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The [infinite] Ramsey theorem states that
Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, for every $F\colon [\omega]^n\to \{1, . . . , k\}$ there exists an infinite $H \subseteq \omega$ such that $F$ is constant ... |
If I want to take the derivative of $ax^n$, I will get $anx^{n-1}$. If I were to take the derivative again, I get $an(n-1)x^{n-2}$.
We can generalize this for integer k easily to get the kth derivative $a\frac{n!}{(n-k)!} x ^{n-k}$. But what about for a more general k?
Does this have some name? Has it been widely stu... |
I am seeking methods to see if a polynomial $f \in O(\mathbb{C^2},0)$ is irreducible. The subject is really new to my and I am studying for myself, for which I don't see about this subject.
would thank them a lot they could provide me some bibliography where to read about this topic
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It seems to be an accepted belief based on decades of experience that naive algorithms are not
adequate to solve NP-complete problems in a reasonable amount of time. Even those who believe P = NP seem to be looking for an algorithm with a very clever representation, so far without success.
Can the observations above b... |
Coming from an understanding of Fermat's primality test, I'm looking for a clear explanation of the Miller-Rabin primality test.
Specifically: I understand that for some reason, having non-trivial square roots of $1 \bmod p$ means that $p$ is definitely composite; and I gather that you can find these non-trivial square... |
I'm just starting to introduce myself to first-order logic, and I find it very confusing. As such, I'm not entirely sure how well I constructed the following. For $L$ a first-order language, let $L$ be equipped with the following nonlogical symbols: a unary function $\dot{+}$, a binary function $\dot{1}$, and a 3-ary r... |
The word "quad" generally means 4. Quadratics don't have 4 of anything. Can anyone explain where the name comes from?
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I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real
numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \rightarrow \infty} n a_n = 0$.
Using the definition of the limit, this is equivalent to showing
\begin{equation}
\forall \var... |
I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data.
Effektively, I ignore that a Gaussian $X$ (with positive mean) may be negative and need mean, standard deviation and full width at half maximum of $\log X$. Unfortunately, t... |
Given a quadratic form with discriminant $D$, what does the class number of $\mathbb{Q}(\sqrt{D})$ tell us?
(This question is inspired by a comment on the question here)
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Please, mention one problem that is NP-Hard but not NP-Complete? And, explain why.
I see some papers assert Degree Constrained Minimum Spanning Tree is an NP-Hard problem and some say it's NP-Complete. Why so? Is it something that we don't have a clear idea about?
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Below are the common set of curves that are used in a DJ mixer on the crossfader. I have a software equivalent and am using the "Transition" type curve but would like to include some different curve types.
On both the x and y axis' my range is from 0-1. What I usually do is have a function that gives me one side of th... |
Notation: $b_{0}+\underset{n=1}{\overset{\infty }{\mathbb{K}}}\left( a_{n}/b_{n}\right) $ is the Gauss Notation for generalized continued fractions.
Description of the Bauer-Muir transformation
(Based on pp. 76-77 of Lisa Lorentzen and Haakon Waadeland's book (A), chapter II, theorems 11, and Theorem 7; also section... |
Hey, I'm looking for a guide on how I find Q given the following, where $a$ and $b$ are constants:
\begin{equation}
\frac{dQ}{dt} = \frac{a + Q}{b}
\end{equation}
I have the answer and working for specific case I'm trying to solve but do not understand the steps involved. A guide on how I can solve this, with an explan... |
Why are differential equations called differential equations?
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I am trying to identify the general case algorithm for counting the different ways dice can add to a given number. For instance, there are six ways to roll a seven with two 6-dice.
I've spent quite a bit of time working on this (for a while my friend and I were using Figurate numbers, as the early items in the series ... |
I know the four colour theorem was solved by a computer checking a large number of cases. What I don't understand is why there are only a large but finite number of cases. It seems like there should be infinitely many planar graphs... What is the intuition behind cutting this down to a finite number of configurations?
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A polyomino is a connected subset of $\mathbb{Z}^2$ - a set of squares joined along their edges such that the resulting form is connected (or, more shortly, a generalized form of Tetris cube). A polyomino can be easily thought of as graph with vertices being elements of $\mathbb{Z}^2$ and edges between two vertices of ... |
A at $(45,10)$, B at $(10,20)$, $AB=AC$ and angle $C=20$ degree find the coordinates of $C$.suggest the formula so i can write code in Perl.
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When given an algrebraic expression of the form
$$ \ 6A + 7{A^2}\ $$
and asked to simplify, would the correct answer be
a)
$$ \ A(6 + 7A)\ $$
or
b)
That this cannot be simplified further
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There are a number of ways to describe this problem. I shall name a few.
Submultisets
Let $(M, f)$ be a multiset where $M = {x_1, ... x_k}, |M| = k$ and $f(x_i) = m_i$, i ranging from 0 to k inclusive. Find the number of submultisets $(S,g)$ of $(M,f)$ of size n, $|(S, g)| = n$.
Integer Partitions
Let $n, x_1, ... x_k,... |
I made an assertion in What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic? that Q has higher consistency strength than Pres, Presburger arithmetic; i.e., Q proves the consistency sentence for Pres.
But in fact, I only know something weaker, that Q can formalise the p... |
If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability?
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It is a well known theorem that any doubly connected domain can be conformally mapped onto an annulus.
Consider the simpler version :
Suppose $D$ is a bounded domain whose boundary is two non-intersecting circles. Then $D$ can be conformally mapped onto an annulus.
I believe that the proof of this should be easier, tha... |
In measure theory, we have "lambda systems" and "pi systems". Pearl's message passing algorithm has "lambda messages" and "pi messages". Is there a reason that lambda and pi go together?
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Is it possible to quantify the number of dimensions in combinatorial spaces. The space I am particularly interested is all partitions of a set, bounded by the Bell number, where objects in this space are particular partitions.
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Could anyone give an example of a sequence of differentiable (real) functions that uniformly converge to a differentiable function, but the derivatives of which don't converge to the derivative of the limit function?
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Let $G$ be a directed graph which contains $N$ vertices, and which satisfies the following condition: each vertex of $G$ has at least one incoming edge.
(For clarity, a vertex $V_1$ has an "incoming edge" if there is an edge $E$ which connects $V_1$ to some other vertex on the graph, $V_2$, for which $E$ is directed fr... |
As an example, consider the polynomial $f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2)$ which clearly has a root $x = 1$.
But we can also find the roots using Cardano's method, which leads to
$$x = \sqrt[3]{\sqrt{28/27} + 1} - \sqrt[3]{\sqrt{28/27} - 1}$$
and two other roots.
It's easy to check numerically that this express... |
I'm a software engineer and have been asked to research a Voronoi implementation in four dimensions. I'm not asking for "teh codez" but am interested in approachable tutorials on Voronoi decomposition and/or implementations in any dimension. Any help here would be greatly appreciated!
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I know that a continuous function which is a BV may not be absolutely continuous. Is there an example of such a function? I was looking for a BV whose derivative is not Lebesgue integrable but I couldn't find one.
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My frustration started after hours of searching failed to turn up a formula for the vertex of a parabola in the general form
$$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$$
As is already well known, the discriminant $\Delta=B^2-4AC$ can be used to diagnose the type of conic one has if it it given in the general form. Other useful formulae... |
In signal detection, an observer is assigned the task of discerning the presence (or absence) of some signal with accompanying noise. There are four possible outcomes: a hit ($H$), a miss, a false alarm ($FA$), or a correct rejection. An iso-sensitivity function is the function that describes the relation between corre... |
Is there a way to derive the Maclaurin series for $\frac{1}{(1-x)}$ after finding the Maclaurin series for $(1+x)^n$ which is $\displaystyle\sum\limits_{k=0}^\infty \frac{f^k(0)}{k!}*x^k$.
From the original equation I could substitute $-1$ for $n$ and $-x$ for $x$ but I don't see how I could do that in the summation... |
While teaching the concept of vector spaces, my professor mentioned that addition and multiplication aren't necessarily what we normally call addition and multiplication, but any other function that complies with the eight axioms needed by the definition of a vector space (for instance, associativity, commutativity of ... |
Among the smooth 1-manifolds (with or without boundary) which embed into $\mathbb{R}^2$, which ones can be represented by a single parametrization $z = (x,y) = f(t)$, for $t \in I$, where $I$ is an interval (not necessarily open or closed), and $f$ is smooth (i.e. infinitely differentiable)?
The reason I ask is that in... |
Here's the problem (from little Spivak):
Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by
$$f(x)=\begin{cases}
e^{-x^{-2}} & x\ne 0 \\
0 & x=0
\end{cases}$$
Show that $f$ is a $C^\infty$ function, and $f^{(i)}(0)=0$ for all $i$.
Anyhow, here's my proof. It gets pretty ugly at the end and I do a little handwaving (every ... |
I am looking for an intuitive explanation of $\liminf$ and $\limsup$ for sequence of sets and how it corresponds to $\liminf$ and $\limsup$ for sets of real numbers. I researched online but cannot find a good comparison. Any link, reference or answer very much appreciated.
For example, what is $\liminf$ and $\limsup$ o... |
Under the condition that the spaces (or maybe just the total) are connected and locally path connected, is then the a covering the same as a homeomorphism?
|
If I have an arbitrary shape, I would like to fill it only with $45-45-90$
triangles.
The aim is to get a Tangram look, so it's related to this question.
Starting with $45-45-90$ triangles would be an amazing start. After
the shape if filled I imagine I could pick adjacent triangles, either
$2$ or $4$ and draw squares... |
I've just come up with this question as I'm studying for a number theory midterm. If $p$ and $q$ are different prime numbers, and it's known that $2^p \equiv 1 \bmod{q}$, then $q\equiv 1 \bmod{p}$. I've tried a few cases and it seems to be true. How can it be proved?
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Is there a reliable method for testing how invertible a Hermitian Toeplitz matrix is without going through the work of actually inverting it?
A determinant is obviously easy to compute, but I'm not sure what threshold to compare it against.
The algorithm I am using is Levinson recursion. The function can detect the di... |
Apologies for the simple question. Is it possible to analytically minimize a function such as
$$f(x) = a^x + c x (x-1) \qquad x > 0, \quad 0 < a,c < 1$$
Nothing else can be said about the constants $a$ or $c$.
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I'm taking a course in differential geometry, and have here been introduced to the wedge product of two vectors defined (in Differential Geometry of Curves and Surfaces by Manfredo Perdigão do Carmo) by:
Let $\mathbf{u}$, $\mathbf{v}$ be in $\mathbb{R}^3$. $\mathbf{u}\wedge\mathbf{v}$ in $\mathbb{R}^3$ is the unique ve... |
I have a stochastic event (say a weighted coin toss) that produces a positive outcome (heads) according to some unknown probability $P$.
Given $N$ total events (coin tosses) and n positive outcomes (heads), how can I measure the likelihood that $\frac nN$ is a good approximation of $P$?
Obviously, as $N$ grows, it beco... |
For functions on $\mathbb{R}^2$, it lets you classify extreme points. When function is log-likelihood, it gives volume of the region where maximum likelihood estimates tend to fall. Are there other places it comes up?
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I am trying to solve a ODE that arises from a Hamilton-Jacobi-Bellman (HJB) equation. The equation is $$\frac{1}{2}b^2(1-\rho_s^2)\psi''-\frac{1}{2}\left(\frac{\mu-r}{\sigma}\right)^2\frac{(\psi')^2}{\psi''}+[ru+\theta a+b\rho_s(\mu-r)(1-\frac{2}{\sigma})]\psi'=0,$$ where $\mu, r, \sigma, \theta, a, \rho_s, b$ are con... |
Let $f$ be defined as
$$f(x)=\begin{cases} x \sin(\frac{1}{x}) & x\ne 0 \\ 0 & x=0 \end{cases}$$
$f(x)$ is not absolutely continuous so it cannot might not satisfy the Lusin N condition.
Is there a direct proof of that it does not? i.e. I wanted to know how to construct a set of zero measure which does not satisfy L... |
Does there exist a family of discs $\lbrace D_{n}\rbrace_{n=1}^{\infty}$ in the Euclidean plane such that
the radius of $D_{n}$ is $\frac{1}{n+1}$,
each $D_{n}$ is contained in the unit disc, and
$D_{n}\cap D_{m} = \emptyset$ for each $n\neq m$ ?
(I'm not sure what tags are appropriate for this kind question, so if... |
Suppose $X$ is a metric space. A real valued function $f$ defined on a metric space $X$ is said to be of first baire class, if $f$ is a pointwise, limit of a sequence of continuous functions on $X$.
A famous well know theorem asserts that if $f$ is in the first baire class, then the set of points of discontinuity is o... |
Suppose $\alpha: \mathbb{R} \rightarrow \mathbb{R}^3$ is a $C^\infty$
curve, parameterized by arc length ($\left\|\alpha'(t)\right\| = 1$),
and with $\alpha(0) = \alpha(\ell)$. Show that
there exists a $t_0 \in [0,\ell]$ such that
$
\left\| \alpha''(t_0)\right\| \geq \frac{2\pi}{\ell}.
$
If we had that $\alpha'(0) = ... |
My professor wants us to do this problem to refresh ourselves with substitution. We have to solve:
$$\int\sqrt{1 + \sqrt{x}}\,\mathrm dx$$
$$\int\sqrt{1 + \sqrt{1 + \sqrt{x}}}\,\mathrm dx$$
...etc...
If someone could point me in the right direction with the first one, I think I can handle the others.
Thanks for t... |
Given that I have two formal power series:
$$ A(x) = \sum_{k \ge 0} a_k x^k $$
$$ B(x) = \sum_{k \ge 0} b_k x^k $$
The Cauchy Product gives a series
$$ C(x) = \sum_{k \ge 0} c_k x^k $$
$$ c_k = \sum_{n=0}^k a_n b_{k-n} $$
Which comes from taking the product of the two series $C(x)=A(x)B(x)$. What then, in terms of $A(... |
For plane curves, twice the quadratic term of the Taylor expansion at a point on that curve is precisely the curvature. I know of one proof as featured in Hubbard and Hubbard, but I was wondering if there was another more elegant solution.
Thanks!
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In a sense, I did just ask this question. However, since the question is about how to write a beautiful looking proof and my proof is entirely rewritten, it seems like it should be a new question. What I'm looking for here are (aside from any actual mathematical problems in the proof of course) is advice on how to be... |
I first noticed this is true for the integers in the sequence $9, 99, 999, 9999,\dots$, since for some term $a_n=10^n-1$ in the sequence and $p$ a prime other than $2$ or $5$, we have $a_n\equiv 0 \pmod p\!\iff\! 10^n\equiv 1 \pmod p$ and we can find infinitely many $n=k(p-1)$ so that $10^n=(10^{p-1})^k\equiv 1^k\equiv... |
I wasnt sure where to ask this but since this is an algorithmic question here it goes. I've come face to face with a math problem and can't seem to get over it for the last couple of days. It goes like this:
You are given an adding machine that sums a set of N+1 digits
consisting of positive integers 1 to N
as it'... |
We know that the sine function takes it values between $[-1,1]$. So is the set $$A = \{ \sin{n} \ : \ n \in \mathbb{N}\}$$ dense in $[-1,1]$. Generally, for showing the set is dense, one proceeds, by finding out what is $\overline{A}$ of this given set. And if $\overline{A} = [-1,1]$, we are through with the proof, but... |
I have a grid made up of overlapping $3\times 3$ squares like so:
The numbers on the grid indicate the number of overlapping squares. Given that we know the maximum number of overlapping squares ($9$ at the middle), and the size of the squares ($3\times 3$), is there a simple way to calculate the rest of the number of... |
How we can show every permutation is either even or odd,but not both......I can't arrive at a proof for this ..... Can anybody give me the proof...
Thanks in advance...
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I have a suspicion that the following expression is true, however my algebra skills aren't brilliant, so any help would be appreciated:
Is it possible to infer $x = \frac{y + z}{2}$ from $a^{\frac{1}{x}} = b^{\frac{1}{y}} + c^{\frac{1}{z}}$ and $a = b + c$
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A sequence of sets is defined as $A_n=\{x \in [0,1] : |\sum_{i=0}^{n-1} 1_{[\frac{i}{2n},\frac{2i+1}{4n})} - 1_{[\frac{2i+1}{4n},\frac{i+1}{2n})}| \geq p\}$ for some positive $p\geq0$. What is $\limsup_{n\to\infty}A_n(p)$ and $\liminf_{n\to\infty}A_n(p)$?
For any $x$, the function $f_n(x) = \sum_{i=0}^{n-1} 1_{[\frac{i... |
Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a set?
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The complexity class RL is described at the complexity zoo as: Has the same relation to L as RP does to P. The randomized machine must halt with probability 1 on any input. It must also run in polynomial time (since otherwise we would just get NL).
The question is - how can we get NL when omitting the demand for polyno... |
We know that $$ 2^n= (1+1)^n = \sum_{k=0}^n {n \choose k}$$ I was asked to solve this limit, $$\lim_{n \to \infty} \ \sum_{k=0}^n {n \choose k}^{-1}=? \quad \text{for} \ n \geq 1$$
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For a sequence of positive integers $a_1, \ldots, a_n$ and a base ring $R$ there is a graded ring $R[x_1,\ldots, x_n]$ where $x_i$ is in degree $a_i$. We can then apply Proj and get a scheme, and this is usually called a weighted projective space; if all of the $a_i$ are 1, then the resulting scheme really is projecti... |
Let's take the numbers 0-10. Their mean is 5, and the individual deviations from 5 are
-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5
And so the average (magnitude of) deviation from the mean is $30/11 \approx 2.72$.
However, this is not the standard deviation. The standard deviation is $\sqrt{10} \approx 3.16$.
The first mean... |
Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial of degree n has at most n roots) is not considered fundamental by algebraists as it's not needed for the development of... |
Suppose A is a Hadamard matrix of size $d$. Let A be in normalized in a sense that first row and first column are all ones. What is the sum of rows? I tried random Hadamard matrices and seem to get $d,0,0,0,0,\ldots$
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Suppose I have an number of distances from an unknown location to a known location. I can use these distances and the known locations to draw a number of circles. The point where all the circles intersect is my unknown location. This is easy to solve. However, if my distance measurements have some error, I will get a c... |
If $n$ is an integer, is there a nice way to write the partial fraction expansion of $\frac{1}{x^n - 1}$? I figure that if $\zeta$ is the $n$-th root of unity, then for some coefficients $a_0, a_1, \ldots, a_{n-1}$ we may write
$$
\frac1{x^n - 1} = \frac{a_0}{x - 1} + \frac{a_1}{x - \zeta} + \frac{a_2}{x - \zeta^2} + ... |
What do you call this function?
$$f(x) = x(1 - x)$$
Does it have a name? It’s cropped up in at least two statistical formulas I've been asked to program so far. (In both cases, $x$ was something called a $p$-value (for what it's worth).)
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While learning about limits and derivatives, I came accross the following problem on one of Stewart's exercises book. I've been trying to wrap my head around it but I haven't got anywhere useful:
$ \forall x f''(x) $ exists
$ \exists c \in \mathbb{R} $ / $\forall x \neq c, f'(x) > 0 \wedge f'(c) = 0 $
Then, $(c, f(c))... |
In this wikipedia, article http://en.wikipedia.org/wiki/Circle#Area_enclosed its stated that the circle is the closed curve which has the maximum area for a given arc length. First, of all, I would like to see different proofs, for this result. (If there are any elementary ones!)
One, interesting observation, which one... |
I know that many types graphs have unique names, ie. a directed graph where each node has exactly one outbound edge is known as a functional graph. Do the graphs with exactly $2$ inbound/outbound edges have a name?
Bonus: If the $2$ inbound/outbound graph is named, is the $n$th inbound/outbound graph named?
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Given an infinite dimensional normed linear space, how would one show that it is possible to fit an infinite collection of non-overlapping balls of radius $\frac{1}{4}$ in the unit ball?
I guess one can immediately reduce the problem to a normed linear space of countably infinite dimensions. The solution seems clear if... |
If we know the rational homology of $X$ is $0,$ can we get some information about the rational homology of $X/G,$ where $G$ is a finite group? Thank you very much for the answers!
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I've somewhat got this question down but I'm only half way.
How many $7$-digit even numbers less than $3,000,000$ can be formed using all the digits $1,2,2,3,5,5,6$?
So I figured that there's about $4$ possible approaches
$1$ _ _ _ _ _ $6$
$1$ _ _ _ _ _ $2$
$2$ _ _ _ _ _ $6$
$2$ _ _ _ _ _ $2$
How do I fill in the ... |
Here's the setup: Let $U,V$ open $\subset \mathbb{R}^n$, and let $u:V\rightarrow \mathbb{R}$ be harmonic, and $v:U\rightarrow V$ be conformal, i.e. $v$ is $C^1$ and the Jacobian $J_v(x)$ is a scalar multiple of an orthogonal transformation for all [; x\in U ;].
I'm trying to prove $u\circ v$ is harmonic. [I've seen t... |
I came across the two congruences that for $p$ an odd prime,
$$1^2\cdot 3^2\cdot 5^2\cdots (p-2)^2\equiv (-1)^{(p+1)/(2)}\pmod p$$
and
$$2^2\cdot 4^2\cdot 6^2\cdots (p-1)^2\equiv (-1)^{(p+1)/(2)}\pmod p$$
What is the reason for this? It seems to me to be closely related to Wilson's Theorem, and possibly Fermat's lit... |
I forget for a while, we don't need the compactness condition here right?
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If I want to find the number of all possible groupings of a set of numbers, including all groupings with only a single number and groupings will all numbers is there a better method than this. For this use I don't want to count empty groups and order is important.
Take the summation of the number of permutations with r... |
Let $M$ be a $2n$ dimensional manifold, and let $A$ be a $n$-dimensional cycle on $M.$ I want to compute the self-intersection $(A.A)$ of $A$ with itself. Let $\eta_A$ be the form in $H^n(M, \mathbb{R})$ given by the Poincare duality isomorphism, i.e. $\int_B \eta_A=(A.B)$ for an $n$-cycle $B.$ Then by the rule that in... |
I have a terrain, which is represented by one mesh with a lot of polygons as shown below:
This terrain will be cut by a plane at a certain level. So there are volumes of the terrains that are located above the plane ( cut volume), and volumes that are located below the plane ( fill volume).
The question is, how do I... |
How can we prove that the inverse of an upper (lower) triangular matrix is upper (lower) triangular?
|
Problem
$N$ point charges are distributed in the unit ball in $\mathbb{R}^k$, $k=2,3$. Given locations of the particles $x_1,\ldots,x_N$ the potential energy is
$E=\sum_{j=1}^{N-1}\sum_{k=j+1}^N |x_j-x_k|^{-1}$
where $|x_j-x_k|$ is Euclidean distance between $x_j$ and $x_k$. I'm interested in both the minimal value of ... |
This paper arxiv.org/pdf/0911.1933 discusses, regarding the irrationality of certain trigonometric functions. Recently, i encountered this problem which says states the given function, $$ f(n)=\frac{1}{\pi} \arccos{\frac{1}{\sqrt{n}}}$$ is irrational for every odd $n \geq 3$. But i couldn't find the proof anywhere. Can... |
Everyone in this community I think would be familiar with International Mathematical Olympiad, which is an International Mathematics Competition held for high school students, with many countries participating from around the world.
What's interesting to note is that many of the IMO participants have gone to win the F... |
Let $p : E \rightarrow B$ be a cover s.t. Aut(E) acts transitively on $p^{-1}(b)$ for some fix $b \in B$. Then $E/Aut(E)$ is homeomorphic to $B$ where $Aut(E) \subset Cov(E,E)$. The problem is that I don't even understand how the map $~E/Aut(E) \to B~$ is defined, would someone please give me this definition?
Apparent... |
I am obsessed with the nine point circle. I was thinking, is there a generalisation to aribtrary tetrahedra and spheres? What about higher dimensions? For each face of the tetrahedron, there is a nine point circle. Do these circles all lie on a sphere?
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Let $S$ be a $(n-1)$-submanifold of a $n$-manifold $M$ and that be a submanifold of $(n+1)$-manifold V. (All of the manifolds are assumed orientable) Let $V$ carry a metric of signature $(1,n)$. Using terminology as in Einstein's Theory let $n$ be a future directed unit normal to $M$ and $m$ be an unit normal to $S$ po... |
Let $C$ be a small pre-additive category. Let $R(C)$ denote its category ring, that is,
$$
R(C)=\bigoplus_{a,b\in \mathrm{Ob}(C)} C(a,b)
$$
as Abelian group, where the direct sum runs over all object $a$, $b$ of $C$. The multiplication in $R(C)$ is given by composition of composable morphisms and 0 for uncomposable mor... |
Ok this is a bit of an unanswerable question, but hopefully someone will answer. As I have been going through college & high school there has been a kind of "path" through which you learn math. For example I went through basic math, prealgebra, algebra, precalc, calculus 1-3. My question is where to go after calculus t... |
Let $A \subset \mathbb{C} $ be an open set containing the closed unit disc.
Let $f$ be an analytic function from $A$ to $\mathbb{C}$ such that $|f(z)|=1$ if $|z|=1$.
Does it follow that $f(z) = a z^{n} \frac{cz^{m}-b}{1-cz^{m}\bar{b}} $ for some $a,b,c \in \mathbb{C}$ s.t. $|c|=1$ $|a|=1$, $|b|<1$ and some $n,m\ge 0 $?... |
In real analysis, we can define a line integral, but we also define (earlier) the regular definite integral.
Why is it that in complex analysis we are interested only in a line integral?
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Let's assume $X,Y$ are arbitrary sets, and $f$ is a function between them. What must those sets, and $f$, satisfy, in order for us to be able to define an integral of $f$ in a way that "makes sense"?
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I've read through the elementary proof of why there exists a solution $x$ to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$ for $p$ an odd prime. Is there a group theory generalization for this fact as well?
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Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ that is given by $Tf := f * g$ where $g$ is in $L^2$.
How do I now find that the spectrum of $T$ is equal to the essential range of $\hat{g}$? How is the spectrum of $T$ related to the invertibility of the operator $G\hat{f} = \hat{f}\hat{g}$?
The ha... |
Define a binary step-order set as a finite set S of positive integers si, i = 0 to n-1, where si ≡ 2i (mod 2i+1) for each i from 0 to n-1. So, i is the power of 2 appearing in the prime factorization of si. It’s elementary that binary step-order sets have distinct subset sums.
What is the least upper bound on the numbe... |
In the book Introduction to Linear Optimization by Bertsimas Dimitri, a polyhedron is defined as a set $ \lbrace x \in \mathbb{R^n} | Ax \geq b \rbrace $, where A is an m x n matrix and b is a vector in $\mathbb{R^m}$. What it means is that a polyhedron is the intersection of several halfspaces.
A ball can also be view... |
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