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I got a problem. I don't know how to solve this kind of problem:
Find an integer $n_0$ that for each $n>n_0$ this inequation would be always true:
$$\dfrac{n^{4}-n^{2}+1}{n^{3}-n}>10000 $$
I don't know how to solve this kind of problems, and I hope you could help me understanding how to proceed step by step.
Thank you... |
http://arxiv.org/pdf/math/0205003v1
In around equation (1.1) the author says
"By necessity all authors have been led in one way or another to the natural approximation
$$F(n) := \sum_{a=1}^n \mu(a) \rho_a$$
which tends to $-\chi$ both a.e. and in L1 norm when restricted to (0,1) but which has been shown to diverge in H... |
It is well known that the Euler totient function $\varphi$ satisfies the formula $n = \sum_{d | n}\varphi(d)$. This follows for example from the fact that $\mathbb Z / n \mathbb Z$ can be written (as monoid) as the disjoint union $\coprod_{d|n} (\mathbb Z / d \mathbb Z)^\times$.
My question is now if this can be gener... |
My problem is proving $x^2 - 13y^2 = 1$ has integers solutions. I can find easily see that (+-1, 0) are trivial solution. My question is: is it sufficient to complete the proof? How can I approach this problem?
Thanks,
Chan
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The negation of :
∃x ∀y (P(x,y) ⇒ Q(x,y))
is:
∀x ∃y ¬(P(x,y) ⇒ Q(x,y))
But I am not sure about the last part (¬(...)). Is that negation well done, in the sense that couldn't be done more concisely?
|
This is a question from my homework for a real analysis course. Please hint only.
Let $M$ be a compact metric space. Let $\mathbb{K}$ be the class of non-empty compact subsets of $M$. The $r$-neighbourhood of $A \in \mathbb{K}$ is
$$ M_r A = \lbrace x \in M : \exists a \in A \text{ and } d(x,a) < r \rbrace = \bigcup_{... |
Let $f, g: S^{n-1} \to X$ be a pair of homotopic continuous maps. Let $X \cup_f B^n$ and $X \cup_g B^n$ be the respective adjunction spaces (pushouts of $B^n \hookleftarrow S^{n-1} \rightarrow X$). I need to show that $X \cup_f B^n$ and $X \cup_g B^n$ are homotopy-equivalent.
I can just about convince myself that it's ... |
I've got $8n^2 \lt 64n\log(n)$ and I need to find the $n$ range if $n\gt 0$ to satisfy the inequality.
|
I need to prove the following statment (actually a special case of it).
Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $I$ an ideal of $R$. Then
$\operatorname{grade}(I,M)\geq 2$ if and only if the homomorphism $M\rightarrow$Hom$_R(I,M)$ given by $m\mapsto(i\mapsto im)$ is an isomorphism.
This is Exercis... |
Problem: Given ${a_n}$ and that 2008 and 2009 are partial limits of ${a_n}$. Also, for all $n$, $|a_{n+1} - a_n| \le 1/2$. Prove that ${a_n}$ has at least 3 partial limits.
I attempted to prove it using the definition of partial limits of a sequence and assuming that there are only two partial limits and show some cont... |
Playing with integrals on the form $$\int \frac{1}{1-x^n}\,dx$$ I noticed that for odd values of n > 5, it doesn't appear to be possible to express the integral as a closed-form expression. Is this true? How can I prove it?
|
How many string exist of length 2M with 9 characters and without repeating a character more than M times? You can suppose that M is greater than 4.
I know that my english is really bad so i'll give you some examples.
Suppose M=5.
AABBCCDDEE is a valid string
AAAAAADEEE is not a valid string (A is repeated more then 5 t... |
Let $S$ be a finite sequence of positive integers of length $n$. Let $G$ be a finite graph containing $n$ not null edgeless subgraphs. Call these subgraphs $G_0,G_1,\ldots,G_{n-1}$ and let each $G_i$ have order $S_i$. Define the edges of $G$ such that, for all $i>0$, $G_i$ forms a complete bipartite graph with $G_{i-1}... |
The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, which satisfy the following operations $+$ and $\cdot$:
$ (x_1,y_1) + (x_2,y_2) = (x_1+x_2,y_1+x_2) $
$(x_1,y_1)(x_2,y_... |
I have some random integers separated by space read from a text file like
99 20 30 1 100 400 5 10
I have to find a sum from any combination of these integers that is closest to a specifiq number like
183
what is the fastest and accurate way of doing this?
|
I need to estimate $\pi$ using the following integration:
$$\int_{0}^{1} \!\sqrt{1-x^2} \ dx$$
using monte carlo
Any help would be greatly appreciated, please note that I'm a student trying to learn this stuff so if you can please please be more indulging and try to explain in depth..
|
I'm trying to assemble a calculation that takes a known number of items (30) and places them into a set number of containers (3) and to calculate that items 1-10 go into container 1, 11-20 go into container 2 and 21-30 go into container 3... over time the number of items will grow infinitely and the number of container... |
I've attempted to solve the problem, but I got $\langle \frac{1}{\sqrt{\frac{29}{4}}}, \frac{5}{\sqrt{\frac{29}{4}}}\rangle$, which is incorrect. There is not a similar problem in my textbook that I can reference.
I know that to find a unit vector, we first find the length/magnitude of the given vector, and multiply $$... |
Hello :)
I hit a problem.
If $\sin x + \cos x = \frac{\sqrt{3} + 1}{2}$, then how much is $\tan x + \cot x$?
|
I am trying to prove a result, for which I have got one part, but I am not able to get the converse part.
Theorem. Let $R$ be a commutative ring with $1$. Then $f(X)=a_{0}+a_{1}X+a_{2}X^{2} + \cdots + a_{n}X^{n}$ is a unit in $R[X]$ if and only if $a_{0}$ is a unit in $R$ and $a_{1},a_{2},\dots,a_{n}$ are all nilpotent... |
Given the Butcher table $$ \begin{array}{ccc} \Theta & \Big | & \Theta \\\ --&--&-- \\\ & | & 1\end{array}$$
I would like to
a) write down the corresponding evolution $\Psi$
b) find the stability function $S(z)$ of the "$\Theta$-method"
Somehow, I end up getting the Euler method but I don't know why. I would be very gr... |
I've recently been studying the functional analysis and I think I need some help with one exercise in the book.
We have the Lusin's theorem, which is stated the following way:
Given some measurable $E$ where $\mu(E)<\infty$ and some function $f$, which is measurable and finite almost everywhere in $E$, the following st... |
Say X and Y are normally distributed random variables, i.e., X,Y ~ N(0,1). For some fixed t and c I know that Pr[X≥t]=c and Pr[Y≥t]=c. Moreover, I also know that X and Y have covariance 0.999.
How can I deduce what Pr[X≥t and Y≥t] roughly is?
Bonus question: what are good references to learn more on related topics for ... |
Is the order of random variables important in the chain rule? I mean, is this true: $P(A,B,C) = P(A)\times P(B|A)\times P(C|A,B) = P(C)\times P(B|C)\times P(A|B,C) = P(C,B,A)$? If it is, what is the meaning of such order? Thank you.
|
Is there some easy way to compute number of particular permutations? Namely let $S_i$ be the number of permutations $(a_1,...,a_n)$, $\{a_1,\ldots,a_n\}=\{1,...,n\}$, satisfying the following conditions:
$a_i=n$ for some $i\in \{1,\ldots,n\}$.
$a_1<a_2<\ldots <a_i$.
Compute $\sum_{i=1}^n S_i$.
|
I would like to know if there a closed form solution for the sum:
$$ S_n(t) = \sum_{k=0}^{n} \cos( t \sqrt{k} ) $$
There is obviously an easy answer when the sum is replaced by an integral so this question is really asking for the exact form that $S_n(t)$ takes.
Any hints or references on how to approach this problem w... |
It is well known that, if HM denotes the harmonic mean and AM the arithmetic mean, we have
$$ AM(x) \ge HM(x) $$
Now I am dealing with the expression
$$ \frac{1}{HM(x)} - \frac{1}{AM(x)} $$
A trivial lower bound for this expression is $0$, but is there also a nice upper bound?
Cheers!
EDIT: Or, if there's no general u... |
Find all functions $f:R\rightarrow R$ which satisfy $f(x+f(y))=f(x-f(y))+4xf(y)$ $\forall x,y \in R$.
I strongly suspect $0$ and $x^2+C$ to be the only solutions but, as is almost the case with functional equations, finding the set the solutions is easy; it is proving that a certain set of solutions represents ALL sol... |
Inputs:
1) A set of $M \times N$ matrices, $\text{{$A,B,C...N$}}$ containing only integers.
2) A single $1 \times N$ matrix of floats, $W$ (weights).
I need to pull one row from each input matrix and sum values for each column, then take the dot product of the resulting $1 \times N$ matrix with $W$, yielding a sc... |
I'm having some trouble with the formatting here, so here's a screenshot of the problem we were given:
Let $\{ A_n \}_{n \geq 1}$ be a sequence of sets. Show that
$$
\lim\sup\limits_{n\to \infty} \mathbf{1}_{A_n} - \lim\inf\limits_{n\to \infty} \mathbf{1}_{A_n}
=
\mathbf{1}_{\{\lim\sup_{n \to \infty} A_n \backsl... |
There is not an example problem in my textbook that I can reference (I have said this on two other questions so far, but it's the truth. The example problems and solutions manual contain physics / engineering examples rather than abstract examples).
For this one, I did not know how to begin. Normally I would give the w... |
A and B throw a fair $6$-sided die in turn. The die is not regular because the faces are ${1,2,2,4,4,6}$. The winner is whoever first gets a cumulated sum equal or greater than $10$. Ask for the probability that A wins, and the expected number of throws when the game generates a winner.
|
I'm stuck in two parts of Ahlfors' proof of Cauchy's theorem in a disk (page 113), that is, if $f$ is holomorphic in an open disc $D$ then $\int_\gamma f(z)dz=0$ over every closed curve $\gamma$ in $D$.
First part:
Fix $z_0\in D$. We define $F(z)=\int_\sigma f(\zeta)d\zeta$ where $\sigma$ is the path joining $z_0$ with... |
Suppose $G$ is a compact connected Lie group and let $\{X_i\}$ be a basis for its Lie algebra $\mathfrak g$. We know that the exponential $\exp:\mathfrak g \to G$ is surjective but when is it the case that $G$ is generated by $\{\exp(tX_i) : t\in \mathbb R\}$?
|
Suppose I wanted to solve (for x and y) an equation of the form $x^2-dp^2y^2=1$ where d is squarefree and p is a prime. Of course I could simply generate the solutions to the Pell equation $x^2-dy^2=1$ and check if the value of y was divisible by $p^2,$ but that would be slow. Any better ideas?
It would be useful to ... |
Given a list of point cloud in terms of $(x,y,z)$ how to determine abnormal points?
The motivation is this. We need to reconstruct a terrain surface out from those point cloud, which the surveyors obtain when doing field survey. The surveyors would take an equipment and record a sufficient sample of the $x,y,z$ of a t... |
Is there such a thing as a countable set with an uncountable subset?
Actually I know the answer. Well, I believe I know the answer, which is NO.
Unfortunately, the professor in a Theory of Computation class said that yes, there is such a subset.
This is to settle a discussion with fellow students. A discussion that i... |
I'm in the process of reading my first Linear Algebra textbook, and was just wondering...Is the standard basis of a vector space in n dimensions equivalent to the row space of the n x n identity matrix?
|
8 cubes can form a torus by gluing them together face-to-face, with each cube sharing a face with each of two other cubes, by torus I just mean a closed loop with a hole. Is the same possible with the dodecahedron? What is the minimum number of dodecahedrons needed for this?
Edit:
Is it possible with the tetrahedron ... |
I have two polygons $P_1 \subseteq P_2$ in the plane, and I would like to determine to which polygon a given point $p\in P_1 \setminus P_2$ is "closest", by means of the following measure: $$r(p)=d(p,P_1)/d_p(P_1,P_2),$$ where $d(p,P_1)$ is the distance between $p$ and $P_1$ and $d_p(P_1,P_2)$ is the distance between $... |
Given a polytope of dimension $n$, is there some general way to determine if it can tile $\mathbb{R}^n$?
|
It is known that the solution of the linear ODE $\frac{d\vec{y}}{dt}=A\vec{y}$ is the span of the columns of $e^{xA}$. This gives us a formula for solving the ODE by solving $e^{xA}$. However, is the reverse also possible? That is, can I calculate $e^{xA}$ by solving the ODE?
|
I have two equations. In this example I'll make them linear, and later see if the solution generalises to different forms of equation:
$$y_1=-0.1x + 0.65$$
$$y_2=0.15x - 0.825$$
I'm interesting in small ranges of these graphs, say 0.5-2.5 of $y_1$ and the same range +7 of $y_2$. We assume the value of the equation outs... |
Given a vector V of n integers and an integer k, k <= n, you want a subvector (a sequence of consecutive elements of the vector ) of maximum length containing at most k distinct elements.
The technique that I use for the resolution of the problem is dynamic programming. The complexity of this algorithm must be O(n*k).
... |
I am wondering how to show that the ideal (x,y) is prime and maximal in $\mathbb{Q}[x,y]$.
|
Given any quadratic equation of the form $y=ax^2+bx+c$, I want to find the minimum value for a specific range of $x$.
My programmer brain can do it in a branchy, algorithmic way as follows, but is there a more elegant solution?
if the $a$ coefficient is positive,
and the end of my range is before the lowest point of ... |
I am relatively very new to probability distribution and after understanding the concept of Expected value of a discrete random variable,I am trying to understand the variance of the same here is an extract from my book:
$Var(X) = E[(X - E(X))^2] = \sum_{\text{all }X} (X - E(X))^2 P_x(X) = E(X^2) - (E(X))^2$
What my ... |
A box contain $A$ white and $B$ black balls and $C$ balls are drawn, then the expected value of the number of white balls drawn is ?
The answer is $\large \frac{ca}{a+b}$. How to approach this one?
|
Consider a given vector $a$ and scalar $d$. What is the set $X$ such that for any $x \in X$ their dot product equals $d$ : $\forall x \in X: x \cdot a = d$ ?
|
Find the sum of the series:
$ \displaystyle \cdots + \frac{1}{z^{3}} + \frac{1}{z^{2}} + \frac{1}{z} + 1 + z + z^{2} + z^{3} \cdots$
This series can be summed in the following way:
$$\cdots + \frac{1}{z^{3}} + \frac{1}{z^{2}} + \frac{1}{z} + 1 = \frac{z}{z-1}$$ and $$ z + z^{2} + z^{3} + \cdots = \frac{z}{1-z}$$
So t... |
I came across this puzzle at some online contest(now completed)
Alexander, son of Phillip of Macedon, has ascended the throne of his father following his assassination. In these tumultuous times, he appoints you as head of the architectural division of his mighty army. The general of the army wants all the catapults t... |
The question is simple. I would like to implement the Gamma function in my calculator written in C; however, I have not been able to find an easy way to programmatically compute an approximation to arbitrary precision.
Is there a good algorithm to compute approximations of the Gamma function?
Thanks!
|
Let $R$ be a discrete valuation ring with field of fraction $K$ and residue field $k$ and let $K'$ be a finite and separable extension of $K$.
If $R$ is henselian ("Hensel's lemma holds", e.g. if it is complete) and $k$ is algebraically closed then the extension $K'|K$ is totally ramified (from Hensel's lemma if follow... |
How to determine all the ideals of the ring $\mathbb{Z}[X]/(2,X^3+1)$?
|
A naive question from a physicist, so forgive the lack of rigor. Consider a Lie group, acting on its Lie algebra by the adjoint action. Does every orbit go through the Cartan subalgebra? Alternatively, for which Lie groups is this true? (I view the statement as a generalization of the elementary fact that every Hermiti... |
Problem: Given $f:[0,1] \rightarrow \mathbb{R}$ ($f$ continuous ) and $f(x^2) = f(x)$ $\forall x \in [0,1]$. Show that function $f$ is a const.
|
Prove that there is no arithmetic progression that consists only of primes.
A question that I've been set; I'm guessing it makes use of primes being written in the form 4k+1 and 4k+3? Not sure where to start.
Thanks.
|
Let $G = GL_n(\mathbb{R})$ be the general linear group. Does $G$ have any subgroups of finite index?
Thanks!
|
A probability problem was posed to me, and I am terrible at probability. It was first posed to me like this:
I have 5 coins. One of them is a
two-headed coin, and the others are
normal coins. I pick a coin, and then
I flip that coin three times; each
time is heads. What is the probability
that the coin is tw... |
I am currently studying for an Algorithms class. In my notes from class and in the my textbook, there is not a similar example I can reference. I have worked on the problem and gone to tutoring, but I can not seem to get the correct answer.
Can anyone give me any tips on how to solve this problem?
Evaluate the summati... |
A Stirling number of the second kind, $S (r,k)$, is defined to be the number of ways one can partition an $r$-element set into $k$ subsets.
Consider the following problem:
You have $r$ distinguishable balls and $n$ undistinguishable boxes. If you are allowed to place balls into these boxes with no restrictions, how m... |
So I'm trying to solve this problem stated like this:
Using Green's Theorem, find the area of the elipse defined by (where $a,b \gt 0$):
$$\frac{x^2}{a^2} + \frac{y^2}{b^2} \leqq 1$$
I'm having trouble doing this,
$$\int_{FrD} \!F \cdot T = {\int\int}_D \left(\frac{df_2}{dx} - \frac{df_1}{dy}\right)$$
Where $F$ is a ... |
Let $S_n = \frac{1}{n}\sum_{i=1}^n X_i$, and $T_n = \frac{1}{n}\sum_{j=1}^nY_j$, where
The $X_i$ are iid, the $Y_i$ are iid (with a different law)
$X_i$, and $Y_j$ are dependent for $i =j$.
For $i\neq j$, $X_i$ and $Y_j$ are independent.
Is there a central limit type result for $S_n^2 - T_n^2$?
|
For $\alpha \geq 0$ the transformation $x \mapsto \log(x) - \alpha \log(1-x)$
maps the unit interval to the real line (in fact for $\alpha = 0$ the transformation is not surjective). For $\alpha=1$ this is the logit transformation, which has a well-known inverse. It is also not difficult to find an inverse for $\alpha ... |
I was reading a paper, and I encountered a definition of some concept. The definition was of the form:
(+) $\ \qquad\qquad (\exists x) \phi(x) \quad \Rightarrow \quad (\exists y) \psi(y)$
where $\phi$ and $\psi$ are two formulae.
I was wondering if the author's could write the above formula in the form:
(++) $\qquad\q... |
Let $R$ be the relation defined on $\mathbb{Z}$ where $a\; R\; b$ means that $a + b^2 \equiv 0\pmod{2}$.
How would I go about finding the equivalence class $[-13]$?
|
Find all continuous $f:[0,1] \rightarrow [0,1]$ such that $f(1-f(x))=f(x)$.
|
I am trying to learn graph theory and the linear algebra used to analyse graphs. The texts I have read through have lots of lemmas and theorems proved. The proofs are convincing but I fail to see the intuition behind them. How do they connect to the properties of the graphs?
I understand eigenvectors very clearly when ... |
I'm trying to figure out a lecture example given on our Analysis course. We are currently going through Riemann integrals.
Let $g:[0,1] \to R, g(x) = 1$ when $x \in [0, \frac{1}{2}]$ and $g(x) = 2$ when $x \in ]\frac{1}{2},1]$.
Is g integrable?
The example goes on to prove that g is indeed integrable by choosing $P_n =... |
In Milnor's famous book "Topology from the Differential Viewpoint" he proves the following on page 11:
If $f: M\to N$ is a smooth map between manifolds of dimension $m\geq n$ and if $y\in N$ is a regular value, then the set $f^{-1}(y) \subset M$ is a smooth manifold of dimension $m-n$.
Right, so I've read this proof p... |
I was working out some problems. This is giving me trouble.
If $p$ is a prime number of the form $4n+1$ then how do i show that:
$$ \sum\limits_{i=1}^{p-1} \Biggl( \biggl\lfloor{\frac{2i^{2}}{p}\biggr\rfloor}-2\biggl\lfloor{\frac{i^{2}}{p}\biggr\rfloor}\Biggr)= \frac{p-1}{2}$$
Two things which i know are:
If $p$ is ... |
A sigma algebra is closed under
countable intersections.
This is a property of a sigma algebra, but I don't understand what this means in plain English. Can someone explain this to me in terms a n00b like me would understand? The part I don't understand the most is what "closed" means in this quote.
Addendum:
The... |
If I compute the eigenvalues and eigenvectors using numpy.linalg.eig (from Python), the eigenvalues returned seem to be all over the place. Using, for example, the Iris dataset, the normalized Eigenvalues are [2.9108 0.9212 0.1474 0.0206], but the ones I currently have are [9206.53059607 314.10307292 12.03601935 3.5303... |
Shafarevich offers the following theorem-definition:
"At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every rational function $u$ that is not identically $0$ on the curve can be written in the form $u = t^k v$, with $v$ regular at $P... |
Maybe this is well-known, but suppose you have an invertible sheaf $\mathcal{L}$ on a scheme $X$. If $X$ is a projective space (or even projective bundle over an integral scheme, by a similar argument...I think), then if $H^0(X, \mathcal{L})\neq 0$ and $H^0(X, \mathcal{L}^\vee)\neq 0$ we get that $\mathcal{L}\simeq \ma... |
In the book "Structure of Groups of Prime Power order (Charles Richard Leedham-Green, Susan McKay), there is an exercise (2.1.10), which asks to show that the automorphism group of ($\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes \mathbb{Z}_p$ is ($\mathbb{Z}_p \times \mathbb{Z}_p) \rtimes GL(2,p)$. To prove this, first, i... |
Someone asked me today, "Why we should care about groups at all?" I realized that I have absolutely no idea how to respond.
One way to treat this might be to reduce "why should we care about groups" to "why should we care about pure math", but I don't think this would be a satisfying approach for many people. So here... |
I need to demonstrate that Ker(AB)=${\vec{0}}$, just like Ker(A) and Ker(B). I have a not-so-elegant way of showing that (I think): A and B are n * p and p * m, and it has to be the case that n $\geq$ p $\geq$ m, and rref's of A and B are either equal to the identity matrix, or are of form I and row(s) of 0's on the bo... |
If $f \colon [a,b] \rightarrow [c,d]$ is a bijection, $f\in \mathcal{R}$ and $f^{-1}$ exists, then prove or disprove that $f^{-1} \in \mathcal{R} [c,d]$.
Remark: I tried to use integration by parts to find $\int_{c}^{d} f^{-1}$ and to prove that was the right limit of Riemann sum, but failed. But I think this idea migh... |
In neutral plane geometry, Euclidean geometry without the parallel postulate, I want to show that the family of parallel lines all perpendicular to a given line pass through all of the plane, leaving no holes.
My present formulation of this idea is as follows: Given two lines $l$ and $m$ and a transversal $t$ perpendic... |
A subspace V in $R^4$ is defined by the equation $x_{1}$-$x_{2}$+$2x_{3}$+$4x_{4}$=0. I need to find T such that Ker(T)=zero vector, and Im(T)=V. How do I approach this problem? As I understand, the equation given to me is a set of points that are solutions to Im(T)=V, so in a sense they vectors of that plane are eleme... |
$N$ leaves a remainder of $4$ when divided by $33$, what are the possible remainders when $N$ is divided by $55$?
My approach :
Number has to be $33k+4$
Possible remainders will be $(33k+4)/55 = 33k/55 + 4/55 = \text{remainder of $\left(3k/5 + 4/55\right)$} =
\text{remainder of $(3k/5)$ + Remainder of $(4/55)$}$
Since... |
I've been trying to prove that every Polish Space is homeomorphic to a $G_\delta$ subspace of the Hilbert Cube. There is a hint saying that given a countable dense subset of the Polish space $\{x_n : n\in\mathbb{N}\}$ define the function $f(x)=(d(x,x_n))_{n\in\mathbb{N}}$ (where $d$ is a metric on the Polish space). I ... |
I have a theory that in order to best compress 1D data between 0 and 1 in a lossy way, you should map each data point $i$ to:
$\int_0^i \! P(x) \, \mathrm{d}x$
Where $P(x)$ is the probability density function of the data. The probability density function over the mapped data should be level and being level when you sto... |
Assume that there are n snakes. Any 2 ends (tail or head) of the "2n" available have to be picked up and tied together and this process has to be repeated infinitely.
If p/q (gcd(p,q) = 1) is the probability of you getting a single long “snake” in the end, what would be the sum of (p+q) for all 2 <= n <= 40?
|
Does there exists other structures in math, which are seemingly random, but deterministic, and follow rules similar to the prime numbers, by rules I mean there must be statements similar to goldbach's conjecture or twin-prime-conjecture etc, for instance i dont consider the digits of pi to have this sort of structure. ... |
What is $Aut_{Modk}(A)$ if A is a finite generated algebra, k is a field?
Is it a group of automorphisms of A as a vector space over k or what?
|
I have a dataset of trajectories. These trajectories are represented in 3D space (x,y,z). All trajectories of this dataset are similar in their shape, but they are not exactly the same, I mean, there is some variation along the points. The trajectories are nonlinear.
What I need is a kind of regression (polynomial?) on... |
I am learning algorithms, and there is a chapter which uses linear optimization methods to solve a matching problem. This is the problem definition:
I find the abbreviations AP for the constraints of the primal problem and APD for the constraints of the dual problem confusing. Could somebody please explain what they... |
Considering the construction of a matrix as follows.
The $n$th row in the matrix is filled with the coeffcients of $x^r$ in the expansion of $(1+x)^n$ from the columns $2n$ to $3n$ inclusive and circle all the numbers that are divisible by $n$ in the same row
How would I find the number of columns for which all the el... |
How can I calculate $a^{(P-1)/2}\pmod{P}$?
for example $3^{500001}\bmod{1000003}$ given that $1000003$ is prime.
I know that if we square the number $3^{500001}$ the result will be either $1$ or $-1$ modulo $1000003$.
but my question is how to continue?
I have this question as bonus in previous exam.
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Demonstrate by induction: $(1+a)^n ≥1+an$ is true, given a real number $a$, for any $n ∈ \mathbb N$. With $a > 0$
I need to demostre this using the induction principle. My doubt is in the second part of the demonstration.
This is what I have:
In order to demonstrate the predicate these two points must be true:
$P(1)$ ... |
I want to show some users of a piece of software some solutions of one-dimension boundary value problems (can also be initial value problems). I'm after a collection of problems whose solutions are very interesting or the BVP are of a particularly interesting nature. The problems can be linear or non-linear. Does anyo... |
What's $\mathrm{Spec}\mathbf{Z}[\mu_{p-1},\frac{1}{(p-1)p}] \cap \mathbf{Z}_p$, where $\mu_{p-1}$ denotes the $(p-1)$-th roots of unity and $\mathbf{Z}_p$ the $p$-adic integers?
It should be $\mathbf{Z}[\mu_{p-1}]$ without (the places above the primes dividing $p-1$ and all but one places above $p$).
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How can I use prime avoidance to prove that if $R$ is a Noetherian ring and $J\subset R$ an ideal with $\mathrm{height}(J)=n$ then there exists $x_1,\ldots,x_n\in J$ such that $\mathrm{height}((x_1,\ldots,x_i))=i$ for all $i=1,\ldots,n$?
I know that if $R$ is local and Noetherian of dimension $d$ then there exist $x_... |
Is it possible to form a closed loop by joining regular (platonic) tetrahedrons together side-to-side, with each tetrahedron having two neighbours? It should be a loop with a hole in, as can be done with 8 cubes, or 8 dodecahedrons as shown below. What is the minimum number of tetrahedrons needed?
Edit:
Could it be th... |
can someone tell me a way of integrating functions like $e^{f(x)}$
I have a specific case: $\int e^{-3x}\,\mathrm{d}x$
PS: I'm not looking for the answer of this, but the way of doing it.
Thanks for your help
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I found an interesting topic on this site with regards to the series I am trying to evaluate:
Summing $\frac{1}{e^{2\pi}-1} + \frac{2}{e^{4\pi}-1} + \frac{3}{e^{6\pi}-1} + \cdots \text{ad inf}$
I was wondering if there is a closed form for even m when we have:
$$\sum_{n=1}^{\infty}\frac{n^{2m-1}}{e^{2\pi n}-1}$$
I have... |
I was recently wondering where the word `kernel' comes from in mathematics. I am sure the internet must know. I did manage to find
http://www.pballew.net/etyindex.html#k
which contains the origin of many math words but it doesn't have `kernel'; it also only seems to address math words you might run into in at most a... |
I've been giving a public talk about Art and Mathematics for a few years now as part of my University's outreach program. Audience members are usually well-educated but may not have much knowledge of math. I've been concentrating on explaining Richard Taylor's analysis of Jackson Pollock's work using fractal dimension,... |
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