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For a 2D Gaussian distribution with
$$ \mu = \begin{pmatrix} \mu_x \\ \mu_y \end{pmatrix}, \quad \Sigma = \begin{pmatrix} \sigma_x^2 & \rho \sigma_x \sigma_y \\ \rho \sigma_x \sigma_y & \sigma_y^2 \end{pmatrix},
$$
its probability density function is
$$
f(x,y) = \frac{1}{2 \pi \sigma_x \sigma_y \sqrt{1-\rho^2}} ... |
$$\lim_{x\to 0 } \frac{x}{\ln x}.$$
This was wrong, I got a big red wrong! Why doesn't L'Hôpital work on this one?
The problem is that $\ln$ is not defined for 0. It needs to be rewritten?
(Thanks to everyone helping me out with my homework, due anxiety I'm not able attend the class workshops.)
Edit: I did get the co... |
Let $\hat{\mathbb{C}} = \mathbb{C} \cup \{ \infty \}$ denote the Riemann Sphere. While reading the wikipedia article about it I found a passage that said that every rational function on the complex plane can be extended to a continuous function on the Riemann Sphere.
The particular construction is as follows, let $R(z... |
How can we prove the following trigonometric identity?
$$\displaystyle \tan(3\pi/11) + 4\sin(2\pi/11) =\sqrt{11}$$
|
I was wondering if $f(x)=O(x^{c+a})$ for all $a>0$ then is it necessarily true that $f(x)=O(x^c\log x)$?
I suspect it's not true but want to know why. (I know the converse is true.)
Any help is much appreciated,
Thank you.
|
I have two questions and I hope it is alright to ask them both at the same time. I'm currently trying to solve an online-exercise about differential calculus and the Taylor series and I'm having some trouble there.
Let $f: (a,b) \rightarrow \mathbb{R}$ be differentiable at the point $x_0$. Which of the following state... |
What is your recommended symbolic computation program/software for free and commercial respectively?
What are its strength and weakness? For example, efficiency, comprehensiveness, etc
Thanks!
|
John Von Neumann once said to Felix Smith, "Young man, in mathematics you don't understand things. You just get used to them." This was a response to Smith's fear about the method of characteristics.
Did he mean that with experience and practice, one obtains understanding?
|
Let $X$, $Y$ be pointed CW complexes, $Y$ connected and $f:X\to Y$ a mapping.
Does the assertion '$\Sigma f:\Sigma X\to\Sigma Y$ is a homotopy equivalence' imply that $f$ is a homotopy equivalence? '$\Sigma$' is the reduced suspension.
If not, is it true with some additional hypotheses on $Y$?
Addition:
Does the assert... |
Is there an easy way to see that the $n$-dimensional Fourier transform of $1/|x|^a$ is equal to $1/|x|^{n-a}$ (up to multiplicative constant) where $x$ is an $n$ dimensional vector (assuming that $ 0 < a < n $) ?
|
I'm reading Serre's "Lie algebra and Lie groups" now, and I found a description of Lie algebra of an algebraic matrix group, given for a free algebra (free as a module) $k'$ with a basis $\lbrace 1, \varepsilon\rbrace$, $\varepsilon^2 = 0$ as all $X \in M(n,k)$ such that $1 + \varepsilon X \in G(k')$, where $G(k')$ is ... |
I am being more stupid than usual: embarrassingly so. Let $G$ be a group. Let $\phi:G \rightarrow G$ be an automorphism with inverse $\psi$. Is it true (and why is it true) that $\psi(x)=\phi(x^{-1})$.
Right now, I got myself in a gumption trap and have not thought my way out of it.
|
This isn't homework.
Prove or dis-prove the statement: If $R$ is a regular language and both $L$ \ $R$ and $L \cup R$ are context-free, then $L$ is context-free.
|
Let $f(x)= \displaystyle \sum \limits_{n=1}^\infty \frac{\sin(nx)}{n^3}.$ Show that $f(x)$ is differentiable and that the derivative $f'(x)$ is continuous.
In class we solved a similar problem, and I think we had to show that both $f(x)$ and $f'(x)$ converge uniformly, but I am not really sure why that is what we have ... |
Gaussian integers are the set:
$$\mathbb{Z}[\imath] =\{a+b\imath : a,b \in\mathbb{Z} \}$$
With norm:
$$\mathrm{N}(a+b\imath)=a^{2}+b^{2}.$$
It satisfies $\mathrm{N}(\alpha\cdot \beta )=\mathrm{N}(\alpha)\cdot\mathrm{N}( \beta )$
The units of $\mathbb{Z}[\imath]$ are precisely: $1,-1,\imath,-\imath$
Result: If $p=4k+1$... |
Background: Let $k$ be an even integer. The Eisenstein series are defined by $$E_{k} = 1 - \frac{2k}{B_{k}}\sum_{n=1}^{\infty} \sigma_{k-1}(n)q^{n}$$ where
$$\sigma_{k-1}(n)= \sum\limits_{d \mid n,\;d \geq 1} d^{k-1}$$ and $B_{k}$ is the $k$-th Bernoulli number. If $k \gt 2$ then $E_{k}$ is a modular form of weight $... |
I'm working through a little proof of this statement, and here is an excerpt of the problem I'm following:
Suppose $p \equiv 1\pmod{4}$. Show that if $x^2-py^2=1$ the $x$ is odd and $y$ is even. Suppose that ${x_0}^2-p{y_0}^2=1$ with $y_0>0$, $y_0$ minimal. Show that $gcd(x_0+1,x_0-1)=2$. Deduce that one of two cases ... |
Motivation
Suppose that $u \in \mathbb{R}^d$ is a unit-norm vector, $\|u\| = 1$, $a, b, c$ are some positive constants and $\xi \in [0,1]$ is another constant (usually chosen close to 1). I am interested in solving the following problem
$$
\sup_{v \in \mathbb{R}^d}\ (\xi + (1-\xi)\|v\|_1^2)\left(\sqrt{
\frac{a \lang... |
If someone could give answers and explain, it would be greatly appreciated. Help required studying for a final.
One hundred tickets, numbered $1,2,3,…,100,$ are sold to $100$ different people for a drawing.
Four different prizes are awarded, including a grand prize (a trip to Tahiti).
A) How many ways are there to aw... |
Suppose you are given a question that goes like this:
Consider three planets that revolve
around a star on separate circular
orbits sharing the same orbital plane.
The first planet, A, takes 385 days to
complete its orbit. The second planet,
B, takes 814 days, while the third
planet, C, takes 1672 days. Su... |
The question :
Prove that :$$ \text{ if } y = 2x^2 - 1,\text{ then } \biggl[ \frac{1}{y} + \frac{1}{3y^3} + \frac{1}{5y^5}+ \cdots \biggr]$$ is equal to $$\frac{1}{2} \biggl[ \frac{1}{x^2} + \frac{1}{2x^4} + \frac{1}{3x^6} + \cdots \biggr]$$
Here I have modified the question, in the actual question (from my paper) th... |
This question might turn out to be really trivial.
$f$ is a one-to-one function from the interval $[0,1]$ to $\mathbb{R}$. Is it necessary that $\exists q \in \mathbb{Q}$ such that $f(x) = q$ for some $x \in [0,1]$ i.e. is it necessary that the image of $f$ contains a rational number?
I came across this question when I... |
I'm interested because I want to show that $x^2-34y^2\equiv -1\pmod{m}$ has solutions for all integers $m$. I started by using the following reasoning:
If $3\nmid m$, then $gcd(m,3)=1$. Then there exists a multiplicative inverse $\bar{3}$ modulo $m$. I note that $5^2-34=-(3^2)$, and thus $\bar{3}^2(5^2-34)\equiv (\bar{... |
I've got a multiplicative-with-noise model $F(x,y)=S(x)*R(y)*D(x,y)+N$, where $S(x)$ and $R(y)$ are unknown functions, $D(x,y)$ is a distance function, that is, a function that depends only on $|x-y|$ and decreases quickly when distance increases. $N$ is an uncorrelated "small random noise" function. All of the functi... |
Is there a proof in the spirit of Euclid to prove Dirichlet's theorem on primes in arithmetic progression? (By the spirit of Euclid, I mean assuming finite number of primes we try to construct another number which has a prime factor which falls in the same equivalence class as the other primes but the number is not div... |
Suppose that in the plane a given conic curve is compelled to pass through two fixed points of that plane.
What are the curves covered by a fixed point of the conic, its center (for an ellipse), its focus, etc. ?
(I apologize for the bad English ...)
|
Suppose $T_1, T_2 : \ell^2(Z_N)-> \ell^2(Z_N)$ are translation invariant linear transformation. How to prove that the composition $T_2 \circ T_1$ is translation invariant?
Recall the translation operator $R_k$ defined by
$(R_k z)(n)=z(n-k) $
Also recall definition:
Let $T : \ell^2(Z_N)-> \ell^2(Z_N)$ be a linear tran... |
I ask this in a partly recreational, and partly research-related spirit, and I realize my problem might be ill-posed, so any suggestions for clarification might go a long way.
Succinctly, my problem can be stated as
Find the period of a "black box"
pseudorandom number generator
but I suppose the following formulati... |
How to prove that convolution is associative, that is
$(x*y)*z = x*(y*z)$, for $x,y,z \in l^2(Z_N)$. Either directly from the definition of convolution( Definition of convolution: For $z,w \in l^2(Z_N)$, the convolution $z*w \in l^2(Z_N)$ is the vector with components
$z*w(m) = \sum_{n=0}^{N-1} z(m-n)w(n)$ ) or using... |
Suppose $b_1, b_2 \in l^2(Z_N)$. How to prove that the composition $T_{b_2} \circ T_{b_1} $ of the convolution operators $T_{b_2}$ and $T_{b_1}$ is the convolution operator $T_b$ with $b=b_2 \cdot b_1$? Hint use previous question where I asked: How to prove that convolution is associative? and also
$b_2 \cdot b_1(m) =... |
$$\sum_{i=1}^n \frac1{4i-1}$$
I know I have to integrate the function but from what to find lower and upper bound.
|
Given two vectors $u, v \in \mathbb{R}^d$ represented the spherical coordinates is there a simple formula to compute the angle between the two vectors? Without loss of generality, we can assume that the vectors $u$ and $v$ have unit norm.
I am not sure that the following notation is standard, but I assume that the vect... |
I was recently studying some literature on chaos theory and non-linear equations . where in various ciphers were created using non- linear equations like Lorenz equation . Are the data generated from these equations truly random , is it true that the cipher created using non -linear equations are much stronger than AES... |
I'm looking for a program or utility to help construct DFA/NFA/PDA etc. and the like.
The main reason I'm interested in this, is because I want to be able to move the various states around without breaking their component connections, or having to redraw all the connections to their new place (otherwise I'd be doing t... |
I want to show that if $X$ and $Y$ are the two loss variables such that $X\leq Y$, then $\text{VaR}_\delta(X)\leq\text{VaR}_\delta(Y)$.
|
It is known that for any locally compact Hausdorff space X, we can define a Hausdorff one-point compactification containing X.
In the case of the (differentiable) manifold $\mathbb R^n$ this one-point compactification turns out to be (homeomorphic to) $\mathbb S^n$, which is again a (differentiable) manifold.
This lead... |
I've come across a problem with the size of coefficients of a series expansion. Here's an example of an expression that I expand into a power series:
$\displaystyle\frac{1-e^{i3n\cdot t}}{1-e^{i\cdot n\cdot t}}$ where $n \in \mathbb{N}$ and $t \in \mathbb{R}$
The problem is that I'd like to work with small coefficien... |
Let $\mu\left(n\right)$ be the Möbius function. Let $\phi\left(n\right)$ be Euler's totient function. Let $\sigma\left(n\right)$ be the sum of divisors and $\tau\left(n\right)$ be the number of divisors functions. I am curious to know whether or not the system:
$\mu\left(n\right)=a$
$\phi\left(n\right)=b$
$\sigma\left(... |
I'm searching for a statistical method to determine if a player is cheating in an online game.
Given a number of positive points and a number of negative points per player (score) and given $n$ players ($n <= 64$). Each player can cheat independent of each other. I used standard deviation, which works great on a sing... |
Let $s = (\frac{1}{\sqrt{d}}, \ldots, \frac{1}{\sqrt{d}})$ and $u \in \mathbb{R}^d$ be two distinct unit norm vectors in the first orthant. Consider moving along the great circle defined by $s$ and $u$ (in the direction from $s$ to $u$) until the first intersection of the great circle and one of the following hyperplan... |
I just want to make sure that I understand these two objects correctly.
Let $R$ be a ring. A subset $R'\subseteq R$ is a subring provided $R'$ is an additive subgroup that is closed with respect to multiplication.
A subset $I\subseteq R$ is an ideal provided $I$ is an additive subgroup such that $rI\subseteq I$ and $I... |
Suppose you have two options for making a payment:
(A) Pay $90 \%$ of the purchase price two months after the date of the sale.
(B) Deduct $X \%$ off the purchase price and pay cash on the date of sale.
Determine $X$ so that a customer is indifferent between the two options when valuing them using an effective annual i... |
How to prove that a power series in more than one variable is invertible?
Is this still true if the variables are noncommutative?
|
I'm hoping that this isn't such a basic question that it gets completely laughed off the site, but why would I want to multiply two polynomials together?
I flipped through some algebra books and have googled around a bit, and whenever they introduce polynomial multiplication they just say 'Suppose you have two polynom... |
What is the order of the set of distinct (up to similarity) nxn matrices over $\mathbb{R}$ with determinant equal to some non-zero scalar... say 6? (eg. countable, uncountable etc.)
The set of matrices over $\mathbb{R}$ is uncountable. So is the order of the set consisting of classes of matrices with the same determina... |
Let ( L, ≤ ) be a lattice, x, y, z ∈ L
I am unable to understand why ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x ) is mentioned as fact in Y.N. Singh's "Mathematical Foundation of Computer Science" - Pg 157
Could I prove this somehow, or is it an axiom?
|
It is well known that $\text{Ext}(\mathbb Z_p,\mathbb Z)$ is the trivial group, because $\mathbb Z_p$ is projective; this seems to be in contradiction with the Exercise 1.1 in Hilton - Stammbach, pag. 88: the two sequences
$$
0 \to \mathbb Z \xrightarrow{\mu} \mathbb Z \xrightarrow{\epsilon}\mathbb Z_3 \to 0
$$
and
... |
Suppose I can calculate the extraordinary cohomology encoded in topological $K$-groups of a topological space $X$ with CW structure. What information does this give me about $C^{*}$-algebras associated with $X$? What is the algebraic analogue of topological suspension or the algebraic version of Bott Periodicity?
Bigge... |
The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit tangent vector at $v$. How to prove the following statement:
There exists a unique geodesic $\gamma$ on $\mathbb H$ such... |
This question is a follow up to this excellent mathematics stackexchange question.
Let $\mu(n)$ be the Möbius function, $\phi(n)$ Euler's totient function, $\sigma(n)$ the sum of divisors function and $\tau(n)$ the number of divisors function. Define the set $S_N,$ for a natural number $N,$ by
$$S_N = \lbrace (m,n) \i... |
In a few examples i noted that the existence of $k$-regular graph on n vertices is :
True , for k or n even.
False , for k and n odd . But we can find a graph with $n-1$ vertices with
degree k and one vertex with degree $k-1$. There doesn't exists a k-regular graph for
k and n odd because $k=\deg(G) = 2*|E(G)| / |V... |
I was re-reading this old book of mine; and I noticed that in defining the rules of differential forms, it "makes sense" that we have the rule $dx \wedge dx=0$ because if $dx$ is infinitesimal, then to first order approximations we can ignore powers of $dx$. Similarly, the definition for the exterior derivative $d$, o... |
Let $B$ be a Stone space (compact, Hausdorff, and totally disconnected). Then I am basically certain (because of Stone's representation theorem) that if $a, b \in B$ are two distinct points in $B$, then $B$ can be written as a disjoint union $U \cup V$ of open sets where $a \in U, b \in V$.
However, I can't seem to ... |
The chance to throw a 6 with one die is 1/6
And 6 times 1/6 = 1
So, if I throw with 6 dice, the chance to throw at least 1 six should be 1.
But when I throw 6 dice, I sometimes don't throw any 6 at all..
How come?
|
Show that the term ZZ where Z is λz.λx. x(z z x) satisfies the requirement for fixed point combinators that ZZM =β M(ZZM).
|
How to solve $y''-\displaystyle\frac{y'}{x}=4x^{2}y$ ?
I know that the solution of this equation is: $y = e^{x^{2}}$, but I cannot resolve.
First I thought that $z=y'$ could be, but was not.
|
A loan for $50,000$ has level payments to be made at the end of each year for 10 years at an annual rate of $9 \%$. (a) Find the balance at the end of $3$ years. (b) Find the interest paid in the third payment.
So I think this involves using amortization. That is, one pays the interest due first and then the principal ... |
Today I was thinking about composition of functions. It has nice properties, its always associative, there is an identity, and if we restrict to bijective functions then we have an inverse.
But then I thought about commutativity. My first intuition was that bijective self maps of a space should commute but then I saw... |
In studying for an exam, I had difficulty with these two questions:
Give a recursive form (including bases) for the following functions.
$$f(n) = 5 + (-1)^n$$
$$f(n) = n(n+3)$$
|
Question: What is the correct notion of a product of integral (or rational) polytopes which induces a factorization of its Ehrhart (quasi-)polynomial into two primitive Ehrhart (quasi-)polynomials corresponding to its constituent polytopes, viz., $L_{P \times Q}(t) = L_{P}(t) L_{Q}(t)$?
(Motivation) Given two closed i... |
I am trying to compute the Kolmogorov distance between two univariate gaussian distributions $\mathcal{N}(0,n)$ and $\mathcal{N}(0,2n)$ for large $n$. I have a feeling this should be simple but whatever I have tried so far doesn't work. Could anyone give me some hints?
By the Kolmogorov distance between two distribut... |
Let $n$ be a positive integer. Suppose we have an equilateral polygon in the Euclidean plane with the property that all angles except possibly two consecutive ones are an integral multiple of $\frac{\pi}{n}$, then all angles are an integral multiple of $\frac{\pi}{n}$.
This problem is #28 on page 61 in these notes rest... |
Let $n$ and $k$ be integers with $1 \leq k \leq n$. Show that:
$$\sum_{k=1}^n {n \choose k}{n \choose k-1} = \frac12{2n+2 \choose n+1} - {2n \choose n}$$
I was told this is supposed to use a combinatorial proof and while I'm not that comfortable with that, many similar proofs use a mathematical proof that shows equalit... |
I need help studying for an exam. How do I find "Big Oh," "Big Omega," and "Big Theta"?
How do I combine this with Induction?
Prove that $f(x)$ is $O(x^n)$, where
$$f(x) = a_n x^n + a_{n-1} x^{n-1} + a_{n-2} x^{n-2} + \cdots + a_2 x^2 + a_1 x + a_0$$
Find "Big O": (Usually log in Computer Science is assumed to be bas... |
I have a formula which is G-force = velocity*omega/9.8. Omega is the angular velocity.
I've seen on the internet that G force is actually acceleration/9.8. I'm confused as to which formula is correct. For simulating the motion of particle taking a turn, would omega simply be velocity divided by radius of turn? Assuming... |
I found this puzzle posted on a student website I frequent, but no one including the poster nor I could solve it. So I'm posting this puzzle here with hope that some of math whiz in here could illuminate on how to approach it:
There is an alien on the surface of a
spherical planet. The alien can run at a top
speed... |
Help would be appreciated. The notes are poor on the subject, and I am clueless.
Verify the following equalities:
Verify the equality
$$\mathsf{SIII}=_β\mathsf{I}$$
where $\mathsf{S} = λxyz.(xz)(yz) \quad \mathsf{I}= λx.x$
Verify the equality $$\operatorname{twice} (\operatorname{twice}) f x =_β f(f(f(f x)))$$
where ... |
How to proceed in this problem ?
|
Given $$1\cdot 2^1 + 2\cdot 2^2 + 3\cdot 2^3 + 4\cdot 2^4 + \cdots + d \cdot 2^d = \sum_{r=1}^d r \cdot 2^r,$$
how can we infer to the following solution? $$2 (d-1) \cdot 2^d + 2. $$
Thank you
|
I am an undergraduate learning about gauge theory and I have been tasked with working through the two examples given on pages 65 and 66 of "Characteristic forms and geometric invariants" by Chern and Simon. I will recount the examples and my progress at a solution. For ease here is the relevant text:
Example 1. Let ... |
Need help studying for an exam.
Practice Question:
If there are $200$ students in the library, how many ways are there for them to be split among the floors of the library if there are $6$ floors?
Hint: The students can not be told apart (they are indistinguishable).
The answer must be in terms of $P(n,r), C(n,r)$, pow... |
I recall reading somewhere that the supremum of a family of topologies on a set is simply the topology generated by the union of all topologies in the family. My question is, what does "generated" mean in this case? For instance, I know when dealing with filter bases, the generated filter is the set of all subsets cont... |
Question: Let $F_n$ the sequence of Fibonacci numbers, given by $F_0 = 0, F_1 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \geq 2$. Show for $n, m \in \mathbb{N}$: $$F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$$
My (very limited) attempt so far: after creating a small list of the values $F_0=0, F_1=1, F_2=1, F_3=2, F_4=3, F_5=5, ... |
$Y = X+N$ where $X,N$ are normally distributed with zero mean.
I know that $Y$ is also normally distributed with zero mean and sum of variances.
I don't know how to get $f(X|y)$ in order to calculate ${\rm E}(X|y)$.
|
I would like to show that the quantity:
$-2\sigma\left(rx^{2}+y^{2}+b\left(z-r\right)^{2}-br^{2}\right)$
is negative on the surface:
$rx^{2}+\sigma y^{2}+\sigma\left(z-2r\right)^{2}=C$
for some sufficiently large value of $C$.
I was not able to massage the first quantity any more in order to make it look like the s... |
I'm in Italian guy almost in my 30. I have a regular job as a programmer. When in school I've never been much interested nor good in math and even at the university I've been studying languages, hence something more on the literary side that on the scientific one.
When in high school I've had mostly literary classes, s... |
I'm really embarrassed to ask this - my apologies.
In a piece of software I'm writing, I am using the following:
$Y = X+\frac{X}{10}N$
That's all fine - but how do I calculate X when I know Y and N?
I know it's really basic, but for some reason I get stuck half-way through moving the equation around.
|
Edit:
The answers I seem to be getting here are that this is not possible, but I have come across a formulae for subdividing an icosahedron (which I don't have to hand, but goes something like this)
$V = 10 u^2+2$
$E = 30 u^2$
$F = 20 u^2$
Where $u=\text{ frequency of subdivision}$
(I may have it the wrong way aroun... |
I'm trying to teach myself mathematics. I only started last year with a previous knowledge of basic arithmetic. I often make a detour into regions featuring problems that are a bit too deep for me. I'm often too curious to wait until I was able to catch up on the necessary bedding. I hope you'll excuse me :-)
Wolfram|A... |
Is there any "algorithm" or steps to follow to get a formula from a table of values.
Example:
Using this values:
X Result
1 3
2 5
3 7
4 9
I'd like to obtain:
Result = 2X+1
Edit
Maybe using excel?
Edit 2
Additional info:
It is not going to be a... |
I've got a question concerning how to proof the Pythagoras theorem using the following assumption:
$x$ is perpendicular to $y$ (if and only if) $||x+y||^2 = ||x||^2 + ||y||^2$,
where $x$ and $y$ are vectors.
I have a basic understanding of linear algebra, however I'm a beginner with this. The question provides hints ho... |
Let's say that we have three points: $p = (x_p,y_p)$, $q = (x_q,y_q)$ and $a = (x_a,y_a)$. How can i find point $b$ which is reflection of $a$ across a line drawn through $p$ and $q$? I know it's simple to calculate, when we have $p$, $q$ etc. But I want to do this in my program, and I'm not sure, how to compute this.
... |
How to prove that $\mathcal{O}_{\sqrt[3]{3}}$ is an euclidean domain? I heard that one should prove the following but why it is enough?
For any $ a,b,c\in\mathbb{R}$, prove that there are $ x,y,z\in\mathbb{R}$ such that $ x-a,y-b,z-c\in\mathbb{Z}$ and that
$$-1\leq x^3+3y^3+9z^3-9xyz\leq 1.$$
|
Let X be a smooth projective variety with canonical class K.
Let a be defined to be the maximum dimension of the image of X under the rational map induced by the linear system |nK| as n ranges over all positive integers.
Let b be defined to be the smallest positive integer c such that the sequence (dim(H^0 (X, nK))/n^c... |
I'm asking here because no textbook or website that I know of gives a definition of the above mentioned term.
Since there's no obvious way (that I can think of) to define a normal subloop, I don't see how the definition of a simple group can be modified for the case of loops.
So - what is the definition of a simple loo... |
What is the next term of the following sequences ?
$3,8,13,24,41, \ldots$ Options: $\{ 70,75,80,85 \}$.
$4,23,60,121, \ldots$ Options: $\{212,221,241,242 \}$.
This two question is taken from a test paper of mine, I am no able to figure out the correct answer.
|
According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces.
Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative linear map?
Or, is there some obstruction, ensuring that every algebraic homomorphism between two Banach algebras is cont... |
I understand the formal definition of uniform continuity of a function,
and how it is different from standard continuity.
My question is: Is there an intuitive way to classify a function
on $\mathbb{R}$ as being
uniformly continuous, just like there is for regular continuity?
I mean, that for a "nice" function $f:\math... |
Consider a function f:[0,1] -> R with f(0)=0
having a finite derivative at each x in (0,1)
Prove that if f' is an increasing function (at least on the interval (0,1))
then h(x)=f(x)/x is also increasing
|
I'm trying to understand a proof in "Combinatorial Group Theory" by Lyndon & Schupp, page 26, Lemma 4.7.
Here's my problem:
$F$ is a free group with $X$ as basis. $x \in X$. $\alpha \in IA(F)$ - meaning it's an automorphism of $F$ which induces the identity automorphism on $F/[F,F]$. $k \in F_n$ - a term in the descend... |
Why in a Hausdorff sequentially compact space the size of the closure of a countable subset is less or equal than $c$ ? I can see why this is true when the space if first countable but we are not assuming so.
|
$f\colon (a,b) \to \mathbb{R}$ is continuous, with finite derivatives everywhere in $(a,b)$, except maybe at $c$.
If
$$\lim_{x\to c}f'(x) = B,$$
show that $f'(c)$ exists and equals $B$.
I'm not sure where to start on this. I've tried using the definitions of continuity and f' but it isn't working out
Well I started... |
I have 2 matrices and have been trying to multiply them but to no avail. Then I found this online site and trying feeding it the values but yet no success.
- R' . T
is what i would like to do but R is a 3x3 (3 rows, 3 cols) matrix and T is a 1 x 3 matrix (1 row, 3 cols).
Any idea how to go ahead with this?
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This is initially a funny question, because I've found this on old notes but I do not find/recover my own derivation... But then the question is more general.
Q1:
I considered the function
$ f(x) = - \frac {2x^2+3x}{(x+1)^2} $
I expressed this by a powerseries $ f_1(x) = -3x + 4x^2 - 5x^3 + 6x^4 - \ldots $
and ... |
Possible Duplicate: Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
$b$, $x$, $y$, $z$, $\ldots$ are integers greater than 1. How can we prove that
$$
\gcd (b ^ x - 1, b ^ y - 1, b ^ z - 1 ,\dots)= b ^ {\gcd (x, y, z, \dots)} - 1\ ?
$$
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Let $f\colon A\to B$, and $g\colon B\to C$.
Prove: If $g\circ f$ is onto and $g$ is 1-1, then $f$ is onto.
Here is all I can think of:
$g\circ f\colon A\to C$, and for all $x\in B$ there is a unique $y\in C$.
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This is just a simple doubt I wanted to clear. I'm calculating the Hamming and Euclidean distance between two columns in a matrix, where each column is storing time information in seconds of when a particular event occurred. When I calculate the Hamming and Euclidean distance between the two columns, I get back a certa... |
Let's say I have an "S" shaped curve made up of 50 points. I am only given the data points, and there is no simple or obvious equation for each point on the curve.
So given only the data points, how would I be able to calculate the slope of the curve at a particular data point?
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Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous and has a finite derivative $f'$ everywhere on $(a,b)$.
If $f(a)=f(b)=0$ prove that for every $y\in\mathbb{R}$ there is some $c$ in $(a,b)$ with $f'(c) = yf(c)$.
We were given a hint to apply Rolle's theorem to $h(x)f(x)$ for an $h$ that depends on $y$ but I don't ... |
How to show :
If $A \subseteq \mathbb{I}$ is not
countable, then $A$ is not closed set
in $\mathbb{R}$.
I think that if $A$ is closed, then $A^{c}$ is open thus: $A^{c}$ is an union of open intervals, but...
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