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Sorry for this uninteresting question but hopefully someone can provide some help.
Is there a way to simplify the following expression?
$$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\frac{n-m-v}{n}\right)^{r}\displaystyle \frac{m}{m+v}-\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\f... |
Sorry for this uninteresting question but hopefully someone can provide some help.
Is there a way to simplify the following expression?
$$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\frac{n-m-v}{n}\right)^{r}\displaystyle \frac{m}{m+v}$$
$$-\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}... |
The idea behind the axiom of choice is to tell you how to choose when you can't necessarily distinguish between the items.
When you take out a pair of socks from the closet (or drawer, etc) you can't tell which one you had on your left foot and which on your right, or which one is this and which one is that. If you ... |
How many Borel-measurable function from R to R are there? |
How many Borel-measurable functions from $\mathbb{R}$ to $\mathbb{R}$ are there? The motivation is from [this][1] answer of mine on MathOverflow.
[1]: http://mathoverflow.net/questions/32126/function-with-range-equal-to-whole-reals-on-every-open-set/49137#49137 |
How many Borel-measurable functions from R to R are there? |
Given Kn define sequence S: writing all vertices in order of traversing it by Euler circle
How many distinct Hamilton cycles are in this sequence S? |
### Problem ###
Given:
$\vec r = r(sin \theta cos \phi, sin \theta sin \phi, cos \theta)$
$S: \left| \vec r \right| = a$, with $\hat n$ outwards
$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z) = \frac{3a}{5}(1,1,-1)$
$\vec F = k \frac{\vec r - \vec r0}{ {\left| \vec r - \vec r_0 \right|}^3 }$
Cal... |
Sorry for this uninteresting question but hopefully someone can provide some help.
Is there a way to simplify the following expression?
$$\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}\left(\frac{n-m-v}{n}\right)^{r}\displaystyle \frac{m}{m+v}$$
$$-\binom{n}{m}\sum_{\nu=0}^{n-m}(-1)^{\nu}\binom{n-m}{v}... |
Just dropping points into the polygon according to a uniform distribution will not, unfortunately, generate very uniform solutions: with extremely high probability there will be large gaps and tight clusters.
Obtaining a highly uniform pattern within an arbitrary (convex) polygon is a difficult problem to solve; exa... |
Let $\tau \in \mathbb{S}^1$ be such that $\tau$ is not a root of unity. Let $E_\tau$ be the quotient space $S^1/\tau^{\mathbb Z}$. Consider it as a pointed space with basepoint the equivalence class of $1$. This is a path-connected space.
What is the fundamental group of this space? |
I have a polygon in 2D (defined by a series of Vertex $V$ with coordinates). The polygon can be convex or concave. I have $n$ number of fix points I can put inside the polygon.
The question is, how can I distribute the fix points as uniformly as possible inside the polygon?
The motivation for this question is I ... |
I have to explain the problem I am thinking first:
The solutions of x^3+6*x-20 are x = 2, -1+3i and -1-3i.
The cubic formula for these solutions is:
$$x = \sqrt[3]{10 + \sqrt{108}} + \sqrt[3]{10 + \sqrt{108}}$$
First $\sqrt{108} = -10.392304845413, 10.392304845413$ has two values.
And $\sqrt[3]{10 + -10.... |
Different notations for roots? |
### Problem ###
Given:
$\vec r = r(sin \theta cos \phi, sin \theta sin \phi, cos \theta)$
$S: \left| \vec r \right| = a$, with $\hat n$ outwards
$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z) = \frac{3a}{5}(1,1,-1)$
$\vec F = k \frac{\vec r - \vec r0}{ {\left| \vec r - \vec r_0 \right|}^3 }$
Cal... |
Does anyone remember a movie about four mathematicians studying infinity, who ended up committing a suicide? It was aired online somewhere, and I can't find it (it's a documentary, if that helps). |
I have to explain the problem I am thinking first:
The solutions of $x^3+6x-20=0$ are $x = 2$, $-1+3i$ and $-1-3i$.
The cubic formula for these solutions is:
$$x = \sqrt[3]{10 + \sqrt{108}} + \sqrt[3]{10 + \sqrt{108}}$$
First $\sqrt{108} \approx -10.392304845413$, $10.392304845413$ has two values.
And $\... |
Are you talking about **Dangerous Knowledge** by BBC? Though it's not quite about four mathematicians (at least one of the people was physicist), and not all of them were studying infinity directly. That's the only one that comes to my mind as I watched it a couple of weeks ago. |
**Dangerous Knowledge** it is!
Cantor - Infinity
Boltzmann - Entropy
Godel and Turing - Logic
http://video.google.com/videoplay?docid=-5122859998068380459# |
I have to explain the problem I am thinking first:
The solutions of $x^3+6x-20=0$ are $x = 2$, $-1+3i$ and $-1-3i$.
The cubic formula for these solutions is:
$$x = \sqrt[3]{10 + \sqrt{108}} + \sqrt[3]{10 + \sqrt{108}}$$
First $\sqrt{108} \approx -10.392304845413$, $10.392304845413$ has two values.
And $\... |
Charlie, this is a result of Sierpiński. It appears in " Sur un problème de la théorie des relations", Fundamenta Mathematicae, vol 28, (1937), 71-74. You can download the paper [here][1].
The general result of which this is a particular case is due to Hajnal (rather than "independent", one usually talks of "free")... |
I have an input array of line segments (a,b)_1 ... (a,b)_n. Let's call it "S" so (a,b)_S_1 ... (a,b)_S_n.
I have another array of the same construct, let's call it "I", so (a,b)_I_1 ... (a,b)_I_n.
I wish to mathematically express the shortest distance of each of I against S. In other words, the matrix formed from... |
### Problem ###
Given:
$$\vec r = r(\sin \theta \cos \phi, \sin \theta \sin \phi, \cos \theta)$$
$S: | \vec r | = a$, with $\hat n$ outwards
$$\vec r_0 = \frac{3a}{5}(\hat x + \hat y - \hat z) = \frac{3a}{5}(1,1,-1)$$
$$\vec F = k \frac{\vec r - \vec r_0}{ {| \vec r - \vec r_0 |}^3 }$$
Calculate:
$$... |
Denote $(a,b)_{S_i}$ by $S_i$ and $(a,b)_{I_k}$ by $I_k$.
Distance between $i^{th}$ line segment of $S$ and $j^{th}$ segment of $I$ is $|| (S_i,\ I_j) ||$.
You can now denote the minimum by
$$ \text{min}_{1 \le i \le n,1 \le j \le n} \{|| (S_i,\ I_j)||\}$$
Or you can denote the matrix by
$M_{ij} = ||... |
It can certainly vary. As an Eulerean path uses all n(n-1)/2 edges and a Hamiltonian path uses n, you can't have more than (n-1)/2. Can you have that many? There aren't any Eulerean paths if n is even and >2. Depending upon your definition, I can define an Eulearean path that has no segment that is Hamiltonian |
The answer to both questions (a) and (b) is yes, and is a consequence of the fact that Riemann integrable functions can be approximated from above and below by [step functions][1]. The proof follows in a fairly straightforward manner once you realize that this allows you to reduce the problem to that of indicator funct... |
One way to prove that a field $K$ has no ideals except the entire field and the trivial ideal is to note the fact that every element $x$ has an inverse. By the definition of an ideal, if $x$ is in the ideal then $x^{-1}x$ is because $x^{-1} \in K$. But now we have that 1 is in the ideal, and so again by the definition ... |
The question does not seem to be very precise.
In any case,
It is [well known][1] that $K_{2k+1}$ can be decomposed into $k$ hamiltonian cycles.
So we can find an Euler tour with exactly $\dfrac{n-1}{2}$ hamiltonian cycles for odd $n$.
For even $n$, we can do $\dfrac{n}{2}-1$ (See link above).
[1]: h... |
The question does not seem to be very precise.
In any case,
It is [well known][1] that $K_{2k+1}$ can be decomposed into $k$ hamiltonian cycles.
So we can find an Euler tour with exactly $\dfrac{n-1}{2}$ hamiltonian cycles for odd $n$.
For even $n$, we can do an "incomplete tour" with $\dfrac{n}{2}-1$ (See... |
For which functions f:[0,1] -> [0,1] is the function f(x)f(y) convex over (x,y) \in [0,1]x[0,1] ? Is there a nice characterization of such functions f?
The obvious examples are exponentials of the form e^(ax + b) and their convex combinations. Anything else?
|
For what functions f(x) is f(x)f(y) convex? |
For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$?
The obvious examples are exponentials of the form $e^{ax + b}$ and their convex combinations. Anything else?
|
For what functions $f(x)$ is $f(x)f(y)$ convex? |
For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$?
The obvious examples are exponentials of the form $e^{ax}$ and their convex combinations. Anything else?
|
P325-326
《Conceptual Mathematics》 1st edition
It gives a sketch of a proof, but I can't carry out the complete proof.
"... This also follows from the appropriate universal mapping properties, which imply that
the two composites satisfy properties that only the corresponding identity maps satisfy."
I can't fi... |
In *Conceptual Mathematics* 1st edition, p. 325-236, there is a sketch of a proof, but I can't carry out the complete proof.
> "... This also follows from the appropriate universal mapping properties, which imply that the two composites satisfy properties that only the corresponding identity maps satisfy."
I ca... |
Why $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completness is not a topological invariant, so why is the above statement true?
|
Why $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space? |
For which functions $f\colon [0,1] \to [0,1]$ is the function $g(x,y)=f(x)f(y)$ convex over $(x,y) \in [0,1]\times [0,1]$ ? Is there a nice characterization of such functions $f$?
The obvious examples are exponentials of the form $e^{ax+b}$ and their convex combinations. Anything else?
|
Can you see what the topology of the quotient $S^1/(\tau)$ is? Its open sets "are" the open sets in $S^1$ which are invariant under multiplication by $\tau$.
Once you see what the topology is, your question becomes easy. |
Why $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the above statement true?
|
Why is it that $\mathbb{Q}$ cannot be homeomorphic to any complete metric space?
Certainly $\mathbb{Q}$ is not a complete metric space. But completeness is not a topological invariant, so why is the above statement true?
|
Why is it that $\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space? |
A subspace of a separable complete metric space is homeomorphic to a complete metric space if and only if it is a $G_\delta$. (I have a reference that says the direction relevant to your question is proved on page 197 of Bourbaki's *[General topology][1]*, volume 2, but I don't currently have access to that page. The ... |
A subspace of a separable complete metric space is homeomorphic to a complete metric space if and only if it is a $G_\delta$. (I have a reference that says the direction relevant to your question is proved on page 197 of Bourbaki's *[General topology][1]*, volume 2, but I don't currently have access to that page. The ... |
Why is $\tau(n) \equiv \sigma_{11}(n) \textrm{ mod } 691$? |
Why is $\tau(n) \equiv \sigma_{11}(n) \pmod{691}$? |
Can the number of solutions $xy(x-y-1)=n$ for $x,y,n \in Z$ be unbounded as n varies?
x,y are integral points on an Elliptic Curve and are easy to find using enumeration of divisors of n (assuming n can be factored).
If yes, will large number of solutions give moderate rank EC?
If one drops $-1$ i.e. $xy(x-y)=... |
$\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$ This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone did) -- how would you go about it? |
How do I integrate the following? |
$$\int{\frac{(1+x^{2})}{(1-x^{2})\sqrt{1+x^{4}}}}\ \text{dx}$$
This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone did) -- how would you go about it? |
How do I integrate the following? $\int{\frac{(1+x^{2})dx}{(1-x^{2})\sqrt{1+x^{4}}}}$ |
$$\int{\frac{1+x^2}{(1-x^2)\sqrt{1+x^4}}}\mathrm dx$$
This was a Calc 2 problem for extra credit (we have done hyperbolic trig functions too, if that helps) and I didn't get it (don't think anyone did) -- how would you go about it? |
How do I integrate the following? $\int{\frac{(1+x^{2})\mathrm dx}{(1-x^{2})\sqrt{1+x^{4}}}}$ |
I have a background in math, but I think the following construction appears often in physics.
Is there a term for a functional which maps one smooth function $f: R^n \rightarrow R$ to another $L(f):R \rightarrow R$ which represents the integral along each level set? That is,
$L(f)(a) = \int_{f^{-1}(a)} d\mu $
... |
How can I find solution for $x^3 = 1 (mod \hspace{5pt} p)$ (Prime(p)) efficiently?
Trivial root is $x_1 = 1$. I need to find other roots $x_2, x_3$. |
How can I find solution for $x^3 \equiv 1 \pmod p$ ($p$ a prime) efficiently?
Trivial root is $x_1 = 1$. I need to find other roots $x_2, x_3$. |
You may read the paper "[123 theorem and its extensions][1]" by Noga Alon
[1]: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.57.8591 |
is it possible to solve it by master method btw ? |
How do we solve the recurrence
$$T(n) = 2T(n/3) + n\log n$$
?
Also, is it possible to solve this recurrence by the Master method? |
The modular discriminant $\Delta (z) \in M_{12,0},$ the linear space of cusp forms of weight $12.$ Now dimension $M_{12,0} = 1$ (see, for example, [T.M. Apostol: Moldular Functions and Dirichlet Series in Number Theory][Apostol])
Every modular form for the full modular group is a polynomial in $E_4$ and $E_6,$ wher... |
The modular discriminant $\Delta (z) \in M_{12,0},$ the linear space of cusp forms of weight $12.$ Now dimension $M_{12,0} = 1$ (see, for example, [T.M. Apostol: Moldular Functions and Dirichlet Series in Number Theory][Apostol])
Every modular form for the full modular group is a polynomial in $E_4$ and $E_6,$ wher... |
The modular discriminant $\Delta (z) \in M_{12,0},$ the linear space of cusp forms of weight $12.$ Now dimension $M_{12,0} = 1$ (see, for example, [T.M. Apostol: Moldular Functions and Dirichlet Series in Number Theory][Apostol])
Every modular form for the full modular group is a polynomial in $E_4$ and $E_6,$ wher... |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where each terminal appears 0 or 2 times?
EDIT: Solving this may give interesting extension to <a href="http://math... |
How do we solve the recurrence $T(n) = 2T(n/3) + n\log n$?
Also, is it possible to solve this recurrence by the Master method? |
1. Can a CFG over large alphabet describe a language, where each terminal appears even number of times?
2. If yes, would the Chomsky Normal Form be polynomial in |Σ| ?
EDIT: What about a language where each terminal appears 0 or 2 times?
EDIT: Solving this may give interesting extension to <a href="http://math... |
Suppose A is a linear transformation from R^3 to R^3 and |det(A)| = 1. I know that A is volume preserving, but is it also area preserving? For example, if a and b are two vectors in R^3 that span a parallelogram, is the area of this parallelogram equal to the area of the paralellogram spanned by A(a) and A(b)?
THank... |
Let $f\colon\mathbb{R}^n\to\mathbb{R}$ be a function with $n\geq 2$, and define $S_{f}=\{(x_1,x_2, \ldots x_n) \in\mathbb{R}^n \mid f(x_1,x_2, \ldots x_n)=1\} = f^{-1}(1)$.
> Is $S_f$ path-connected when $f$ is $C^{\infty}$? If not, what about if $f \in \mathbb{Q}[x_1,\ldots,x_n]$?
**Context**: I had to establish... |
\begin{align}
\newcommand{\mba}{\mathbf{a}}
\newcommand{\mbb}{\mathbf{b}}
\newcommand{\mbl}{\mathbf{l}}
\newcommand{\mbp}{\mathbf{p}}
\newcommand{\mbv}{\mathbf{v}}
\newcommand{\mbx}{\mathbf{x}}
\int_{l_1}^0 \dotsi \int_{l_k}^0 \exp(\mbx \cdot (\mba - \mbb) + \mbl \cdot \mbb)d\mbx
&=\exp(\mbl\cdot\mbb)\prod_{i=... |
In
http://math.stackexchange.com/questions/11945/limit-of-sqrt7-sqrt7-sqrt7-cdots
Timothy Wagner gave a correct answer that was questioned for not having shown that the limit exists in the first place. My question is, are there any examples were a a value of limit can be derived although the limit does not exist? Ple... |
Several papers give poly-time algorithms for constrained paths on labelled graphs, e.g. [1]
Quote:
----------
Given an alphabet Σ, a (directed) graph G whose edges are weighted and Σ-labeled,
and a formal language L ⊆ Σ∗ , the formal-language-constrained shortest/simple path problem con-
sists of finding a short... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please help me to solve the most important of those problems.
**For example:** ***infinity*** !
What is infinity ? Is th... |
In
http://math.stackexchange.com/questions/11945/limit-of-sqrt7-sqrt7-sqrt7-cdots
Timothy Wagner gave a correct answer that was questioned for not having shown that the limit exists in the first place. My question is, are there any examples were a a value of limit can be derived although the limit does not exist? Ple... |
In
http://math.stackexchange.com/questions/11945/limit-of-sqrt7-sqrt7-sqrt7-cdots
Timothy Wagner gave a correct answer that was questioned for not having shown that the limit exists in the first place. My question is, are there any examples were a value of limit can be derived although the limit does not exist? Pleas... |
I am referring to the Polyhedron Volume [formula here][1]:
![alt text][2]
My question is, does the above formula applicable for the Polyhedron below?
<img src="https://docs.google.com/drawings/pub?id=155vFKEaDL3ZL8sy-zhil60zlL9wA07sJBOuZRHVvrKQ&w=960&h=720">
The given faces for the polyhedron are:
... |
Does Green Theorem Polyhedron Volume Calculation Applicable for a Cut Through Polygon? |
The proof is not at all obvious if you begin simply with the formula
$$\Delta(q) = q \prod_{n=1}^{\infty} (1-q^n)^{24}.$$ However, as Derek Jennings explains in his answer, if you use the (absolutely crucial!) fact that $\Delta$ is a cusp form of weight twelve and level one, the proof is actually not very difficult.
... |
The simplest idea is to generate a sequence of points $(X_i,Y_i)$, where $X_i,Y_i$ are independent uniform random variables, in a rectangle containing the polygon (the minimal rectangle bounding the polygon). Then, you simply drop the points which fall outside the polygon. You continue until you get $n$ unrejected poin... |
(This should be a comment under Carl's [answer](http://math.stackexchange.com/questions/15734/request-for-example-of-a-value-of-limit-of-a-sequence-where-sequence-does-not-con/15735#15735).)
@Arjan: One very general way in which to formalize your idea of "possible definitions of convergence" is that of [Banach limit... |
I'm not going to prove something, this is just a question. From the first day which I went to University until now I had some root problems in some basic mathematical assumptions and concepts. Please help me to solve the most important of those problems.
**For example:** ***infinity*** !
What is infinity ? Is tha... |
My question is on the convergence of the riemann sum, when the value spaces are square-integrable random variables. The convergence does depend on the evaluation point we choose, why is the case. here is some background to make this clearer
suppose $f\colon \Re \mapsto \Re $ is some continous function on $[a,b]$, the ... |
After seeing [this][1] recent question asking how to calculate the following integral
$$ \int \frac{1 + x^2}{(1 - x^2) \sqrt{1 + x^4}} \, dx $$
and some of the comments that suggested that it was an elliptic integral, I tried reading a little bit on the Wikipedia article about [elliptic integrals][2].
It seems t... |
When is an elliptic integral expressible in terms of elementary functions? |
Tonelli's algorithm for square-roots can easily be generalized to compute d'th roots in finite fields, e.g. see Adleman; Manders; Miller: [On taking roots in finite fields](http://www.cs.cmu.edu/~glmiller/Publications/AMM77.pdf), and Bach; Shallit: [Algorithmic number theory, section 7.3.](http://books.google.com/book... |
I think there may be a simpler algorithm.
If I understand you correctly, you're interested in a BDD with $2n+1$ levels, the first and last containing the root and sink, respectively. Edges only go between adjacent levels, with the correct annotation.
Let's concentrate on the middle three levels $A,B,C$. Thus all ... |
I am interested in finding out about the current status of the planned book: The Triangle Book by John H. Conway and Steve Sigur. I understand that Steve Sigur died some time back. I got no reply from Prof. Conway; is there someone who knows the fate of this project. Is it abandoned or is it still under preparation? He... |
Let's consider the sequence space $E =\mathbb R^{\mathbb N}$. If I believe in Choice, I have an isomorphism $E \simeq \mathbb R^{(\mathfrak c)}$ for some cardinal $\mathfrak c$. I further have some inequalities about $\mathfrak c$: first, $\mathfrak c = \dim \mathbb R^{\mathbb N} \leq |\mathbb R^{\mathbb N}| \leq (2^{\... |
The proof is not at all obvious if you begin simply with the formula
$$\Delta(q) = q \prod_{n=1}^{\infty} (1-q^n)^{24}.$$ However, as Derek Jennings explains in his answer, if you use the (absolutely crucial!) fact that $\Delta$ is a cusp form of weight twelve and level one, the proof is actually not very difficult.
... |
The proof is not at all obvious if you begin simply with the formula
$$\Delta(q) = q \prod_{n=1}^{\infty} (1-q^n)^{24}.$$ However, as Derek Jennings explains in his answer, if you use the (absolutely crucial!) fact that $\Delta$ is a cusp form of weight twelve and level one, the proof is actually not very difficult.
... |
The proof is not at all obvious if you begin simply with the formula
$$\Delta(q) = q \prod_{n=1}^{\infty} (1-q^n)^{24}.$$ However, as Derek Jennings explains in his answer, if you use the (absolutely crucial!) fact that $\Delta$ is a cusp form of weight twelve and level one, the proof is actually not very difficult.
... |
Limits of $R_n$ and $L_n$ coincide when Stieltjes integral exists. Existence of the Stieltjes integral does not follow from the existence of these limits. In general existence and definition of Stieltjes integral can be messy business as the Figure 2.1 in page 6 (page 10 of the ps file) of [this document can attest][1]... |
A simple answer that IMO is easy to justify using your definition of orientation goes like this.
Given *any* manifold $M$ and a point $p \in M$ there is a homomorphism $O : \pi_1(M,p) \to \mathbb Z_2$ and the idea is this: if $\phi : [0,1] \to M$ is a path such that $\phi(0)=\phi(1)=p$, given any basis for the tang... |
From Rudin's *Principles of Mathematical Analysis* (Chapter 2, Exercise 6)
Let $E'$ be the set of all limit points of a set $E$. Prove that $E'$ is closed.
I think I got it but my argument is a bit hand wavy:
If $x$ is a limit point of $E'$, then every neighborhood of $x$ contains some $y\in E'$, and every nei... |
From my understanding :
Since $$n^{log_3 2+0.4} = n^0.797$$ .(taking $\epsilon = 0.4$ in this case)
being smaller then $n \log(n)$ it qualifies for the case 3 .
Hence we check if ,
$$2\frac{n}{3} \log{n/3} \leq cn \log(n)$$
=$$\frac{2}{3} \log{n/3} \leq c \log(n)$$
=$$ \frac{n^\frac{2c}{3}}{3} \leq \log(n)$$
... |
How can I solve the recurrence
$$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$
|
how to solve $T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}$ ? |
How can I solve the recurrence
$$T (n) = T \left(\frac{n}{2} +\sqrt{n}\right) +\sqrt{6046}\ ?$$
Please don't just write the name of the method, as I just started learning this stuff and things are a little hard for me to figure out on my own.
Thanks in advance.
|
I have seen two proofs of the simplicity of $A_n,~ n \geq 5$ (Dummit & Foote, Hungerford). But, neither of them are such that they 'stick' to the head (atleast my head). In a sense, I still do not have the feeling that I know why they are simple and why should it be 5 and not any other number (perhaps this is only beca... |
To use Feller's original notation. Let $E_m(r,n)$ be the number of distributions leaving exactly $m$ cells empty, then we have $$p_m(r,n)=n^{-r}\binom{n}{k}A(r,n-m)$$
as the corresponding probability ($A$ is to be defined)
Now, we need compute the probability $x_m(r,n)$ of finding $m$ or more cells empty.
Now, ... |
I am aware that compactness fails on finite models, but the common counter-example uses models of arbitrary big finite size. So if we bound the size what results can we get?
Assume we have an infinite set of sentences $\Sigma$, of a language of first order logic $\mathcal{L}$. If there is a natural number $n$ such ... |
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