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I'm a beginner to maths and have trouble simplyfying the following function:
$$\frac{p^y \cdot (pq)^o}{p^{2y+o} \cdot q^{o-2}}$$
The final answer is
$$p^{-y} \cdot q^2$$
But I'm not sure how to get there.
Any help is appreciated.
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Motivation: In S.J. Patterson's An introduction to the theory of the Riemann
Zeta-Function it is proved (p.132) that
$\displaystyle -\Gamma ^{\prime }(t)/\Gamma (t)=\gamma +t^{-1}+\underset{j\geq 1}{\sum }[(j+t)^{-1}-j^{-1}]$.
In Exercise A4.1 (p.135) one is asked to show that if $|t|<1$
$\displaystyle -\Gamma ^{\prime... |
In $b^a = x$, $b$ is the base, a is the exponent and $x$ is the result of the operation. But in its logarithm counterpart, $\log_{b}(x) = a$, $b$ is still the base, and $a$ is now the result. What is $x$ called here? The exponent?
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I suspect it is impossible to split a (any) 3d solid into two, such that each of the pieces is identical in shape (but not volume) to the original. How can I prove this?
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In the course of analyzing a certain Markov chain, I once had to prove the following algebraic identity.
Is there a slick or known proof?
For $n$-tuples $(x_1,x_2,\dots, x_n)$ of positive real numbers define
$$\mu(x_1,x_2,\dots, x_n)=\prod_{j=1}^n {x_j\over x_j+x_{j+1}+\cdots+x_n}.$$
Then if $x^\ast$ is another positi... |
Say $P$ is a convex Euclidean polytope, where the origin is not contained in any bounding hyperplane containing a facet of $P$, with $n$ facets given by $\langle f_i , x\rangle = 1$ and $m$ vertices $v_j$ with $1\leq i\leq n$ and $1\leq j\leq m$. (That is, each facet equation has been multiplied through as needed to ob... |
Suppose ${X_t}$ is a random walk with mean zero. (either discrete or continuous time)
Fix a time $T$.
What is: $P[X_t < 0 \text{ for all } t \leq T]$?
In words, what's the probability the random walk from $0$ to $T$ has never been above $0$?
Thank you.
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Note: logs below are base 2. (Not sure how to do subscripts here)
Wondering if the below equation is true when thinking asymptotically (Computer Science)
$log_2((n!)^n) = O(n \sin(n \frac{\pi}{2}) + \log_2{n})$
But I'm not sure how to compute this.
I'm guessing we need to take the log of both sides of the following eq... |
The two element group has a centre (of course it does, it's abelian) but no outer automorphisms. I can come up with a (convoluted) countably infinite non-abelian example. Is there a finite one? If so what is the smallest?
Is there a name for this sort of group? ("Outerless?" Since a group without a centre is "centrele... |
Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to have two lower bounds that are not comparable, and hence be unable to say whether one lower bound is really the greates... |
I was preparing for an area exam in analysis and came across a problem in the book Real Analysis by Haaser & Sullivan. From p.34 Q 2.4.3, If the field F is isomorphic to the subset S' of F', show that S' is a subfield of F'. I would appreciate any hints on how to solve this problem as I'm stuck, but that's not my actua... |
This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf E$ exists, and in particular, $L(E)=\langle\inf E]$.
I'm simply curious about showing that $L(E)\subseteq\langle\inf E]$... |
Could someone give me some tips?
Let $e_1,e_2,\dots$ be iid normal mean 0 variance 1. Let $X_t := e_1+\cdots+e_t$, for $t=1,2,\dots$ and $X_0 := 0$. (So we have a discrete-time random walk whose steps are iid $N(0,1)$)
Define first-hitting time $T_0 := \inf\{t>0 : X_t <0\}$.
What is the CDF of $T_0$, i.e. what is $\m... |
This problem is taken from Vojtěch Jarník International Mathematical Competition 2010, Category I, Problem 1. — edit by KennyTM
On going through this post Does there exist a bijective $f:\mathbb{N} \to \mathbb{N}$ such that $\sum f(n)/n^2$ converges? i happened to get the following 2 problems into my mind:
Let $f: \ma... |
I have a question which should be pretty basic commutative algebra, but I can't find a reference and I'm stuck on proving a result myself, so here it goes:
Let $\varphi: S \to R$ be a morphism of commutative rings and let $M$ be a finitely generated $S$-module. Then, we can "extend the scalars of $M$" by considering $M... |
Mathematica tells me that
$$\sum _{k=0}^n { n \choose k} \frac{(-1)^k}{2k+1} = \frac{(2n)!!}{(2n+1)!!}.$$
Although I have not been able to come up with a proof.
Proofs, hints, or references are all welcome.
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In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$.
One of which was that $\kappa$ is inaccessible and has the tree property (that is if any tree of cardinality $\kappa$ for which every level is o... |
I know the definition of prime number when dealing with integers, but I can't understand why the following definition also works:
A prime is a quantity $p$ such that whenever $p$ is a factor of some product $a\cdot b$, then either $p$ is a factor of $a$ or $p$ is a factor of $b$.
For example, take $4$ (which clearly ... |
I was wondering if a field is an algebra over itself (http://en.wikipedia.org/wiki/Algebra_over_a_field)?
Also is a ring an algebra over itself (http://en.wikipedia.org/wiki/Algebra_(ring_theory)? If not, does the ring require to be commutative?
Thanks and regards!
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I'm trying to express the integral
$$I = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(x_1, x_2) \; g(x_1, x_2) \; \mathrm{d}x_1 \mathrm{d}x_2$$
of two real valued functions $f(x_1,x_2)$ and $g(x_1,x_2)$ in terms of their Fourier transforms $\tilde{f}(\omega_1, \omega_2)$ and $\tilde{g}(\omega_1, \omega_2)$. Howe... |
Consider a situation where a person has partial knowledge, but we have a more complete picture. For example, suppose that we want to know the probability that a fish is red. Suppose that the person with partial information knows 1/3 of all fish are red, but we know that the particular species is actually red 2/3 of the... |
What is the area of the region bounded by $y = f_n(x)$ and the $x$-axis as $n$ gets large?
$f_0=1-|x|$
$f_n=1-|1-2f_{n-1}(x)|$
Assume the domain of $f$ to be the real numbers.
By trying various values for $n$ and $x$ I got that $f_n(x) = 2^{n}\cdot f_0(x)$
By integrating this I got that the area is given by $2^{n-1}\c... |
We can divide $7^{17} - 7^{15}$ by?
The answer is $6$, but how?
Thanks in advance.
|
How do I evaluate:
$$\lim_{\tau\rightarrow \infty} \int_{-c/\tau}^0 e^{-\beta x}(\tau x+c)^{(\lambda/\tau) -1}dx$$ to give $$1/\lambda$$
I suspect I need to convert to gamma function but cannot simplify it to get the answer.
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I'm currently getting confused about indices in some spectral sequences. Assume we work in the category of modules for simplicity. Let $A^\cdot$ be a (bounded on the right) complex and let $B^\cdot$ (I don't think we have to assume anything about the boundedness of $B$). I want to compute $Ext^n(A^\cdot,B^\cdot)$, whic... |
On a lark, I wanted to know if one can use Padé approximants to compute the exponential $\exp(z)$ of a quaternion $z=a+b\mathbf{i}+c\mathbf{j}+d\mathbf{k}$. Since Mathematica has a package meant for manipulating quaternions, as well as computing the exponential of a quaternion, I decided to try things out.
The numerato... |
Suppose I have a 4-index, real-valued tensor $\lambda_{ij}^{ab}$ where $i$ and $j$ are from some subspace $O$ of the whole numbers, and $a$ and $b$ are from some other subspace $V,$ which is disjunct from $O.$ Suppose also that we know that $\lambda$ is antisymmetric. If we impose the constraint that $\lambda$ is Her... |
Let $n \in \mathbb{N}$. How to find the least positive integer $\gamma$ such that $$ \sum\limits_{i=1}^{n} \cos{\theta_i} \leq \gamma$$ provided $$\prod\limits_{i=1}^{n} \tan{\theta_i} = 2^{\frac{n}{2}}$$ for any $\theta_i \in (0,\frac{\pi}{2})$, $i=1, 2, \ldots, n$?
I had trouble solving this problem, some time ago. I... |
Suppose $K$ is the splitting field of $x^3-2$ over $\mathbb{Q}$, and let $\mathcal{O}_K$ denote its ring of integers. I want to show that for any prime number $p$, $p\mathcal{O}_K$ is not a prime ideal of $\mathcal{O}_K$. The hint I was given was to consider the factorization of $x^3-2$ over $F_p$.
My question(s)
My ... |
I am trying to understand pairwise testing.
How many combinations of tests would be there for example, if
a can take values from 1 to m
b can take values from 1 to n
c can take values from 1 to p
a, b and c can take m, n and p distinct values respectively. What are the total number of pairwise combinations possible?
... |
Given a sequence of row sums $r_1, \ldots, r_m$ and column sums $c_1, \ldots, c_n$, all positive, I'd like to find a matrix $A_{m\times n}$ consistent with the given row and column sums that has the fewest nonzero entries. That is, we want to minimize $|\{a_{ij} \ne 0\}|$ under the constraint that $\sum_j a_{ij} = r_i$... |
For computing the present worth of an infinite sequence of equally spaced payments $(n^{2})$ I had the need to evaluate
$$\displaystyle\sum_{n=1}^{\infty}\frac{n^{2}}{x^{n}}=\dfrac{x(x+1)}{(x-1)^{3}}\qquad x>1.$$
The method I used was based on the geometric series $\displaystyle\sum_{n=1}^{\infty}x^{n}=\dfrac{x}{1-x}$ ... |
I have a 3D plane defined by three points: $P_0$ , $P_1$ and $P_2$. How to check whether a point $P$ is located right on and inside the 3D triangle?
So, for example, if I have a plane defined by $({0,0,0})$, $({10,0,0})$ and $({0,10,0})$, then the point $({50,0,0})$ is considered not located on the plane, whereas the p... |
If we have an algebraic number $\alpha$ with (complex) absolute value $1$, it does not follow that $\alpha$ is a root of unity (i.e., that $\alpha^n = 1$ for some $n$). For example, $(3/5 + 4/5 i)$ is not a root of unity.
But if we assume that $\alpha$ is an algebraic integer with absolute value $1$, does it follow th... |
I'm starting a very long quest to learn about math, so that I can program games. I'm mostly a corporate developer, and it's somewhat boring and non exciting. When I began my career, I chose it because I wanted to create games.
I'm told that Linear Algebra is the best place to start. Where should I go?
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If the matrix is positive definite, then all its eigenvalues are strictly positive.
Is the converse also true?
That is, if the eigenvalues are strictly positive, then matrix is positive definite?
Can you give example of $2 \times 2$ matrix with $2$ positive eigenvalues but is not positive definite?
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In this thread, the question was to find a $f: \mathbb{R} \to \mathbb{R}$ such that
$$f(f(x)) = f(x) + x$$
(which was revealed in the comments to be solved by $f(x) = \varphi x$ where $\varphi$ is the golden ratio $\frac{1+\sqrt{5}}{2}$).
Having read about iterated functions shortly before though, I came up with this ... |
I'm not sure if this a proper recurance relation per se but I'd be interested in the methodology in solving a recurrence relation of the following form:
$Z_0 = 1$
$Z_1 = x_1$
$Z_2 = x_1Z_1 + x_2 = x_1^2 + x_2$
$Z_3 = x_1Z_2 + x_2Z_1 + x_3 = x_1^3 + x_1x_2 + x_1x_2 + x_3$
$Z_n = x_1 Z_{n-1} + x_2 Z_{n-2} + ... + x_n Z_0... |
I'm having a hard time solving this problem. Would appreciate any hint! (and special thanks to Byron Schmuland for answering my previous 2 questions. This 3rd question is different.)
Let $e_t$: $e_1$,$e_2$,$\ldots$ be i.i.d. normal mean 0 and variance 1. Let $X_t := e_1+\ldots+e_t$, for $t=1,2,\ldots$ and $X_0 := 0$. ... |
I have a constant area $A$ , and I can mold that area into a regular $n$ -side polygon, where $n\geq 3$. The issue now is how to find the $n$ such that it can contain the most number of circles, each with a constant radius $R$ ?
Edit to clarify the question: the area must be a regular $n$ -side polygon. Sorry for no... |
I've been reading a bit about how the set of bounds changes for a set depending on what superset one works with. I considered the sets $S\subseteq T\subseteq\mathbb{Q}$ and worked out a few contrived examples:
If $S=T=$ {$x\in\mathbb{Q}\ | \ x^2\lt 2$}, so here $S$ is not bounded above in $T$, but it is bounded about i... |
Consider the following sequence $a_n$:
$a_1 = 0$
$a_n = 1 + \frac{1}{2^n-2} \sum_{i=1}^{n-1} \binom{n}{i} a_i$
The first few terms are $0,1,\frac{3}{2},\frac{13}{7},\frac{15}{7}$.
The sequence comes out of the analysis of a certain process, whose asymptotics we expect to be about $\log_2 n$. Numerical computations show... |
If $\varphi:U\subset \mathbb{R}^n \to \mathbb{R}^m$ is $C^1$, let $\mathrm{T}\varphi:\mathrm{T}U \to \mathrm{T}R^m$ be its tangent map. The inverse function theorem tells us that if $\ker(\mathrm{T}\varphi(x))$ is zero, $\varphi$ is injective in some neighborhood of $x$. If the kernel is non-zero, what can we say about... |
A number $n$ is perfect if it's equal to the sum of its divisors (smaller than itself). A well known theorem by Euler states that every even perfect number is of the form $2^{p-1}(2^p-1)$ where $2^p-1$ is prime (this is what is called a Mersenne prime).
How is this theorem proved? The converse (showing that every numbe... |
I am trying to compute the asymptotic growth-rate in a specific combinatorial problem depending on a parameter w, using the Transfer-Matrix method. This amounts to computing the largest eigenvalue of the corresponding matrix.
For small values of w, the corresponding matrix is small and I can use the so-called power met... |
If $h(x_{t\wedge \tau})$ is a submartingale such that $h(0)=1$ and $\tau$ is a stopping time. Why is it that $h(x_\tau)=1_{{\tau<\infty}}$?. Also $h(x)$ is decreasing.
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I would like to learn Graph Theory from the beginning. It seems to me that one does not need to be familiar with many abstract type subjects to be able to understand the more basic concepts of graphs.
Which subjects should one know prior to learn Graph Theory at the introductory level?
And which book or lecture notes ... |
If $f$ is a positive function, the intuitive interpretation of the Riemann integral
$\int_a^b f(x) dx$
is the area under the curve $f$ between $a$ and $b$.
Suppose $f$ and $g$ are smooth positive functions. Is it correct to interpret the Riemann-Stieltjes integral
$\int_a^b f(x) d g(x)$ as the volume under a "ribbon", ... |
If you look at Wolfram Alpha
or this Wikipedia page List of countries by median age
Clearly median seems to be the statistic of choice when it comes to ages.
I am not able to explain to myself why arithmetic mean would be a worse statistic. Why is it so?
|
I was doing a physics problem today and I needed to find when some movement is accelerating and when it is decelerating. So after some calculations, I got the equation I needed. I know that it is correct because when plotted, I can see from graph that I got the expected solution.
Unfortunately I don't know how to actu... |
I'm a noob, and I'm not a mathematician (Although I will be a Math major next semester). My question is:
I have 68 maps I would like to use as posters on my wall at home. They are all rectangles, and all different dimensions. Is there an algorithm (or better yet, a website or other simple computer program I can use)... |
The following probability question appeared in an earlier thread:
I have two children. One is a boy born on a Tuesday. What is the probability I have two boys?
The claim was that it is not actually a mathematical problem and it is only a language problem.
If one wanted to restate this problem formally the obvious wa... |
I've been struggling with the following problem I found for a while now:
Suppose $(T,\preceq)$ is a partially ordered subset of $(U,\preceq)$ and $S\subseteq T$. If $\inf_T S$ and $u=\inf_U S$ both exist, and there exists a subset $S'\subseteq T$ such that $u=\sup_U S'$, then $\inf_U S=\inf_T S$.
First I noted that $\... |
What is the precise difference or relation between these terms in logic:
Inference, Reasoning, Deduction, and Induction?
|
We know that if $f : \mathbb{R} \to \mathbb{R}$ is a continuous function, then $f$ carries connected sets to connected sets and compact sets to compact sets. That is if $A \subset \mathbb{R}$ is connected then $f(A)$ is connected, and if $A$ is compact then $f(A)$ is compact.
Question: Suppose $f: \mathbb{R} \to \mathb... |
In a proof or deduction system, there are some axioms and some inference rules. To have a small proof system, axioms are defined for the primitives (for example for negation and implication in propositional logic). Now, we want to have a proof system that proves all provable formulae including the formulae contain non-... |
I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the point. Further, I am not terribly interested in applications and would rather be exposed to how Fourier Analysis fits into t... |
In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if
(1) Given $\epsilon > 0$ there is a $\delta > 0$ such that $\sum_{i=1}^n |f(y_i)-f(x_i)|< \epsilon$ for every finite collect... |
What's the difference between a bar chart and a histogram?
|
Question
Do you know any application or in which field this trajectories can be applied?
Comment
If you know any book or any site that can show me the applications, please let me know. Thanks for your time
|
Problem taken from the link: http://web.mit.edu/rwbarton/Public/func-eq.pdf I am stating the question here
For which $\alpha$ does there exists a nonconstant function $f: \mathbb{R} \to \mathbb{R}$ such that
\begin{equation*}
f(\alpha(x+y))=f(x)+f(y)~\forall x,y \in \mathbb{R}.
\end{equation*}
Clearly for $\alpha=1$ ... |
Doing a little practice exercise, and came across the following question:
There are three functions: $f(x)$, $g(x)$, and $h(x)$.
Find $f(x)$ and $g(x)$, such that $h(x) = f \circ g(x)$ and $g(x) = \sqrt{4} + 8$.
The value of $h(x)$ is defined as $h(x) = (\sqrt{4} + 8) ^ 4$.
I'm not really sure how to approach the pro... |
To prove the Cayley-Hamilton theorem in linear algebra, my professor said that a lemma was necessary:
Lemma: Let $A \in M_n(\mathbb{K})$ be an $n\times n$ matrix over a field $\mathbb{K}$, let $b(t) \in M_n(\mathbb{K})[t]$ and $P(t) = b(t)[A-tI]$, then $P(A) = 0$
The theorem (which says that if $f$ is an endomorphism... |
I've been reading Ravi Vakil's notes on algebraic geometry notes and am having a hard time connecting the standard definition of the localization of a module to the definition in terms of universal properties (page 24).
Can anybody either explain the connection or provide some kind of 'roadmap' of the material necessa... |
From any textbook on fourier analysis:
"It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = \int_{-\pi}^{\pi}f(z)g(x-z)\;dz = (g \ast f)(x),$$ by using the substitution $z = x-y.$"
I don't doubt that this is true, but I ... |
This is a simple terminology question. Let $S$ be a (let's say smooth)
surface in $\mathbb{R}^3$, and $p$ a point
on $S$. Suppose the principle curvatures $\kappa_1$ and $\kappa_2$ at $p$ are both negative.
I am imagining $p$ sitting at the bottom of a dent in the surface.
Is there an accepted term to describe such a... |
I fear I over simplified the following problem:
For any partially ordered set $(A,\leq)$, let $A^* = A- \{\max A,\min A\}$ if $\max A$ and $\min A$ exist. Show the inclusion $(A^*,\leq)\hookrightarrow (A,\leq)$ is $\sup$-continuous.
So I took any set $B\subseteq A^*$ such that $\sup B$ exists. Then $\iota(B)=B$, and $\... |
My goal here is to make my own custom "polyhedral map" of Earth. If you print out something from the "Map Fold-outs" page, you will have something almost exactly like what I'm trying to make.
I have 2 triangles. One is a spherical triangle drawn on a 3D globe. By definition, each edge of a spherical triangle is part of... |
This page contains a result which it refers to as the UFD field lemma. I was wondering if anybody knew of any other references which discuss this result--this page is the only place I've seen it.
The UFD field lemma appears to assert that if $R$ is a unique factorization domain containing infinitely many prime elements... |
I'm working my way through Niven's Introduction to Number Theory, and the wording of the following problem is making me unsure of my answer:
Show that there is a one-to-one correspondence between twin primes and numbers $n$ such that $n^2-1$ has just four positive divisors.
I felt an obvious bijection would be $f\colo... |
It is well known that $\sqrt{2}$ is irrational, and by modifying the proof (replacing even with divisible by $3$), one can prove that $\sqrt{3}$ is irrational, as well. On the other hand, clearly $\sqrt{n^2} = n$ for any positive integer $n$. It seems that any positive integer has a square root that is either an inte... |
In Wikipedia it is stated that the volume of the parallelepiped given its edge lengths $a,b,c$, and the internal angles between the edges $\alpha ,\beta ,\gamma $ is:
\begin{equation*}
V=abc\sqrt{1+2\cos \alpha \cos \beta \cos \gamma -\cos ^{2}\alpha
- \cos^{2}\beta - \cos ^{2}\gamma }\qquad(*).
\end{equation*}
I was... |
The Fourier transform can be defined on $L^1(\mathbb{R}^n) \cap L^2(\mathbb{R}^n)$, and we can extend this to $X:=L^2(\mathbb{R}^n)$ by a density argument.
Now, by Plancherel we know that $\|\widehat{f}\|_{L^2(\mathbb{R}^n)} = \|f\|_{L^2(\mathbb{R}^n)}$, so the Fourier transform is an isometry on this space.
My questio... |
Let $\mathcal{M}$ be an n-dimensional manifold endowed with an affine connection $\nabla$. Let $\gamma_1:[a,b]\rightarrow M$ and $\gamma_2:[c,d]\rightarrow \mathcal{M}$ be two curves with the same initial and final points, that is,
$p=\gamma_1(a)=\gamma_2(c), q=\gamma_1(b)=\gamma_2(d)$. Take $X\in T_p\mathcal{M}$. Par... |
Find the radius of the circles if the size of the larger square is 1x1.
Enjoy!
(read about the origin of sangaku)
|
Any linear map between two finite-dimensional vector spaces can be represented as a matrix under the bases of the two spaces.
But if one or all of the vector spaces is infinite dimensional, is the linear map still represented as a matrix under their bases?
If there is matrix of infinite dimension, what is it used for ... |
An ordinary eigenvector can be viewed as a vector on which the operator acts by only stretching (without rotating) it.
Is there a similar intuition behind generalized eigenvectors?
EDIT: By generalized eigenvectors I'm referring to vectors in the kernel of $(T-\lambda I)^k$ for some $k$.
|
What is a conditional double limit and how to compute it for
\begin{equation*}
\lim_{(x,y)\rightarrow\infty,x\leq y}\frac{2x-1}{x-1}+\frac{x}{y}?
\end{equation*}
|
I take it there isn't a classification of finite complete groups yet. Has someone put together a classification of small complete groups? I.e. $S_4$, $\text{Aut}(G)$ for $G$ simple, $\text{Hol}(C_p)$ for $p$ odd prime are complete. The smallest complete group not of one of these forms is $H$ of order $\left|H\right|$, ... |
In a inner product space with inner product $\langle\ ,\ \rangle$ and real or complex line as its base field, for each point $x$ in the space, is $\langle x,-\rangle$ continuous function on the second argument, and is $\langle - ,x\rangle$ continuous function on the first argument? "Continuous" is defined respect to th... |
We, know that there are infinitely many primes of the form $4n-1,4n+1,5n-1,\cdots, \text{etc}$. I saw these things in Apostol's Introduction to Analytic Number theory textbook. I would like to have an argument working for $n^{2}$. The first expression which came to my mind was $4n^{2}+3$, which gives a prime for $n=1,2... |
In many combinatorial enumeration problems it is possible to find a rational generating function (i.e. the quotient of two polynomials) for the sequence in question. The question is - given the generating function, how can we find (algorithmically) the values of the sequence, i.e. the coefficients of the corresponding ... |
I found the following in the wikipedia page for Euler class.
«If the rank $r$ is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$.
Under the splitting principle, this corresponds to the square of the Vandermonde polynomial equaling the discriminant: the Euler class correspo... |
I came across the following theorem that for any class $K$ of truth assignments, $K\subseteq \operatorname{Mod}(\operatorname{Th}(K))$, and if $K$ is axiomatizable, then $K=\operatorname{Mod}(\operatorname{Th}(K))$. Together these seem to me to imply that there is some class $K$ such that $K\neq \operatorname{Mod}(\ope... |
I've been using the sentence:
If a series converges then the limit of the sequence is zero
as a criterion to prove that a series diverges (when $\lim \neq 0$) and I can understand the rationale behind it, but I can't find a formal proof.
Can you help me?
|
Let $X$ be a contractible space. If $x_0 \in X$, it is not necessarily true that the pointed space $(X,x_0)$ is contractible (i.e., it is possible that any contracting homotopy will move $x_0$). An example is given in 1.4 of Spanier: the comb space. However, this space is contractible as a pointed space if the basepo... |
$$\arccos\left(\frac{1-x^2}{1+x^2}\right) = 2\arctan{x}$$ for $x \geq 0$.
I'm not even sure what kind of math to try? :(
|
I would like to collect a list of mathematical concepts that have been named after mathematicians, which are now used in lowercase form (such as "abelian"). This question is partly motivated by my former supervisor, who joked (something like):
You know you've made it as a mathematician when they start using your name... |
I've got two questions:
(1). I would like to know if by definition the Sobolev space norm is defined only by differentials with respect to one variable i.e.
$\parallel f \parallel \; = \; \left[ \int_{\mathbb{R}} \left( \left| f \right|^p \; + \; \left| \frac{\partial f}{\partial x} \right|^p \; + \ldots + \; \left| \f... |
I'm looking at the random sequence $x_n$ with $x_0=x_1=1$ and
\begin{equation}
x_{n+1}=2x_n\pm x_{n-1}
\end{equation}
where we choose the $\pm$ sign independently with equal probability. Now considering the recurrence relations $x_{n+1}=2x_n+x_{n-1}$ and $x_{n+1}=2x_n-x_{n-1}$ separately, it is clear that $x_n\rightar... |
In India, generally during the graduate years, we follow a course work pattern, unlike many places in the U.S where students are exposed to research during their undergraduate years itself.
As, a graduate student, we have basic courses like, Algebra, Analysis, Rings and Modules, Measure theory, Topology, functional an... |
The fact that Ramanujan's Constant $e^{\pi \sqrt{163}}$ is almost an integer ($262 537 412 640 768 743.99999999999925...$) doesn't seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it's almost-but-not-quite an integer in layman's terms (I'm not a mathematician)?
|
If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
|
Let $c$ be a positive integer that is not prime. Show that there is some positive integer $b$ such that $b \mid c$ and $b \leq \sqrt{c}$.
I know this can be proved by contradiction, but I'm not sure how to approach it. Usually I write the proof in the form $P \rightarrow Q$, and then if we can prove $P \land \neg Q$ ... |
If $A, B$ are two non empty sets of real numbers and for every $a$ from $A$ and $b$ from $B$
$a < b$,
how can I prove that
$\sup A \leq \inf B$?
|
In the question How to prove: if $a,b \in \mathbb N$, then $a^{1/b}$ is an integer or an irrational number?
there was an answer (revision 0) by Douglas S Stones which said in complete:
"
If $\sqrt{a}=x/y$ where $y$ does not divide $x$, then $a=(\sqrt{a})^2=x^2/y^2$ is not an integer (since $y^2$ does not divide $x^2$... |
The problem: Let $X$ be a product of equidimensional spheres of arbitrary dimension, say $k$, and $G$ a finite group acting freely on $X$. Assume that the induced $G$-action on the $Z_2$-cohomology ring of $X$ is trivial. We then have a Serre spectral sequence with the $E_2$-page given by the expression $H^*(G; H^*(X; ... |
The following is an example given by my professor, but there is an equality that I don't understand.
Let a partially ordered set $(S,\preceq)$ is the union of three sets such that $S=X\cup Y\cup Z$ such that $x\preceq y$ and $x\preceq z$ for all $x\in X, y\in Y, z\in Z$, but no $y\in Y$ and no $z\in Z$ are comparable.... |
The following is a problem in an algebra textbook, probably a well-known fact, but I just don't know how to Google it.
Let $K/k$ be a finite field extension. Then $K(X)/k(X)$ is also finite with the same degree as $K/k$.
Obviously if $v_1,...,v_n$ is a $k$-Basis of $K$, any polynomial in $K(X)$ can be written as a ... |
I am reviewing Calc $2$ material and I came across a problem which asked me to explain why $x^\pi$ does not have a Taylor Series expansion around $x=0$. To me it seems that it would have an expansion but it would just be $0$, so maybe it's not a suitable expansion. It doesn't have any holes and it is infinitely differ... |
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