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Consider a sequence of functions $f_n$ where $f_n : \mathbb{R} \to \mathbb{R}$
and $f_n$ are all differentiable with derivatives $f^\prime_n$. The sequence $f_n$ and the sequence $f^\prime_n$ both converge uniformly to functions $f$ and $g$ respectively. According to the definition given in this wiki page on uniform ... |
The thought came from the following problem:
Let $V$ be a Euclidean space. Let $T$ be an inner product on $V$. Let $f$ be a linear transformation $f:V \to V$ such that $T(x,f(y))=T(f(x),y)$ for $x,y\in V$. Let $v_1,\dots,v_n$ be an orthonormal basis, and let $A=(a_{ij})$ be the matrix of $f$ with respect to this bas... |
I have some questions about the definition of splines and in particular periodic splines.
So in non periodic case splines are piecewise polynomials of degree $<=m$ which are $m-1$ times differentiable.(I took $m-1$ time differentiability for simplicity)
By this definition we can call linear or quadratic polynomials ... |
There is a sequence $a_n$ and $a_n \to 0$ as $n \to \infty$. Let $S_N = \sum_{n=0}^{N} a_n$. Let the sequence $S_N$ be converging to a limit $l$ where $l \in \mathbb{R}$. What is $\lim_{K\to\infty}\lim_{N\to\infty}\sum_{n=K}^{N} a_n $ ?
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The theorem of Seifert-Van-Kampen states that the fundamental group $\pi_1$ commutes with certain colimits. There is a beautiful and conceptual proof in Peter May's "A Concise Course in Algebraic Topology", stating the Theorem first for groupoids and then gives a formal argument how to deduce the result for $\pi_1$. Ho... |
Suppose I have an integral:
$F \equiv \int_{a}^{b} f(x) dx$
I know that $f(x)$ is real-valued, finite and non-negative everywhere on the interval $(a, b)$ where $a$, $b$ are real numbers or $\pm \infty$ and $a \leq b$. I know that in the obvious, well-behaved cases this implies $F \geq 0$.
Is there any obscure patho... |
I know that $\int f(x) \overline{g(x)} dx$ is an inner product on $L^2$. But is it one on $L^1$? I think it isn't, but I am have had difficulty figuring out which defining property is violated.
Thanks in advance for any pointers!
|
I have an equation in the form :
$$
M(x,y) + N(x,y)\frac{dy}{dx}
$$
I know the integrating factor for this equation. My problem is I can not rearrange my equation in to the form :
$$
\frac{dy}{dx} + p(x)y = Q(x)
$$
Do I need to rearrange if the integrating factor is known?
|
I tried balancing this chemical equation
$$\mathrm{Al(OH)_3 + H_2SO_4 \to Al_2(SO_4)_3 + H_2O}$$
with a system of equations, but the answer doesn't seem to map well. I get a negative coefficient which is prohibitive in this equations. How do I interpret the answer?
|
I'm actually a programmer so I'm sorry if this is a stupid question.
I am trying to solve for $x$, but I got stuck at this step:
$$e^{\frac{q1 + x}{b}} = e^{\frac{p - x\cdot o}{mb}} - e^{\frac{q2}{b}}$$
I'm not sure how to get x by itself because it's in the exponent in both places if I do: $\ln(e^{\cdots})$
I'll end u... |
This is a followup to A very simple discrete dynamical system with pebbles
The setup: we have slots $n$ slots labeled $1, \ldots, n$ and $k$ pebbles, each of which is initially placed in some slot. Now the pebbles want to space themselves out as evenly as possible, and so they do the following. At each time step $t$, e... |
What are other faster methods of calculating $e^x$ up to any number of decimals other than using the Taylor series formula?
|
I am having trouble following the argument in Massey for the proof of Theorem 8.3, which states that if $f: X \rightarrow Y$ is a homotopy equivalence, then $f_*: \pi(X, x) \rightarrow \pi(Y, f(x))$ is an isomorphism for any $x \in X$.
The proof in the book assumes Theorem 8.2., which states that if $\phi_0, \phi_1: X ... |
I'm looking for good examples where double induction is necessary. What I mean by double induction is induction on $\omega^2$. These are intended as examples in an "Automatas and Formal Languages" course.
One standard example is the following: in order to cut an $n\times m$ chocolate bar into its constituents, we need ... |
Show that if $V$ is a finite-dimensional vector space with a dot product $\langle-,-\rangle$, and $f: V \rightarrow V$ linear with $\forall v,w \in V: \langle v,w \rangle=0 \Rightarrow \langle f(v),f(w) \rangle=0$ then $\exists C \in \mathbb{R}$ such that $(C\cdot f)$ is a linear isometry.
Notes & Thoughts:
$g$ is a li... |
I would assume the answer to my question is yes, but I want to make sure because my book uses both terminologies. Please also indicate where zero falls into the mix.
UPDATE:
Here is an excerpt from my book:
The definition of $\Theta(g(n))$ requires
that every member $f(n) \in \Theta(g(n))$ be
asymptotically non-n... |
I am curios about the decidability or undecidability of the following decision problem.
INPUT: a first-order formula $\varphi$
OUTPUT:
Yes, if $\varphi$ is satisfiable but not finitely-satisfiable
No, otherwise
The problem is very close to Church's and Trakhtenbrot's Theorems, but somehow I do not see how to modify ... |
Is it true that:
If the intersection of the Sylow p-subgroups is trivial, then the intersection of their normalizers is equal to the intersection of their centralizers?
I half remember this being true for odd p, but I cannot find the reference. I have not found a counterexample for p=2 or p=3.
|
I had made up this problem a while back, and I think I had a tedious, uninsightful proof. Also, I am not able to reconstruct the proof I had.
Here is the problem:
Let $\displaystyle \pi: \{1,2, \dots, 13\} \to \{1,2,\dots, 13\}$ be a bijection, i.e. it is basically a permutation of the first $\displaystyle 13$ positive... |
Can anyone tell me how to calculate the Riemann integral of the characteristic function of the Cantor set? It's probably obvious but I don't see how to write it down.
Many thanks for your help!
|
Let $a$ and $b$ be positive integers both greater than $1$. I'd like to compute the average value of the summation
\begin{eqnarray}
\sum_{k = 0}^{\lfloor \log_{a} x \rfloor} \left \lbrace \log_{b} a^{-k} x \right \rbrace,
\end{eqnarray}
where $\{ \cdot \}$ denotes the fractional part function and $\lfloor \cdot \rflo... |
This is a pretty dumb question, but it's been a while since I had to do math like this and it's escaping me at the moment (actually, I'm not sure I ever knew how to do this. I remember the basic trigonometric identities, but not anything like this).
I have a simple equation of one unknown, but the unknown occurs twice ... |
Inverse of a sparse matrix could be dense, but what if I'm only interested in the main diagonal of the result? Is there a method that is more efficient than computing the full inverse?
|
What are the automorphism groups of non-abelian groups of order 27? (there are two non-abelian groups of order 27).
|
I'm trying to prove a result similar to Runge's theorem and Mergelyan's theorem (link at the bottom of the previous link), but without the condition of analyticity. The problem is as follows:
Let γ : [0,1] → $\mathbb{C}$ be a C$^1$ curve and K a compact subset disjoint from Γ = {γ(t) : t ∈ [0,1]}. Let f : Γ→ $\mathbb{C... |
A little stumped!
This is probably a very basic probability question, but I am lost.
At work I was asked the probability of a user hitting an outage on the website. I have some following metrics.
Total system downtime = 500,000 seconds a year.
Total amount of seconds a year = 31,556,926 seconds.
Thus, p of system down... |
Can someone give me an elementary proof of this fact?
Edit: This is an exercise in Marker's text, right after he defines
$$\text{acl}(A)=\{x:x \text{ is algebraic over }A\}.$$
The question and the full definition is here.
|
Question
I am looking for a unique characterization of a convex polygon with $n$ vertices, relative to a feature point $p$ in the interior of the polygon. This characterization would be a vector of numbers. Suppose a polygon is described by its feature point and vertices $P=(p,v_1,\ldots,v_n)$, then the characterizatio... |
Which books teach numerical method for system of differential equations?
|
Suppose I have found an irrational number, say e^x upto m decimal places....
Now I want to find upto say m+n decimal places. Is there any way that I can use the previous value of e^x upto m decimal places to help me find the remaining n decimal places? (I can save some number, say, like remainder, when finding upto m d... |
I'm trying to prove the following claim:
$f_n \in C_c$, $C_c$ being the set of continuous functions with compact support, then $\mathrm{lim}_{n \rightarrow \infty} || f_n - f||_{\infty} = 0$ implies $f_n(x) \rightarrow f(x)$ uniformly.
So, according to my understanding,
$$ \mathrm{lim}_{n \rightarrow \infty} || f_n - ... |
How do I compute $\text{(irrational)}^{\text{(irrational)}}$ up to a required number of decimals say m, in the fastest way ? (one way is of course compute both the irrational numbers to a precision much larger than m and then solve it... but you never know how much excess of m you will need to calculate the irrationals... |
As exam preparation we were trying to proof the following task:
Let $V=\mathbb{R}^2$ and let $\phi$ be an endomorphism of $V$ with $\phi \circ \phi = id$ and $\phi \neq id$ and $\phi \neq -id$. Proof that this implies the existence of a basis $B=(b_1,b_2)$ of $V$ with $\phi(b_1) = b_1$ and $\phi(b_2)=-b_2$
Unfortunate... |
The problem is 10.6a from Computational Geometry: Algorithms and Applications.
We want to solve the following query
problem: Given a set $S$ of $n$
disjoint line segments in the plane,
determine those segments that
intersect a vertical ray running from
a point $(q_x, q_y)$ vertically
upwards to infinity. D... |
I'm trying to learn something about mathematical logic. I think, I partly understand definitions of free variables, bound variables and quantification, but it's not easy topic for me and it's not enough to solve problems. I mean, for example I have such problem:
mark sets on a graph:
$\{ z : \mathbb{R} | \forall{x}\ex... |
I have done some higher math, being an engineer. They usually focus on getting the correct answer as supposed to actually understanding math. Glorified calculators, basically. I know how to do stuff with $\log x$, even calculate it with different bases, yay me. So my next question might come as a bit of surprise.
Can s... |
Sorry about the title, I have no idea how to describe these types of problems.
Problem statement:
$A(S)$ is the set of 1-1 mappings of $S$ onto itself. Let $S \supset T$ and consider the subset $U(T) = $ { $f \in A(S)$ | $f(t) \in T$ for every $t \in T$ }. $S$ has $n$ elements and $T$ has $m$ elements. Show that the... |
I'm trying to find $$f(x)=\sum_{n=2}^{\infty} \frac{x^n}{n(n-1)}$$
I found the radius of convergence of the above series which is $R=1$. Checking $x=\pm 1$ also yields a convergent series. Therefore the above series converges for all $x\in [-1, 1]$.
Using differentiation of the series term by term we get: $$f'(x)=\sum_... |
[ In first, I say "I'm sorry!", because I am not a Englishman and I don't know your language terms very well. ]
OK, I have some polynomials (like $a^2 +2ab +b^2$ ). And I can't decompress these (for example $a^2 +2ab +b^2 = (a+b)^2$).
Can you help me? (if you can, please write the name or formula of combination (like $... |
I am trying to describe the edges of an undirected graph that contains loops. On Wikipedia they are characterized as 2-multisets, meaning it has two elements which can be identical, and the order is not important. I never heard of that term, and find no reference to it anywhere else, so I am not sure whether it is OK t... |
If I understand correctly, Gauss proved that given any oriented Riemannian surface, one can find a complex structure on the surface so that the metric on the charts is just $f|dz|$, where $f>0$.
I've heard these coordinates referred as "conformal coordinates," which makes sense, but I've also heard of them referred to ... |
Possible Duplicate:
Intuitive explanation for the identity $\sum\limits_{k=1}^n {k^3} = \left(\sum\limits_{k=1}^n k\right)^2$
There is the interesting identity:
$$\left ( \sum_{i=1}^n i \right )^2 = \left ( \sum_{i=1}^n i^3 \right ) $$
which holds for any positive integer $n$.
I know several was of proving this (fi... |
One of my pals asked me to look at this question
Let $f: [0,1] \to \mathbb{R}$ be differentiable. Suppose that $f(0) = 0$
and $0 < f'(x) < 1$ for all $x \in (0, 1)$, where $f'(x)$ is the derivative of $f$. Prove that
$$ \left(\int_{0}^{1}f(x) \ dx \right)^{2} \geq \int_{0}^{1}(f(x))^{3} \ dx$$
I don't really have a cl... |
Latest EDIT : This question in its latest form has been solved here
I want to know the existence of a class of functions with the following properties. If they exist, how to construct them ? In case they do not exist, please suggest a proof.
Let $ f : \mathbb{R} \to \mathbb{R} $ be a differentiable function with compa... |
Problem: count the number of distinct ways to write number X as the sum of numbers {a, b, c...} with replacement.
For instance, there are 3 ways to write 11:
2+2+7
2+2+2+2+3
2+3+3+3
And if I remember correctly, isn't there a way to solve this using generating functions or polynomials of some sort?
(this question inspi... |
I want to get acquainted with approaches made to solve P vs NP problem.
1) What is already achieved in solving the P vs NP problem?
2) What articles are most cited/famous in that field? Please, provide links for them.
3) How one can divide existing approaches to different classes?
|
Given $A = B \times C$
Using any $9$ digits form two numbers $B$ and $C$.All digits must be used exactly once.
What is the maximum possible value of $A$?
|
Let
$GL_{n}(\mathbb{R}) = \{ A \in M_{n}(\mathbb{R}) \ | \ \text{det}(A) \neq 0 \}$
$SL_{n}(\mathbb{R}) = \{ A \in M_{n}(\mathbb{R}) \ | \ \text{det}(A) = 1 \}$
We know that $$GL_{n}(\mathbb{R})/SL_{n}(\mathbb{R}) \cong \mathbb{R}^{\ast}$$
I am interested to know the answer for this question.
Does there exist a subg... |
Determine in How many ways $N!$ can be expressed as sum of consecutive numbers.For example for $3!$ can be as $1+2+3$ only $1$ ways.
Please suggest an elegant approach.
|
I have the following group:
$$A = \mathbb{Z}/p_1^{r_1} \times \mathbb{Z}/p_2^{r_2} \mathbb{Z} \times \cdots \times \mathbb{Z}/p_n^{r_n} \mathbb{Z}$$
each $p_i$ being a prime, but they need not be distinct primes.
I want to solve the following isomorphism of abelian groups,
$$\frac{X}{\mathbb{Z}/p\mathbb{Z}} \cong A$$
I... |
$$1 + \sum_{j=1}^{n} j!j$$
I want to find a formula for the above and then prove it by induction. The answer according to Wolfram is $(n+1)!-1$, however I have no idea how to get there. Any hints or ideas on how I should tackle this one?
|
Given that we shouldn't say that "$f(z)$ is a function", shouldn't we also not write "$p \in k[X_1, \ldots, X_n]$ is a polynomial"? Along those lines, I usually write $p(X_1, \ldots, X_n) \in k[X_1, \ldots, X_n]$ in order to balance the "free variables" on both sides of the relation, but that gets unwieldy when you sta... |
Sorry this is a really simple problem but I couldnt figure it out. I'm trying to find the value of $3^{1000}\bmod 7$. I know the actual value is 4, but I used a calculator. How would I simplify this problem to get the correct answer without a calculator? I dont want a direct answer, but maybe a nudge in the right direc... |
The Gaussian $\mathbb{Z}[i]$ and Eisenstein $\mathbb{Z}[\omega]$ integers have been used to solve some diophantine equations. I have never seen any examples of the golden integers $\mathbb{Z}[\varphi]$ used in number theory though. If anyone happens to know some equations we can apply this in and how it's done I would ... |
A property of characteristic classes $c$ is that $c(f^* E) = f^* c(E)$ where $E\to M$ is a bundle and $f^* E$ is the pullback of $E$ by some map $f: N\to M$. In Bott and Tu it is stated (for Chern classes) that this implies that if $E \simeq F$ then $c(E) = c(F)$. However I am unable to see why this is true (from the... |
Let $V$ be a vector subspace of $R^N$, and $l:V \to R$ a linear mapping such that $l(V\bigcap R_{+}^N)\subseteq R_{+}$ (i.e., $l$ is positive).
I have heard that there exists a separating hyperplane sort of argument that allows us to show that $l$ extends to a positive linear functional on $R^N$. I have my own proof, b... |
Let $V$ be a vector space with basis $e_1, \ldots, e_n$ and $V^*$ be its dual space with dual basis $e_1^*, \ldots, e_n^*$. Let $k$ be an integer between $1$ and $n$. Why $\wedge^{n-k}V=\wedge^{k}V^*$? Thank you very much.
|
I haven't taken a math class in a few years and I have come to realize, as I am taking one again, that I do not know/remember how to factor equations.
One example from my textbook is:
$3x^3-x^2+3x+1$
Looking for a very detailed description of how to factor this, and just factoring in general.
EDIT: Isaac gave a very g... |
I have a problem understanding second-order conditions at a critical point in finding critical points of two-variable functions.
Let's consider $f(x,y)$.
The first-order conditions are $\frac{\partial f}{\partial x}=0$ and $\frac{\partial f}{\partial y}=0$. So the rate of change of $f$ in respect to both $x$ and $y$ is... |
What is the difference between max, min and sup, inf?
|
I need help proving this inequality.
$$\int_{1}^{2} \left(\frac{x+1}{2} \right)^{x+1} \ dx < \int_{1}^{2} x^x \ dx$$
Normally I would define $f(x) = \left(\displaystyle\frac{x+1}{2} \right)^{x+1}$ and $g(x) = x^x$ and try to compare those two.
Any suggestions?
|
In my differential geometry course we had the following
Theorem (Cartan-Hadamard): Let $M$ be a connected, simply connected, complete Riemannian manifold. Then the following are equivalent:
$M$ has nonpositive curvature
$|d\exp_p(v)\hat v| \ge |\hat v|$ for all $p \in M$, $v, \hat v \in T_pM$
$d(\exp_p(v), \exp_p(w)) ... |
What are some great math social activities for students? I'm looking for things that bring people together with a "light" mathematical touch. The goal is to create a stronger mathematical community in a small college math department.
Examples I've heard so far:
Math Stack Exchange Party: Students get together and surf ... |
There is an urn of N balls, each of a unique color. In each step one takes out 2 balls without replacement, changes the color on the 2nd ball to that of the first ball and returns them back to the urn. What is the expected number of steps until all the balls in the urn are of the same color?
|
Given $R$ an integral domain (commutative ring with no zero divisors), and $\mathfrak P$ a prime ideal in $R$, is there a relation between the field of fractions of $R$ and the field of fractions of $R/\mathfrak P$?
It's trivial to see that whenever $\mathfrak P$ is also maximal, then $\text{Frac}(R/\mathfrak P)\cong R... |
In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these particular spirals are derived from the definition of the logarithm?
|
I was told this question by a friend, who said that their friend had thought about it on and off for six months without any luck. I have then had it for a while without any luck either. It is in the style of a contest question, but it is unlike any other, as really nothing seems to gain any ground.
Here's the questio... |
What study practices have led you to the best success in learning mathematics (or applied math, or theoretical CS) in the classroom (undergraduate/lower graduate level), especially in courses that would otherwise be "too hard" for you?
Alternatively, if you teach math, what practices have your best students devised tha... |
I'm studying Shelah's proof (actually written by Uri Abraham) that adding one generic real implies the existence of a Suslin tree (available in this link, I think that freely for everyone.)
The notion of forcing is the set finite functions from $\omega$ to $\omega$, with stronger being "extending", then we construct a ... |
Prove that if a differentiable curve $g:[0,1] \to \mathbb{C}$ (complex plane) parametrizes counterclockwise the boundary of an open set $O$ in $\mathbb{C}$, then under suitable conditions area of $O$ is
$$
{1 \over 2i } \int_g \overline{z} dz $$
computed over the boundary.
|
I know that the product of two Gaussians is a Gaussian, and I know that the convolution of two Gaussians is also a Gaussian. I guess I was just wondering if there's a proof out there to show that the convolution of two Gaussians is a Gaussian.
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Let $\mathcal{F}$ be a $\sigma$-algebra of subsets of $\Omega$ and $B \in \mathcal{F}$. Show that $E = \{A \cap B: A \in \mathcal{F}\}$ is a $\sigma$-algebra of subsets of $B$. Is it still true when $B$ is a subset of $\Omega$ that does not belong to $\mathcal{F}$?
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Show that the set of all orthogonal matrices in the set of all $n \times n$ matrices endowed with any norm topology is compact.
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I have this problem. I want to prove that if I'm choosing a basis for polynomials that their exponent is from $1$ until $n$ and they are all $F(2)=0$, they are linear independent.
To be more spefic, let's have a vector space which contains $N$ function, first is with $1$ degree and so on, all are $0$ in $x=2$, why shou... |
A regular local ring is a domain. Is a regular ring (a ring whose localization at every prime ideal is regular) also a domain? I am unable to find/construct a proof or a counterexample. Any help would be appreciated.
|
I'm looking for a fast algorithm for generating all the partitions of an integer up to a certain maximum length; ideally, I don't want to have to generate all of them and then discard the ones that are too long, as this will take around 5 times longer in my case.
Specifically, given $L = N(N+1)$, I need to generate all... |
I just wrote a blog post and wasn't sure how to word a particular sentence. Say I have the following function:
\begin{equation}
f(x) = x^2
\end{equation}
Then I can say that the value of f(x) grows quadratically with x*. Similarly with this function:
\begin{equation}
f(x) = e^x
\end{equation}
...I could say that f(x) g... |
I'm currently learning the chain, product and quotient rules of differenentiation. Only I'm lacking some exercises. I'm looking for a website, or something that's available online with some more difficult examples of those rules combined. For example things like this:
Find the derivative of:
$$\ln\frac{(x+1)^2}{x^2+1}$... |
I found this question in a book (Topology II: homotopy and homology: classical manifolds)
Show that the quotient space $X = S^2 \times S^2 / [(x_1,x_2) \sim (Rx_1,Rx_2)]$ where R is the reflection in the equatorial plane, is homeomorphic to $S^4$.
I am still in the process of learning topology and I really don't think ... |
Let $O$ be a non-empty subset of $\mathbb{C}$ (the complex plane), and let $g : O \to \mathbb{C}$ be a holomorphic function that satisfies the differential equation $g^{\prime\prime}(z) - zg(z) = 0$ for every $z \in O$. Prove that for all differentiable curves $U$ in $O$ with the same endpoints, the value $$\int_U g(x)... |
1.Let f be a real-valued differentiable function defined on [0, 1]. If $f(0)=0$ and
$f(1)=1$, prove that there exists two numbers $x_1,x_2 \in [0, 1]$ such that
$\frac{1}{f'(x_1)}+\frac{1}{f'(x_2)}=2$.
2.Let f be a real-valued differentiable function defined on (0,$\infty$). If $\lim_{x\rightarrow0^+}f(x)=\lim_{x\right... |
I was working on something on Matlab couple of months back and found a tweak to speed up Gauss elimination on Matlab by dividing the original matrix into 4 block matrices and then solving them in a block-wise fashion.
I am however wondering about couple of things.
What is the reason for the optimal "split parameter" t... |
If we are given some linearly dependent vectors, would the T of those vectors necessarily be dependent (given a transformation from $R^n$ to $R^p$)?
And if we are given some linearly independent vectors, would T of those vectors necessarily be independent (given a transformation from $R^n$ to $R^p$)?
The answers are pr... |
This is in reference to page 153 of the notes found here
http://www.math.lsa.umich.edu/~hochster/614F10/614.pdf
In the second paragraph, the author is considering inverse limit of a family of subsets of a certain set where the defining maps of the inverse systems are inclusions. I am not sure why the inverse limit is t... |
Suppose you have an arbitrary triangle with vertices $A$, $B$, and $C$. This paper (section 4.2) says that you can generate a random point, $P$, uniformly from within triangle $ABC$ by the following convex combination of the vertices:
$P = (1 - \sqrt{r_1}) A + (\sqrt{r_1} (1 - r_2)) B + (r_2 \sqrt{r_1}) C$
where $r_1,... |
I want to ask if there is a slick way to prove:
$$\sum_{j=0}^n{(-1)^j{{n}\choose{j}}\left(1-\frac{j}{n}\right)^n}=\frac{n!}{n^n}$$
Edit:
I know Yuval has given a proof, but that one is not direct. I am requesting for a direct algebraic proof of this identity.
Thanks.
|
I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$.
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I know this is kind of minor. But I want to know if I understand this correctly. In "baby rudin", on page 42, is the condition
$$3^{-m}\lt \frac{\beta-\alpha}{6}$$
tight? I thought
$$3^{-m}\lt \frac{\beta-\alpha}{4}$$
is sufficient, am I correct?
Thanks.
Edit:
Okay, the background is: the Cantor set (i.e. the union of... |
I have been on this one for hours, cant figure out how to write this in the proper format/wording.
Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. For the sum and product of $f$ and $g$, determine which statements are true. If true provide a proof; if false, provide a counterexample.
If $f$ and $g$ are... |
I have the following statement in a paper:
Let $\Psi$ be the formal power series defined over the alphabet $\Omega$ and the log semiring by: $(\Psi, (a, b)) = -log(c((a,b)))$ for $(a,b) \in \Omega$, and let $S$ be the formal pwer series $S$ over the log semiring defined by: $S=\Omega^*+\Psi+\Omega^*$ (an alphabet is ... |
Is there any $\sigma$-algebra that is strictly between the Borel $\sigma$-algebra and the Lebesgue $\sigma$-algebra?
How about not in between the two, but in general, are there any other $\sigma$ algebra(s)?
What can be concluded about measure too, e.g. is Lebesgue measure the only measure for Lebesgue $\sigma$ algebr... |
The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$
For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using the above formula. As $n$ tends to infinity, it becomes an integration: $$\int_{x_0}^{x_1} f(x)~dx \over x_1 - x_0$$
On... |
Here is a quote from Derbyshire's "Unknown Quantity"
The rational numbers are "dense." This means that between any two of them you can always find another one.
This being a pop. math book, this is not meant as a formal definition of dense sets but it got me wondering whether the dense subsets of $\mathbb{R}$ could be... |
The first part of my question asked : State all the irreducible Polynomials in $\mathbb{Z}_2[x]$ of order 3.
I was able to do this and get the following polynomials :
$x^3 + x^2 + x + 1 \Rightarrow$ reducible
$x^3 + x^2 + 1 \Rightarrow$ irreducible
$x^3 + x + 1 \Rightarrow$ irreducible
$x^3 + 1 \Rightarrow$ reduci... |
How do I find the solutions of tanx = x upto any number of decimals?
(Of course, there is the graphical method but it just helps in finding the approximate value...)
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I have learnt that the intersection of pure subgroups of a group $G$ is not necessarily pure. Can someone show me an example when such a case exists?
I'm aware that if $G$ is torsion-free, then the intersection of pure subgroup ls of $G$ is necessarily pure. So the example above must involve for which the group $G$ is ... |
I'm currently practicing differentiation. The exercise I currently have is the following
Find the derivative of:
$(x + 6)^3 (9 x^3 - 2)^5$
Okay, well I can do that now. When I do this I uses the chain/product rule to get the following result:
$3(x+6)^2 (9x^3 -2)^5 + 45(x+6)^3 (3x^2) (9x^3 - 2)^4$
However, when I put th... |
I have been puzzling over this for a few days now. (It's not homework.) Suppose $f$ is a positive, non-decreasing, continuous, integrable function. Suppose there are two finite sequences of positive real numbers $\{a_i\}$ and $\{b_i\}$ where
$$\sum^n_{i=1}{f(a_i)} \leq \sum^n_{i=1}{f(b_i)}.$$
Is it true that
$$\sum^n_... |
I picked this problem from here http://mathalon.in/?page=show_problem.php&pid=106 . I don't know how to solve it accurately. The number seems too to be done by any computation method.
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I'm having trouble showing the following implication:
Let $M$ be a Riemannian manifold, let $L\subset M$ be a submanifold such that the following holds:
If $\gamma: I \to M$ is a geodesic s.t. $\gamma(0) \in L$, $\dot{\gamma}(0) \in T_{\gamma(0)}L$, then there is an $\epsilon >0$ such that for $|t| < \epsilon$ we have... |
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