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Question: Besides generally learning Algebraic Topology, what are some prerequisites for studying Computational Topology? Are there any accessible papers which introduce the field and the methods being used?
Motivation: A number of my peers do applied mathematics where I go to school, and upon hearing how they solved ... |
Suppose in an atomless space, $f\in L^1$ and $||f||_1=1$
I want to prove there exists $g$ and $h$, $g\neq h$, such that $||g||_1=||h||_1=1$ and $f=(1-\lambda)g+\lambda h, 0 \lt \lambda \lt 1$
I'm not sure how to start this question, any tips would be greatly appreciated
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(I think this question is acceptable for this site.)
I ask Maple to evaluate integral(1/(exp(i*t)-z),t,-π,π), and I get [if |z| = 1 then undefined else -(2*π)/z]. I understand the first part, the integrand is singular. However, for |z| ≠ 1, this is an integral of a finite continuous function over a finite closed inte... |
Following are some corollaries regarding the weak duality theorem.
Consider a constrained problem,
$\min_{x \in X} f(x),$ subject to
$g(x) \leq 0$ and $h(x) =0$.
Its dual problem is $\sup_{u \geq 0,
v} \theta(u,v)$, where the dual
Lagrangian function is defined as
$\theta(u,v):= \inf_{x \in X} \, f(x) +
u^T ... |
Possible Duplicate:
Card doubling paradox
I don't know how to resolve the following paradoxon: Assume two finite sets A, B, one being twice as big than the other, but you don't know which. You want to guess the bigger one. You want to choose A, but before that, you reason like this: "With probability 0.5 the other s... |
An AF-algebra is a $C^* $-algebra which is the inductive limit of an inductive sequence of finite-dimensional $C^*$-algebras.
Elliott's theorem concerning the classification of AF-algebras says that two AF-algebras $\mathfrak{A}$ and $\mathfrak{B}$ are isomorphic as $C^* $-algebras if and only if $(K_0(\mathfrak{A}),K_... |
Where can I learn more about the implications, meta discussions, history and the foundations of mathematics?
Is Russell's Introduction to Mathematical Philosophy a good start?
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I am reading a proof of there are infinitely many primes of the form $4m+3$, but have trouble understanding it. The proof goes like this:
Assume there are finitely many primes, and take $p_k$ to be the largest prime of the form $4m+3$.
Let $N_k = 2^2 \cdot 3 \cdot 5 \cdots p_k - 1$, where $p_1=2, p_2=3, p_3=5, \dots$... |
I can increase and decrease the frequency of a signal using the combination of fft and a Fourier series expansion FOR loop in the code below but if the signal/array is to large it becomes extremely slow (an array that's 1x44100 takes about 2 mins to complete) I'm sure it has to do with the for loop but I'm not exactly ... |
I'm having great difficulty solving the following problem, and even figuring out where to start with the proof.
$$
\neg A\lor\neg(\neg B\land(\neg A\lor B))
$$
Please see the following examples of how to do proofs, I would appreciate it if you could attempt to give me guidance using the tools and the line numbers that ... |
Can anyone help me with this simple grade 12 math question on joint variation? I'm stuck, I'm the WORST at word problems.
Thanks!
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I know from linear algebra that for a non-homogeneous consistent linear system that has an infinite number of solutions, if the last row of the augmented matrix has all zeros, then we have an infinite number of particular solutions.
I was wondering if there is a similar parallel with differential equations, namely say ... |
What is the formula for the total variation of a step function on [a,b]? I understand how to write a formula for the total variation of a general function of bounded variation. Any ideas?
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The question is to "prove the reduction formula"
$$ \int{ \frac{ x^2 }{ \left(a^2 + x^2\right)^n } dx } = \frac{ 1 }{ 2n-2 } \left( -\frac{x}{ \left( a^2+x^2 \right)^{n-1} } + \int{ \frac{dx}{ \left( a^2 + x^2 \right)^{n-1} } } \right) $$
What I got is
Set
$ u = x $
$ du = dx $
$\displaystyle{ dv = \frac{ x }{... |
Here is a question that is similar to my last one. I've been trying to learn about Grothendieck's Existence Theorem, but it seems that there aren't very many places that talk about formal schemes and even less that come up with examples.
Suppose $(\mathfrak{X}, \mathcal{O}_\mathfrak{X})$ is a Noetherian formal scheme a... |
We know the solution of the Sultan's dowry problem:
To reject the first $n/e$ candidates and then to select the first who exceeds the best of the sample.
How to find the best strategy if we want select the 2, or 3, or $k$ best instead of only one?
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let $f(z)$ be the analytic component of $\sqrt{z}$ such that $0\leq arg f(z)<\pi$. Let $F(z)=\frac{2}{3}z^{\frac{3}{2}}$. Then $F'(z)=f(z)$ and $F(z)$ is analytic in $0\leq arg z<2\pi$. Let $\gamma$ be the contour: square with vertices $1+i, -1+i, -1-i, 1-i$. Why $\int_{\gamma} f(z)dz = \lim_{\epsilon \to 0}(F(1-\epsil... |
I am solving some practice questions while revising probabilities and wanted to confirm my methodology. The question is as follows:
A magician has five animals: 2 doves
and 3 rabbits- If he pulls 2 animals
out at random, what is the chance that
he will get a matched pair.
The way I solved it:
Probability of ma... |
There's multiple products and multiple buyers, each with their own quote for the product, and some not willing to buy the product at all. Some buyers require a minimum dollar amount for the order or they won't buy anything. I'm trying to find a series of equations or mathematical steps to take to find the most rewardin... |
What are leading short and lagging long moving average indicators and how do we calculate them?
e.g. for the following data set, and let window size be 2. Can you show me what the running leading and lagging moving averages are?
4 5 2 10 3 8 9
Reference: http://www.mathworks.com/help/toolbox/finance/movavg.html
Thanks... |
What is the equation for plotting points on an exponential curve with fixed end points?
For example, if I want to plot 10 point along a curve that starts with 10,000 (x=1, y=10000) and ends with 30,000 (x=10, y=30000) the formula is y=10000*(1.129831^(x-1)).
But if I change 1.129831 to some different value, 2 for examp... |
Given:
$f(x) + f(x+T) = 2$ ; where $T$ is a fixed positive number.
The solution is given as:
put $x = x+T$
then given equation becomes
$f(x+T) + f(x+2T) = 2$
subtract given equation from above. You'll get: $f(x) = f(x+2T)$.
Hence $2T$ is the period of $f(x)$.
I don't get it. wouldn't putting $x = x+T$ change the va... |
Let $a,x,b$ be positive integers satisfying $x^{a+b} = a^b \cdot b$. How can I prove that $a=x$ and $b=x^x$?
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I was wondering what "vice versa" means using the language of logic?
For example,
If P is unbounded, D is infeasible, and vice versa.
Does the vice versa part mean that
"If D is infeasible, P is unbounded", "If D is unbounded, P is
infeasible", or something else?
How to understand "vice versa"
generally?
Thanks!
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I have just learned in probability that picking a specific value from an uncountably infinite set (continuous) has a probability of zero, and that we thus estimate such things over an interval using integrals. This clearly does not apply to finite sets (discrete), where you can easily calculate the probability. But doe... |
Good morning,
Let T be a linear transformation, which acts on a vector space V, over a field F.
Let W be a sub-space which invariants of T, and f,g polynomials from the same field F.
I need to prove that W is invariant of g(T) and f(T)(W) invariant of g(T).
obviously, g(T) is also a linear mapping and therefore is al... |
I am having trouble solving recurrence relations, probably because I am missing the basics.
Is there any web reference/book I can read to help me cover the basics?
I watched some lectures and read the chapter but it seems like it's not enough.
I don't have the slightest clue about how to solve that does not fit in the ... |
Hi I want to study harmonic measure for planar domain and its extension to higher dimension,could you tell me some books/papers on this topic starting from elementary?
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I'm reading these lecture notes here about the tensor product and the author uses $M^\vee$ where $M$ is an $R$-module. Look at page 2 for an example (example 2.1). What does it mean? I looked through the notes but I couldn't find the definition. Many thanks for your help!
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Given a linear Diophantine equation, how can I count the number of positive solutions?
More specifically, I am interested in the number of positive solutions for the following linear Diophantine equation:
$3w + 2x + y + z = 47$
Update: I am only interested in non-zero solutions.
|
I'm wondering how could one transform an IFF statement in clauses for proof by resolution.
Taking a <-> b for IFF a THEN b, I've come up with
a <-> b = (a -> b) && (b -> a)
which then can be turned in the following 2 clauses
!a || b
!b || a
But applying resolution here would obviously break everything.
Is it possible t... |
If A is a matrix in SL(2,Z) is it directly obvious the stabilizer of A ,
G_A = {B in SL(2,Z) | A.B = A}, is the set containig only the Identity? or is this not true?
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Given a functor $F:A\to B$ of abelian categories we may say that $F$ is left exact if it maps exact sequences to left exact sequences, and similarily for right. For arbitrary categories, we may say that $F$ is left exact if it preserves finite limits (supposedly, this was introduced in SGAIV, but I don't have it). The ... |
Suppose I have a p-dimensional integral:
$$\int_{0}^{\infty}\int_{0}^{\infty}\dots \int_{0}^{\infty}f(x_1,x_2,\dots,x_p)dx_1dx_2\dots dx_p$$
And then I make a rotation + translation transform:
$$W=A^{T}(X-b)$$
Question: How will the region of integration $X>0$ change in the $W$ space?
Can assume $A$ is a matrix of eige... |
The relation $\sim$ is defined on $\mathbb{Z}^+$ (all positive integers). We say $x\sim y$ if and only if $y=3^kx$ for some real number $k$.
I need to prove that $\sim$ is an equivalence relation.
To prove an equivalence relation we must certify that:
It is reflexive
It is symmetric
It is transitive
I am not sure how... |
I need to calculate $\pi$ -- in base: 4, 12, 32, and 128 -- to an arbitrary number of digits. (It's for an artist friend).
I remember Taylor series and I've found miscellaneous "BBP" formulas, but so far, nothing that points to calculating the digits in arbitrary bases.
How can this be done in a practical manner?
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Consider the set X={a,b}, and the collection £ given by £={ Ø, {a},{b},X }. show that £ is a topology on X.
I know that from the definition of topological space i must consider to show the following three axioms
To show that both the empty set and X belong to £
2.To show that the intersection of any fi... |
How would you describe the Power Law in simple words? The Wikipedia entry is too long and verbose. I would like to understand the concept of the power law and how and why it shows up everywhere.
For example, a recent Economist article Cry havoc! And let slip the maths of war, shows that terrorist attacks follow an inve... |
While I was watching a lecture on Information theory, I found that entropy of an information source is the average amount of information that it provides in terms of bits (or nats, decits or whatever), which's actually the weighted average of information contained in all the symbols that source provides (weighted by pr... |
Does anyone have any ideas towards solving these four equations one at a time?
$a^3 - 3a^2b + b^3 = \pm 1$
$a^3 + 3a^2b - 6 ab^2 + b^3 = \pm 1$
I am guessing that the $1$ might mean we can use units in some algebraic number field to solve these but I have no idea which one or how to find it. Maybe I am wrong entirely... |
If $S$ is a measurable Abelian group, and $\mu$ probability measure on $S$,
how would you prove that
$$ f(x) = \mu (x + B) $$
is measurable function for every fixed measurable set $B$?
|
Let's say that we want to solve a classic transportation problem without capacities using the stepping stone method. (Problem definition: A bipartite graph with supply nodes a1...m, demand nodes b1...bn, and costs wij for the transport of goods from ai to bj. Find a flow which satisfies the demand of all b nodes at min... |
Let $x,y$ and $m$ be integers. Prove if $m | 4x$ + y and $m | 7x+2y$ then $m|x$ and $m|y$
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I need a hint to solve exercise 13.2.9 in Dummit and Foote. Suppose $F$ is a field of char not equal to 2. Suppose $a^2 -b$ is a square where $a,b \in F$ and $b$ is not a square.
Show $\sqrt{a + \sqrt{b}} =\sqrt{m} +\sqrt{n}$ for some $m,n \in F$.
I've reduced to $(a+\sqrt{b})(a-\sqrt{b})=(\sqrt{m}+\sqrt{n})^{2}$ b... |
Please recommend resources to study Idempotence and other similar properties of processes and operations in depth.
I want to know what other properties like Idempotence are there for an operation. I am a computer science student so these kinds of properties of a process intrigue me.
I don't even know which field of m... |
Consider two sets on the plane $A=\mathbb{Q}\times \mathbb{R}$ and $B=\mathbb{R}\times \mathbb{Q}$.
We know that $A\cap B=\mathbb{Q}\times \mathbb{Q}\neq\emptyset$. What about the general cases?
That is, will $A\cap B\neq\emptyset$ if $A,B\subset\mathbb{R}^2$ satisfy that
each vertical fiber $A_y$ of $A$ is dense in ... |
A relation $\mathrm{R}$ is defined on the set of all positive integers by:
$x\mathrm{R}y$ if and only if $y = 3^k\cdot x$ for some non-negative integer $k$.
Prove that $\mathrm{R}$ is a partial order.
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I need some help please with this question:
A player decides whether to stop playing or not just after the first loss -or- after 10 games.
the probability for winning a single game is p.
all the games are independent.
Let X be the amount of total games which the player Participated.
I'm trying to look for the distribu... |
I need to find a way of proving that the square roots of a finite set
of different primes are linearly independent over the field of
rationals.
I've tried to solve the problem using elementary algebra
and also using the theory of field extensions, without success. To
prove linear independence of two primes i... |
Given a set $S$ of $N$ numbers, my aim is to partition it into two sets ($S_1$ and $S_2$), so that
(i) the difference $\sum S_1 - \sum S_2$ is minimized and
(ii) the difference $|S_1| - |S_2|$ is $1$ if $N$ is odd, and $0$ if $N$ is even.
I am able to come up with a dynamic programming algorithm (inspired by dp - knaps... |
The sequence $10000, 121, 100, 31, 24, n, 20$ represents a number $x$ with respect to different bases. What is the missing number, $n$?
This is from my elementary computer aptitude paper. Is there any way to solve this quickly?
|
I need to generate a Deterministic Finite Automata (DFA), selected from all possible DFAs that satisfy the properties below. The DFA must be selected with uniform distribution.
The DFA must have the following four properties:
- The DFA has N nodes.
- Each node has 2 outgoing transitions.
- Each node is reachable from... |
Why is this true?
$$ \int_0^\infty e^{-\frac{1}{2}(b^2+x^2)} I_0(bx) x \,dx = 1 $$
Note that $I_0(x)$ is a modified bessel function of the first kind. The difficulty for me lies in a) translating the bessel function into something basic (the argument is not an integer, hence making it complicated to me), and 2) this do... |
It is well known that the only ring automorphism of $\mathbb{R}$ is the identity. This follows from the fact that all ring automorphisms of $\mathbb{R}$ must fix $0$, and be order preserving, and hence can be proven to be continuous.
It is clear that you cannot follow the same line of reasoning to prove the same thing ... |
I'm looking for a counterexample to pointwise convergence implying weak convergence.
I.e., I'm looking for $f, f_1, f_2, ... \in L^1$ such that $\displaystyle{\lim_{n \to \infty} f_n=f \ }$ a.e., but there exists a bounded measurable $g$ such that $\displaystyle{\lim_{n\to \infty} \int f_n g \neq \int f g}$.
|
Prove, or give a counter example:
Let $\mu$ be a finite positive borel measure on $\mathbb{R}$. Then $\int (x-y)^{-2} d \mu (y) = \infty $ almost everywhere on $\mu$ (for the selection of x's).
This is a question I had in an exam, and the answer is supposed to be presented in less than 30 words, so there must be someth... |
I am trying to figure out why this is true:
$$ \lim_{p \to 0}\frac{1}{2p}\left((1+p)e^{-\frac{y}{1+p}} - (1-p)e^{-\frac{y}{1-p}}\right) = e^{-y} + ye^{-y}$$
I have already tried L'Hopital's Rule, but it gave me something that I couldn't simplify. The problem seems to be the $\frac{1}{2p}$ term never seems to go away. I... |
Using angle sum-difference, double, and/or triple angle relations with tangent, cosine, and sine, need help proving tangent half-angle relations. Thanks!
|
Suppose I've got two quaternions that each represent an angle. I need to interpolate between these two angles (from 0% to one side to 100% to another side).
Since I work a lot with complex numbers, I'd thought about getting the "arg" of these quaternions, averaging them, and creating a new quaternion. But then I don't ... |
Pitt Theorem says that any bounded linear operator $T\colon \ell^r \to \ell^p$, $1 \leq p < r < \infty$, or $T\colon c_0 \to \ell^p$ is compact.
I know how to prove this in case $\ell^r \to \ell^p$, and $c_0 \to \ell^p$, where $p > 1$. Main idea in the first case is that $\ell^r$ is reflexive and hence closed ball $B_{... |
I'm trying to read a proof in Dummit and Foote that says splitting fields of isomorphic fields are isomorphic. There is a passage that goes
"Recall that an isomorphism $\varphi$ from one field $F$ to another field $F'$ induces a natural isomorphism between the polynomial rings $F[x]$ and $F'[x]$. In particular, if $f... |
Given a directed graph $C$ that only contains a directed cycle of length $L$ (and all resulting sub-cycles), that visits each node at least once,
$$C=(V, D)$$
where $V$ is a fixed set of vertices and $D$ is a set of directed edges, and:
$$|V| = N$$
$$|D| = L$$
$$ 1 \leq \deg^+(v) \leq 2, v \in V$$
How many directed gr... |
Let $L$ be a language whose alphabet has $n$ letters. We have a dictionary listing $k$ equivalent pairs of words, i.e. $A_i \sim B_i$ for $1 \leq i \leq k$, where $A_i$ and $B_i$ are distinct words. We know that one can reduce any word from the language to a word of length at most $100$, by replacing similar words (fro... |
A magic square of order $n$ is an $n \times n$ grid containing each of the numbers $1,2,\dots,n^2$, so that the numbers in each row, column, and diagonal sum to the same number $n(n^2+1)/2$.
This question follows on from Is half-filled magic square problem NP-complete? about completing a partially filled magic square.
... |
I ran into the following problem while working in neural nets.
Given natural numbers $b$ and $r$, uniformly randomly choose $b+r$ points within a unit square. Call the $b$ points the blue points and the $r$ points red points. What is the probability $p(b,r)$ that the convex hull of the blue points, $H_b$ , overlaps wi... |
How can I evaluate
$$\sum_{n=1}^\infty\frac{2n}{3^{n+1}}$$?
I know the answer thanks to Wolfram Alpha, but I'm more concerned with how I can derive that answer. It cites tests to prove that it is convergent, but my class has never learned these before. So I feel that there must be a simpler method.
In general, how can ... |
My questions are regarding a constrained problem,
$$\min_{x \in X \subseteq \mathbb{R}^n} f(x),$$
subject to $g(x) \leq 0 \in \mathbb{R}^m, h(x) =0 \in \mathbb{R}^k$. Its dual problem is $$\sup_{u \geq 0 \in \mathbb{R}^m,v \in \mathbb{R}^k} \theta(u,v),$$
where $\theta(u,v):= \inf_{x \in X} \, f(x) + u^T g(x) + v^T ... |
Excluding maps that can be colored with 2 or 3 colors, how many different four coloring exist for a given regular map?
Naturally, two identical maps have to be regarded as differently colored if the same coloring cannot be obtained by swapping colors inside one of the two maps. It is also intended that the two maps use... |
Suppose I pick 'N' integers over an interval [A, B] without replacement. As a function of 'N' and the interval length, what distribution / average values should I expect for the distances between nearest-neighbors in a sorted array of the selected integers?
Edit: I apologize, an important note is that the distances be... |
Let $$f=\sum f_{s}\exp(2\pi isx)\in C^{(p-1)}[0,1]$$ and $$f^{(p)}\ in\ L_2[0,1]\ \ ( \sum\left|f_{s}\right|^{2}j^{2p}<\infty )$$
Does it imply that $f_s=O(s^{-(p+\psi)})$ for some $\psi>0.5$?
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The problem is to figure out how to solve a real cubic equation of the form $x^3 + px + q = 0$ using trigonometry. The first step is to prove the identity
$$
4\cos^3 \theta - 3\cos \theta - \cos 3\theta = 0
$$
I can do this using de Moivre's formula as per the hint provided. I just equate the real parts of the ex... |
I'm out of college, and trying to self-learn complex analysis. I'm finding Ahlfors' text difficult. Any recommendations? I'm probably at an intermediate sophistication level for an undergrad. (Bonus points if the text has a section on the Riemann Zeta function and the Prime Number Theorem.)
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I am reading a text and I am curious to know how certain approximations were reached.
The first function approximations is: $$ 1- \frac{1}{2p}((1+p)e^{\frac{-y}{x(1+p)}} - (1-p)e^{\frac{-y}{x(1-p)}}) \approx \frac{y^2}{2x^2 (1-p^2)}$$
,when $y \ll x$. Note that I tried using the approximation $e^x \approx 1+x$, when x ... |
Assume $Q \in \mathbb{R}^{n\times n}$, and $b,c,d \in \mathbb{R}^n$. A quadratic programming problem is:
$$ \min_{x \in \mathbb{R}^{n}} \tfrac{1}{2} x^T Q x + c^T x,$$
subject to $A x \leq b, E x = d$.
I was wondering what are some sufficient and/or necessary conditions for the quadratic programming problem to have n... |
It is usually argued (and also joked about) that classifying sets into open and closed is a bit paradoxical, since sets can be open and closed at the same time, or neither. This can be analyzed very clearly by noting that closed is an antonym of open: it means exactly not open (and vice versa). Then, by saying that a s... |
Let $L$ be an extension field of $K$. Suppose that the degree $[L:K]$ is a prime number. How to show that $L$ is a simple extension of $K$?
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Is there a generalization of the Cauchy-Schwarz Inequality for multiple integrals?
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Prompted by this question I was looking for $A \subset (0,1)$ such that for any interval $(a,b)\subset (0,1), A \cap (a,b)$ and $A^c \cap (a,b)$ are both uncountable. One such $A$ is the set of all numbers that have a finite number of $1$'s in their base $3$ expansion. As no choice was used in the construction, it sh... |
Let $\sum a_{n}$ be an absolutely convergent series such that $$\sum a_{kn}=0$$ for all $k\geq 1$. Help me prove that $a_{n}=0$ for all $n$.
Thank you!
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Suppose that $f(x) > 0$ for all $x$, and that $f$ is decreasing. Prove that there is a continuous decreasing function $g$ such that $0 < g(x) \le f(x)$ for all $x$.
To be quite honest, I have no idea how to approach this problem. (I also have no clue about the second part, but I imagine a hint at this solution will h... |
Using the quadratic formula, we have either 0, 1, or 2 solutions. I wonder if we could use it this formula for congruence? Or is there a formula to solve quadratic equation for congruence?
Edit
Assume that $ax^2 + bx + c \equiv 0 \pmod{p}$, where $p$ is prime with $(a, p) = 1$, then is there a formula for this equation... |
I am working on a track editor and have found myself in a situation where I need to define two touching circles. Ideally I would like to know the centre point, and radius of these circles.
The information I have is a point on the circumference of each of the circles, and the tangent to the circle at that point.
On my o... |
How would I do the following question. I know how to do it with two variables (just B(U) and B(U + V) but I do not know how to figure this out with 4 (or even 3) terms) Thanks for the help.
E [B(U)B(U+V)B(U+V+W)B(U+V+W+x)] where U + V + W > U + V > U and x > 0.
Thanks for the help. By the way the answer is 3U^2 + 3UV +... |
I read one theorem in the book, they said there will be exactly $\dfrac{p-1}{2}$ quadratic residues of $p$. So for each $i$,
$$x^2 \equiv a_i \pmod{p} \text{ where } 1 \leq i \leq p - 1$$
But if we sum all $a_i$, then what does this sum equal to?
$$\sum_{i=1}^{\frac{p-1}{2}}a_i = ?$$
I haven't used the condition that ... |
What is the covariance function for $U(t)$ if $U(t) = e^{-t}B(e^{2t})$ for $t \geq 0$ where $B(t)$ is standard Brownian motion.
Thanks for the help!
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Original question: If $I,J$ are ideals and $R$ is a PID why does $\,I+J=R\,\Rightarrow$ $\,IJ=I\cap J\,$?
Updated question: With same hypotheses, why does $IJ= I \cap J\,\Rightarrow$ $I+J=R$?
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If a and b are two quadratic residues of the prime p, then it is easily checked that ab is also a quadratic residue modulo p; if c is a quadratic residue modulo p, and $ {cd \equiv {1} \pmod{p} } $, then since 1 is a quadratic residue of p, d is a quadratic residue of p; so the set of all quadratic residues form a grou... |
Inspiring from this thread Is there a formula for solving the congruence equation $ax^2 + bx + c=0$?
I want to get rid of the confusion when dividing both side of congruence.
I'm always confused about modulo arithmetic, so today I really want to make it clear.
If $a \equiv b \pmod{m}$ then obviously, $ac \equiv bc \pm... |
Can you help prove the functional equation: $$\sum_{n=-\infty}^\infty e^{-\pi n^2x}=x^{-1/2}\sum_{n=-\infty}^\infty e^{-\pi n^2/x}.$$
Specifically, I am looking for a solution using complex analysis, but I am interested in any solutions.
Thanks!
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I've run into an application where I need to compute a bunch of elementary symmetric polynomials. It is trivial to compute a sum or product of quantities, of course, so my concern is with computing the "other" symmetric polynomials.
For instance (I use here the notation $\sigma_n^k$ for the $k$-th symmetric polynomial ... |
One equation that every first-year student wants to be true is $A/B\cong C\implies A\cong B\times C$, where $A$, $B$, and $C$ are algebraic structures of some kind (modules, groups, rngs, . . .). However, this is not true: set $A=\mathbb{Z}$, $B=2\mathbb{Z}$, $C=\mathbb{Z}/2\mathbb{Z}$.
In the above case, though, $B\ti... |
Problem
Prove that $\lfloor \sqrt{p} \rfloor + \lfloor \sqrt{2p} \rfloor +...+ \lfloor \sqrt{\frac{p-1}{4}p} \rfloor = \dfrac{p^2 - 1}{12}$ where $p$ prime such that $p \equiv 1 \pmod{4}$.
I really have no idea how to start :(! The square root part really messed me up. Can anyone give me a hint?
Thank you
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Im currently in a Calc II class at CWRU. I missed a day of class covering Second Order DE's. This topic is not covered in the textbook. Could someone direct me towards a introduction or tutorial to this topic. Here is an example of the type we are covering.
4y'+ 4y" + y = 0 ; y(0)= -12 , y'(0)= 1
|
What would the covariance function be of $V(t) = (1-t) B[t/(1-t)]$ if $B(t)$ is standard Brownian motion. Also $t$ is between $0$ and $1$.
Thanks for the help!
EDIT:
Here is where I am stuck:
I believe that $Cov(V(t),V(s)) = E[V(t)V(s)]-E[V(t)][V(s)]$. We know that $E[V(t)]$ and $E[V(s)]$ are $0$ so the second term di... |
Does $a^2+b^2=1$ have infinitely many solutions for $a,b\in\mathbb{Q}$?
I'm fairly certain it does, but I'm hoping to see a rigorous proof of this statement. Thanks.
Here is my motivation. I'm working on a strange little problem. I'm working in a geometry over an ordered field. Suppose I have a circle $\Gamma$ with ce... |
How would I go about finding the distribution of $B(u) + B(u+v)$ where $u+v > u$?
I know that both $B(u)$ and $B(u+v)$ are normal random variables. The sum of two normal random variables is also normal with parameters equal to the sum of means and variances. However, this only applies for independent normal variables. ... |
While browsing around, I read that any ordered skew-field that satisfies the Archimedean property is commutative, but it was offered without proof. Out of curiosity, is there a quick proof or reference for this fact?
|
Everyone:
I am trying to make sense of these statements (doubts are ones contained within
asterisks **); both $M$,$N$ are surfaces.:
Setup: $N$ is a surface trivially-embedded in $X^4$, a 4-manifold.
$M$ is a compact surface embedded in $X^4$ , with $\partial M$ contained in $N$, and
** $M$ is normal to ... |
Completeness axiom: a non-empty subset of $\mathbb{R}$ which is bounded above, has a supremum in $\mathbb{R}$.
Natural question is, what is $\sup(\emptyset)$?
In many books, the concept of supremum is stated for non-empty subsets of $\mathbb{R}$, and do not mention anything about $\emptyset$. In some books, it is stat... |
Problem
Prove that $a$ is quadratic residue mod $m$ iff $a$ is quadratic residue mod $p$ for each prime $p$ divides $m$ where $m$ is odd, and $(a, m) = 1$.
My attempt was using this property of congruence:
If
$$x^2 \equiv a \pmod{p_1}$$
$$x^2 \equiv a \pmod{p_2}$$
$$ \cdots $$
$$x^2 \equiv a \pmod{p... |
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