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Let the set $S = \{0,1,2,3,4,5,6,7,8,9\}$.
Let T be a family of sets of S. Every element of T has 5 elements.
If two distinct elements A and B of T always satisfy
$|A \cap B| \leq 3 $
what is the maximum number of elements in T?
**Additional context**
The program has shown that the number of elements of... |
I was wondering about the necessary conditions to a warped product metric $dx^2+f(x)^2 dy^2$ be a complete metric on $\mathbb{R}^2$, when $f$ is a positive function defined on $\mathbb{R}$.
Does anyone know anything about it? Is it right, or is there a easy counterexample?
I appreciate it. |
Is $dx^2+f(x)^2 dy^2$ always a complete Riemannian metric on $\mathbb{R}^2$, when $f:\mathbb{R}\to\mathbb{R}_{>0}$? |
Let$f:[a,b]\to \mathbb{R},\forall x_{0}\in[a,b],\lim_{x\to x_0}f(x)=0$,Prove:f(x) is Riemannian integrable on [a,b] and its Riemannian integral$\int_a^bf(x)dx=0$
Using the completeness theorem of real numbers，I have proved that $\forall \varepsilon>0,$there exists at most finite $x\in[a,b]$ su... |
> Let $\alpha > 1$ and $c \in \mathbb{R}_{+}$. If $f: U \rightarrow \mathbb{R}^{n}$, where $U \subseteq \mathbb{R}^{m}$ is an open set, satisfies $|f(x) - f(y)| \leq c|x-y|^{\alpha}$ for any $x,y \in U$, then $f$ is constant on each connected component of $U$.
The theorem above is a slight generalization of the res... |
Given $6$ digit numbers: $0,1,2,3,4,5,6$, how many numbers such that created from these numbers have $5$ digits and divisible by $30$?
For example: $30300, 12360,...$
Here is what I got:
Let a number form: $\overline{a_1a_2a_3a_4a_5}$
Because it divisible $30$ so $a_5=0$
And we have to find: $a_1+a_2+a_3+... |
Here is my attempt to prove it:
Fix X a finite set so either X $= O$ or X $\approx \mathbb{N}_{k}$. Hence card(X) $= 0$ or card(X) $= k$. Fix $k \in \mathbb{N}$
Suppose Y is also a finite set so either Y $= O$ or Y $\approx \mathbb{N}_{n}$. Hence card(Y) $= 0$ or card(Y) $= n$
Let P(n) be the statement: For al... |
I've been working on a proof that states if $X$ is an infinite set, then there exists an injection $f: \mathbb{N} \to X$.
Here is the proof I came up with:
> Let $X$ be an infinite set.
>
> We will construct the injection $f: \mathbb{N} \to X$ inductively.
>
> For the base case, define $f_1: \{1\} \to X$ by... |
I have found the proofs of the argument principle from Ponnusamy and Silverman's Complex variables with applications and Brown and Churchill's Complex variables. But I am not sure how to make sense of the argument function, which is not analytic or continuous in the full circle due to requiring a branch cut.
To s... |
Given $6$ digit numbers: $0,1,2,3,4,5,6$, how many numbers such that created from these numbers have $5$ digits and divisible by $30$?
For example: $30300, 12360,...$
Here is what I got:
Let a number form: $\overline{a_1a_2a_3a_4a_5}$
Because it divisible $30$ so $a_5=0$
And we have to find: $a_1+a_2+a_3+... |
Today I tried to compute $\zeta$(e) but no luck.Here's my approach:-
$$\zeta(e)=\sum_{n=1}^\infty 1/n^e$$= $$\sum_{n=1}^\infty 1/(2n)^e$$+$$\sum_{n=0}^\infty 1/(2n+1)^e$$
After that I have no idea what to do.If there exists a rational constant then what is it? and if not what is the proof?
Here I am not trying to ... |
$y = 0$. For what range of $ x$, will there be a unique, strictly positive $r$?
$x > 0$
$0 = -3x + \frac{5}{(1+r)} + \frac{X}{(1+r)^2}$
let $p = 1+r $
using the quadratic equation:
$p = \frac{-5 \pm \sqrt{5^2 - 4 \times (-3x) (x) }}{2 \times -3x}$
$1+r = \frac{-5 \pm \sqrt{5^2 - 4 \times (-3x) (x) }}... |
solution verification algebra unique quadratic solution? |
I'm currently going through Sheldon Axler's Linear Algebra Done Right. I am confused about three specific lines in Axler's proof that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue. The full proof is [here](https://www.cin.ufpe.br/~jrsl/Books/Linear%20Algebra%20Done%20Right%20-%... |
I'm currently going through Sheldon Axler's Linear Algebra Done Right. I am confused about three specific lines in Axler's proof that every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue. The full proof is [here](https://www.cin.ufpe.br/~jrsl/Books/Linear%20Algebra%20Done%20Right%20-%... |
> Let $\alpha > 1$ and $c \in \mathbb{R}_{+}$. If $f: U \rightarrow \mathbb{R}^{n}$, where $U \subseteq \mathbb{R}^{m}$ is an open set, satisfies $|f(x) - f(y)| \leq c|x-y|^{\alpha}$ for any $x,y \in U$, then $f$ is constant on each connected component of $U$.
The theorem above is a slight generalization of the res... |
First, let me say that I'm impressed by your maturity and wisdom. It's not easy to recognize your own limitations, accept them, and adapt. Most people have to learn the hard way, by living through a few decades of struggle and frustration. Some people actually enjoy struggle and frustration, though. Your choice.
I h... |
My plan as an undergraduate was unequivocally to be a pure mathematician, working as an algebraist as a bigshot professor at a bigshot university. I'm graduating this month, and I didn't get into where I expected to get into. My letters were great and I'm published, but my GRE was bad and my grades were good but not ... |
> Let $f$ be a holomorphic function on $\mathbb{C}$ s.t there exists
> $a>0$ and $f(z)=f(z+a)=f(z+ia)$ for every $z\in\mathbb{C}$. Prove that
> $f$ is constant.
Let $f(z)=u(x,y)+iv(x,y), z=x+yi$. From hypothesis we have $$u(x,y)+iv(x,y)=u(x+a,y)+iv(x+a,y)=u(x,y+a)+iv(x,y+a)\quad\forall x,y\mathbb{R}.$$
Also, sinc... |
Does the size of $N$ determine the size of $2^n$ that can be reached with the Collatz conjecture? |
I have a doubt concerning Lemma 3.7.4 from Topics in Algebra by I. N. Herstein.
The statement of the Lemma is:
> Let $R$ be a Euclidean ring. Then every element in $R$ is either a unit in $R$ or can be written as the product of a finite number of prime elements of $R$.
My question is that $0$ is an element of ... |
The $\text{Pareto}(1, \alpha)$ MLE is $\frac{n}{\sum_{i = 1}^{n} \log x_{i}}$, where $X_{1}, X_{2}, ..., X_{n}$ are i.i.d. with $f(x | \alpha) = \alpha x^{-\alpha - 1}$. Further, add the constraint that $\alpha > 2$.
This MLE may be less than 2. What's the proper way to handle that?
If the parameter space for $\a... |
> Let $f$ be a holomorphic function on $\mathbb{C}$ s.t there exists
> $a>0$ and $f(z)=f(z+a)=f(z+ia)$ for every $z\in\mathbb{C}$. Prove that
> $f$ is constant.
Let $f(z)=u(x,y)+iv(x,y), z=x+yi$. From hypothesis we have $$u(x,y)+iv(x,y)=u(x+a,y)+iv(x+a,y)=u(x,y+a)+iv(x,y+a)\quad\forall x,y\in\mathbb{R}.$$
Also, s... |
A topological space $X$ is [sequentially discrete](https://topology.pi-base.org/properties/P000167) if every convergent sequence in $X$ is eventually constant.
A space $X$ is [totally path disconnected](https://topology.pi-base.org/properties/P000046) if every path $f:[0,1]\to X$ is constant.
It seems the followi... |
PANACEA has 7 letters with 3 As.
There are 4! ways to arrange (P, N, C, E) _ P_N_C_E _. Between these letters, there are 5 slots. So, to arrange 3 As in 5 slots is 5P3. We divide it by 3! for the 3 As to avoid duplicates. We have:
$4!\cdot \frac{5P3}{3!} = 240$
This is the correct answer:
```Total arrangeme... |
In how many ways can the letters of the word PANACEA be arranged so that the three As are NOT all together? |
This has been puzzling me for a few days now. Here is the relevant excerpt from the book mentioned in the title:
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/XSgZl.png
What four primes is he referring to, right at the end of the excerpt? |
What does Schroeder mean by "we have found 4 more primes" in section 3.2 of his book Number Theory in Science and Communication (5th ed.)? |
Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal structures, and the first has a canonicalsymmetric monoidal structure. The second category has (at least) two symmetric mo... |
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems to... |
The following solution is not mine, I found it on [a discussion][1] on HN.
Let $\color{blue}{\textrm{B}} = \{i; \, \chi(S[i \dots j]) = \color{blue}{\text{blue}} \; \forall j \geq i \}$, that means $\color{blue}{\textrm{B}}$ is the set of all indexes $i$ so that all substring of $S$ starting from $i$ is blue. There... |
Solve equation: $\cos x\cos 2x\cos(3x-\frac{\pi}{3} ) = \frac{1}{2}+ \sin x \sin 2x \sin 3x$
I guess $\sin x=0$ may be one of the roots but I can't factor polynomial. Can anyone help me with this? |
The following very nice solution is not mine, apparently it is by [Philipp Czerner][1], I found it on [a discussion][2] on HN.
Let $\color{blue}{\textrm{B}} = \{i; \, \chi(S[i \dots j]) = \color{blue}{\text{blue}} \; \forall j \geq i \}$, that means $\color{blue}{\textrm{B}}$ is the set of all indexes $i$ so that a... |
Solve equation: $\cos x\cos 2x\cos(3x-\frac{\pi}{3} ) = \frac{1}{2}+ \sin x \sin 2x \sin 3x$
I guess $\sin x=0$ may be one of the roots. I've been trying very hard to factor polynomials but was unable to solve it. Can anyone help, please?
Thank you. |
Let $X$ be a topological space. The simplicial set $\operatorname{sing}(X)$ has as its $n$ simplicies all singular $n$-simplices $\Delta^n \to X$. The realization of the singular simplicial set is then
$$
\lvert\operatorname{sing}(X)\rvert = \left.\coprod_{n\in\mathbb{N}}\underbrace{\operatorname{sing}_n(X)}_{\tex... |
Let $k$ be a commutative base ring. We have a category $\operatorname{Mod}_k$ of $k$-modules and a category $\operatorname{grMod}_k$ of $\mathbb{Z}$-graded $k$-modules. Both of these have monoidal structures, and the first has a canonical symmetric monoidal structure. The second category has (at least) two symmetric m... |
I have an equation of the form:
$$y(t)=\frac{Ce^{-t/(\alpha\beta)}}{\alpha} + C\left(1-e^{-t/(\alpha\beta)}\right)$$
then, according to the solution provided, as $\alpha \rightarrow 0$, the expression becomes:
$$y(t)=C\delta(t)\int_{t=0}^{\infty}\frac{e^{-t/(\alpha\beta)}}{\alpha}dt + C$$
which ends up being ... |
How does taking the limit ends up with integration? |
How does taking the limit end up with an integration? |
Suppose we are dealing with a $2$-dimensional sphere.For simplicity we consider $S^2\subset \mathbb R^3$.Now we can give a parametrization on the sphere via the map $\varphi:(0,2\pi)\times(0,\pi)\to \mathbb R$,given by,
$\varphi(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)=(x(\theta,\phi),y(\theta... |
I have an equation of the form:
$$y(t)=\frac{Ce^{-t/(\alpha\beta)}}{\alpha} + C\left(1-e^{-t/(\alpha\beta)}\right)$$
then, according to the solution provided, as $\alpha \rightarrow 0$, the expression becomes:
$$y(t)=C\delta(t)\int_{t=0}^{\infty}\frac{e^{-t/(\alpha\beta)}}{\alpha}dt + C$$
which ends up being ... |
Let $(H, ||\cdot||)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.
I was wondering if it is enough to conclude that $$\sup_{n\in\mathbb N}||f_n||_H <+\infty.$$
I'd say that the answer is yes because being bounded in $H$ means that there exists $R>0$ such that $||f_n||_H \le R$. With this info... |
Assuming that $(f_n)_n$ is bounded, does $\sup_{n\in\mathbb N}||f_n||_H <+\infty$ hold? |
This problem is from the book "Inside Interesting Integrals" written by Prof. Paul J, Nahin, which is stated as follows,
Imagine a circle (let’s call it $C_{1}$) that has radius $a$. We then chose at random (i.e., uniformly distributed) and independently, three points from the interior of that circle. These three po... |
I am currently trying to show that the set $S = \mathbb{Q} \cap [0, 1]$ is not compact by showing the cover
$$
\mathcal{C} = \left\{ \left( \frac{p}{q} - \frac{1}{10^q!}, \frac{p}{q} + \frac{1}{10^q!} \right) \right\}_{p/q \in S}
$$
admits no finite subcover, with the assumption that each $\frac{p}{q} \in S$ is wri... |
Is it possible to show my cover has no finite subcover? |
Let $(H, ||\cdot||)$ be a Hilbert space and let $(f_n)_n$ be a bounded sequence in $H$.
I was wondering if this information is enough to conclude that $$\sup_{n\in\mathbb N}||f_n||_H <+\infty.$$
I'd say that the answer is yes because being bounded in $H$ means that there exists $R>0$ such that $||f_n||_H \le R$. ... |
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/6bunI.png
I understand all of the proof until I get to the part where we divide by 2M. My teacher explained that we divide by 2M because we are aiming to get to epsilon, which makes sense but this does not seem vigorous enough to me as I b... |
I would like to determine whether the polynomial $x^n+5x+3$ is irreducible over $\mathbb{Q}$ for $n\ge 2$. I thought of considering $\bmod{2}$ so that we obtain a polynomial $x^n+x+1$. However the polynomial $x^n+x+1$ is not always irreducible over $\mathbb{F}_2$ ([see here](https://math.stackexchange.com/questions/179... |
I have an equation of the form
$$y(t)=\frac{Ce^{-t/(\alpha\beta)}}{\alpha} + C\left(1-e^{-t/(\alpha\beta)}\right)$$
then, according to the solution provided, as $\alpha \rightarrow 0$, the expression becomes
$$y(t)=C\delta(t)\int_{t=0}^{\infty}\frac{e^{-t/(\alpha\beta)}}{\alpha}dt + C$$
which ends up being eq... |
I am learning numerical linear algebra and curious about one thing. It is possible to reduce any matrix to the Hessenberg form in finite steps with a unitary matrix. But why is it impossible to reduce it further into an upper triangular form in finite step?
What is the fundamental barrier? |
I'm interested in solving Exercise 6.1-5 from the fourth edition of CLRS:
> At which levels in a max-heap might the $k$th largest element reside for $2 \leq k \leq \lfloor n/2 \rfloor$, assuming that all elements are distinct?
**My Attempt:**
I tried solving a more specific problem of finding the possible *ind... |
What are the possible levels of the $k$th largest element in a max-heap of $n$ distinct elements? [CLRS] |
Does the natural number $N$ determine $2^m$ that can be reached with the Collatz conjecture? |
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems to... |
Let $f(x)$ be a cubic whose coefficient of the leading term is positive and $g(x) = e^{\sin(\pi x)} - 1$. The composite function $h(x) = g(f(x))$ is defined in the set of all numbers, and has a local maximum at $x=0$. In the open interval $(0,3)$ the function $h(x)$ intersects the line $y= 1$ seven times.
Given tha... |
Does the natural number $N$ determine an upper limit of $2^m$ that can be reached with the Collatz conjecture? |
Evaluate the limit as $x$ goes to $\infty$ of the following
$$ x(1+\frac{1}{x})^x -kx^2\ln(1+\frac{1}{x})$$
I tried calculating the limits of the $2$ terms separately. By applying L'Hopital,the $2nd$ limit can be solved by writing $x^2\ln(1+\frac{1}{x})=\frac{\ln(1+\frac{1}{x})}{\frac{1}{x^2}}$. Now,Lhopital gives... |
In the Manfred Stoll, Introduction to Real Analysis he states 9.5.3 Theorem as follows :
> **9.5.3** **THEOREM ( Dirichlet )** Let $f$ be a real-valued periodic function of $\mathbb{R}$ of period $2\pi$ with $f\in \mathcal{R}[-\pi ,\pi]$. Suppose $x_0 \in \mathbb{R}$ is such that
(a) $f(x_0^{+})$ and $f(x_0^{... |
Why $h(t) := f(x_0+t)-f(x_0^{+})$ is Riemann integrable on $[0, \pi]$? ( In proof of the Dirichlet's Theorem about convergence of Fourier series ). |
This is still pretty far from an explicit answer, but it includes some theoretical musings and pretty pictures and, really, isn't that all anyone could ask for? (I also establish the convergence of a class of similar values, but the other answers would likely suffice for that too)
We are considering the sequence of ... |
Here is the problem I am doing:
> Take a deep breath before attempting this problem. In the book
> Innumeracy [22], John Allen Paulos writes:
>
> Now for better news of a kind of immortal persistence. First, take a
> deep breath. Assume Shakespeare’s account is accurate and Julius
> Caesar gasped [“Et tu, Br... |
In the Manfred Stoll, Introduction to Real Analysis he states 9.5.3 Theorem as follows :
> **9.5.3** **THEOREM ( Dirichlet )** Let $f$ be a real-valued periodic function of $\mathbb{R}$ of period $2\pi$ with $f\in \mathcal{R}[-\pi ,\pi]$. Suppose $x_0 \in \mathbb{R}$ is such that
(a) $f(x_0^{+})$ and $f(x_0^{... |
**Theorem:1** Let the function
$$f (z) = u(r, θ) + iv(r, θ)$$be defined throughout some neighborhood of a nonzero point $z_0 = r_0 \exp(iθ_0)$ and suppose that
>**(a)** The first-order partial derivatives of the functions u and v with respect to r and θ
exist everywhere in the neighborhood;
>
>**(b)** Those parti... |
Here is the problem I am doing:
> Take a deep breath before attempting this problem. In the book
> Innumeracy [22], John Allen Paulos writes:
>
> Now for better news of a kind of immortal persistence. First, take a
> deep breath. Assume Shakespeare’s account is accurate and Julius
> Caesar gasped [“Et tu, Br... |
> Evaluate the limit as $x$ goes to $\infty$ of the following
$$x\left(1+\frac1x\right)^x-kx^2\ln\left(1+\frac1x\right)$$
I tried calculating the limits of the two terms separately. By applying L'Hopital, the 2nd limit can be solved by writing $x^2\ln\left(1+\frac1x\right)=\frac{\ln\left(1+\frac1x\right)}{\frac{1}... |
PANACEA has 7 letters with 3 As.
There are 4! ways to arrange (P, N, C, E) _ P_N_C_E _. Between these letters, there are 5 slots. So, to arrange 3 As in 5 slots is 5P3. We divide it by 3! for the 3 As to avoid duplicates. We have:
$4!\cdot \frac{5P3}{3!} = 240$
This is the correct answer:
```Total arrangeme... |
Let $\lambda_j = \tfrac{4}{(2j - 1)^2 \pi^2}$ and $\psi_j(x) = \sqrt{2} \sin(\tfrac{2j-1}{2} \pi x)$ denote the eigenfunctions associated to the first-order Sobolev space.
It is claimed in many texts on reproducing kernel Hilbert spaces that we have
$$
\min\{x, y\} = \sum_{j \geq 1} \lambda_j \psi_j(x) \psi_j(y) ... |
Is it true that $\frac{8}{\pi^2} \sum_{n \geq 1, n~\text{odd}} \frac{\sin(\tfrac{n}{2} \pi y) \sin(\tfrac{n}{2} \pi x)}{n^2} = \min\{x, y\}$? |
I saw in the textbook 2<x<5 is described as "x is greater than 2 but less than 5". Is it correct to use "and" instead of "but"?
I think describe the inequality as "x is greater than 2 **and** less than 5" is also correct, is this true? Thank you. |
I saw in the textbook $2<x<5$ is described as *"x is greater than 2 but less than 5"*. Is it correct to use "and" instead of "but"?
I think describe the inequality as *"x is greater than $2$ **and** less than $5$"* is also correct, is this true? Thank you. |
I can see that the map is an homeomorphism for neighbourhoods around non-zero points not including zero. So I need to check only around neighbourhoods of zero.
Can anyone help me out? |
Indeed, there is an apparent paradox as we can see on this example (like yours) with two hyperbolas (featured with their asymptotes) :
[![enter image description here][1]][1]
$$r=\frac{p_0}{1 - e_0 \cos \varphi} \ \text{and} \ r=\frac{p_1}{1 - e_1 \cos \varphi}\tag{1}$$
with $$e_0=1.5, e_1=2, \ p_0=1, \ p_1... |
Suppose we are dealing with a $2$-dimensional sphere.For simplicity we consider $S^2\subset \mathbb R^3$.Now we can give a parametrization on the sphere via the map $\varphi:(0,2\pi)\times(0,\pi)\to \mathbb R^3$,given by,
$\varphi(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi)=(x(\theta,\phi),y(\the... |
Number of elements in set S = {(x,y,z): x,y,z ∈ Z, x+2y+3z=42, x,y,z ≥ 0} is?
My solution:
This is equal to the coefficient of $t^{42}$ in
$(1+t+t^2+t^3....t^{43})(1+t^2+t^4...t^{42})(1+t^3+t^6...t^{42})$
= $\frac{(1-t^{43})}{1-t} * \frac{(1-t^{44})}{1-t^2} * \frac{(1-t^{45})}{1-t^3} $
= $\frac{1} {(1... |
Number of elements in set S = {(x,y,z): x,y,z ∈ Z, x+2y+3z=42, x,y,z ≥ 0} is? |
Here, I **first** provide a proof with required details based on the relative entropy method, and **then** I discuss how your **main issue** on dropping the expectation constraint when using the KKT method can be solved. Indeed, without the expectation constraint, the covariance of the distribution cannot be fixed, and... |
In Mathematica Guidebook for symbolic computations (https://www.amazon.com/dp/0387950206/wolframresearch-20), in the Exercises, 30 (c)(p. 359), there is a question:
[![qseriestoqproduct][1]][1]
[1]: https://i.stack.imgur.com/7lqEq.png
I have no clue how to implement this!. Help me how to deal with this? Thank ... |
I can see that the map $f(z) = z^k$ is an homeomorphism for neighbourhoods around non-zero points not including zero. So do I need to check only around neighbourhoods of zero.
Can anyone help me out? |
Is it possible to find the equation of straight line if the position vector of the point on the line and direction cosines of line a perpenducular to it are given? I tried to solve this problem using dot product but I ended up getting an equation of plane rather than that of a line.
... |
Is it possible to find the equation of straight line if the point on the line and direction cosines of line perpendicular to it are given? |
Let $\lambda_j = \tfrac{4}{(2j - 1)^2 \pi^2}$ and $\psi_j(x) = \sqrt{2} \sin(\tfrac{2j-1}{2} \pi x)$ denote the eigenfunctions associated to the first-order Sobolev space.
It is claimed in many texts on reproducing kernel Hilbert spaces that we have
$$
\min\{x, y\} = \sum_{j \geq 1} \lambda_j \psi_j(x) \psi_j(y) ... |
Is it true that $\min\{x, y\} = 2 \sum_{n \geq 1, n~\text{odd}} \frac{\sin(n x) \sin(ny)}{n^2}$? |
Number of elements in set S = {(x,y,z): x,y,z ∈ Z, x+2y+3z=42, x,y,z ≥ 0} is?
My solution:
This is equal to the coefficient of $t^{42}$ in
$(1+t+t^2+t^3....t^{43})(1+t^2+t^4...t^{42})(1+t^3+t^6...t^{42})$
= $\frac{(1-t^{43})}{1-t} * \frac{(1-t^{44})}{1-t^2} * \frac{(1-t^{45})}{1-t^3} $
= $\frac{1} {(1... |
Number of elements in set $S = \{(x,y,z): x,y,z \in \mathbb{Z}, x+2y+3z=42, x,y,z \ge 0\}$ is?
My solution:
This is equal to the coefficient of $t^{42}$ in
$(1+t+t^2+t^3+...+t^{43})(1+t^2+t^4+...+t^{42})(1+t^3+t^6+...+t^{42})$
= $\frac{(1-t^{43})}{1-t} \times \frac{(1-t^{44})}{1-t^2} \times \frac{(1-t^{... |
Number of elements in set $S = \{(x,y,z): x,y,z \in \mathbb{Z}, x+2y+3z=42, x,y,z \ge 0\}$ is? |
It is standard notation but I define them anyway to avoid ambiguity: $$\Gamma_0(N) := \{\begin{pmatrix} a & b \\
c & d \end{pmatrix} \in \Gamma : c \equiv 0 \text{(mod $N$)}\}$$ and $$\Gamma:=SL_2(\mathbb{Z})$$
\\ With this notation I'm trying to figure out what the subgroups of finite index of $\G... |
It is standard notation but I define them anyway to avoid ambiguity: $$\Gamma_0(N) := \left\{\begin{pmatrix} a & b \\
c & d \end{pmatrix} \in \Gamma : c \equiv 0 \text{(mod $N$)}\right\}$$ and $$\Gamma:=SL_2(\mathbb{Z})$$
With this notation I'm trying to figure out what the subgroups of finite in... |
A von Neumann $M$ is called prime if $M=M_1\bar{\otimes} M_2$ implies that $M_1$ or $M_2$ is a type $I$ von Neumann algebra.
A factor is prime if and inly if the factor cannot be factorized as the tensor product of two diffuse von Neumann algebras.
If a von Neumann algebra $M$ is prime, $N$ is a von Neumann su... |
I'm supposed to compute it by using some of the following limits somehow:
$$\lim_{x\to 0}(1+x)^{\frac{1}{x}}=\lim_{x\to \infty}\left(1+\frac{1}{x}\right)^{x}$$
But I can't find how to make this limit appear in the calculation. Can someone give me a hint? |
How to compute $\lim_{x\to \infty} x(a^{\frac{1}{x}}-1)$? |
From $h(t):= f(x_0+t)-f(x_0^{+})$, since $f(x_0^{+})$ is constant, it suffices to show that $f(x_0 +t)$ is Riemann integrable on $[0, \pi]$. We will use answer of https://math.stackexchange.com/questions/3029512/integrability-of-composite-functions?noredirect=1&lq=1 :
> Lemma. If $f$ is a Riemann integrable function... |
Let a Collatz sequence of a natural number $x$ as follow:
$$\{x\}: \begin{cases}
x_0=x\\
x_{n+1} = \begin{cases}
\frac{3x_n+1}{2} \text{ if $x_n$ is odd}\\
\frac{x_n}{2} \text{ if $x_n$ is even}
\end{cases}
\quad\forall n\in\mathbb{N}^*
\end{cases}$$
The sequence is said to be convergent iff there exists k... |
Let a Collatz sequence of a natural number $x$ as follow:
$$\{x\}: \begin{cases}
x_0=x\\
x_{n+1} = \begin{cases}
\frac{3x_n+1}{2} \text{ if $x_n$ is odd}\\
\frac{x_n}{2} \text{ if $x_n$ is even}
\end{cases}
\quad\forall n\in\mathbb{N}^*
\end{cases}$$
The sequence is said to be convergent iff there exists k... |
Suppose $f(x,y,z)$ is a real-valued function that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
\Delta(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f_{yy} & f_{yz}\\
f_{zx} & f_{zy} & f_{zz}\\
\end{bmatrix},
$$
which is real symmetric. Therefore, its eigenvalues m... |
A fair coin is tossed $n\geq 2$ times. Player 1 wins if there are more HH than HT in the sequence, Player 2 wins if there are more HT than HH. Who has the higher probability of winning ?
Modelling the tosses with iid $X_i$, the number of $HH$ is $S_n = \sum_{i=1}^{n-1} 1_{X_i=H,X_{i+1}=H}$ and the number of HT is $R... |
Suppose $f(x,y,z)$ is a real-valued function of three real variables that satisfies the Laplace equation, $\nabla^2f=0$. Consider the Hessian matrix
$$
\Delta(x,y,z)=
\begin{bmatrix}
f_{xx} & f_{xy} & f_{xz}\\
f_{yx} & f_{yy} & f_{yz}\\
f_{zx} & f_{zy} & f_{zz}\\
\end{bmatrix},
$$
which is real symmetric. Ther... |
Let $P(x)$ be a polynomial with integer coefficients, and $p$ be a large prime. I want to find the number of solutions to the congruence equation $$P(x)\equiv 0(\mathrm{mod}\ p^\alpha).$$
In my practice the degree of $P(x)$ is $4$. We can write
$$ P(x) = \sum_{j=0}^4 a_j x^j.$$
I wonder that whether the number of ... |
In a group of children, each child has a certain number of pencils from 1 to n. If the number of children who have i or more pencils is $2^{n-i}$, for i=1,2,3....,n and the total number of pencils is 511, find the maximum number of pencils with any child.
----------
**My attempt:-**
Number of children with ... |
Hi im having trouble solving this mcq question on econometrics. Could someone help me? Thank you!
[Full question here][1]
[1]: https://i.stack.imgur.com/bquJy.png |
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