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Is this a known special function:
$$\int\nolimits_0^1 a^p(1-a)^{1-p}\\,b^{1-p}\\,(1-b)^p dp\qquad ?$$
I am really only interested in maximizing this over $(a,b)$ in $[0,1] \times [0,1]$, so a pointer to a nice numerical evaluation is appreciated as much or more so than an unstable exact formula.
Thanks for any help
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Given a row of 16 houses where 10 are red and 6 are blue, what is the expected number of neigbors of a different color?
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My text gives a much more complicated proof of this result, which makes me wonder if the argument I have in my head for this has something wrong with it. Does this work, or have I made a bad assumption somewhere along the line?
Let $U \subseteq \mathbb{C}$ be open, $\overline{D}(z_0, r) \subset U$, and $f : U \rightar... |
What are some good ways to motivate the material on infinite series that appears at the end of a typical American Calculus II course?
My students in this course are generally from biochemistry, computer science, economics, business, and physics (with a few humanities folks taking the course for fun) - not just math maj... |
This is in relation to a problem dealing with the three-dimensional analogue of Pell's Equation. I would like to factor
$ x^3+Dy^3+D^2z^3-3Dxyz $ into $\frac{1}{2}(x+Dy+D^2y)$ and another factor.
I would like just a hint on how to go about doing this as I'm pretty much stumped.
EDIT: I think the factor I was looking f... |
arising out of comment made by Yuval Filmus in what is the cardinality of set of all smooth functions in $L^1$? I got this idea (forgive me for my ignorance for if it is nothing but an elementary definition/result in real analysis). The idea is like this Let $f:X\to Y$ is a mapping, where $X$ is ... |
This question was firstly posted on Mathoverflow. Two answers are pretty interesting. The potential counter-example given in the second answer is really interesting, but it is not surely a counter-example.
Consider two partially ordered sets $A = \{a< b,a< c\}$, $B=\{x< z,y< z\}$.
Their linear extensions (here we allow... |
Given $xy=C$ where $x, y$ are integer variables and $C$ is integer constant.
What is the most efficient algorithm that finds $x,y$ such that $x+y$ is minimum?
Providing references is highly appreciated.
Edit: Input integers are reasonably encoded.
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I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification".
To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly those elements $x$ of $A$ for which $S(x)$ holds.
Earlier in the same section, I learned that statements in set theory sh... |
Let $\Sigma$ be an alphabet of size $s$, with which we build strings of length $n$.
The Hamming ball of radius $d$ centered at $x\in\Sigma^n$ is the set $B(x, d)$ of words in $\Sigma^n$ that differ from $x$ in at most $d$ positions.
Similarly, the Hamming circle of radius $d$ centered at $x\in\Sigma^n$ is the set $C(... |
Let $A \in \mathbb{R}^{n \times n}$ be a positive definite matrix and let $a > 0$ be some constant.
I am interested in the following quantity
$$
\hat x = {\rm arg} \max_{x \in R^n}\ x' \left( \frac{1}{a + \|x\|^2_2} A \right) x
$$
Is it possible to relate $\|\hat x\|_2^2$ to the eigenvalue and $\hat x$ to the eigenvec... |
This is from Calculus I test, that I failed to solve correctly. This really bugs me, because for Solve[y == Log[E, 3 - 2x] / Sqrt[x + 1], x], the Mathematica is telling me that: "The equations appear to involve the variables to be solved for in an essentially non-algebraic way.".
I assume that the answer is $x \in \ma... |
$f(a,b) = f(a-1, b) + f(a-1, b-1) + f(a, b-1), ab \neq 0$
$f(a,b) = 1, ab = 0$
So what is $f(a, b)$?
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So, wikipedia has a page about "affine space", where such a space is an object (set, topological space, blah) with a free transitive (blah, continuous, blah) action by the additive group of a free module.
At least when the base ring is an algebraically closed field, we can identify the top. sp. of $\mathbf{A}^n_k$, wit... |
I'd heard of propositional logic for years, but until I came across this question, I'd never heard of predicate logic. Moreover, the fact that Introduction to Logic: Predicate Logic and Introduction to Logic: Propositional Logic (both by Howard Pospesel) are distinct books leads me to believe there are significant dif... |
How can I prove that the Cartier dual of $\alpha_p$ is again $\alpha_p$ (using the Yoneda lemma)? It should be something like $\alpha_p(R) \to (\alpha_p(R) \to \mu_p(R)), x \mapsto (y \mapsto \exp_{p-1}(x + y))$, where $\exp_{p-1}$ is the truncated exponential sequence. My problem is that this isn't a homomorphism.
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how to find the center of an arc given start and end points
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Ok, that's a theorem. As it says in this question.
I just cannot understand how mathematics can prove such a thing; what does exactly this theorem say? How to prove it mathematically?
Thanks!
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From Wikipedia's triangle center article: "Thus every point is potentially a triangle center. However the vast majority of triangle centers are of little interest, just as most continuous functions are of little interest. The Encyclopedia of Triangle Centers is an ever-expanding list of interesting ones."
Hyperbolic Ba... |
During my studies of real analysis I've come across the definition of a field (or algebra) of sets.
My question is: What is the connection between this structure and the structure of a field or algebra known from abstract algebra?
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We have a perfect binary tree on $2^k-1$ nodes. Every node in the tree is marked with probability $\frac{1}{2}$, and a node is either marked or unmarked. We want to find a marked node and return it. Our algorithm is as follows
procedure find(v);
if (v is marked)
print(v)
else if (v is not a leaf)
v1 is the left... |
If there is none, why?
And for the other side, what about open set $(0, 1)$ to closed set $[0, 1]$ with a continuous function?
Thanks
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if $X_n$ converges to $X$ and $Y_n$ converges to $Y$ in distribution, what about $X_n + Y_n $ would that converge to $X+Y$ in distribution ?
any ideas how i could prove or disprove this
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I am looking for an algorithm that "describes" the following situation:
15 - 22 people are combined into groups of 4 or 5 based upon stated preferences of who they wish to be grouped with. Each person will be grouped with at least one of their preferences.
This isn't "homework".
|
I have an interesting math question I've developed while playing cards with my family and wonder if anyone has a way to approach the problem.
Say I have n people who each have a deck of cards. Each deck has a different number of cards in it. We are shuffling the cards for a game and in the end we all want to have the s... |
I am writing a program to solve a Rubik's cube, and would like to know the answer to this question.
There are 12 ways to make one move on a Rubik's cube. How many ways are there to make a sequence of six moves?
From my project's specification: up to six moves may be used to scramble the cube. My job is to write a progr... |
I'm considering a parametrized sigmoid function such as the following logistic:
$$f(x)=\frac{e^{a+bx}}{1+e^{a+bx}}$$
And I'm interested only in the interval $\displaystyle x >= 0$ and $\displaystyle x < x_{max}$ (with a given $x_{max}$).
Two tiny questions regarding its properties:
1) Is the sum of sigmoid functions ... |
I have always taken for granted that expected value is a linear operator. For any random variables $X$ and $Y$: $E(aX + bY) = aE(X) + bE(Y)$. Can anyone point me to a rigorous proof of this?
Also, I know that generally median $Med()$ is not a linear operator, meaning $Med(aX + bY)$ might not be equal to $a Med(X) + b M... |
How does one prove this identity: $$\lim_{n \to \infty} 2^{2n-1} \sqrt{n} \frac{ \Gamma(n)^{2}}{\Gamma(2n)} = \sqrt{\pi}$$
Taken from Gamelin : Complex Analysis
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What are the open sets of a Stone space of a Boolean algebra B?
According to Wikipedia, they are the ultrafilters on B that contain an element of B. However, this cannot be the case since an ultrafilter is a set of subsets of B and contains no elements of B. Using the alternate definition of the Stone space as the sp... |
As an exercise in Python programming I've written a script that finds amicable pairs, and I've had fun doing so. But I'm wondering if amicable pairs have a use, in mathematics or elsewhere?
As a reminder:
Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the ... |
I have a set of functions that map square matrices with $n$ rows and columns to square matrices with $k < n$ rows and columns. Is there a name for this property? I know that 'projection' would be the wrong word. Also I don't mean that the function only changes the size, it might also do other things.
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I have a stochastic variable x with this property: if it's measured at
t1 and again at t2, then x(t2)-x(t1) has a normal distribution with
mean 0 and standard deviation Sqrt[t2-t1].
I want to find the distribution of the maximum value this variable
reaches between t1 and t2, or confirm my intuition that this is not... |
How to solve the recursion:
$$p[n,m] = p[n-1,m-1] + p[n+1,m-1] + p[n,m-1]$$
Ideally in general, but if you need base cases:
$$p[n,0] = 0 \text{ (for } n \neq 0),$$
$$p[0,0] = 1$$
I've asked a similar question previously and believe $n^{m-2}$ is
somehow involved.
Would turning this into a DFQ help?
|
Take a Multiplicative group G, defined by a prime p. Now take an element t from it. Its inverse is t-1. To calculate the inverse you can use the extended Euclidian algorithm or this trick, which I found on wikipedia:
t-1 = tp-2 mod p
Now this makes some sense to me, but the following, which I found in some code, doe... |
I get confused in the following problem. Actually I don't know how to derive the derivative of the Jacobian. Could anybody help me?
Given a smooth vector field $\vec{b}$ on $R^{n}$, let $\vec{X}(s)=\vec{X}(s,x,t)$solves the ODE:
$\dot{\vec{X}}=\vec{b}(\vec{X})(s \in R)$,
$\vec{X}(t)=\vec{X}$
Define the Jacobian $J(s,x... |
Reading a book on fractional calculus reminded me that I'd like to know more on the following method/idea.
Given an integral: $\displaystyle \int_{-\pi}^\pi{\frac{\displaystyle\sum_{j=a}^b{e^{i\cdot c_j \cdot t}}}{\displaystyle\sum_{k=a}^b{e^{i \cdot d_k \cdot t}}} dt}$
I can use recurrences and generating functions to... |
I know the general Black-Scholes formula for Option pricing theory (for calls and puts), however I want to know the other solutions to the Black-Scholes PDE and its various boundary conditions. Can someone start from the B-S PDE and derive its various solutions based on different boundary conditions? Even if you could ... |
Say I have two locally free sheaves $F,G$ on projective variety $X$. I know the cohomology groups $H^i(X,F)$ and $H^i(X,G)$. Is this enough to give me information about $H^i(X,F\otimes G)$? In particular, if $H^i(X,F)=0$, what conditions on $G$ guarantee that also $H^i(X,F\otimes G)=0$?
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Let M be an (n-1)-manifold in R^n . Let M(e) be the set of end-points of normal vectors (in both directions) of length e and suppose e is small enough so that M(e) is also an (n-1)-manifold. Show that M(e) is orientable (even if M is not)
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Let $L^{\infty}$ denote the set of all essentially bounded functions and suppose that
$f \in L^{\infty}$ and $g \in L^{\infty}$. Then the product $fg$ is in $L^{\infty}$. Now the question is: prove or disprove: $\|fg\|_\infty \leq \|f\|_\infty \|g\|_\infty$.
Any hints?
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Our math prof gave us this and we are not sure what to make of the notation, can someone give us a hand?
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Assume a capacitated network flow problem with graph $G = (N,A)$ and capacities $u_{ij}$ (finite), costs for each path of $c_{ij}$ and I/O of $b_{i}$.
How can it be shown there is an equivalent transportation problem with a source node for every arc $(i,j)$ in $N$ having supply $u_{ij}$, and a sink node with demand $\s... |
When I first learned linear algebra, both the professor and the book used brackets like [ and ] to enclose matrices. However, in my current differential equations textbook, matrices are enclosed by parenthesis, and I suddenly realize everybody else are using them too.
So are brackets/parenthesis for enclosing matrices ... |
To motivate my question, recall the following well-known fact: Suppose that $p\equiv 1\pmod 4$ is a prime number. Then the equation $x^2\equiv -1\pmod p$ has a solution.
One can show this as follows: Consider the following polynomial in ${\mathbb Z}_p[x]$: $x^{4k}-1$, where $p=4k+1$. The roots of this polynomial are pr... |
This is a real-analysis homework question so I of course have to be very precise and justify anything or any theorem I use.
If $b_n$ is a bounded sequence and $\lim(a_n) = 0$, show that $\lim(a_nb_n) = 0$
Intuitively, since $b_n$ is bounded, then sup($b_n$) is some finite number and therefore we can take an $N$ natural... |
I'm reading the following proof:
Suppose $\mu(X) < \infty$ and $0 < p< q \leq \infty$.
If $q=\infty$ then $L^{q}(\mu) \subseteq L^{p}(\mu)$.
Proof: If $q=\infty$ then $\int_X |f|^{p}d\mu \leq (||f||_{\infty})^{p} \int_X 1d\mu$.
Why we have the above inequality??
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Let $M$ be a compact, connected smooth manifold. If $p, q$ are points in $M$, is there always a geodesic that goes from $p$ to $q$?
I know that this is certainly not true if $M$ is not compact, but I couldn't find a counterexample for the compact case.
Can anybody help me out?
Thanks,
S.
|
I am a bit confused about the concept of proper subsets,precisely whether to include one or both of the void set and the set itself.
An extract from my module goes like this :
Obviously,every set is the subset of itself and the void set $\emptyset$ is the subset of every set. These two subsets are called improper subse... |
Working on probability theory in an economics class. This was on of the practice problems that I'm struggling to back up with math. Any help would be appreciated! :)
An accident insurance company has reserves to cover claims on no more than 3 accidents. If all accidents have a probability of 0.01 of occurring, if all ... |
As usual denote $L^p$ the quotient space where two integrable functions are identified if they are equal almost everywhere. So I'm using the definition written here:
http://en.wikipedia.org/wiki/Lp_space
Then we have the following result: for each $p \geq 1$ we have $L^{\infty}(X) \subseteq L^{p}(X)$ where X is a finit... |
Suppose $X$ is a Banach space and $T: X \to X$ is a bounded linear operator. If $T$ is onto, is it necessarily 1-1?
|
How do I show
$$\frac{\mathrm d^2 \log(\Gamma(z))}{\mathrm dz^2} = \sum_{n = 0}^{\infty} \frac{1}{(z+n)^2}$$?
$\Gamma(z)$ is the gamma function.
|
I am doing some pre study/revision in maths before i take my next education steps and am working through a hefty engineering maths book from the beginning. The first chapter is a review of algebraic techniques using indices.
The question is to simplify this
\begin{align}
\frac{2xy^2}{(2xy)^2}
\end{align}
and the soluti... |
is it possible to calculate shadow areas of buildings or simulate shadows of buildings in a city, using the heights of these buildings and the sun angle and azimuth? the basic light tracing concept using trigonometry is known. But the Problem is how to find away to come up with simulated shadow of group of buildings in... |
Suppose we are given two real numbers: $a, b \in \mathbb{R}$, $b > a$. Find $m \in \{1, 2, 2.5, 5\}$ and $k \in \mathbb{Z}$, satisfying the following condition: $\frac{b-a}{m10^k} \in [5,10)$.
How to solve this problem without using brute force?
|
When $f$ is a Lebesgue measurable function, is $\Vert f \Vert_p$ continuous in $p$, when $p$ is in a set such that f belongs to $L_p$? If yes, how to show that? Thanks!
Yes. It is an exercise taken from Rudin's book. I have spent quite a while thinking but got no idea.
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For a Lebesgue measurable function $f$, $\Vert f \Vert_p$ converges to $\Vert f \Vert_\inf$ as $p$ goes to infinity.
I was wondering if $\Vert f \Vert_p$ converges to $\exp(\int \log |f|)$ as $p$ goes to $0$ for probability measure. How to prove this?
Thanks!
|
In a parallelogram ABCD. M is the midpoint of CD. Line BM intersects AC at L and it also intersects AD extended at E. Prove that EL=2BL
PS: This is not a homework problem. I was solving geometry for fun. I'm unable to solve this. :(
|
As we know that $L_p \subseteq L_q$ when $0 < p < q$ for probability measure, I was wondering when $L_p = L_q$ is true and why. Is it to impose some restriction on the domain space? Thanks!
|
A Steinmetz solid is the intersection of two cylinders. My question is how to find the surface area of one with both cylinders of radius 1, without parameterizing the solid.
I parameterized it; that is, used $r(u,v)=\langle \sqrt{1-v^2}, u \sqrt{1-v^2}, v \rangle$, $u,v \in [-1,1]$, and integrated the magnitude of the ... |
My daughter is stuck on the concept that $$2^0 = 1,$$ having the intuitive expectation that it be equal to zero. I have tried explaining it, but I guess not well enough.
How would you explain the concept to a child, other than the teachers "that is just the rule" approach?
|
How many subgroups with index two are there of a free group on two generators? What are their generators?
All I know is that the subgroups should have $(2 \times 2) + 1 - 2 = 3$ generators.
|
I received some homework (calculus), which I can't prove:
$f(n) * g(n) \ge f(n) + g(n)$ for some $n \geq 1$ is always true $(f(n), g(n) \ge 1)$
I think that this is true, so I need to prove that there is some $n \geq 1$ ($n$ is a natural number) for which this inequality will be true always, thanks in advance for any ... |
Let $C$ be a convex subset of a Banach space $X$ and $T:C\to C$ a (norm) continuous affine map, i.e. $$T(tx+(1-t)y)=tT(x)+(1-t)T(y)$$ for $0\le t\le 1$. Is $T$ weakly continuous, i.e continuous as a map from $(C,\tau)$ to $(C,\tau)$ where $\tau$ is the topology induced by the weak topology of $X$.
|
I'm teaching first semester calculus and trying to find a way to explain why each hypothesis in l'Hostpial's rule is needed.
If $f$ and $g$ are real differentiable functions on an interval containing a point $c$ then there are three hypotheses one needs to check in order to apply l'Hospitals rule to compute $\lim_{x \t... |
I have proved this using Venn diagram but when I am trying to prove this using the rule that "If $ A \subset B \text{ and } B \subset A $ then $ A = B $", I am having some problems with my understanding of the same,here is how I did so far:
Let $x \in A \cap (B-C) \Rightarrow x \in A \text{ and } x \in (B-C) \Righta... |
Find a necessary and sufficient condition for the linear equation Ax = b to have
no solution. (hint: Use duality to find a strong alternative to Ax = b).
|
Which of the following is the unbounded set ?
$ X = \{ x\mid x = \frac{1}{n},n \in \mathbb{N} \} $
$ Y = \{ x\mid x = \frac{1}{2^n},n \in \mathbb{N} \} $
$ Z = \{ x\mid x = 2^n,n \in \mathbb{N} \} $
$ W = \{ x\mid x \in \mathbb{N}, x \lt 4532 \} $
After reading the definition from Wikipedia , I can only convince... |
Let $g:[0,1] \rightarrow \mathbb{R}$ be twice differentiable with $g''(x)\gt 0$ for all $x\in[0,1]$. If $g(0)>0$ and $g(1)=1$, show that $g(d)=d$ for some point $d\in(0,1)$ if and only if $g'(1)\gt 1$.
Proof.
Suppose there exits $d\in(0,1)$ such that $g(d)=d$. Then by the MVT applied on $[d,1]$, $f'(c)(d-1)=g(d)-g(1) ... |
Imagine I had two sets of samples. The only information I have about them is their
Size
Average
Standard deviation
Is it possible to calculate the standard deviation of a third set that is composed by the union of all samples in both sets?
And what if the original sets had the same size?
|
I'm stuck with this induction proof:
So far, given:
$\begin{align*}
T(1) & = 2 \\
T(n) & = 2T(n/2)+2 \\
& = 2(2T(n/[2^2])+2) + 2 \\
& = [2^2]T(n/[2^2]) + [2^2] + 2 \\
& = [2^2](2T(n/[2^2])+2) + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + [2^3] + [2^2] + 2 \\
& = [2^3]T(n/[2^3]) + 2\{[2^2] + [2^1] + 1\} \\
& \vdots \\
& = [2^k]T... |
Let $f \in L^{1}(\mathbb{R})$ such that $f(x) >0$ for almost all $x \in \mathbb{R}$. Let A be a Lebesgue measurable set such that $\int_{A} f =0$. Prove $m(A) = 0$.
wlog we can assume $f(x)>0$ everywhere define $A_{n}={x \in A: f(x) > \frac{1}{n}}$
so since $f(x) > 0$ everywhere, $\int_{A_n} f dm \leq \int_{A} fdm = 0$... |
It is claimed in some logic books that the compactness theorem of first-order logic can be proven using Tychnoff's theorem from topology.
Now, to me this feels very strange because I consider logic as something "lower" in the sense that the theorems of topology follow from the axioms and theorems in logic. How can we a... |
Let $X$ be a sigma finite measure space with $p>0$. The book states the following:
There exists a map $f\in L^{p}(X)$ such that $f>0$ and $f$ is bounded by $1$.
Proof: Let $C_{n}$ be a sequence of disjoint sets of finite measure such that $X$ is the union
of these sets. Now just put $f(x) = \left(\frac{2^{-n}}{1+\mu(... |
What are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest.
So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it?
I'm aski... |
What is an intuitive explanation of a positive-semidefinite matrix? Or a simple example which gives more intuition for it rather than the bare definition. Say $x$ is some vector in space and $M$ is some operation on vectors.
The definition is:
A $n$ × $n$ Hermitian matrix M is called positive-semidefinite if
$$x^{*} M ... |
We have three counters, $i, j, k$, all initialized to zero. Each step consists of adding or subtracting one from one of the counters, so $(\Delta i, \Delta j, \Delta k)$ is selected among $(\pm1, 0, 0), (0, \pm1, 0), (0, 0, \pm1)$, each with probability 1/6. What is the expected number of steps before all the counters ... |
Injective functions are not the same thing as injective correspondences, correct? Injective functions are a subset of the injective correspondences.
For example in $y^2 = x$, y is not a function of x, but this is still a correspondence between $\mathbb{R} \rightarrow \mathbb{R}$. Can we say that the correspondence, $C... |
Hello
i have to solve the following problem
find an $\displaystyle f(k)$ where $\displaystyle S_k=\theta(f(k))$
where $\displaystyle S_k =\sum_{n=1}^{k^2-1} \sqrt{n}$
I tried first of all to calculate or "limit" my sum using integral rule so i came up with
$\displaystyle \frac{2(k^2-1)^{3/2}}{3} \leq S_k \leq \frac{2(k... |
This may be embarassingly simple, but I can't see it.
Let $M$ be a Riemannian manifold of dimension $n$; fix $x \in M$, and let $B(x,r)$ denote the geodesic ball in $M$ of radius $r$ centered at $x$. Let $V(r) = \operatorname{Vol}(B(x,r))$ be the Riemannian volume of $B(x,r)$. It seems to be the case that for small $... |
I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean?
I've read a few pages on this issue, and they all seem to boil down to the same thing:
Any real number $x$ can be written as $e^{\ln{x}}$ (seems obvious enough.)
Mumble mumble mumble
This is eq... |
How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
|
If A and B are two points on the earth, how could I find any arbitrary point between them along the shortest distance side of their great circle path?
Points are in radians
longitude = $0$ to $2\pi$
latitude = $0$ to $\pi$, $0$ being at north pole
Points are not antipodal
I desire something where I specify a range $0.0... |
Let $\Omega = $ {$\omega_{i}$} be an ordered set of $n$ positive reals in the unit interval, $\omega_{1} \leq \cdots \leq \omega_{n} \leq 1$. Define the $n$-simplex $\Delta(\Omega; (\mathbb{R}^{+})^{n})$ by the non-negative points $(x_{1}, \dots, x_{n}) \subset (\mathbb{R}^{+})^{n}$ which satisfy the inequality
\begin... |
I think the following picture helps explain it easier than words:
I know the coordinates of point a which is at a vertex of the triangle, as well as the coordinates of point b which is somewhere along the edge of the opposite side. I also know the y-coordinate of point c and I'm trying to find its x-coordinate.
What ... |
The definition of small is that they have O(lg n) bits. One way is just to test the integers 2,3,... for primality and keep the first n primes, but this takes at least O(n log n) time (times the cost of primality testing) using a naive algorithm. Is it possible to do it in linear time?
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I am looking for an algorithm to compute the $n$ shortest lattice vectors in $\mathcal{R}^2$. The problem statement is as follows:
Given a lattice $L: \{ m \vec{u}+n\vec{v} \} \in \mathcal{R}^2$, a norm $\left\lVert\cdot\right\rVert$, and an integer $N>0$, compute a set of $N$ shortest lattice vectors $S$ such that if ... |
$G$ and $G \times G$ where $G = \Bbb Z_2 \times \Bbb Z_2 \times \Bbb Z_2 \times\cdots$
The answer says yes but I cannot figure out what homomorphism function I could use.
|
Ok this is a really silly question and I should know this, but I can't seem to figure something out:
for the last step, how do they know that $0 \leq x \leq 4$? If we use the minimum value of theta, which is $-\pi/2$, and plug that into $x=\cos(\theta)$, then we get $0$, and same for $\pi/2$.
|
I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five pieces. My question is :
"Is it possible to construct (mathematically) these five sets or is the proof more of an existence... |
If "$|\;\;|$" denotes the absolute value, then is it always true $|x^{s}|=|x|^{s}$ whenever $s$ is any positive real number?
I know this holds for any positive integer. What about positive reals?
|
Let V be a vector space with a non-trivial subspaces U, W. $ U \neq W$. Consider $u_0 \in V$, with $u_0 \neq 0$ and $w_0 \in V$ with $w_0 \neq 0$ then we have $u_0 + U \in V/U$ and $w_0 + W \in V/W$. Is there any sensible way to define:
$(u_0 + U) + (w_0 + W)$?
I'm thinking "no" $(2,4) + (2,6,7)$ makes no sense. So, w... |
The question is: Given a licence plate that can have either 2 or 3 letters followed by either 2 or 3 numbers, how many different license plates can be printed.
My math is as such $(26^2 + 26^3)*(9^2+9^3)$ which comes to 14,784,120 but the book says the answer is 20,077,200. What am I doing wrong here?
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I've been reading Vakil's notes on algebraic geometry (on my own -- this is not part of a class), and I'm stuck on one problem (number 6.1.H). It goes as follows.
Let $X$ be a scheme. Prove that $X$ is quasicompact and quasiseparated if and only if $X$ can be covered by a finite number of affine open subsets, any two... |
This is a homework question so specifically I am looking for a direction (help). Not an absolute answer.
I understand how to solve 2nd order nonhomogeneous (and I think 3rd order is similiar) recurrence relations. Whats kinking me up with this one is a term with n.
h[n] = 4h[n- 1] - 4h[n-2] + 3n + 1
Here I am using sq... |
Consider a spherical triangle with vertices $A, B$ and $C$, respectively. How to determine its area?
I know the formula:
$A = E R^2$,
where $R$ is radius of sphere, and $E$ is the excess angle of $(a + b + c - \pi)$, but how to determine the angles between $ABC, ACB$ and $BAC$?
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According to the book the following is true:
Let $f$ be a measurable function and suppose $|f| \leq h$ where $h$ is a real-valued function
and $h$ is integrable. Then f is integrable.
My question is: suppose instead we have the inequality $ |f| \leq h$ but almost everywhere and of course $h$ assumed to be integrable. I... |
You have three bags, A, B, and C. Bags A and B hold x red and y blue marbles each; bag C holds 2x red marbles and 2y blue marbles.
Let us say that we pull out z marbles from bag A, z marbles from bag B, and 2z marbles from bag C. Will the expected number of red marbles pulled from bag A and bag B together be the same a... |
As a result of this question, I've been thinking about the following condition on a topological space $Y$:
For every topological space $X$, $E\subseteq X$, and continuous maps $f,g\colon X\to Y$, if $E$ is dense in $X$, and $f$ and $g$ agree on $E$ (that is, $f(e)=g(e)$ for all $e\in E$), then $f=g$.
If $Y$ is Hausdo... |
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