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The function space $H^{\alpha} (\Omega)$ for $0 < \alpha \le 1$, is the set of functions:
$$\{ f \in C^0(\Omega) : \sup_{x \neq y} \dfrac{|f(x) - f(y)|}{|x-y|^{\alpha}} < \infty \}$$
with the metric $d_{H^{\alpha}} = || f - g ||_{H^{\alpha}}$, where
$$||f||_{H^{\alpha}} = ||f||_{sup} + [f]_{H^{\alpha}} \text{ , ... |
I'm trying to solve
$$\lim\limits_{x \to \infty}\frac{\ln^{1000} x}{x^5}$$
Here's what I get:
$$e^{\lim\limits_{x \to \infty}\ln{\frac{\ln^{1000}x}{x^5} }}$$
Dropping the $e$ for ease,
$$\lim\limits_{x \to \infty} 1000\ln{(\ln{(x)})} - 5 \ln{x} $$
Now I have $\infty - \infty$.. I know there must be a next step, but I d... |
I would like so find the function $n(z)$ that solves the following equation:
$$
n(z) = \frac{.2 + .24 z \int_1^{\infty} \frac{n(z)}{z^2}dz}{.5 + .24 z \int_1^{\infty} \frac{n(z)}{z^2}dz}
$$
That is, $n(z)$ is defined in terms of it's own integral. Even if there's no nice analytic solution to this, Numerical approxim... |
Given a sequence $a_n$, I know that if I can find a divergent subsequence of $a_n$, or two subsequences of $a_n$ that converge to different values, $a_n$ diverges, since, if I have understood correctly, a sequence $a_n$ converges to a limit $L$ if and only if every subsequence of $a_n$ converges to that value $L$.
I've... |
Suppose a statement form $\varphi$ always has value T or U. Show $\varphi$ is a classical tautology.
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Total differential for a function of two variables is known as
$df = f_x(x_0, y_0)dx + f_y(x_0,y_0)dy$
It's not clear how to derive this. Thomas' Calculus 11th Ed pg 1021 says to use the definition of linearization L(x,y). The definition of linearization for a function f(x) at page 223 is only for one variable
$f(x) \... |
Definition of closure from here
Boundary (Topology): The points in the
closure of a set which are not in the
interior of that set.
Definition of Exterior from here
Let $T$ be a topological space. Let $H \subseteq T$. The exterior of $H$ is the complement of the closure of $H$ in $T$. Alternatively, the exterior ... |
I need to show that every positive integer $n$ can be written uniquely in the form $n = ab$, where $a$ is square-free and $b$ is a square. Then I need to show that $b$ is then the largest square dividing $n$.
The problem I have here is that I can't even see how this is true. How can $1$ be represented this way? Is ... |
The wikipedia example of Polynomial Long Division starts with:
Divide the first term of the numerator
by the highest term of the denominator
if the denominator is $x-3$, and we don't know $x$, how to know which is highest? $x$ or $3$?
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So I've just got my cheese from the cheesemonger and he's cut it along the axis $x$,$y$, and $z$ so I can do my math homework with it. The lengths of my cheese are:
$x$=6
$y$=4
$z$=2
I'm asked to find the volume using calculus, because using geometry would be cheating, evidently.
The three ways I am asked to "cut up... |
I am trying to prove that morphisms of finite type are stable under base change, but I am having some trouble moving from the case where everything is affine to the general case. Suppose $f:X \rightarrow Y$ is a morphism of finite type and $Y'$ is a $Y$-scheme. I want to show that the morphism $g: X \times_Y Y' \righ... |
I'm working on a homework assignment in which part of the question statement says that each of the classical Lie algebras can be described as the set of all matrices $X \in gl(n,\mathbb{C})$ satisfying
$$XM + MX^T = 0$$
for some particular choice of $M$. Specifically, it says that if $M$ is the $n\times n$ unit matrix ... |
Say our question requires us to use traces to sketch the quadric surface with equation $x^2+y^2/9+z^2/4=1$. This is a worked example in our book, but I'm confused by their steps. So first we set $z=k$ to get $x^2+y^2/9=1-k^2/4$. First of all, is $k$ just a constant? And what is the difference between setting $z=k$ rath... |
I have two random variables $X$ and $Y$. Both are distributed according to $N(0,1)$. If their covariance is 0, are they independent?
I know that this is not true for other distributions, say the Wikipedia example: $X$ chosen uniformly in $[-1,1]$ and $Y=X^2$.
|
Suppose $M$ is a n-dimensional (compact) manifold and $f$ is a differentiable function with exactly three (non-degenerate) critical points. Then one can show, using Morse theory, that $M$ is homeomorphic to a $\frac{n}{2}$ sphere with an $n$ cell attached.
I understand why we will have critical points with index $0$ an... |
If you take a look at WolframAlpha, or other computer algebraic system, you will find that it is able to do symbolic manipulation like real humans.
For example, if you type in an integral, it can show you step by step on how to solve the integration.
What are the algorithms behind all this?
|
Let $A \in \mathbb{R}^{n \times n}$ be a symmetric row-stochastic matrix (that is, for each row, the sum over the columns is $1$) with all elements positive.
Let $B \in \mathbb{R}^{n \times m}$ be a matrix with all elements positive and the sum over columns is $1$.
What can we say about the product $A \times B$? What i... |
I know $\arcsin(\sqrt{2}/2)$ is equal to $\pi/4$. However I don't understand why, I've done some searching on google about arc trig functions and I haven't found any webpages that explain it very well. Can any of you help?
Thanks!
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So our calc teacher is being tricky and sending us off to fend for ourselves in the world of mathematics.
Here's the question:
We need to set up an integral to find the volume of the solid formed by the area bounded by the functions y=ln(x), x=3, and the x-axis revolved around the line x=3.
The problem?
Most of us put ... |
What is the difference between a topological and a metric space?
|
What is the difference between limit point of a sequence and limit of a sequence. Can it be unique?
|
By identity representation I mean the representation sending each element of $SL(2,k)$ to itself. Is there a simple way to see this isomorphism? I feel like I am missing something incredibly basic here.
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Inspired by this question, at MO i am asking this question.
Can anyone list some elementary articles at ArXiv which can be understood by High-School/Undergrad Students. I am asking this because, i would like to see some interesting papers and articles to learn something new. If the paper is very much advanced then its ... |
I was able to prove an extension of the Ham Sandwich Theorem (namely; that given any 3 integrable functions on $\mathbb{R}^3$, there exists a plane which simultaneously divides their total integrals in half), and am attempting to show that this does not carry over to 4 functions. I have reduced the proof to one fact w... |
Earlier today, I was talking with my friend about some "cool" infinite series and the value they converge to like the Basel problem, Madhava-Leibniz formula for $\pi/4, \log 2$ and similar alternating series etc.
One series that popped into our discussion was $\sum\limits_{n=1}^{\infty} \frac{1}{n^n}$.
Proving the conv... |
This is the proof from the book:
Theorem. There are infinitely many primes of the form $4n+3$.
Lemma. If $a$ and $b$ are integers, both of the form $4n + 1$, then the product $ab$ is also in this form.
Proof of Theorem:
Let assume that there are only a finite number of primes of the form $4n + 3$, say
$$p_0, p_1, p... |
I have $\log X \sim Exp(\vartheta - 1)$ and I would like to show
$$ P \Big [ \Big |\frac{1}{\frac{1}{n} \sum_{i=1}^n \log X_i} - (\vartheta - 1) \Big | > \varepsilon \Big ] \rightarrow 0 \hspace{5 mm} \forall \vartheta (n \rightarrow \infty)$$
In the answer to this question it states that all moments of the exponent... |
what does it means that a place divide a prime on an algebraic number field?
|
What does $N.\hat{N}_{fa}(S)$ mean, when $\hat{N}_{fa}(S)$ is a cumulative distribution function? $\hat{N}_{fa}(S) = \sum\limits_{s=0}^{S}\hat{p}_{fa}(s)$
The formula is from this paper (section 4.2): Logo Retrieval with A Contrario Visual Query Expansion
|
Consider an alchemist that has many ($N$) sorts of ingredients in his possession. Initial amounts of each of the ingredients is expressed by vector $C^0=(C_1, C_2, \dots, C_N)$.
Alchemist knows several ($M$) recipes of ingredient transmutation, expressed as a set of recipes: $R=\{ R^1, R^2, \dots, R^M\}$. Each recipe ... |
E.g. 1/8=0.125 has three decimals when written out in base 10, but what is a good example of a simple fraction where the decimal sequence terminates but is very large?
Is there some sort of rule which determines how many decimals the terminating exact decimal expansion can have based on the amount of digits in the nume... |
I am trying to work out the formula used for this graph, I don't really know where to start, does anyone have any suggestions? Its a bit unclear, its showing FAT, Mach No, CIT, FWD B/P Position.
Link to graph
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I found this explanation in a journal paper but I could not understand it. Can someone give me an explanation or possibly a proof that:
If
$$\frac{\mathrm{d}V(t)}{\mathrm{d}t}=\sqrt{2}\sum_{h=1}^{H}h\omega V_{h}\cos\left(h\omega t+\frac{\pi }{2}\right),$$
then why integration over whole period is:
$$\frac{1}{T}\int_{0}... |
I have two urns, urn $1$ contains $k$ white balls and $l$ red balls, urn $2$ contains $n - k$ white balls and $n -l$ red balls. I have the total probability
$ P($ "white ball is drawn" $) = \frac{k}{k + l} \frac{1}{2} + \frac{n - k}{2n - (k + l)} \frac{1}{2}$
and I would like to find $k$ and $l$ such that this probabi... |
I am reading Max Karoubi's "K-Theory" and I think I'm overlooking some trivial fact. We have a vector bundle $E\rightarrow X$ and a morphism $p:E\rightarrow E$ with $p^2=p$. He is showing that $\ker p$ is locally trivial. First he assumes that $E=X\times V$ for a vector space $V$. Here's where I'm stuck:
He defines... |
What do 1, 2, 3 represent in $\operatorname{U}(1)+\operatorname{SU}(2)+\operatorname{SU}(3)$?
If they are dimensions, how they can be added? or plus has another meaning?
|
Let $ y = \min \{ (x + 6), (4 – x) \}$, then find $y$.
How to solve this problem?
|
Which sum shows up most often when you roll 10 dice?
|
I arrived at something during my maths ponderings which is really exciting for me.
It is clearly stated in the book on Riemann Hypothesis by Borwein that the convergence of $\sum_{n=1}^{\infty} \frac{\mu(n)}{n^s}$ for $\Re(s) > 1/2$ is necessary and sufficient for RH. This is ofcourse valid since, $\sum \mu(n)/n^s = 1... |
The symbols $\forall$ and $\exists$ denote "for all" and "there exists" quantifiers. In some papers, I saw the (not so common) quantifiers $Я$ and $\exists^+$, denoting "for a randomly chosen element of" and "for most elements in", respectively.
Are there other symbols for quantifiers?
I'm specially interested in quant... |
I've been suggested to this site by some nice people at mathoverflow.net
Before I get started, let me tell you a little about myself. I’m a fourth year Mechanical Engineering student at the University of Michigan, Dearborn Campus. This said, I am not a mathematician, however I do know a bit about math, and I also enjoy... |
I wish to calculate a power like $$2.14 ^ {2.14}$$
When I ask my calculator to do it, I just get an answer, but I want to see the calculation.
So my question is, how to calculate this with a pen, paper and a bunch of brains.
|
You have six triangles. Two are red, two are blue, and two are green. How many truly different hexagons can you make by combining these triangles?
I have two possible approachtes to solving this question:
In general, you can arrange $n$ objects, of which $a$ are of type one, $b$ are of type two, and $c$ are of type th... |
If a set has $n$ elements then what are maximum number of equivalence classes and equivalence relations possible on it?
|
Problem: What is the cardinality of all lines $l$ on $\mathbb R^{2}$ which do not contain a point $(x,y)\in l$ where $x, y \in \mathbb Q$ (call it $A$).
My solution: I was thinking of using CB theorem for this problem. It's easy to show that the cardinality of all lines in $\mathbb R^{2}$ is $2^{\aleph_0}$, so it's obv... |
consider $f(x)=x^4+5x^3+1$. Let $\alpha$ be a root of this polynomial and consider $K=\mathbb{Q}(\alpha)$. Which are the absolute values on $K$ extending the usual absolute values on $\mathbb{Q}$? Are these finitely many? In general, if $K$ is a number field how to find them?
Thanks
|
I'm trying to show that given $(X_i)$ i.i.d., $E[X_i^2] < \infty$, $\mu := E[X_i]$ then $P\Big [ \lim_{n \rightarrow \infty} \frac{S_n}{n} = \mu \Big ] = 1$ where $ S_n := \sum_{k=1}^n X_k$.
So far, I have rewritten $P\Big [ \lim_{n \rightarrow \infty} \frac{S_n}{n} = \mu \Big ]$ as
$$ \lim_{k \rightarrow \infty} P... |
Let $A$, $B$, and $C$ be nonempty sets. I need to find conditions to guarantee that there is an injective function from $B^A$, the set of all functions from $A$ to $B$, to $\displaystyle (C^B)^{C^A}$.
Afterwards, I need to show this kind of a function and to prove that it is injective one.
I tried to do that with a pow... |
let $G$ be a group (can assume finite) acting on a vector space $V$, and assume that the action is semisimple. Let $H$ a normal subgroup of $G$. Why also the action of $H$ on $V$ is semisimple too?
|
Say I define a probability distribution on $P$ dimensional symmetric matrices such that the diagonals are strictly positive but the off diagonals are unrestricted (except for the symmetry constraint). Does this induce a proper probability distribution on the space of symmetric positive definite matrices?
A simple examp... |
I am studying Linear Algebra II, and I came across several questions in which, for a certain linear transformation ($T\colon\mathbf{V}\to\mathbf{V}$) I was told that:
$$||T(a)|| \leq ||a||.$$
I am not completely certain how to use this information. For instance, consider the following question (please forgive my transl... |
I am doing some background research on sensitivity and elasticity analysis, and I came across the following definitions of elasticity:
$$e_{ij}=\frac{a_{ij}\partial \lambda}{\lambda \partial a_{ij}}$$
$$e_{ij} = \frac{\partial \log \lambda}{\partial \log a_{ij}}$$
Why are these equivalent?
|
I'm trying to prove that an Archimedian field is a subfield of the real numbers, my plan is to use the fact that the rationals are dense within the field and their Dedekind completion is the real numbers.
However, I seem to run into some trouble writing a neat proof for the fact that the two properties of Archimedian f... |
I'd appreciate some help with the following problem form Conway's book on functions of one complex variable:
Let $f$ be analytic in $\overline B (0;R)$ with $|f(z)|\le M$ for $|z|\le R$ and $|f(0)|=a>0$. Show that the number of zeros of f in $B(0;R/3)$ is less than or equal to $$\frac{1}{\log(2)}\log\left(\frac M a\rig... |
Better yet, what I'm asking is how do you actually write your mathematics?
I think I need to give brief background: Through most of my childhood, I'd considered myself pretty good at math, up through the high school level. I easily followed mathematical concepts introduced in my classes and even did a few competition... |
Where does the "naive definition" of the product of two ideals $I$ and $J$, $IJ = \{ ij \mid i \in I, j \in J \}$ fall apart?
(The product of two ideals $I$ and $J$ is defined to be $IJ := \sum_i a_ib_i$, where each $a_i$ is in $I$ and each $b_i$ is in $J$.)
Note: This is a question from a problem set I'm working on.... |
title says everything. How do I evaluate the limit given ?
|
Taken from elsewhere on the web : -
"The derivative of a sine function is
d SIN(ω*t) / dt = ω * COS(ω t)
where ω = angular frequency = 2 π *frequency
and this derivative leads the original function by 90
degrees (a quarter period) because a cosine wave leads a sine wave by
90 degrees. This derivative is also different ... |
Let $A$ be a block upper triangular matrix:
$$A = \begin{pmatrix} A_{1,1}&A_{1,2}\\ 0&A_{2,2} \end{pmatrix}$$
where $A_{1,1} ∈ C^{p \times p}$, $A_{2,2} ∈ C^{(n-p) \times (n-p)}$. Show that the eigenvalues of $A$ are the combined eigenvalues of $A_{1,1}$ and $A_{2,2}$
I've been pretty much stuck looking at this for a... |
This is Exercise I.11.4 of Bourbaki's General Topology.
Let $X$ be a connected space.
a) Let $A$ be a connected subset of $X$, $B$ a subset of $\complement A$ which is both open and closed in $\complement A$. Show that $A\cup B$ is connected.
b) Let $A$ be a connected subset of $X$ and $B$ a component of the set $\com... |
Suppose $1<p_1<p_2<\infty$, then show $L^{p_1}[a,b] \supset L^{p_2}[a,b]$.
I was able to show $\|f\|_{p_1} \le \|f\|_{p_2} (b-a)^{1/p_1 - 1/p_2}$ but I'm not sure how to proceed from here.
|
I'm having a really hard time understanding how to figure out if a mapping is linear or not. Here is my homework question:
Determine which of the following mappings F are linear.
(a) $F: \mathbb{R}^3 \to \mathbb{R}^2$ defined by $F(x,y,z) = (x, z)$
(b) $F: \mathbb{R}^4 \to \mathbb{R}^4$ defined by $F(X) = -X$
(c) $F: \... |
I am wondering the validity and the procedure to condense/group several states into one in markov chain, namely, if it is possible, how to transform the state vector and the transition matrix? Many thanks.
|
Notation: $|A|$ is the Lebesgue measure of $A \subset \mathbb{R}^d$, and $A_\delta = \{ x : \text{dist}(x,A) \leq \delta \} $ is the $\delta$-neighborhood of $A$.
I want to show that there is a constant $C$ such that $|E_{2\delta}| \leq C |E_{\delta}|$ for all compact sets $E \subseteq \mathbb{R}^d$ and all $\delta > 0... |
I'm hoping the userbase here doesn't mind if I do a little crowd-sourcing.
I'm curious to find out about popular general-interest mathematics or statistics classes that are offered universities that you know about. By popular I mean crowd pleasing, in that these courses are suitable to large audiences, the course is ... |
After reading about Russell's paradox, I see that the set of all sets does not exist, so instead it is called a class.
What other commonly known classes exist that are not sets? I know the class of all singleton sets is not a set, because you can unite that class to get the class of all sets. This seems like the class ... |
I am trying to estimate, in either direction, the following integral:
$\int_{B} a^{i,j} X_j D_ig dx$ . We are integrating over a unit ball, $B$, and $[a^{i,j}]$ is positive.
I have tried to use ellipticity, but it didn't work. I was wondering if some one can show me:
a. How to use ellipticity to estimate the integral,... |
Given that $y'(x)+p(x)y(x)\geq 0$ and $y(x_0)\geq 0$, how does one go about showing that $y(x)\geq0$ for all $x\in [x_0,\infty)$
|
The book says that $$\lim_{x \rightarrow 0^{-}} \left( \frac{1}{x} - \frac{1}{|x|} \right) \mbox{does not exist}$$
But, given any $M \lt 0$ of large magnitude, if I choose $\delta = \frac{-x^{2}M}{2}$ then any value of x where $|x-0|< \delta$ and $x <0$ (as we are coming from the left) will lead to $\left( \frac{1}{x} ... |
The goal of this problem is to solve the initial value problem
$$y' = -f(x)y;\quad y(0) = 1;$$
where
$$f(x)=
\begin{cases}
1,&\text{if }x\leq 2\\
\frac{3}{2},& \text {if } x>2.
\end{cases}
$$
Since $f$ is discontinuous, it is necessary to solve the above ODE separately in each of
the intervals where $f$ is continuous.... |
I have a question related to this one. In my answer I was going to try and say something about the possible manifolds that might arise in this way, i.e. as mapping cones of elements of $\pi_{2n-1}(S^n)$. Certainly not all of them will be manifolds. At first I felt like there should be a reason why we'd need to be us... |
Haskell is a programming language which uses some concepts from category theory like functor, monad, etc. My question is: Learning intuitive concepts about category from Haskell will ruin my intuition when I learn category theory as a mathematician or it could help developing it?
|
Consider an $n$-sided convex polygon $P$ that contains the origin in the complex plane. Let the $j$-th vertex be denoted $z_j = r_j e^{i\theta_j}$ ($0 \leq \theta_j < 2 \pi$) for $j= 1 \dots n$. I'm interested in non-zero values of
$$ a_k(P)= \sum_{j=1}^{n} \frac{z_{j}^{k}}{|z_{j}|^{k-1}}=\sum_{j=1}^{n} r_j e^{ik\thet... |
The question is prompted by change of basis problems -- the book keeps multiplying the bases by matrix $S$ from the left in order to keep subscripts nice and obviously matching, but in examples bases are multiplied by $S$ (the change of basis matrix) from whatever side. So is matrix multiplication commutative if at lea... |
Show that if a $3\times 3$ matrix $A$ represents the reflection about a plane, then $A$ is similar to the matrix $\begin{bmatrix}
1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & -1
\end{bmatrix}$
This is not homework, just me preparing for a test -- but how can this be solved? I know the definition of similarity, but can't f... |
There are $m$ integer points on a line of length $m-1$ (including the end point of the line). We randomly mark $n$ points of the $m$ points ($n\leq m$), dividing the line into several segments, so each segment have an integer length. What is the probability the max segment's length is equal to $k$?
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Consider a subset $K\subset \mathbb C^2$ consisting of pairs $(z,\bar z)$ such that $|z|=1$. Is there an easy way to see that continuous functions on $K$ can be uniformly approximated by polynomials in two variables?
The problem is not so trivial because if you take the set $L\subset \mathbb C^2$ of pairs $(z,0)$ with ... |
Is there a good source of undergraduates linear algebra problems available on line?
Do you have any better idea of how to resharpen my linear algebra skills?
|
First of all, my background in Homology and Algebraic Topology is very limited and doesn't include much except for the basic concepts; I'm trying to understand some Homology techniques as applications relevant to a course in combinatorics I'm taking.
Anyway, in class we've seen the example of computing the homology gro... |
Wikipedia gives this evaluation:
$$ \int x^ne^{cx}\,\mathrm dx=\frac1cx^ne^{cx}-\frac nc\int x^{n-1}e^{cx}\,\mathrm dx=\left(\frac{\partial}{\partial c}\right)^n\frac{e^{cx}}{c}$$
But I have no idea how I should exactly understand the partial part: $\left(\frac{\partial}{\partial c}\right)\frac{e^{cx}}{c}$
EDIT
Thanks ... |
I need to figure out what is the power of the group of functions from R to R that for each x that is not from Q, it's f(x) belongs to {x,x+1}
I can make a mapping for each function to be Q-->R * (R/Q)--> {x,x+1}
Does it ok?
Thanks
|
Given integers a,b,c and d; is the absolute value of the determinant of [a c;b d] less than or equal to the absolute value of [a e;b f] + absolute value of [e c;f d] for all a,b,c,d?
In other words, is |ad-bc| less than or equal to |af-be| + |de-cf| for all integers a,b,c and ; with the additional constraint that the o... |
what is the general definition for some partial differential equation being called elliptic, parabolic or hyperbolic - in particular, if the PDE is nonlinear and above second-order.
So far, I have not found any precise definition in literature.
|
How can I calculate the cardinality of
$$\left(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}}\right)\cap\mathcal{P}(\mathbb{N}\times\mathbb{N}).$$
where $A^B$ is the set of all functions $f\colon B\to A$.
thanks
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given p>1, whats an example of f where $\int_{-\infty}^{\infty} |f| < \infty$ but $\int_{-\infty}^{\infty} |f|^p = \infty$
what about vica-versa? that is whats an example of g where $\int_{-\infty}^{\infty} |g| = \infty$ but $\int_{-\infty}^{\infty} |g|^p < \infty$
|
Show that if $n$, $a$, $b$, and $c$ are positive integers with $\gcd(a, b) = 1$ and $ab = c^n$, then there are positive integers $d$, and $e$ such that $a = d^n$ and $b = e^n$.
I know that (by Bezout) $\gcd\left(a,b\right) = 1$ implies $ax + by = 1$ for some integers $x$ and $y$, and also that $\gcd\left(a^n,b^n\righ... |
How to prove that in an isosceles triangle circumcenter, centroid, orthocenter & incentre are collinear?
|
I need tricks or shortcuts to find the inverse of $2 \times 2$ and $3 \times 3$ matrices. I have to take a time-based exam, in which I have to find the inverse of square matrices.
|
I need to get the axis angle from a matrix of the form:
$$
\begin{matrix}
\cos \theta & -\sin \theta & tx\\
\sin \theta & \cos \theta & ty\\
0 & 0 & 1\\
\end{matrix}
$$
Into an axis angle. I've tried the standard formula:
$\theta$ = $\arccos$ ( (m[0][0] + m[1][1] + m[2][2] - 1) / 2 )
However this formula inverts ... |
I have the 7 following reccurence relations:
$A_n = B_{n-1} + C_{n-1}$
$B_n = A_n + C_{n-1}$
$C_n = B_n + C_{n-1}$
$D_n = E_{n-1} + G_{n-1}$
$E_n = D_n + F_{n-1}$
$F_n = G_n + C_n$
$G_n = E_n + F_{n-1}$
which I would like to solve, with the goal of eventually finding an explicit form of $E_n$. I started out by looking ... |
We know that Stirling numbers of the second kind is the number of ways to partition a set of $n$ elements into $m$ nonempty sets.
My question is what's the number ways if the max cardinality of all the partitioned sets is $k$.
|
I was trying to answer this question - whether a subsequence of a regular sequence is regular in a Noetherian ring which is not local. In the local case, regular sequences can be permuted and so a subsequence can be considered to be the initial subsequence of a regular sequence and hence regular.
Let's consider the fo... |
Show that if $a$, $b$ are positive integers, then we have:
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$.
|
Using congruences, show that the following is always an integer for every integer
value of $n$:
$$\frac{1}{5}n^5 + \frac{1}{3}n^3 + \frac{7}{15}n.$$
|
I'm working on to prove the following:
Show that the convergence in the space $\omega$ (space of all sequences with respect to the Fréchet metric) is the coordinate convergence.
Any hint is appreciated, thanks.
|
I have a basic question :
a) Suppose $\gamma $ and $ \gamma' $ be conjugate Fuchsian groups acting freely and properly discontinuously on the upper half-plane H to produce two Riemann surfaces $ S $ and $S'$. How can we prove that $S$ and $S$ are isometric ? A detailed answer would be appreciated .Do I need to have $S$... |
Let's say we have a basis $B$ of $\mathbb{R}^n$ consisting of vectors $\vec{v}_1$ through $\vec{v}_n$, and some other basis $C$ of $\mathbb{R}^n$. Then, would $[\vec{v}_1]_C$ through $[\vec{v}_n]_C$ be a basis for $\mathbb{R}^n$ as well? It seems obvious, but I am not sure how to go about this.
I am trying to prove thi... |
For functions $f: \mathbb{R} \rightarrow \mathbb{C}$, define
$$
M f(x) = \sup_{t >0} \frac{1}{2}| f(x+t) + f(x-t) |.
$$
Given $p \geq 1$, I want to construct a sequence of smooth functions $f_n$ such that $||f_n||_p \leq 1$, $|| M f_n ||_p < \infty$, but $|| M f_n ||_p \rightarrow \infty$.
Is it even possible? ... |
I want a very quick non recursive method for finding the HCF (Highest Common Factor) of 2 very large numbers
|
This is exercise 38 from Chapter 3 (Modules and Vector Spaces) in Algebra by Adkins and Weintraub (GTM). How do you solve this problem?
Let
\begin{equation*}
R = \lbrace f : [0, 1] \to \Re : f \;\text{ is continuous and} \; f (0) = f (1) \rbrace
\end{equation*}
and let
\begin{equation*}
M = \lbrace f : [0, 1]\to \R... |
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