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How does one find the number of integral solutions of equations like,
$2x + 5y + 4z = 20$
I know how to do this counting if the coefficient of every variable on the LHS is $1$.
But I don't know how to do this counting with arbitrary coefficients.
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If I have say $2$ A matrices, $2$ B matrices and $1$ C matrix then I would like to know how many distinct traces of products of all of them can be formed.
Assume that $A$, $B$ and $C$ don't commute with each other.
Like $AABBC$, $CAABB$ and $BCAAB$ are distinct products which will have the same trace but $ABABC$ has ... |
Why does the following hold:
\begin{equation*}
\displaystyle \sum\limits_{n=0}^{\infty} 0.7^n=\frac{1}{1-0.7} = 10/3\quad ?
\end{equation*}
Can we generalize the above to
$\displaystyle \sum_{n=0}^{\infty} x^n = \frac{1}{1-x}$ ?
Are there some values of $x$ for which the above formula is invalid?
What about if we ta... |
I am following a proof from a book, but it seems to have a typo or use some strange notation I have never seen, and I have already spent a lot of time trying to decipher this, but cannot find the solution.
It is basically showing one how to construct general real numbers from Dedekind's definition by expressing those a... |
I want a triangle composed of points A, B and C in Cartesian 3D space.
I currently know the positions of points A and B, but I need point C. I have the line segment AB, and thus its magnitude. I have only the magnitudes of line segments AC and BC.
From this data, how do I derive point C? Please explain your logic. Than... |
both of Lie algebras of type $B_2$ and $C_2$ have dimension 10 and we can find two basis of them on page 3 in the book: Introduction to Lie algebras and representation theory . How could we show that these two Lie algebras are isomorphic by constructing an explicit correspondence between two basis of them. Thank you v... |
Let $V$ be a finite dimensional vector space over $\mathbb{R}$ of dimension $n$.
The vector space $\underbrace{V \oplus V \oplus \cdots \oplus V}_k$ has dimension $kn$, and the vector space $\underbrace{V \otimes V \otimes \cdots \otimes V}_k$ has dimension $n^k$.
Is there a similar construction (that does not depend ... |
In a recent talk, Marcus du Sautoy says there are 2125922464947725402112000 (2.1*10^24) symmetries of a Rubik's cube, but doesn't explicitly identify what qualifies as a symmetry.
What counts as a symmetry of the Rubik's cube? Is it a thing like, "turn the top face once clockwise, then once counterclockwise"?
How ar... |
I can visualize the exponential function with the graph $y = e^x$, but I can do that for almost any function.
In addition to its graph, the function $f(x) = x^n$ can be visualized as the volume of a box with sides of length $x$ in n-dimensional space, and the trigonometric functions can be interpreted as side lengths o... |
Original problem $Ax=b$ (let's say A is 400x5 matrix)
$x$ can be found through SVD. Now we can create a new problem that will provide us with the same solution for $x'$:
$I_5 x' = x$
Where $I_5$ is a 5x5 identity matrix
Let's now extend the problem by adding more rows to $A$ and $b$. Calling the new variables A' and b... |
what are the differences between classical Yang-Baxter Equation and quantum Yang-Baxter Equation? Thank you very much.
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We already have a large list of the Best book ever on Number Theory, but I'm looking for a more targeted response for analytic number theory.
Specifically, I'm taking a trip on which I may or may not have access to internet resources, nor my University's library. I'm starting to work through Montgomery and Vaughan's M... |
This question might be easy.
The hard question is this: prove that if $a,b,c\geq3$ then there are no solutions in positive integers $x,y,z$ to $x^a+y^b=z^c$ with $x,y,z$ coprime. This implies Fermat, most cases of Catalan, etc., and is an open problem.
But it's really crucial that $x,y,z$ are coprime to make this quest... |
Similar to a previous question here, I wonder if cyclic permutations are the only relations amongst traces of (non-commutative) monomials. Since the evaluations $\operatorname{tr}:k\langle x,y,\dots \rangle \to k$ take an infinite dimensional vector space to a one-dimensional vector space there must be quite a few rel... |
Solve the differential equation:
$$x \frac{dy}{dx} = y(3-y)$$
where $x=2$ when $y=2$, giving $y$ as a function of $x$.
Can someone solve this and then explain what the second line about $x=2$, $y=2$ means?
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How do you show that the column space of a matrix A is orthogonal to its nullspace?
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I've attached an image of my book (Theorem 4.4.1 is at the bottom of the image). I need help understanding what this book is saying.
In the first sentence on p.113:
"If (I) holds, then the primal is
feasible, and its optimal objective is
obviously zero",
They are talking about the scalar value resulting from tak... |
In this question, Jack Schmidt asks to prove a certain identity for $2\times 2$ matrices A and B.
In fact he asks to show that tr(AABABB−AABBAB) = 0. In an answer by user7406, he shows that 3 times this expression must be 0, solving the problem at least when the characteristic of the ground field isn't 3.
In a comment... |
When one defines the splitting field for an arbitrary collection of polynomials, how does one show the uniqueness of such a splitting field? (I'm guessing it is still unique.) The induction argument used for the case of a single polynomial obviously doesn't work. I'm not so keen on using transfinite induction, either. ... |
I'm having trouble establishing this identity below.
Suppose $Y$ and $Z$ are random variables on $(\Omega,F,P)$ and $Y \in L^1$. Suppose $Z$ is bounded and let $G \subset F$ be a sub-$\sigma$-field. Show that $$\mathbb{E}(\mathbb{E}(Y|G)Z)=\mathbb{E}(\mathbb{E}(Z|G)Y).$$
Is there a way I can attack it using t... |
There are a lot of common misconceptions when it comes to math. A common one that has already been addressed on this site is $1 \neq .999\cdots$, as is that imaginary numbers "do not exist". Another one which I have encountered is that one can prove $2 = 1$ using calculus as follows:
$$2 = \frac{\frac{d}{dx}(x^2)}{x} =... |
How to solve this nonlinear partial differential equation?
$$\displaystyle\frac{\partial^2}{\partial x^2} f(x,t) +b \frac{\partial^2}{\partial x^2} f(x,t) \cdot \frac{\partial^2}{\partial t^2} f(x,t) + a = 0,$$
where $a,b$ are constant.
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It's probably a vey silly question, but I'm confused. Does o(1) simply mean $\lim_{n \to \infty} \frac{f(n)}{\epsilon}=0$ for some $n>N$?
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The following is a question from lecture notes and although not assigned homework, I am trying to solve it.
Assume that we have a collection $C$ of $N$ documents and a query $q$. There are $R_{q}$ relevant documents in the collection for this query, with $R_{q} < N$. If we select $R_{q}$ documents randomly, what is the... |
Let $M$ be a $d$-dimensional compact manifold and $f:M\to M$ be a diffeomorphism on $M$. Whitney embedding theorem says that we can embed $M$ into $\mathbb{R}^{2d+1}$. Let $T$ be a tubular neighborhood of $M$ with respect to the embedding.
In some papers they say that there exists a diffeomorphism $F:T\to F(T)\Subset ... |
Thanks for your time. I am interested in various ways/techniques/tricks/methods (induction, convexity, concavity, maximum, minimum, geometry, trigonometry, ...) for proving the inequalities and their generalizations
(1) $\sqrt{1-x_1^2 -y_1^2} + \sqrt{1-x_2^2 -y_2^2} + \sqrt{1-x_3^2 -y_3^2} \le 3\,\sqrt{1-\left(\frac... |
Consider a system of linear equations $Ax=b$. Can it be that for a generic choice of $A$ and $b$, elements of $x$ are distinct?
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How can you prove that for $n\geq1$, you have $n^n\geq e^{n-1}$? (no induction)
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This is a continuation of a previous question .
Define that elements of $x=(x_1,x_2,...,x_n)$ are distinct if for each $i≠j, x_i≠x_j.$ Consider a system of linear equation $Ax=b$. We want to understand the property of $(A,b)$ such that the solution $x$ given $A$ and $b$ is distinct in the sense defined above.
In a pr... |
This is a continuation of a previous question .
Define that elements of $x=(x_1,x_2,...,x_n)$ are distinct if for each $i≠j, x_i≠x_j.$ consider a system of linear equation $Ax=b$. We want to understand the property of $(A,b)$ such that the solution $x$ given $A$ and $b$ is distinct in the sense defined above.
In a pr... |
can someone provide explicit charts for non-standard differentiable structures on, for instance $\mathbb{R}^4$ (or some other manifold)?
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This is part 2 of the question I asked here. Can you tell me if this proof is correct? I thought it's better to ask this in a new separate question, the old post was getting a bit long. Many thanks for your help!
Claim: $f$ restricts to a homotopy equivalence from each path-component of $X$ to the corresponding path-co... |
In my work on lattice point enumeration of polytopes, I stumbled upon the following sequence:
\begin{eqnarray}
1, 120, 579, 1600, 3405, 6216, 10255, 15744, 22905, 31960, 43131, ...
\end{eqnarray}
which counts the Structured great rhombicosidodecahedral numbers (A100145) by the formula
\begin{eqnarray}
a(n)=\tfrac{1}{6... |
I would like to show that for smooth, nice variety $X$, $\operatorname{Pic}(X \times A^{1})=\operatorname{Pic}(X)$.
I only need to prove this in the generality where the Picard group and the divisor class group are the same.
The pullback homomorphism $p^{*}:Cl(X)\rightarrow Cl(X \times A^{1})$ of the projection is my b... |
Let $A=\begin{bmatrix}a & b\\ c & d\end{bmatrix}$.
How could we show that $ad-bc$ is the area of a parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+b, c+d)$?
Are the areas of the following parallelograms the same?
$(1)$ parallelogram with vertices $(0, 0),\ (a, b),\ (c, d),\ (a+c, b+d)$.
$(2)$ parallelogram ... |
Consider a class of functions on the real line defined by a following property: for each $\alpha \in [0,1]$, for each $x,y \in [0,1]$, there exists $z \in [0,1]$ such that $ \alpha f(x) + (1-\alpha) f(y)= f(z). $ How general is this class of functions?
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I'm just curious, considering how similar the graphs of the sine & cosine functions are in shape, why is the shape of the tangent's graph so different, despite being used in very similar types of problem?
Thanks!
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We learned that if $V$ is a finite inner product space then for every linear transformation $T:V\to V$, there exists a unique linear transformation $T^*:V\to V$ such that $\forall u, v \in V: (Tv, u)=(v, T^*u)$.
The construction of $T^*$ used the fact that $V$ is finite and therefore has an orthonormal basis, which is ... |
Suppose the conditional distribution of $\theta$ and $x$ satisfies the monotone likelihood ratio property: for every real numbers $\bar{\theta} \ge \theta$ and $\bar{x} \ge x$, we have $ f(\bar{\theta}|\bar{x})/f(\theta|\bar{x}) \ge f(\bar{\theta}|x)/f(\theta|x).$
Now suppose I extend this property as follows: for ea... |
In what context should I use $=$ and $\equiv$?
What is the precise difference?
Thanks!
(I wasn't sure what to tag this with, any suggestions?)
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From reading the answers to different questions on category theory, it seems that category theory is useful as a framework for thinking about mathematics. Also, from the book Algebra by Saunders Mac Lane, in the preface for first edition there is a passage,
"... It is now clear that we study not just a single algebrai... |
I've been reading around the net and everything I find is really confusing. I just need a formula that will get me 95% there. I have a tool that outputs the distance between two lat/long points.
Point 1: 32.773178, -79.920094
Point 2: 32.781666666666666, -79.916666666666671
Distance: 0.0091526545913161624
I would like... |
Let $A$ be a matrix from $\mathbb{M}_{n \times n}(F)$ and $f(x) \in F[x]$.
How does one prove the following: $f(A)$ is invertible iff $\gcd(Ma,f)=1$ where $Ma$ is the minimal polynomial of $A$.
Thanks.
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Simplify: $\sin^4x + \sin^2x \cdot \cos^2x$
The textbook states the answer as $\sin^2x$ and I understand the reasoning: Take a factor of $\sin^2x$ out and you are left with $\sin^2x \cdot 1$
However I can't work out why my method is wrong (it produces the answer of 1):
Divide everything by $\sin^2x$
$(\sin^4x / \sin^2... |
I'm trying to find $\int{\sqrt{25 - x^2} dx}$
Now I know that $\int{\frac{dx}{\sqrt{25 - x^2}}}$ would have been $\arcsin{\frac{x}{5}} + C$, but this integral I'm asking about has the rooted term in the numerator.
What are some techniques to evaluate this indefinite integral?
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I'm getting better at using trig functions, but this problem has me up against a wall.
As I understand it, to find if a function is increasing or decreasing over a given interval, select a number inside that interval and if the result is greater or less than zero, the interval is increasing and decreasing, respectively... |
Recently I am learning ergodic theory and reading several books about it.
Usually Poincaré recurrence theorem is stated and proved before ergodicity and ergodic theorems. But ergodic theorem does not rely on the result of Poincaré recurrence theorem. So I am wondering why the authors always mention Poincaré recurrence ... |
I understand that, over $\mathbb{C}$, the rank of the Picard group of a K3 surface $X$ is bounded above by $20$ because we can use the exponential sheaf sequence:
$0 \to 2\pi i \mathbb{Z} \to \mathcal{O}_X \to \mathcal{O}_X^\times \to 0$,
and because $H^1(X,\mathcal{O}_X)$ is trivial this gives an injective homomorphis... |
I had a quick look around here and google before, and didn't find any answer to this particular question, and it's beginning to really irritate me now, so I'm asking here:
How is one supposed to write l (little L), 1 (one) and | (pipe) so that they don't all look the same? One of my teachers draws them all as vertical ... |
I ran across this infinite product:
$$\lim_{n\to\infty}\prod_{k=2}^n\left(1-\frac1{\binom{k+1}{2}}\right)$$
I easily found that it converges to 1/3. Using my calculator, I found that
$$1-\frac1{\binom{k+1}{2}}=\frac{(k-1)(k+2)}{k(k+1)}$$
Then, here is my question
$$\prod_{k=2}^n\frac{(k-1)(k+2)}{k(k+1)}=\frac{n+2}{3n}$... |
I encountered this problem while studying for an analysis exam. Here is a related question I asked some days ago.
The problem is as follows: Suppose $a_n$ is a decreasing sequence of positive real numbers and that$$\sum_{n = 0}^{\infty}{a_n \sin{(nx)}}$$ converges uniformly on $\mathbb{R}$, show that $$\lim_{n \to \inf... |
An experimental drug is being tested for side effects. The producers of the drug claim that only 5% of users will experience side effects. If 20 people are tested what is the probability that 2 or more will experience side effects?
Answer: 0.2641
How do you do this problem? Looking at the Probability Wikibook didn't ... |
Let $n,k$ be positive integers. Is there a closed form of the sum
$$\sum_{s=0}^{k} \binom{n}{s} \binom{s}{k-s}\text{?}$$
By that I mean a representation which is free of sums and hypergeometric functions or alike.
Combinatoric interpretation: This is the number of possibilities to distribute $k$ balls in $n$ urns, wher... |
I am taking a Numerical Computation class, and we are currently learning about Newton's Method for finding the roots of a system of non-linear equations. I have no problems understanding how the algorithm works, but as an engineer I work better when I can think of a concrete application for concepts.
Can anyone give a ... |
I have 24 values for $Y$ and corresponding 24 values for $t$. The Y values are measured experimentally, while t has values $t=1,2,\dots 24$.
I want to find the relationship between Y and t as an equation using Fourier analysis.
I wrote the following MATLAB code:
Y=[10.6534
9.6646
8.7137
8.2863
8.2863
... |
I am having trouble figuring out how I can solve this log:
$$6 \log(x^2+1)-x=0$$
The steps i've thought to take so far are as follows:
step 1: subtract the right most x to the other side of the equation:
$$6 \log(x^2+1) = x$$
step 2: divide by 6:
$$\log(x^2+1)=\frac{x}{6}$$
step 3: make both sides an exponent of 10 to... |
Let T be a linear opertaor which acts on the vector space of square $N\times N$ matrices and A a matrix from this vector space.
T acts this way:
T(X)=AX.
How can I calculate the Characteristic and minimal polynomials of T?
In addition, I need to show that if A is diagonalizable than so does T.
I know that In order to ... |
-----------------Updated after answer----------------
Given the Asia net.
I am computing something like:
Tuberculosis Present:
$P(T)=P(V)P(T|V)+ P( \lnot V)*P(T| \lnot V)$ = (0.1)(0.05)+(0.99)(0.01)=0.0104
Tuberculosis Absent:
1.0 - 0.0104 =0.99
Lung Cancer Present:
$P(L)=P(S)P(L|S)+ P( \lnot S)*P(L| \lnot S)$ = (0.... |
I'm trying to find the convolution $(A \ast A)(x)$, where
$$A(x) = \begin{cases}
1 + x, & -1\leq x \leq 0,\\
1 - x, & 0\leq x \leq 1,\\
0, & \text{otherwise},
\end{cases}$$
so far without success. Can someone help?
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My textbook says that:
$$
\frac{(n+1)^n}{n!}=\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)^2\cdots\left(1+\frac{1}{n}\right)^n<e^n
$$
But I do not understand this. Can you please enlighten me?
Edit: How do you show that
$$
\frac{(n+1)^n}{n!}=\left(1+\frac{1}{1}\right)\left(1+\frac{1}{2}\right)^2\cdots\left(1+... |
Let $\ell^2 =\ell^2(\mathbb{Z})$. Choose $\theta \in ]0,1[$ and set:
$$Tx=(\theta x_{n-1} +(1-\theta)x_{n+1})_{n\in \mathbb{Z}}$$
for each $x=(x_n)_{n\in \mathbb{Z}}\in \ell^2$ (thus $T$ is a convex combination of the right and left shift operators).
It is easy to prove that, for every $\theta$, $T$ is a bounded linear... |
I am trying to implement a solution (working code) for the 4.1 paragraph in this paper.
The problem:
We have words with lengths for instance:
$l_1$ = 1, $l_2$ = 2, $l_3$ = 3, $l_4$ = 8 and $l_5$ = 7.
These words will be part of the white-list.
We calculate the sample mean and the variance of the lengths of these wor... |
I want to show that $\displaystyle\frac{1}{2}(n+1)<\frac{n^{n+\frac{1}{2}}}{e^{n-1}}$. But except induction, I do not know how I could prove this?
|
A couple problems are giving me trouble in finding the relative maxima/minima of the function. I think the problem stems from me possibly not finding all of the critical numbers of the function, but I don't see what I missed.
Given $f(x)= 5x + 10 \sin x$, I calculated the derivative as $5 + 10 \cos x$, and found the f... |
In my current lecture I regularly encounter usage of the real part of, say, a scalar product of two vectors similar to angles in classical geometry. For example in Hilbert space theory:
Let $H$ be a Hilbert space, $C \subset H$ be convex and closed. For $x_0 \in H$, $x \in C$ is the best approximation of $x_0$ by $C$, ... |
Let $(B_i)_{i\in I}$ be an indexed family of closed, connected sets in a compact space X. Suppose $I$ is ordered, sucht that $i < j \implies B_i \supset B_j$.
Is $B = \bigcap_i B_i$ necessarily connected?
I can prove it, if I assume $X$ to be Hausdorff as well: If $B$ is not connected, then there are two disjoint, clos... |
I am trying to understand the difference between the following two equations:
$$\bar{P} = \limsup_{t \to \infty}\frac{1}{t} \sum_{\tau = 0}^{t-1}E\{P[\tau]\} < \infty$$
and
$$\bar{P} = \lim_{t \to \infty}\frac{1}{t} \sum_{\tau = 0}^{t-1}E\{P[\tau]\} < \infty$$
where $\bar{P}$ denotes the average value of P and E stand... |
I was given this problem 30 years ago by a coworker, posted it 15 years ago to rec.puzzles, and got a solution from Barry Wolk, but have never seen it again. Consider the series:
$$1, \frac{1}{2},\frac{\displaystyle\frac{1}{2}}{\displaystyle\frac{3}{4}},\frac{\displaystyle\frac{\displaystyle\frac{1}{2}}{\displaystyle\... |
What is the relation between a dimension and a component in a Gaussian Mixture Model? And what is the meaning of dimension and component? Thank you.
Please correct me if I'm wrong: my understanding is the observed data have many dimensions. Each dimension represents a feature/aspect of the collected data and has its ow... |
Here is the statement of The Well Ordering Principle: If $A$ is a nonempty set, then there exists a linear ordering of A such that the set is well ordered.
In the book, it says that the chief advantage of the well ordering principle is that it enables us to extend the principle of mathematical induction for positive in... |
The following is a homework question for which I am asking guidance.
Let $A$, $B$, $C$ be independent random variables uniformly distributed between $(0,1)$. What is the probability that the polynomial $Ax^2 + Bx + C$ has real roots?
That means I need $P(B^2 -4AC \geq 0$). I've tried calling $X=B^2 -4AC$ and finding ... |
Using the logic definition of a structure as a set coupled with finitary functions and relations, what is the definition of the union of two structures $\mathfrak{A}_1 \cup \mathfrak{A}_2$?
I encountered this in a discussion of preservation, where $\mathfrak{A}_1$ is an elementary substructure of $\mathfrak{A}_2$, if t... |
By the inequality here or using integration I can get the following lower bound for $\log(n!)$. How can I get a better lower bound for $\log(n!)$?
$$\log(n!)>\log\left(\frac{(n+1)^n}{e^n}\right)$$
|
If the player is immediately to the left of the dealer, I think the answer is:
52 x (52-p) x (52-2p) x ... x (52-(n-1)p),
and if the player is the kth player, I think the answer is:
(52-k+1) x (52-p-k+1) x (52-2p-k+1) x ... x (52-(n-1)p-k+1).
Am I right?
I'm assuming that the pack has 52 cards, and that dealing start... |
sorry! am not clear with these questions
why an empty set is open as well as closed?
why the set of all real numbers is open as well as closed?
|
I have a 1x3 matrix representing a point in space:
[x]
[y]
[1]
And a 3x3 matrix representing an affine 2d transformation matrix
[a][b][c]
[d][e][f]
[g][h][i]
How to I multiply the matrices so that I am given the matrix?
[newX]
[newY]
[w]
|
is/are there a closed form for
$\sin{(a)}+\sin{(a+d)}+\cdots+\sin{(a+n\,d)}$
$\cos{(a)}+\cos{(a+d)}+\cdots+\cos{(a+n\,d)}$
$\tan{(a)}+\tan{(a+d)}+\cdots+\tan{(a+n\,d)}$
$\sin{(a)}+\sin{(a^2)}+\cdots+\sin{(a^n)}$
$\sin{(\frac{1}{a})}+\sin{(\frac{1}{a+d})}+\cdots+\sin{(\frac{1}{a+n\,d})}$
|
While reading a paper about the modular group $\Gamma = PSL_{2}(\mathbb{Z})$, I read that $PSL_{2}(\mathbb{Z}) \cong C_{2} * C_{3}$ and consequently, all the torsion elements in $\Gamma$ are of order 2 or 3. While I understand the isomorphism, I don't know how to prove the second statement.
More in general, is it true... |
Given the ring of polynomials $\mathbb{Z}_n[X]$, consider $$\mathbb{P}_n = \{a_0 +a_1x+a_2x^2+\cdots+a_{n-1}x^{n-1}| a_i \in \mathbb{Z}_n\},$$ i.e.
$\mathbb{P}_n$ is the set of all polynomials in $\mathbb{Z}_n[X]$ with exponents in $\mathbb{Z}_n$.
So, $\mathbb{P}_2 = \{0,1,x,1+x\},$
$$\mathbb{P}_3 = \{0, x^2, 2x^2, x... |
For the following calculation, why $E[Z_1^2] = \pi/4$ ? In general, how to calculate $E[Z^2]$ ? Thanks much.
3 down vote
Let's also elaborate on Ross Millikan's answer, adapted to the case f(x)=1−x2−−−−−√, 0≤x≤1. Suppose that $(X1,Y1),(X2,Y2),\ldots$is a sequence of independent uniform vectors on $[0,1]\times [0,1]$, s... |
Is there a geometric explanation for why a sphere has surface area $4 \pi r^2$ ?
Ie equal to 4 times its cross-section (a circle of radius r).
|
We know that when $n$ is odd, $\operatorname{O}_n(\mathbb R) \simeq \operatorname{SO}_n (\mathbb R) \times \mathbb Z_2$.
However, this seems not true when $n$ is even. But I have no idea how to prove something is not a direct product.
I have tried to verify some basic properties of direct product. For example, $\operat... |
Let $X,Y$ be two firms. An MBA applies for the job in $X$ and $Y$. The probability he being selected in firm $X$ is $0.7$ and being rejected at $Y$ is $0.5$. Probability of at least one of his applications being rejected is $0.6$.
What is the probability that he will be selected in one of the firms?
i.e.,
$P(X)=0.7$, $... |
Why $\frac{1}{n+1}<\log(n+1)-\log(n)<\frac{1}{n}$?
|
This question raised from this one.
Solving $i^i=x$ we get $x=e^{(i \frac{\pi}{2} + i2k\pi)i}=e^{-\frac{\pi}{2} - 2k\pi}$ $(k \in \mathbb{Z})$, than what about the values of $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$? Is there an argument to show that $i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}} \neq \mathbb{C} \setminus\{0\}$?... |
Let me consider a continuous function $y=f(x)$ for $x \in [0,1]$. Now consider its inverse $f^{-1}(y)= \{x:f(x)=y \}$. How can I characterize continuity property of $f^{-1}(y)$ in terms of $y$?
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$|G|=pqr^2$ where $3\leq p<q<r$ prime. Show that if $r>\frac{1}{2}(pq-1)$ then $G$ is solvable.
I took $H\leq G$ $r$-sylow subgroup of $G$, there is a theorem claiming that there exists a homomorphism from $G$ to $S_{pq}$ ($pq$ is the index of $H$).
If the homomorphism is injective then $G$ is a subgroup of $S_{... |
I need to prove that this is divergent:
$$\prod_{\stackrel{p\leq N}{p\text{ prime}}}\left(1 + \frac{1}{p} + \frac{1}{p^2} + \cdots + \frac{1}{p^k} + \cdots\right),$$
where the expression inside of the product is:
$$\frac{1}{(1-\frac{1}{p})} = (1 + \frac{1}{p} + \frac{1}{p^2} + \cdots + \frac{1}{p^k} + \cdots).$$
|
The standard normal distribution
$$f(x) = \frac{1}{\sqrt{2\pi}} e^{\frac{-x^2}{2}},$$
has the characteristic function
$$\int_{-\infty}^\infty f(x) e^{itx} dx = e^{-\frac{t^2}{2}}$$
and this can be proved by obtaining the moments.
However, is there a more direct method of proving that the standard normal has the stated ... |
I have read "let $f:X \rightarrow Y$ be a $C^{\infty}$ manifold without a boundary and let $Z \subset Y$ be a submanifold of $Y$. Geometrically, we say that $f$ is transverse to $Z$ if $X \times Z$ and $Gr(f) = \{(x, y) \in X \times Y |f(x) = y \}$ are in general position." What does it mean?
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OK, so, given that $(X, \rho)$ is a metric space, endow $\mathbb{R}$ with the ordinary Euclidean metric $\varepsilon$ and $X \times X$ by
$(\rho \times \rho) \left((x_1,y_1),(x_2,y_2)\right) = \sqrt{\rho(x_1,x_2)^2+\rho(y_1,y_2)^2}$.
Then, prove that
$\rho: X \times X \rightarrow \mathbb{R}_0^+, \quad (x,y)\mapsto \rh... |
How I can compute explicitly a set of differential forms generating the de Rham cohomology of a 2-torus of revolution in $\mathbb{R}^3$ ?
A 2-torus is embedded in $\mathbb{R}^3$ by
$\psi(u,v):=\left(x(u,v),y(u,v),z(u,n)\right),$ where
$x(u, v) = (R + r \cos{v}) \cos{u},y(u, v) = (R + r \cos{v}) \sin{u} , z(u, v) = ... |
I want to ask for two examples in the following cases:
1) Given a bounded sequence $\{a_n\}$, $$\lim_{n\to \infty}{(a_{n+1}-a_n)}=0$$ but $\{a_n\}$ diverges.
2) A function defined on real-line $f(x)$'s Taylor series converges at a point $x_0$ but does not equal to $f(x_0)$.
Thanks for your help.
Edit
in 2), I was think... |
Given the vector space $\begin{equation} \text{P}_2[t] = \{ a + b\,\, t + c \,\,t^2 \mid a, b, c \in \mathbb R\,\,\,\, \} \end{equation}$ and linear functionals $\phi_1, \phi_2, \phi_3 \in \left( \text{P}_2[t] \right) ^* $ defined as below;
$ \begin{equation} \phi_1\left(\,\,p\,\,\left(\,\,t\right)\,\,\right) = \int_0^... |
Let $f : \mathbb{R}^n \rightarrow \mathbb{R}, f \in C^2$. I have to prove that for every nondegenerate critical point of $f$, there exists a neighbourhood which does not contain any further critical points.
I know that I will have to look at the Taylor formula and I'm pretty sure that I have to use that $\det (\partial... |
Let $f: \mathbb{R}\to\mathbb{R}$ and the essential supremum of $f$
$$e=\inf\{\alpha\ge 0 \mid |f|\le\alpha\text{ almost everywhere}\}$$
I can't see why $$\lambda^*\left(\{x\in\mathbb{R} \mid |f(x)|\ge e-\epsilon\}\right)>0$$
for $a<b$ and $\epsilon>0$, where $\lambda$ is the Lebesgue measure.
Could someone help me ?
Fi... |
Are the following series convergent? If yes, where will they converge?
$$\sum_{n=1}^\infty{\frac{\sin{e^n}} {n}}$$
$$\sum_{n=1}^\infty\left({\frac{\sin{e^n}} {n}}\right)^2$$
$$\sum_{n=1}^\infty{\frac{\arctan{e^n}} {n}}$$
$$\sum_{n=1}^\infty{\frac{\cos{e^n}} {n}}$$
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Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$
How to prove this series converges/diverges?
$$\sum_{n=1}^\infty \cos{p_n}$$
|
I am working through Ireland/Rosen at the moment, and I cannot solve a (probably) simple exercise. Any nudges you can give me in the right direction are appreciated.
How does one show that $x^2 \equiv 1 \pmod{p^{\ell}}$ has only two solutions (namely $\pm 1$)? Here, $\ell$ is a positive integer (which we can take to ... |
Is there a way to obtain the Fourier transform of a function (e.g., sinc) without the Gibbs phenomenon?
|
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