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Statement
\begin{equation}
\int_{\mathbb R} \exp \left( -2\pi (\frac{x}{\sqrt{2}})^2 \right) \exp\left( -i2 \pi \frac {x}{\sqrt{2}} \cdot f \right) dx
= \exp \left( -\pi f^{2} / 2 \right)
\end{equation}
My attempt based on J.G.'s tips
Call the integral I(f).
\begin{equation}
I(f) = \int_{\mathbb R} \left( -\pi x^{2} ... |
I have a theorem for the hamming bound or the sphere packing bound.
A q-ary $(n, m, 2e+1)$ code satisfies
$$M \bigg\{ \binom {n}{0} + \binom{n}{1} (q-1)+...+\binom{n}{e}(q-1)^e\bigg\} \leq q^n $$
What is $q^n$?
What is this theorem trying to say?
|
I am trying to prove the following statement: Let $R$ be a real closed field (such as the real numbers). Let $M\subseteq R^n$ be a semi-algebraic set, i.e. a set which is defined by a Boolean combination of finitely many polynomial inequalities. Suppose that $M$ is unbounded, closed and star-shaped, i.e. there exists $... |
I'm still a high school student and very interested in maths but none of my school books describe these kind of things, and only say how to do it not the whys. My question is very simple, for example:
19
+25
= 44
because the one from adding 9 and 5 goes on to add with 1 and two. How did this rule ... |
Question: $xRy$ iff $|x-y| \leq 2$
I think I've found this to be reflexive and symmetric, but I'm stuck on transitivity. Can someone assist me with testing transitivity?
|
$$
\sum_{x = 1}^{\infty}\frac{3x^{2} + 12x + 16}
{\left[\vphantom{A^{A}}x\left(x + 1\right)\left(x + 2\right)
\left(x + 3\right)\left(x + 4\right)\right]^{\, 3}} =
\frac{1}{4\left(a!\right)^{b}}
$$
Compute $b-a$ if $b$ and $a$ are positive integers.
I asked my teacher to help me in solving this sum. But, unfortunatel... |
In a fictional deck of cards, there are 30 cards, 15 different ones (each card has an identical pair, so 15 pairs = 30 cards).
I want to answer the question:
I am dealt 10 cards. I wish to receive 5 specific and identical cards (so 1 card of each 5 pairs).
What is the probability of receiving these 5 specific cards?
I ... |
I saw this example given as a $\sigma$-algebra in various places. It goes like this: Let $X$ be a set and assume that the collection $\{A_1,\dots, A_N\}$ is a partition of $X$. Then the collection $\mathcal{F}$ of all unions of sets $A_j$ forms a $\sigma$-algebra.
I have no problems with checking the steps of $\sigma$-... |
Estimate from above the series:
$$\sum_{i=1}^n \frac{1}{(i-j)^2}$$
for $n\in\mathbb N$, $j=1,2,\dots,n$ and $j\neq i$. I'd like to know an estimate dependent by $j$. This question comes from a simple curiosity but I have no idea to solve it. Do anyone have any idea? Thank you.
|
If the random variable $X$ is binomially distributed with parameters $n=6$ and $p=0.3$, what is $$E(4+3X^2)$$
I know $E(X) = np = 1.8$. I solved this problem by finding $P(X)$ of all $X$ using $(^6_x)(0.3)^x(0.7)^{6-x}$ and then apply them with $4+3X^2$ to solve for $E(4+3X^2)$. But this takes too long and I think th... |
Problem:
Let $\,A,\,B,\,C,\,D\,$ be four distinct spheres in a space.
Suppose the spheres $A$ and $B$ intersect along a circle which belongs to some plane $P$, the spheres $B$ and $C$ intersect along a circle which belongs to some plane $Q$, the spheres $C$ and $D$ intersect along a circle which belongs to some plan... |
I have the following system:
$$
\dot x(t)=\begin{bmatrix}
-2 & 1 & 2 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}x(t)+\begin{bmatrix}
0 \\ 1 \\ 0 \end{bmatrix}u(t)
$$
The controllability matrix is:
$$
P=\begin{bmatrix}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{bmatrix}
$$
I have only been able to find the following rule so f... |
$$
T \begin{bmatrix}
x_1 \\
x_2 \\
\end{bmatrix} =
\begin{bmatrix}
x_1 +3x_2\\
x_2 \\
\end{bmatrix}
$$
By considering $ker(T)$ first
$$ker(T)=\{(x_1,x_2) |T(x_1,x_2)=0\} \\
x_1+3x_2=0 \implies x_1=-3x_2 \\
ker(T)=(-3x_2,x_2)=x_2(-3,1) \\
\therefore basis(ker(T... |
I have a difficult integral to compute.I know the result, but need to know the method of calculation.
How prove this result? $$\int_0^{1/\sqrt{2}}\frac{\arcsin({x^2})}{\sqrt{1+x^2}(1+2x^2)}dx=\frac{\pi^2}{144}$$
|
Given the following equation
$$\frac{J_{n - 1} (u)}{uJ_n (u)} - \frac{K_{n-1}(w)}{wK_n(w)} = 0$$
(where $J$ is the Bessel function of the first kind, $K$ is the modified Bessel function of the second kind, $u, w \in \mathbb{R}$, $u \neq w$ in general)
it should be proved that it is equivalent to
$$\frac{J_{n - 1} (u)}{... |
Let $Z_i = \frac{X_i}{Y_i + C}$ with $i = 1, 2, \dotsc, N$ denote the sequence of the random variables, where $X_i$ and $Y_i$ are exponentially distributed independent random variables with different means $\lambda_x$ and $\lambda_y$. What will be the distribution of $\min\limits_{i = 1, 2, \dotsc, N}(Z_i)$ as $N \rig... |
If given this function:
$$f(x) =
\begin{cases}
e^x, & x \le 0 \\[2ex]
-e^{-x}+2, & \text{x > 0}
\end{cases} $$
How do I calculate the derivative at $x=0$?
Shall I calculate by the normal way or should I use the limit definition?
|
I have 3 Images. A, B and C. if I place it on graph, its look something like this. Now main image is A and I place B and C on that image's (A) center point.
For easy understanding, let's consider only 2 images, A and B.
Now I rotate image B to 10 degree as per below image. so how can I found value of line (Red or Yell... |
I have been trying to find
$$\lim_{x\to -2}{\frac{x+2}{\sqrt{-x-1}-1}}$$
without L'Hospital's Rule, but I am stuck.
I tried
Rationalizationg the denominator
Factoring out $\,x$
But it did not work. Finally, I used L'Hospital's Theorem and I got the answer $-2$.
Is there any way to evaluate this without this concept?... |
Let $(X_n)$ be a sequence of independent random variables, such that $P(X_i=1) = P(X_i=-1) = 1/2$. Then, the reflection principle states that for all $a > 0$,
$$P(\max_{1\leq k\leq n} S_k \geq a) = P(S_n \geq a) + P(S_n \geq a + 1).$$
I have looked at many places but I have rarely found a full proof for the simple rand... |
Let us consider two topological spaces $X$ and $Y$. Now let us consider projection mappings $p_1$ and $p_2$ defined from the product set of $X$ and $Y$ to $X$ and $Y$ respectively .The lecture notes I am going through says that if $U$ is an open set in $X$ and $V$ is an open set in $Y$ then the set product of $U$ and $... |
Does the following variation of Fermat Little Theorem hold? How do you prove it?
Let $p$ be a prime number greater than $3$. Then there exist a natural
non-prime $m > 1$ such that $$p^{\phi(m)} \equiv p \quad
(\mathrm{mod}(m)),$$ where $\phi$ is Euler's function.
|
This is a differential equation question. Since I might not be able to explain is well, I will attach a link to the question as well as a screenshot of the mark scheme.
Question:
Mark Scheme:
Here's what I've done so far:
$\frac {dh}{dt} = kh^2$
when $h = 1, \frac {dh}{dt} = 4.95$
$k = 4.95$
$\frac {dh}{dt} = 4.95h... |
Consider the three equations
$$
y-x^2=0,\quad z+xy=0,\quad -y-z+x^2-xy+y^2+z^2-x^4=0.
$$
How can I find multiple roots of this?
Is it allowed to reduce the system as far as possible and then to find multiple roots of one equation?
For example, we have
$$
y=x^2,\quad z=-x^3
$$
from the first two equations, and plugging ... |
I was reading on Wikipedia about quotient ideals. It mentions that if $W$ and $V$ are affine varieties (assume $V$ is) and $I(V)$ and $I(W)$ are the ideals for $V$ and $W$, then $$I(V):I(W) = I(V\setminus W).$$
They also talk about $Z(I:J) = \text{cl}(Z(I)-Z(J))$. Is it not right if we remove closure? That is, it is no... |
I am slightly interested in discrete wavelet transforms (DWT), but so far I have mostly used already-derived and existing well-known wavelets, such as Daubechies, Cohen-Daubechies-Faveau, Symlets and so on. As far as I know most existing wavelets tend to focus on one or a few of the following properties, compact suppor... |
Problem:
A recruiting firm finds that $20$% of the applicants for a particular sales position are fluent in both English and Spanish. Applicants are selected at random from the pool and interviewed sequentially.
Suppose that the first applicant who is fluent in both English and Spanish is offered the position, and the... |
In a vijantkhand sports stadium, athletes choose from $4$ different
racket games (apart from athletes which is compulsory for all)
These are tennis, table tennis, squash and badminton. It is known
that $20\%$ of thee athletes practising there are not choosing any
of the racket sports. The four games given he... |
Any idea on how I can get an expression in the form of sum of products from the following one?:
\begin{equation}
\prod_{i=1}^M \left(\sum_{n=1}^i x_n\right)
\end{equation}
|
I was trying to find the nature (maxima, minima, inflection points) of the function $$\frac{x^5}{20}-\frac{x^4}{12}+5=0$$
But I faced a conceptual problem. It is given in the solution to the problem that $f''(0)=0$ and $f'''(0) \neq 0$ so $0$ is not an inflection point. But why should we check the third derivative?
Isn... |
I'm used to dealing with binomial expansion in the form $(1+x)^n$. I understand that if the number is not $1$ then you have to divide the whole bracket by something which would make it $1$. However I'm not too sure how to deal with an $x^2$ in the bracket. Can I just work through it as usual using the binomial expansio... |
For a $3 \times 3$ matrix:
$
$[A]$ = \begin{bmatrix}
6 & -2 & 2 \\
-2 & 3 & -1 \\
2 & -1 & 3
\end{bmatrix}
$
I have the eigenvalues: $\lambda = 2, 2, 8$
Now for each value I need to find eigenvectors:
Now, when solving for when $\lambda = 2$ and after applying the row operations,... |
I have a list of n objects
say [ apple, orange, carrot, cherry, banana ]
Now I am trying to come up with an equation which will generate an unique number for each combinations from this list.
eg :
say I assign following values for each items
apple = 1
orange = 2
carrot = 3
cherry = 4
banana = 5
If I directly add the... |
I am interested in the orders of random permutations. Since the law of the log of the order of a permutation converges to a normal law (for instance Erdös-Turan Statistical group theory III), one expects that the probability for two permutations of $S_n$ to have the same order goes to 0 as n goes to infinity. Indeed ex... |
I have a question about the first part of the proof for the statement "If $x$ and $y$ are odd integers, then $xy$ is odd" if we are using a direct proof.
Now i've gotten the proof roughly correct but I have an issue with what we write at the beginning of the proof. If we were to put the statement into first order logic... |
I read the following problem in the book Introduction to Algorithms by Cormen, Leiserson, Rives, Stein and I couldn't make any progress with it.
There is a set of $n$ numbers. Give an algorithm that decides in $O(n\times\log \space n)$ steps if there are two numbers in the set whose sum is a given number ($x$). Could y... |
Suppose you had the function
$$
f(x) = \; \text{ the integer part of } x
$$
I wish to show that this is not continuous at the point $x=1$, which I will try to do by showing that $\lim_{x \rightarrow 1} f(x)$ does not exist. To do this, I use a direct $\varepsilon - \delta$ proof of convergence. That is, show that for s... |
Use Banach contraction theorem to find a solution of x+e^x=0 correct to 3 decimal places.
Any suggestions how I should begin? Banach contraction theorem is new to me.
|
First of all: yes, there is already a post about it, but I missread retract as strong deformation retract and wanted to know if this solution is right if we really do assume the stronger assumption of a derformation retract.
Let $T^2$ be the two-dimensional torus and $D$ an embedded disk. We will show by contraposition... |
I'm trying to find the most correct way of ranking countries based on multiple factors with measurements in different units. Take the following example:
I am comparing $4$ countries nl.:
United States,
China,
Australia,
South Africa
And trying to create a composite ranking system based on their values of the following ... |
Let $R$ be a commutative ring and let $I \subset R$ be an ideal.
For any $n \ge 1$, the ring homomorphism $R \rightarrow R/I^n$ is surjective, hence an epimorphism in the category of rings. What about the natural map $R \rightarrow \hat{R_I}:=\lim_n R/I^n$ to the $I$-adic completion of $R$? This map is no longer surjec... |
Show from first principles, that is using the definition of limit, that
$$\lim_{z\to i}\frac{z-1}{z^2+1} = \infty$$
Please can someone actually show me the procedure, struggling to understand it when people describe with words
|
Let $X$ be a normed space, $f \in X'$, $\alpha \in \mathbb{R}$;
Prove that: $$\text{int}\{x \in X: f(x) \ge \alpha\}=\{x \in X:f(x)>\alpha \}$$
I noted the set in the left by $A$ and the set in the right by $B$; so, I have to prove that: $\text{int}(A)=B$.
I thought to prove it by double inclusion; let $x \in \text{int... |
over $GF(q)$ where $q\in\mathbb{N}$, we build a string of size $q^n-1$. now, how can I show that it is always possible to construct that string so it contains all sub-strings of size $n$ exactly once, but the one sub-string missing is $x^n$ for some $x\in GF(q)$?
|
How can one do the problem 1.3.11 b in Algebraic Geometry and Arithmetic Curves? I have read basics of commutative algebra but this one seems to be too difficult.
Let $A$ be a commutative ring with unit which is a complete ring for the $I$-adic topology, where $I$ is an ideal of $A$. Let $(M_n)_{n\geq 0}$ be $A$-module... |
Let $\{x, y\}$ be a linearly independent set in a Euclidean space V.
Define $f : \mathbb{R} \rightarrow \mathbb{R}$ by $f(a) = \|x − ay\|$
I do not understand the question. How should I define that function? Compute that norm and write the value of f(a)?
|
Right so I'm trying to understand truth tables in the context of digital logic. And paticularly with lettered boolean expresssions.
Now I do understand truth tables, you have either true or false as the value, and you have some operators that determine that true or false answer. And, Or, Not etc.
But the set of questio... |
I'm quite new to Lie Algebras, and so there's a lot of easy stuff that I'm probably missing. Anyway following Kac notes I'm asked to compute the center of $\mathfrak{gl}(n,\Bbb K)$, and I've done it in the usual way (defining a bases for two matrices and working it out with brute force).
Anyway I was wondering: since ... |
Assume $A \cap B \neq \emptyset$. Find a counter-example to the claim:
$\sup(A \cap B) = \min\{\sup(A), \:\sup(B)\}$
I cant seem to find a counter example to the above claim, can anyone provide
a counter example to this ?
Thanks for reading my post.
|
I have heard that the sequence $$x_{n+1}=rx_n(1-x_n)$$ for $r$ between $3$ and $4$ does not recur i.e. there is no $a>0$ such that $x_{n+a} = x_n$ and $x_0$ is any number between 0 and 1 exclusive.
I have tried it by brute force up to $x_{120}$ and it does not seem to recur, but is it possible that the 'apparent' non-r... |
Is the function $f$, defined by
$$ f(z) =
\begin{cases}
\frac{z^2+2iz-1}{2z^2+iz+1} & \text{ if } z \not \in \{-i\}\\
0 & \text{ if } z = -i
\end{cases}$$
continuous at $−i$? Explain your answer.
|
A few times, I wanted to highlight that a variable in the right-hand side of an equation depended on other variable. I can't recall a good example right now, but consider this one:
$\tau = F d(\vec{r})\cos(\theta)$.
Probably, the person who hypotetically wrote probably wanted to mean that $d$ depends on $\vec{r}$. In m... |
In the book by A. Pascucci (PDE and Martingale Methods in Option Pricing) I have found the following definition of $\mathbb{L}^2_{\text{loc}}$ process.
Later (pp. 329-330) for a process $\lambda\in\mathbb{L}^2_{loc}$ the exponential martingale associated to $\lambda$ is defined as
$$
Z_t^{\lambda} = \exp\left(-\int_... |
So question is :
For any $m\times n$ matrix $A$,let $T_A$ be the linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ defined by
$T_A(x) = Ax$ for all $x \in \mathbb{R}^n$.Let $A$ and $B$ be $n \times n$ matrices.
Prove that $\operatorname{im}(T_A) \subseteq \operatorname{im}(T_B)$ if and only if every column of... |
I am trying to use the method from my previous question to solve this PDE:
$$ 3u_x + 2u_t = \cos x $$
with initial condition $u(x,0) = x^2$.
So I need to solve these:
\begin{align}
\frac{dx}{ds} &= 3 &\implies x &= 3s + x_0 \\
\frac{dt}{ds} &= 2 &\implies t &= 2s + t_0\\
\frac{du}{ds} &= \cos x
\end{align}
To integrat... |
I know that if I have had $y = x^{x+0} $ aka $y = x^x$ I could do
$y = x^x$ // $x = e^{\ln(x)}$
$y=x^{e^{\ln(x)}}$ // $\ln$()
$\ln(y) = \ln(x)e^{\ln(x)}$ then using Lambert's W function I could directly solve for $x$.
$x=e^{W(\ln(y))}$
What do I do in the case I've stated in the title above where $x^x$ becomes $... |
I'm a Psychologist and Neuroscientist with interest in math and I just started reading about Topology. I have to say it's not easy to grasp the concepts without a practical example, so I'm trying to understand topology in a practical (psychologically applicable) way.
I was thinking for example about the concept of some... |
Let $\mathbb Q\subset\mathbb Q(\sqrt{3}+\sqrt[3]{5})$. I want to find the Galois group of the given field extension.
It would be easy for me if I could find a basis of the given field extension but how does one find a basis of that field extension ? Are there any better ways of finding the Galois group in this case ... |
Prove / disprove: Let $G$ be a finite group of order $n$, if $m\in \mathbb N$ such that $g^m=1$ for all $g\in G$, then $m=n$.
I can't find a group that will satisfy this condition so I think it's wrong, it's also kind of similar to the little theorem but it would have to be $m = n-1$. What is the counter example to d... |
Consider the system
$$
\dot{x}=y,\qquad\dot{y}=-x+y^2.
$$
Then, it is said that the system is reversible $(t\to -t, y\to -y)$. What does this mean?
If I put this into the equations, I get
$$
\dot{x}(-t)=y(-t),\qquad \dot{y}(-t)=-x(-t)+y(-t)^2.
$$
So does reversible mean here that when replacing $t$ by $-t$ and $y$ by $... |
$\newcommand{\End}{\mathrm{End}}$ I have been struggling with this Representation Theory question for the past week:
Let $R$ be a ring and $S_1, \ldots, S_r$ simple $R$-modules with $S_i$ not isomorphic to $S_j$ whenever $i \neq j$ and fix positive integers $n_1, \ldots, n_r$.
Show that we have ring isomorphisms
\begin... |
"
Mathematical induction If p(n) is a statement involving the natural number n such that:
p(1) is true, and
p(k)$\Rightarrow$p(k+1) for any arbitrary natural number k, then p(n) is true for any natural number n.
Example 12 Prove by mathematical induction that
1+2+3+...+n= n(n+1)/2 for every natural number n
[Proof]... |
I've seen the definition of a quasicoherent sheaf $\mathcal{F}$ on an arbitrary functor
$$
X : CRing \to Sets
$$
as a specification of an $R$-module $\mathcal{F}(x)$ for each $R$-point $x \in X(R)$ with a collection of isomorphisms $\alpha_{x,x'}: R' \otimes_R \mathcal{F}(x) \to \mathcal{F}(x') $ where $f : R\to R'$ is... |
Assume $b$ is going to replace member $a$ in set $S$. That is, the set S is initially like this $S=\{a\}$, but now the new member $b$ is going to replace $b$ to have $S=\{b\}$ at the end.
How do you express that in mathematical notation? Is it right to say $a\leftarrow b$
|
I'm having trouble integrating
$$\int_{-\infty}^{\infty} \left(\frac{1}{\alpha + ix} + \frac{1}{\alpha - ix}\right)^2 \, dx$$
where $\alpha$ is a real number and $i = \sqrt{-1}$. I'm guessing that I would have to integrate it using methods on complex analysis wherein I would use a semicircle contour enclosing a pole ($... |
I have $3$ lists of things. I want to choose things randomly from this lists.
But have percentage limitation on the lists.
Like I can select only $60$% things from $1$st list, $30$% from $2$nd list & $10$% from $3$rd list.
I want to achieve this by a mathematical formula which can provide me a random selection for thi... |
This might sound like a very trivial question but I found different answers on the web.
Assume one has a rational function $$\frac{f(x)}{g(x)} ,$$ where $f(x)$ and $g(x)$ are polynomials. What is the degree of the rational function?
Is it the maximum degree of $f$ and $g$? Or is it $\deg(f) – \deg(g)$?
Thanks
|
If:
$$P(A) = 0.26, P(B) = 0.39, P(C) = 0.15, P(D) = 0.20$$
$$P(A \cap B) = 0.21$$
$$P(A \cap D) + P(B \cap C) = 0.11$$
and
$$P(A \cap C) = 0$$
$$P(B \cap D) = 0$$
Is it possible to find or make an estimate for the value of $P(A \cap D)$?
I've tried it with venn diagrams and couldn't get anywhere. This is a real world p... |
The eigenvalues of $AB$ and of $BA$ are identical for all square $A$ and $B$. I have done the proof in a easy way…
If $ABv = λv$, then $B Aw = λw$, where $w = B v$. Thus, as long as $w \neq 0$, it is an eigen vector of $B A$ with eigenvalue $λ$ and the other case when $w = 0$ can be easily handled.
But I have seen the ... |
Prove continuity of function with the delta-epsilon definition in point $x_o=0$
$$f:\mathbb{R}\rightarrow \mathbb{R}$$
$$f(x) =
\begin{cases}
x^2+1, & x \in \mathbb{Q} \\[2ex]
2^x, & x \in \mathbb{R}-\mathbb{Q}
\end{cases} $$
What I've done:
I know that I know that in order to do so I must find that for any $\epsilo... |
I have read in a number of sources (including here) that a holomorphic function on a compact Riemann surface must be constant. The reason given has always been the Maximum Modulus principle, but without explaining why. I could use some help in proving this result.
|
Let $D_{2n}$ be the dihedral group of order $2n$. I would like to show that if $4$ divides $n$, then the normaliser $N_{D_{2n}}(D_{4})=D_{8}$. I know that
$$|D_{2n}:N_{D_{2n}}(D_{4})|=|\mathcal{C}|,$$
where $\mathcal{C}$ is the conjugacy class of $D_{4}$ in $D_{2n}$. I am not sure how I would find the size of $\mathca... |
I'm not a math genius so please consider that when posting your explanation.
I have the following sets, arbitrarily named:
a [a1, a2, a3]
b [b1, b2]
c [c1, c2, c3, c4]
d [d1, d2, d3, d4, d5]
e [e1, e2, e3]
f [f1, f2]
g [g1]
Question #1) What is the total number of combinations if I have to select one element from each... |
The question arise from the definition of conditional probability, that is defined as
$$\Pr[A\cap B]=\Pr[A|B]\cdot\Pr[B]$$
Alternatively, in the context of cummulative joint probability distributions we can write
$$F_{XY}(x,y)=\Pr[X\le x, Y\le y]\overset{?}{=}\begin{cases}\Pr[X\le x \cap Y\le y] & \text{it is this expr... |
I am trying to solve this Representation Theory question:
Let $F$ be a field and $G$ a finite group. Let $N_G = {\sum}_{g \in G} {g} \in F[G]$. Show that the left ideal $(N_G) \subset F[G]$ is a simple submodule of $F[G]$.
Any help would be highly beneficial. Thanks :)
|
Suppose I have a sequence of real numbers $\{a_n\}_n$ and I'm told that $\{a_n\}_n$ is bounded away from $0$ and $2$.
(1) What does it mean exactly? My thinking is that it means $a_n\neq 0$ and $a_n \neq 2$ $\forall n$.
(2) Does it imply that $\lim_{n\rightarrow \infty}a_n \neq 0$ and $\lim_{n\rightarrow \infty}a_n \... |
Let $F$ be a field of order 32. Find number of solutions $(x,y)\in F\times F$ for $x^2 +y^2 +xy =0$.
I have figured out that non zero elements of this field forms a cyclic multiplicative group of order 31 and charecteristic of $ F $ must be 2. Have no idea how to proceed further.
|
Suppose I have processes $X_n(t)\overset{d}{\longrightarrow} X(t)$, and $Y_n(t)\overset{p}{\longrightarrow} ct$ for some constant $c$. Then, can I conclude like in Slutsky's theorem that $(X_n(t),Y_n(t))\overset{d}{\longrightarrow} (X(t),ct)$?
I tried to adapt the proof for normal random variables, so proving that for... |
Hello I have a problem with solving Trigonometric equations. Why this is not true for $0\le\theta\le360$
$$2\sin\theta\cos\theta=\sin\theta$$
$$2\cos\theta=1$$
Set of solutions $\theta=60,360$
and this is ?
$$2\sin\theta\cos\theta=\sin\theta$$
$$\sin\theta(2\cos\theta-1)=0$$
Set of solutions $\theta=0,60,180,300,360$
B... |
(Convention. I include $0$ in the natural numbers, i.e. $0 \in \mathbb{N}$)
Definition. Whenever $n$ is a natural number, define that $$\langle n\rangle : \{0,\ldots,9\}^\mathbb{N}$$ is the unique such sequence satisfying $$n = \sum_{k:\mathbb{N}} \langle n \rangle_k 10^k.$$
For example, if $n = 475$, then $$\langle ... |
Enumerate the irreducible polynomials with degree $d$ and integer coefficients
$[-l,l]$ and check how many of them have a solvable galois-group. We can assume that the leading coefficient of the polynomials is positive.
Denote
$q(d,l) : $ number of irreducible polynomials with solvable galois-group
$r(d,l) : $ number ... |
Given a set A with median Am = 10 and set B with median Bm = 20 is it true that the median of the combined set C is $10 \le$ Cm$\le 20$ ?
My first thought was that this wasn't true so I tried to find a counter example but I wasnt able to so I am assuming that it likely is true but I havent been able to find any theorem... |
Let's say we have a algebraically closed field $k$. Let $H=\{(x_1,x_2)\in k^2 : x_1x_2=1\}$ be an affine variety. My question is, can $k^1$ be embedded into $H \times H$ as a closed subset?
My idea to approach this problem would be to look at the rings of regular functions on $\mathbb k^1$ (which is $k [x]$) and of $H ... |
Find a power series centered at the origin that satisfies the Bessel differential equation $$zf''(z)+f'(z)+zf(z)=0$$ with initial condition $f(0)=1$. Show that this series converges for all z in C.
I have worked with Diff EQs before I am unsure how to work in the power series.
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Let $S_k(N)$ be the space of weight $k$ cusp forms with respect to $\Gamma_0(N)$ for $1\leq N\leq 100$. We have a decomposition:
$$S_k(N)= S_k^{\text{new}}\oplus S_k^{\text{old}}$$
Suppose that we know the eigenbasis of new forms of $S_k(N)$, for all $1\leq N\leq 100$. How do we find eigenbasis of old forms?
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Suppose there is random variable $K= X + Z$
$$P(X=x)=\frac{T\lambda^{x/\alpha}}{(k/\alpha)!}e^{-T\lambda}\quad(x=0,\alpha,2\alpha,......)$$
$$P(Z=z)=\frac{\lambda^{z/\beta}}{(z/\beta)!}e^{-\lambda}\quad(z=0,\beta,2\beta,......)$$
$$X=\alpha U,\,Z=\beta V \quad (U\,and\,V\,is\,poisson )$$
In case of $\alpha=3,\,\beta=1$... |
In the question Smooth transition between two lines (2d) there is an example of a composite curve which has a point where it is non-smooth.
In general, what is the name for that transition point?
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If you have a rotation matrix (or an attitude/direct cosine matrix, which are all synonyms). This matrix actually transforms vectors from one reference frame to another. But if your goal is to know/calculate the orientation of an object. Why do you need to be able to transform those vectors in order to know the orienta... |
Consider a decreasing function $f(x)$, resulting from the sum of two other decreasing functions: $f(x)=g(x)+h(x)$. All these 3 functions are positive. In addition, we have $g(0)=h(0)=1$. Further, we assume that $h(x) < g(x)$ over $(0,1]$ . The interval of $x$ is $[0,1]$.
How can I bound (as tight as possible) the firs... |
I have this task to do:
A machine is composed of $3$ components, which function independently of
each other with probabilities $p_1$, $p_2$ and $p_3$, respectively. Calculate the probability that the machine works when
the machine only works when all the components are working,
the machine works when at least one ... |
Question: It was found that the mean length of $100$ parts produced by a
lathe was $20.05$ mm with a standard deviation of $0.02$ mm. Find the
probability that a part selected at random would have a length
(a) between $20.03$ mm and $20.08$ mm.
The link for this question is here at example 5. http://www.intmath.c... |
Could anybody give an example of a finite abelian $2$-group with more than one characteristic subgroup of order $2$ ? (In other words, a finite abelian $2$-group $G$ with a Klein subgroup $V$ such that every element of $V$ is fixed by every automorphism of $G$.) Thanks in advance.
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This is about correct operation in a multiplicative cyclic group. Say we have the group $\mathbb Z_4^{\times} = \{1,2,3\}$, if we multiply the elements, say $3\cdot 2\cdot 2 $ we get $12 \mod 4 = 0$ but $0\notin \mathbb Z_4^{\times}$ so that can't be right, so what is the correct result of this operation?
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Suppose I have a star-shaped region in the plane with a particular point marked. The marked vertex is in the kernel of the star.
In the image the left most point is marked in blue. Yes I drew this in paint-- don't judge me. Just by looking it would appear that the largest (area) convex region inside this star that has... |
Let $A \neq B \in M_{n\times n}$ be linear maps.
I'd like to prove that there exists $n\in \mathbb N$ such that $e^{A/n} \neq e^{B/n}$.
I tried assuming that $e^{\frac{A}{n}} = e^{\frac{B}{n}}$ for every $n\in \mathbb N$ and get using uniqueness of a solution to an ODE that $A = B$, but that didn't work out. Any help... |
How does one prove that $$e^x \ge x^e$$ for all $x \ge 0$?
I tried to do this by setting $f(x)=e^x-x^e$
Plotting this function shows this easily, as seen here.
However, when I tried to prove this, it proved quite difficult. It seems to be increasing for $0 \le x \le e$, and seems to be increasing for all $x \ge e$. ... |
In a linear programming context, does the primal optimal solution yield an explicit way to find the primal dual solution? I vaguely remember something like this from an optimization class but can't recall the precise statement.
The closest I've found is that if $x, y$ are primal and dual feasible respectively, and the... |
Im looking for distance function that describes N prism. Im looking for pentagon prism, heptagon prism and octagon prism functions.
Function accepts vec3 position, which is observer position. Function returns one value, which is shortest distance to the surface of the prism.
Function doesnt provide where the pierce po... |
I have an incompressible, inviscid fluid, under the influence of gravity, with a velocity potential:
$$ \mathbf{u} = (-\cos(x)\sin(y), \sin(x)\cos(y), 0) $$
Using Euler's equations,
$$ \mathbf{u} \cdot (\mathbf{\nabla} \cdot \mathbf{u}) = \frac{-1}{\rho} \mathbf{\nabla}p - g\mathbf{\hat{z}} $$
that is,
$$ \frac{\partia... |
I'm trying to prove backward induction, which I'll state as follows:
Consider the set $\mathsf{A}$, where $n\in{\mathsf{A}}$, and $m+1\in{\mathsf{A}}$ $\implies$$m\in{\mathsf{A}}$. Then $\mathsf{A}=\{0,...,n\}$.
I'm attempting to proceed with the well-ordering principle, since I'm a little confused when trying to prove... |
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