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In many elementary (and not-so-elementary) Euclidean geometry texts, a (simple) polygon is said to be tangential if it is convex and has an inscribed circle (i.e., a circle that intersects and is tangent to each side of the polygon). The assumption of convexity is not needed: I've come up with a rather laborious proo... |
Why is saying $\frac{dy}{dx}\frac{dy}{dx}=y\frac{d^2y}{(dx)^2 }$ not valid? Does Leibniz notation (and thinking of it as an infinitesimal quotient) not work for higher-order derivatives?
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It is not quite intuitive , at least from its origin. Could any one can give me an intuitive explanation?Thank you!
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Say we have a undirected, connected graph that represents a network of streets. How would you prove that there always exists a tour of the network, where one ``drives'' in both directions on every street exactly once?
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From an olympic book:
Let $m$ be an odd number and $a_1, ... , a_m$ integers. $x=(x_1,...,x_m)$ is a permutation of the integers $1, ..., m$ and $f$ at $x$ is defined by $f(x)=x_1a_1 + ... + x_m a_m$. Prove that there exist two permutations $x$ and $y$ such that $f(x)-f(y)$ is divisible by $m!$.
I'm aware of a few fact... |
The following seems to hold in numerical simulations, is it true?
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\sqrt{x(1-x)}}=2$$
It's a combination of two known integrals
$$\lim_{n\to \infty} \int_0^1 dx \frac{n! 2^{-n} n}{(n x)!(n-n x)!\}=1$$
$$\int_0^1 dx \frac{1}{\sqrt{x(1-x)}}=\pi$$
First one ... |
Sometime back i watched the documentary of Andrew Wiles proving the Fermat's Last theorem. A truly inspiring video and i still watch it whenever i am in a depressed mood. There are certain things(infact many) which i couldn't follow and i would like it to be explained here.
The first is:
The Taniyama-Shimura conjectur... |
Find the Cartesian equation for the perpendicular bisector of the line joining A(2,3) and B(0,6)
How do I do this?
Thank you!
|
According to my textbook, the second derivative of
\begin{equation*}
y^{2}+xy-x^{2}=9
\end{equation*}
is
\begin{equation*}
\frac{90}{(2y+x)^{3}}.
\end{equation*}
The problem states "Express $\frac{d^{2}y}{dx^{2}}$ in terms of $x$ and $y$." I've tried for two days straight now, and I can't get that answer. I am convi... |
How to "show", that if $x=1$, then its powers will end with $1$?
Or, if $x=5$, how do we show it's powers end with $5$?
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Show that BE is the perpendicular bisector to AC.
I tried to prove this through Pythagoras, but the answer I got did not prove it was at a right angle, and therefore said it was not the perpendicular bisector. I'm probably making a silly mistake, so any advice would be helpful!
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I was reviewing my class notes and found the following:
"The name 'torsion' comes from topology and refers to spaces that are twisted, ex. Möbius band"
In our notes we used the following definition for torsion element and torsion module:
An element m of an R-module M is called a torsion element if $rm=0$ for some $r\in... |
The perpendicular bisector of the line
joining $A(0,1)$ and $C(-4,7)$ intersects
the $x$-axis at $B$ and the $y$-axis at $D$.
Find the area of the quadrilateral.
Thank you in advance!
|
I have to show that any line parallel to $Ax+ By + c =0$ is of the form $Ax + By + k =0$
How do I show this?
Thank you!
|
The Questions
$70\%$ of all vehicles pass inspection. Assuming vehicles pass or fail independently. What is the probability:
a) exactly one of the next $3$ vehicles passes
b) at most $1$ of the next $3$ vehicles passes
The answer to a) is $.189.$ The way I calculated it was:
$P(\text{success}) \cdot P(\text{fail}) ... |
In the area of convex analysis and the area of optimization in their general sense, are convex subsets assumed to be in vector spaces or topological vector spaces? Are convex functions defined to be from convex subsets in vector spaces or topological vector spaces to general field or some special field such as $\mathbb... |
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals $\int e^{ax}\cos{bx}\ dx$ and $\int e^{ax}\sin{bx}\ dx$ by taking the real and imaginary parts of the "complex" integral $\... |
Suppose $A$ is a commutative algebra over $\mathbb{R}$ with unity. $\mathbb{R}$-linear map $\xi\colon A\to A$ is a derivation of $A$ iff $\xi(ab)=a\xi(b)+\xi(a)b$ for any $a,b\in A$.
If $\gamma\colon \mathbb{R}\to A$, $a\in A$, then we say, that $a=\frac{\partial}{\partial t}| _ {t=\tau} \gamma(t)$ iff $h(a) =\frac{\pa... |
We know this is true for commutative ring, but if $S\subset R$ is a left and right Ore set, and $S^{-1}R$ its localization by this Ore set, is this always a flat $R$-module?
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For context I am developing a ray-tracer for a computer science class, and want to implement some more advanced shapes than just spheres. So while this is related to schoolwork, I'm not asking you to do my work for me, the work is implementing the programming, and it's the math I don't understand, so I'm just looking ... |
My (maybe wrong) impression is that while probability is widely used in science (for example, in statistical mechanics), it is rarely seen in pure mathematics. Which leads me to the question -
Are there some interesting application of Probability Theory in pure mathematics, outside Probability Theory itself?
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I have a problem, which has $n$ variables. Let's call these variables as $a_1$, $a_2$, $a_3$ ... $a_n$. Each of these variables have their own range (i.e, $a_{i\min} \leq a_i \leq a_{i\max}$ for all $i$). And most importantly, each $a_n$ is ranked in terms of importance. Without a loss of generality one can assume tha... |
This question may seem simple, but I have no idea where to start.
I'm in design phase of a hobby electronics project and need your help. Basically, I'm going to have two platforms with electronics on both of them. I'm planning to have a stepper motor on the bottom platform which will be connected to top platform by an... |
The birational change of variables $(u,v) = (\frac{36+y}{6x},\frac{36-y}{6x})$ maps $u^3+v^3=1$ to $y^2 = x^3 - 432$ which has discriminant $-2^{12}\cdot 3^9$.
Using pari/gp we can compute the torsion subgroup:
? elltors(ellinit([0,0,0,0,-432]))
%1 = [3, [3], [[12, 36]]]
This says the torsion subgroup has order 3, is ... |
Let $I$ be a finite nonempty set, and $D$ a nonvoid compact subset of an N-dimensional Euclidean space. The functions $V:D\rightarrow\mathbb{R}$ and $u_i:D\rightarrow\mathbb{R}$ are given. They are all continuous, and, moreover, each $u_i$ satisfies $u_i(\lambda p + (1-\lambda) q) = \lambda u_i( p) + (1-\lambda) u_i(q)... |
Possible Duplicate:
Square root of a function (in the sense of composition)
I'm interested in solving equations of the form $f(f(x))=g(x)$ for $x\in\mathbb{R}$ where $g(x)$ is a known function.
For example: if $g(x)=x$ then we $f(f(x))=x$ so one solution is $f(x)=x$, $\forall x\in\mathbb{R}$. However, another is
$f... |
In my daughter's class of $23$, three students and the teacher all share the same birthday. Of course, there are $365$ days in the year, and the first case of the shared birthday is not counted in the probability. But is the likelihood of this just $$\frac{1}{365}\times 3 = 0.0082?$$ For my curiosity, leap years can be... |
Seems straight-forward but I've been unable to get it right. Here are my steps:
$$y'(x) = \sqrt{-2y(x) + 28},\hspace{20 pt} y(-4)=-4$$
$$\int {1 \over \sqrt{28-2y} }\hspace{2 pt}\text{d}y = \int \text{d}x$$
$$-\sqrt{28-2y} = x + c$$
$$(28-2y) = (x+c)^2$$
$$y = -1/2x^2 - cx - c^2/2 + 14$$
$$c = {-2, 10}$$
$$\Rightarrow ... |
What does the following term mean?
category of geometrically reduced schemes of finite type over some field
(I know what a category and a field is but I cannot translate any of the middle bits)
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I'm sure this is a solved problem but I don't know what to search for:
I have a set of positive numbers ($256$ at the moment but it could get bigger) and I need to find another set of positive numbers such that for each number in the original set, there is a sub set of the new set that sums to within some error bound o... |
Suppose $\mathcal{F}$ is a famliy of analytic functions of the unit disc. Suppose also that
$( Re(f(z)) )^2 \ne ( Im(f(z)) ) $for all $|z|<1$ and all $f \in \mathcal{F}$.
It follows from the Fundamental Normality test that $\mathcal{F}$ is a normal family.
Is there a for elementary way of showing $\mathcal{F}$ ... |
I fell in love with mathematics a bit too late when I've already taken decisions regarding my future, career-wise. Now I would like to learn math on my own but I'm a bit confused as where to start. My knowledge of mathematics is comparable to that of a 15-16 year old highschool freshman. I would like to know how would ... |
Why are row reductions so useful in linear algebra? It is easy to get lost in the mechanical solving of equations. I know I can get a matrix into reduced row echelon form. But what are the outcomes of this? What can this mean?
|
Let $e(n)$ be the number of partitions of $n$ with even number of even parts and let $o(n)$ denote the number of partitions with odd number of even parts. In Enumerative Combinatorics 1, it is claimed that it is easy to see that $\sum_{n \geq 0} (e(n) - o(n)) x^n = \frac{1}{(1-x) \times (1+x^2) \times (1 - x^3 ) \times... |
I was reading about isomorphisms and homomorphisms on general structures, and first came across the definition of an injective homomorphism, or an embedding. This made me curious, is it possible for two structures $A$ and $B$ to be embeddable in each other, yet no isomorphism exists between them?
After some looking aro... |
Is there any formula for this series?
$$1 + \frac12 + \frac13 + \cdots + \frac 1 n .$$
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For the definition of moduli space of Riemann surface, it seems that it is not based on the definiton in terms of functor(representable...) but rather put a topology on the set of isormophism classes. Is there any difference between these? Or if I miss something? Is there any reference ? Thank you!
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According this problem/solution set from an MIT class (http://ocw.mit.edu/courses/mathematics/18-100c-analysis-i-spring-2006/exams/exam1_sol.pdf), the assertion:
"Every collection of disjoint intervals in R is countable."
is True, because "every interval contains a rational number", and the rationals are countable.
It ... |
I have heard that Legendre functions are important in number theory. Can any one tell me how?
The Legendre function of the first kind $P_s$ is defined by
\begin{eqnarray*}P_s(x) =& \frac{1}{2\pi}\int_{-\pi}^{\pi}\left(x+\sqrt{x^2-1}\cos\theta\right)^s d\theta
\newline
=&\frac{1}{\pi}\int_0^1\left(x+\sqrt{x^2-1}(2t-1)\r... |
The German mathematician Martin Eichler once stated that there were five fundamental operations of mathematics: addition, subtraction, multiplication, division, and modular forms. This was also mentioned in Simon Singh's book on Fermat's last theorem.
I don't know much about modular forms beyond the Wikipedia entry, an... |
My question was inspired by this stackexchange question. For the last 90 minutes I have been trying to prove this formula due to Gregorio Fontana:
$$H_n = \gamma + \log n + {1 \over 2n} - \sum_{k=2}^\infty { (k-1)! C_k \over n(n+1)\ldots(n+k-1)},
\qquad \textrm{ for } n=1,2,3,\ldots,$$
where $H_n = \sum\limits_{k=1}^n ... |
I managed to prove that if $\phi(0)$ and $\forall n(n\in\mathbb{N},\phi(n)\rightarrow\phi(n+1))$ then $\forall n(n\in\mathbb{N}\rightarrow\phi(n))$. The proof is based on the fact that $\{m\in\mathbb{N}:\phi(m)\}$ is an inductive set. Now I'd like to show (but I can't) that if you take a $k\in\mathbb{N}$, if $\phi(k)$ ... |
Ok this is for a HW but I'm not looking for a handout...just a hint to get me on the right track.
I have no idea where to begin proving this:
Show that if A is a symmetric matrix, then
$$x^TAy = y^TAx$$
|
For a continuous scalar field on a circle, there is a diameter of the circle such that the endpoints of the diameter have the same value. If you think of the scalar field as "temperature", then what this says is that there are points on opposite sides of the circle that are the same temperature: isothermic antipodes.
S... |
Given $f(x) = \frac{1}{2}x^TAx + b^Tx + \alpha $
where A is an nxn symmetric matrix, b is an n-dimensional vector, and alpha a scalar. Show that
$\bigtriangledown _{x}f(x) = Ax + b$
and
$H = \bigtriangledown ^{2}_{x}f(x) = A$
Is this simply a matter of taking a derivative with respect to X, how would you attack this... |
Is there any rectangle, that if you divide it into another rectangle (or any other quadrilateral), the relation between the two different edges is the same in both rectangles?
For example - Orginal rectangle:
Divided Rectangle:
Note that these illustrations aren't really exact, it's just to clear the point.
Thanks.
|
I know how to partition given $p(x)$ using a generating function as my textbook on discrete mathematics explains it in detail. However, I want to know if it is possible to restrict the source elements of the partition? So, given I want to find the partition of $x$ is it possible to do it using only elements from $S={y_... |
Given a set of vectors $\{x_1,\ldots,x_n\}$, how can I find a set of orthonormal vectors $\{y_1,\ldots,y_m\colon m\leq n\}$ spanning the same space spanned by $\{x_i\}$?
|
Just out of pure curiosity ...
Suppose I want to connect the two points $(0,0)$ and $(1,1)$ with the graph of some continuous and differentiable function
$$f : [0; 1] \to [0; 1]$$
and let $s$ be the arc length of that function in $[0; 1]$.
Of course, the function with minimum $s$ that satisfies the above conditions is... |
I studied applied math, so each course (except abstract algebra) was dedicated to solution of a similar problems. After those courses it seems that every branch of mathematics has a developed theory with a few unsolved problems. I believe that is wrong. I found out that there is no general approach to Diophantine equat... |
I think this is obvious and I am just missing something but:
Given some finitely presentable algebra, can we always reduce the relations to commutator relations?
And moreover, if yes, then is there a reasonable algorithm?
As I said, I feel like I should know this, so I apologize if it is obvious.
Thanks,
|
I know the limits in the categories of groups, abelian groups and topological spaces and was wondering about the same thing.
|
For the equation
$f(x,y) = x^2/y$
given that $x$ and $y$ are integers
What is the easiest/fastest way to find at least one tuple $(x_0,y_0)$ solution to $f(x,y)=4$ or any other number really. I know that $(2,1)$ would work, but is there a process for finding the solution to any integer that is equal to $f(x,y)$?
Is t... |
I'm learning about proof by contrapositive and by mathematical induction in a computer science class. I'm banging my head trying to solve this problem and would like some help:
Prove that the sum of the first $n$ even numbers is $n^2 + n$
(a) indirectly by assuming that the sum of the first $n$ odd numbers is $n^2$
... |
When studying group theory, I was interested to learn that a the quotient set $G/N$ of a group $G$ is again a group with $\pi\colon G\to G/N$ a homomorphism when $N$ is a normal subgroup of $G$.
Is there a more general set of criteria for this situation outside of group theory? As in, for any set $U$ and an equivalenc... |
I'm taking a course in Complex Analysis, and the teacher mentioned that if we do not restrict our attention to analytic functions, we would just be looking at functions from $\mathbb{R}^2$ to $\mathbb{R}^2$.
What I don't understand is why this is not true when we do restrict our attention to analytic functions. I und... |
So for the formula $\dfrac {1}{n}$, If you were to add up all $y$ values from $n=1$ to $n=∞$, wouldn't the sum
$$\sum_{n=1}^\infty \frac{1}{n}$$
approach a number because even though you are always adding, aren't you just adding smaller and smaller numbers? Wouldn't this mean that it approached a certain number?
|
If $\mathcal{F}_1 \subset \mathcal{F}_2 \subset \dotsb$ are sigma algebras, what is wrong with claiming that $\cup_i\mathcal{F}_i$ is a sigma algebra?
It seems closed under complement since for all $x$ in the union, $x$ has to belong to some $\mathcal{F}_i$, and so must its complement.
It seems closed under countable u... |
I was considering the group of automorphisms on a vector space $\mathbb{Q}^3$ with the binary operation of addition. Is there a general description of the elements of this group? At first, it seemed to be that it would consist of bijective linear transformations, but I'm not sure how informative that is. Is there a spe... |
I'm actually looking for more intuition behind what's going on than a simple answer. I understand that the Borel sets are the smallest sigma algebra that contain all of the open sets. I can see how open intervals and their unions are in this set, and how it makes sense to measure them. It's hard to see why you'd wan... |
Can anyone recommend good, free online articles or books to learn Discrete mathematics? When I google'd for them, I came across few resources..but don't know whether they are good to start learning with. It's better to know a tutorial or a book that get you started in Discrete Mathematics. Can readers of SE Mathematics... |
I have the Cartesian coordinates of the hypotenuse 'corners' in a right angle triangle. I also have the length of the sides of the triangle. What is the method of determining the coordinates of the third vertex where the opposite & adjacent sides meet.
Thanks, Kevin.
|
Here is a question that I can't find the answer to in my notes or my textbook.
Is there any relation between $\text{Sym}_n$ and $\text{Sym}_k$ where $k < n$? Is $\text{Sym}_k \subset \text{Sym}_n$? Since $k! | n!\text{ } \forall k < n $ they satisfy Lagrange's theorem, but of course that doesnt guarantee the existenc... |
This an Exercise from Apostol's analytic number theory. I have been struggling with this problem for quite some time. Looks elementary though.
The question is to prove that this quantity
$$\Biggl[\frac{8n+13}{25}\Biggr] - \Biggl[\frac{n-12 - \Bigl[\frac{n-17}{25}\Bigr]}{3}\Biggr]$$ is independent of $n$.
|
I have a series of inequalities:
$$y_{1min} \leq y_{1}(x) \leq y_{1max}$$
$$y_{2min} \leq y_{2}(x) \leq y_{2max}$$
$$..$$
$$y_{nmin} \leq y_{n}(x) \leq y_{nmax}$$
Note that $x\in\mathbb{R}$
The question is, is there a general method that allows me to find the range of $x$ that satisfies the above inequalities?
Things a... |
The CASIO fx-7400G calculator is very quick at doing $(1+1/1000000)^{1000000}$, giving an output of $2.718280469$.
Mathematica (v7) takes over a minute, running on an Intel Core2 Duo processor, and its output is this:
With $n=10000000$, my patience ran out. The calculator is quick again with an output of $2.718281693$... |
An affine transformation is a linear transformation followed by a translation. They are morphism between affine spaces.
A rigid transformation consists of a rotation and a translation. I was wondering what are the spaces between which the rigid transformations can become morphisms?
what are the spaces between which the... |
Goursat's Lemma characterizes the subgroups of direct products. Is there a similar characterization for the subgroups of semidirect products? What about if I'm only interested in the normal subgroups?
|
Let $C$ be the curve in $\mathbb{C}^2$ given by $f(x,y)=x^3+y^3-y=0$ and let g:$C \to \mathbb{C}$ be the projection $g(x,y)=x$. I want to find where this map $g$ is ramified, and do this by computing the sheaf of relative differentials. I have $(3y^2-1)dy=-3x^2dx$. Now, by the definition of relative differentials I tho... |
I am reading about active contours models. It is said that this can be broadly classified as either parametric active contour models and geometric active contour models. The first one are represented explicitly as parameterized curves in a Lagragian framework, while the geometric active contours are represented implici... |
I'd like to know the name of this kind of polynomial
$p(x)=x^n+a_{1}x^{n-1}+\ldots+a_{n-1}x+1$
where the $a_{i}\in\lbrace0,1\rbrace$.
Thanks.
|
So first off all, this WAS homework. Submitted it about 3 hours ago.
Let $A$ be an $m \times m$ matrix. then show that if rank$ A = m$, then $Ax = 0$ has a unique solution.
My roommate said this is wrong but here is how I solved it.
very simply, i stated that if the rank of A is m that means the number of leading one... |
I have all points where my curve pass through, but I need to get the coordinates of the control points to be able to draw the curve. How can I do to calculate this points?
|
I was reading this book "Rational points on Elliptic curves" by J.Silverman, and J.Tate, 2 prominent figures in Number theory and was very intrigued after reading the first couple of pages.
The connection between Algebra and Geometry is displayed in a professional manner in that book. One can generally ask this questio... |
So I have some circles that look kind of like this:
I'm given the radius of the circle with center point $A$ which is also the distance $AB$, the distance $AB$ between the two center points on the x axis (they share same $y$ values for the centers), and the distance $CD$ which is the height of the shape created by the... |
I hope you will excuse my vague title. Let's define
$$\varsigma \ \colon\ \mathbb N\to \mathbb Z\ \colon\ k\mapsto (-1)^{\lfloor \frac{k-1}{2}\rfloor}$$
and
$$S(m,n)=\displaystyle\sum_{k=m}^n k\cdot \varsigma(k)$$
Find every integer $n\ge 1$ such that $S(1,\lfloor n/2\rfloor)=S(\lfloor n/2\rfloor+1,n)$
Show that $-(n+... |
Do generalized hypergeometric functions
$${}_p F_q(a_1,\ldots,a_p; b_1, \ldots,b_q; z) $$
with $p = q+1$ always possess a singularity at $z=1$, independent of the their parameters $a_1,\ldots,a_p$ and $b_1,\ldots,b_q$ under the provision that all cases of finite polynomials are excluded? Can someone mention a reliabl... |
What are the benefits of using a conjugate linear inner product in a complex vector space vs a simple linear inner product? That is, why do we demand that $(y,x) = \overline{(x,y)}$ as opposed to $(y,x)=(x,y)$? Of course, this ensures that $(x,x)$ is real and thus makes an easy definition of norm, but is that necessa... |
Prompted by today's Minute Math question on the MAA site (http://amc.maa.org/mathclub/5-0,problems/T-problems/T-web,ia/2005web/tb05-12-ia.shtml), I started thinking about the probability that the sum of the numbers rolled on a set of $n$ dice is prime, particularly the asymptotics as $n\rightarrow\infty$. Heuristics s... |
A common way to define a group is as the group of structure-preserving transformations on some structured set. For example, the symmetric group on a set $X$ preserves no structure: or, in other words, it preserves only the structure of being a set. When $X$ is finite, what structure can the alternating group be said ... |
The torsion of a curve in $\mathbb{R}^3$ indicates how much it twists around. The curl of a vector field indicates how much the vector field twists around. Is there a relation between the curl of a vector field and the torsion of a curve through that vector field at a given point?
|
Fix a finite group $G$, and look at all its irreducible representations/$\mathbb{C}$. It is said in Serre's book that "there cannot be any $\mathbb{C}$-linear relation between the matrix coefficients of these irreducible representations (of course we fix a set of bases first), because of the orthogonality formulas". I ... |
How does one find a simple expression of the one below applying the binomial theorem:
$$\sum_{k=1}^n k \cdot 2^k{ n \choose k}$$
Edit:
$\frac{d}{dx}(x^k)=kx^{k-1}$
$(1 + x)^n = \sum_{k=0}^n { n \choose k}x^k = \sum_{k=0}^n { n \choose k}\frac{d}{dx}(x^k) = \sum_{k=0}^n { n \choose k}kx^{k-1} $
$n(1+x)^{n-1} = \sum_{k=1... |
Before I begin, note that this is significantly outside what I've studied, so if it's a load of crap just let me know.
Up till now, everything I've covered assumes the same domain for all variables.
I'm trying to define an arbitrary one-to-one relationship between elements in two different sets, and want each of the va... |
I apologise for the really basic question. This didn't really fit on any other StackExchange website so the Maths one was the closest one where I could ask.
Really Basic Question- What are $4$ sided shapes called again?
Like how triangles are $3$ sided shapes, octagons are $8$ sided shapes, ... What are the $4$ sided ... |
Given $X_i, Y_i \in SO_3, i=1..n, n>2$, find $A,B$ that satisfies $AX_iB=Y_i$.
For $n=1$, there are an infinite number of solutions, a trivial one is:
$B=I$
$A=Y_1X_1^T$
For $n=2$, there should be exactly one solution. One could do the following:
Compute $A$ from the equation where $i=1$: $A=Y_1B^{-1}X_1^{-1}$,
Sub... |
I have an array of "lines" each defined by 2 points. I am working with only the line segments lying between those points. I need to search lines that could continue one another (relative to some angle) and lie on the same line (with some offset)
I mean I had something like 3 lines
I solved some mathematical problem (f... |
The problem asks us to calculate:
$$ \sum_{i = 0}^{n}(-1)^i \binom{n}{i} \binom{n}{n-i}$$
The way I tried solving is:
The given sum is the coefficient of $x^n$ in $ (1+x)^n(1-x)^n $, which is $ (1 - x^2)^n $.
The coefficient of $x^n$ in $(1 -x^2)^n$ is $$ (-1)^{n/2} \binom{n}{n/2}. $$
Am I doing it right?
|
Assume that $k,n\in\mathbb{Z}^+$. Prove that the sum
\begin{equation*}
\dfrac{1}{k+1}+\dfrac{1}{k+2}+\dfrac{1}{k+3}+\ldots +\dfrac{1}{k+n-1}+\dfrac{1}{k+n}
\end{equation*}
is not an integer.
The case $k=0$ is proved in this question: "Is there an elementary proof that $\sum_{k=1}^{n}\frac{1}{k}$ is never an integer?"
|
Background:
I was studying Theorem 2.3.3 from Introduction to Graph Theory by W. B. West. The main idea of his proof is as follows:
T, resulting tree.
T*, spanning tree of minimum weight.
Let, $T \neq T^*$, then there are edges in $T^*$ that are not in $T.$
We first consider only $1$ edge. Let's name it $e.$ Hence, ... |
Call a finite group $A$ affine if it has a normal, self-centralizing, complemented, elementary abelian subgroup $V$. Such a group $A$ is a semi-direct product $G\ltimes V$ where $V$ is a vector space of dimension $n$ over $\mathbb{F}_p$ and $G$ is a group of matrices in $\operatorname{GL}(n,p)$. The elements of $A$ c... |
Suppose $\mu$ is a measure. Is there any difference in meaning between the notation
$$\int f(x)d\mu(x)$$
and the notation
$$\int f(x) \mu(dx)$$?
|
I have coordinates for 4 vertices/points that define a plane and the normal/perpendicular.
The plane has an arbitrary rotation applied to it.
How can I 'un-rotate'/translate the points so that the plane has rotation 0 on x,y,z ?
I've tried to get the plane rotation from the plane's normal:
rotationX = atan2(normal.z,no... |
I spent a little time looking around the Wikipedia and Wolfram articles on Penrose Tiling, the Domino Problem, Wang Tiles, etc., but I'm having a little trouble getting into them. A lot of these resources are fairly non-linear in that they often have circular dependencies of concepts that you need to know.
For example... |
When teaching abstract vector spaces for the first time, it is handy to have some really weird examples at hand, or even some really weird non-examples that may illustrate the concept. For example, a physicist friend of mine uses "color space" as a (non) example, with two different bases given essentially {red, green, ... |
For my homework, I have been asked to rationalise and simplify this surd;
$$\frac{11}{3\sqrt{3}+7}$$
Each time I do this I get the wrong answer. The method I am using is;
$$ \frac{11}{3\sqrt3+7} \times \frac{3\sqrt3-7}{3\sqrt3-7} $$
I ended up with $$\frac{33+11\sqrt3-77}{9+3+21+7\sqrt3-21-7\sqrt3}$$
This ends up no wh... |
A famous exercise which one encounters while doing Complex Analysis (Residue theory) is to prove that the given integral:
$$\int\limits_0^\infty \frac{\sin x} x \,\mathrm dx = \frac \pi 2$$
Well, can anyone prove this without using Residue theory? I actually thought of using the series representation of $\sin x$:
$$\... |
Simplicial sequences generalize the familiar "linear", "triangular", and "tetrahedral" number sequences. (A line segment is a $1$-simplex, a triangle is a $2$-simplex, a tetrahedron is a $3$-simplex, and so on, so I'm calling these simplicial sequences.)
In general, the $n$-th $d$-simplicial number is the binomial coef... |
I'm trying to approximate $\pi$ using Monte Carlo integration; I am approximating the integral
$$\int\limits_0^1\!\frac{4}{1+x^2}\;\mathrm{d}x=\pi$$
This is working fine, and so is estimating the error (variance), $\sigma$. However, when I then try to use importance sampling with a Cauchy(0,1) distribution, things star... |
I have $2$ numbers $a, b$. I need a formula (or a how to) to find which $2$ numbers $c,d$ will add together to give a and times together to give $b$.
So
$c + d = a$
$c \cdot d = b$
|
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