ ppm/gallon
Solution B: $2$ ppm/gallon
How much solvent ($0$ ppm/gallon) is required? Say $X$
$\frac{1 \text{gallon} \times 400 \text{ppm/gallon}}{(1 + X) \text{gallons}} = 2 \text{ppm/gallon}$
Solve for $X$.
[Answer: $199 \text{gallons}$]
$\frac{V_o\times C_o}{V_o + V_d} = C_e$ (f... |
I am aware of the formula for the area of a regular polygon:
$A=([Side Count] \times [Side Length] \times [Apothem Length])/2$
However, I could not find an equation for the area of a non-regular polygon form the list of its (different) side lengths like the equation for the area of any triangle:
$A=\sqrt{Semiperim... |
Let $E/F$ be a quadratic field extension with Galois group $\{1,c\}$, where $c$ is the nontrivial automorphism of $E$. Consider $V$ to be a $E$-vector space of dimension $n$. We equip it with a $\epsilon$-Hermitian form $\phi: V \times V \rightarrow E$. Then in [MVW, p91], it is claimed that there exists an element $\d... |
Let $E/F$ be a quadratic field extension with Galois group $\{1,c\}$, where $c$ is the nontrivial automorphism of $E$. Consider $V$ to be a $E$-vector space of dimension $n$. We equip it with a $\epsilon$-Hermitian form $\phi: V \times V \rightarrow E$. Then in [Page 90 of the book of Mœglin-Vignéras-Waldspurger][1], i... |
> **Problem:** Let $a,b,c$ are non-negatives such that $ab+bc+ca=2$. Prove:
$$\frac{1}{\sqrt{a+b^2}}+\frac{1}{\sqrt{b+c^2}}+\frac{1}{\sqrt{c+a^2}} \geq \frac{2\sqrt{3}+9}{6}$$
Equality holds iff $(a,b,c) =(2,1,0)$
**Some thoughts**
Let:$\frac{1}{\sqrt{a+b^2}}=x, \frac{1}{\sqrt{b+c^2}}=y, \frac{1}{\sqrt{c+a^2}... |
Determine whether the series $\sum_{k=2}^{\infty}\frac{1}{k(\ln{k})(\ln{(\ln{k}}))}$ converges or diverges without the use of an integral test. I know that $\sum_{k=2}^{\infty}\frac{1}{k\ln{k}}$ diverges and that $k(\ln{k})(\ln{\ln{k}})\geq k\ln{k}$ when $k\geq e^e$. However, showing the reciprocal does not prove anyth... |
>Consider a circle where $50$ people are, all of different ages. We can create a minimal pair out of two people X and Y. X and Y are not sitting next to each other, and they form two arcs on the circle of the people between them. They satisfy the conditions for a minimal pair if in at least one of the arcs, everybody ... |
Problem 4, Chapter 9 in "Special Functions and Their Applications" (N.N. Lebedev)is to show that
$$w(z,t)=(1-t)^{\beta -\gamma }
(1-t+z t)^{-\beta }=\sum
_{n=0}^{\infty } \frac{(\gamma)
_n \, _2F_1(-n,\beta ;\gamma
;z) t^n}{n!}$$ $| t| <\min \left\{1,\frac{1}{|
z-1| }\right\}$
Following th... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ is $f_1(z)$ is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defini... |
A compact set A is defined st given a cover of open sets of A, there exist a finte subcover with elements in that original cover. And so A is defined as a finite union of open sets, and so A should be open (def of a topological space, the intersection of two opens is open itself). By heine borel, in R^n a compact set i... |
How can a compact set on R^2 be closed (heine borel), when by definition a compact set has to be finitely covered by open sets? |
How can a compact set on R^n be closed (heine borel), when by definition a compact set has to be finitely covered by open sets? |
The version of Hall's Marriage Theorem I'd like to consider is the following:
---
**Theorem** Let $G$ be a finite bipartite graph with bipartition $\{X,Y\}$ and edge set $E$, so that we can view $E$ as a relation from $X$ to $Y$. Suppose that, for all $S \subseteq X$,
$$\lvert \{y \in Y : \exists x \in S (xEy)\}... |
Can Hall's Marriage Theorem be reduced to the $d$-regular case? |
Is (Q,+) elementary equivalent to (R,+)? |
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