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H: 1-form is exact
Let $\omega = f_1 dx_1 +f_2dx_2 + \cdots + f_ndx_n$ be a closed $ C^{\infty}$ $1-$form on $ \mathbb R ^n$. Define a function $g$ by
$\displaystyle{ g(x_1, x_2,\cdots, x_n) = \int_{0}^{x_1} f_1(t,x_2 , x_3, \cdots ,x_n) +\int_{0}^{x_2} f_1(0,t, x_3, x_4, \cdots ,x_n) + \int_{0}^{x_3} f_1(0,0,t, x_4 ... |
H: Calculate: $\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$
Calculate the following sum for integers $n\ge2$:
$$\sum_{k=0}^{n-2} 2^{k} \tan \left(\frac{\pi}{2^{n-k}}\right)$$
I'm trying to obtain a closed form if that is possible.
AI: We have this nice identity $$\tan(\theta) = \cot(\theta)-2 \cot(2 ... |
H: Eigenvalues and Diagonalization
The typical definition given for a diagonalizable matrix is:
Given $A\in M^F_{n\times n}$
$A$ is diagonalizable $\iff$ A has $n$ linearly independent eigenvectors.
Is it also true that
$A$ is diagonalizable $\iff$ A has $n$ unique eigenvalues.
AI: No. It is not true that $A$ is diag... |
H: Expectation value of a product of an Ito integral and a function of a Brownian motion
this problem has come up in my research and is confusing me immensely, any light you can shed would be deeply appreciated.
Let $B(t)$ denote a standard Brownian motion (Wiener process), such that the difference $B(t)-B(s)$ has a n... |
H: Quantile and percentile terminology
Note: This is answered by user974514 below, but there was some discussion outside of the "answer", so I paraphrased the final answers inline here.
I've asked around for the exact usages of the terms "quantile" and "percentile" and "rank" and I'm getting conflicting answers from m... |
H: How to define a bijection between $(0,1)$ and $(0,1]$?
How to define a bijection between $(0,1)$ and $(0,1]$?
Or any other open and closed intervals?
If the intervals are both open like $(-1,2)\text{ and }(-5,4)$ I do a cheap trick (don't know if that's how you're supposed to do it):
I make a function $f : (-1,... |
H: Compute: $\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$
I try to solve the following sum:
$$\sum_{k=1}^{\infty}\sum_{n=1}^{\infty} \frac{1}{k^2n+2nk+n^2k}$$
I'm very curious about the possible approaching ways that lead us to solve it. I'm not experienced with these sums, and any hint, suggestion ... |
H: Equality of Voronoi diagram
What can we say about two sets $A$ and $B$ if both of them have the same Voronoi diagram.
First, I thought if the Voronoi diagram are equal so the sets also should be equal, but by definition, Voronoi diagram is determined by distances to a specified family of objects (subsets) in the ... |
H: Is there a topology on the countable set which makes the space is not first countable but has countable pseudocharacter?
I want to know is there a topology on the countable set which makes the space is not first countable but has countable pseudocharacter?
Thanks for any help:)
AI: The pseudo-character is defined ... |
H: generalised inverse function
Let $f:\mathbb{R} \rightarrow [0, 1]$ be increasing (edit: i.e., non-decreasing).
Define $f^-(y) = \inf \{x \in \mathbb{R} : f(x) \geq y \}$, $y \in [0, 1]$.
Is the following line true?
$$x \leq f^-(y) \quad\leftrightarrow\quad f(x) \leq y$$
AI: The definition doesn't make sense for $y$... |
H: norm of a variant of Fejer 's kernel
Let $K_N$ the Fejer's kernel on $\mathbb{T}$. Let $l$ be a positive integer. Let $Q$ the function defined by
$$
Q(t)=K_N(lt).
$$
In Hewitt/Ross "Abstract Harmonic Analysis 2" page 438, I can read that if $1<p<2$ we have
$$
||Q||_{L_p}=||K_N||_{L_p}.
$$
Why?
AI: In general, if yo... |
H: How did Ramanujan get this result?
We know Ramanujan got this result
$$\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots }}}=3$$
and he used the formula
$$x+n+a=\sqrt{ax+{{(n+a)}^{2}}+x\sqrt{a(x+n)+{{(n+a)}^{2}}+(x+n)\sqrt{\cdots }}}$$
where $x=2,n=1,a=0$ ,we get the first result, but I don't know how to prove it, can you help me?
... |
H: "Theorem of Witt" for modules
For modules $M_1 \oplus N \cong M_2 \oplus N$, why is $M_1 \cong M_2$, if $Hom(M_1,N)=\{0\}=Hom(M_2,N)$?
It seams similar to Witt's cancellation theorem for quadratic forms.
Regards, Khanna
AI: Let $\phi\colon M_1 \oplus N \to M_2 \oplus N$ an isomorphism. Denote by $i_1\colon M_1 \... |
H: Quotient of a Regular local ring.
Is the quotient of a regular local ring by a prime ideal Cohen-Macaulay? If so, how can we see this, if not, is there a counterexample?
We know that a regular local ring is a UFD, so $0$ is a prime ideal, so in this case, the quotient is a regular local ring and hence a CM ring.... |
H: Question about functions in $L^2$
Let $u(x,y),v(x,y)\in L^2$. What can we say about $\int_{-\infty}^\infty \int_{-\infty}^\infty \frac{d}{dy}(uv) \, dy \, dx$? Does it equal zero? If so, why?
AI: I think your integral might not make sense. First of all, I would write
$$\lim_{a\to\infty}\left[u(x,y)v(x,y)\right]_{y=... |
H: Do filters on a Boolean algebra also make a Boolean algebra?
Let $\mathfrak{B}=(B,\bot,\top,\lnot,\wedge,\vee)$ be a boolean algebra. $B_F$ be the set of all filters on $\mathfrak B$. And for all filter $F$, $G$, $F \wedge_{B_F} G \colon= \mathbf C(F \cup G)$ in which $\mathbf C$ denotes the filter closure operator... |
H: Galois Group over Finite Field
I am having a bit of difficulty trying to answer the following question:
What is the Galois group of $X^8-1$ over $\mathbb{F}_{11}$?
So far I have factored $X^8-1$ as
$$X^8-1=(X+10)(X+1)(X^2+1)(X^4+1).$$
I know $X^2+1$ is irreducible over $\mathbb{F}_{11}$ since $10$ is not a squa... |
H: Homework help with projectile motion
I would please like help on the following question related to projectile motion.
A horizontal drainpipe 6 metres above sea level empties stormwater into the sea. If the water comes out horizontally and reaches the sea 2 metres out from the pipe, find the initial velocity of the... |
H: Multiple singular values
In a lot of texts I have seen involving the singulare value decomposition, it only says, that there are as many nonzero singular value as is the rank of the matrix $A$, which is to be decomposed.
Now I have looked at different examples throughout the net and everywhere these singular values... |
H: Has anyone ever tried to develop a theory based on a negation of a commonly believed conjecture?
I know that plenty of theorems have been published assuming the Riemann hypothesis to be true. I understand that the main goal of such research is to have a theory ready when someone finally proves the Riemann hypothesi... |
H: Dedekind complete ⇒ Sequentially complete
Let F be an ordered field with least upper bound property.
1.Let $\alpha: \mathbb{N} \to F$ be a Cauchy sequence.
Since F is an ordered field, $x$ is bounded both above and below.
2.By assumption and dual of it, $A$={$\alpha(n)$|$n\in \mathbb{N}$} has a inf $a_0$ and sup $b... |
H: what is the use of derivatives
Can any one explain me what is the use of derivatives in real life. When and where we use derivative, i know it can be used to find rate of change but why?. My logic was in real life most of the things we do are not linear functions and derivatives helps to make a real life functions... |
H: How do I handle image gradient calculation at the edge of images?
The image gradient is the rate of change over any given pixel of an image, either in the horizontal or vertical direction. An image can be thought of as a large matrix of values [0, 255]. A common horizontal matrix for taking an image gradient is
... |
H: Polynomials irreducible over $\mathbb{Q}$ but reducible over $\mathbb{F}_p$ for every prime $p$
Let $f(x) \in \mathbb{Z}[x]$. If we reduce the coefficents of $f(x)$ modulo $p$, where $p$ is prime, we get a polynomial $f^*(x) \in \mathbb{F}_p[x]$. Then if $f^*(x)$ is irreducible and has the same degree as $f(x)$, th... |
H: A sufficient condition for order isomorphism of posets?
Let $\mathfrak{A}$ be a poset. For $a, b \in \mathfrak{A}$ we will denote $a
\not\asymp b$ if only if there are a non-least element $c$ such that $c
\leqslant a \wedge c \leqslant b$.
Let $\mathfrak{A}$, $\mathfrak{B}$ are posets. I call a pointfree
funcoid a ... |
H: Can anyone give any insight on this group given these generators and relations?
$G = \langle x,y | x^3 = 1, y^3 = 1, (xy)^3 = 1, (xy^2)^n = 1 \rangle$
I am studying this group and I can't seem to get anywhere with it. I've tried making a Cayley Table but it's getting pretty big. This makes me think I'm doing some... |
H: $PGL_2(q)$ acts on $\Omega$ $3-$transitively?
Anyone who studies Permutation Groups will be encountering the following definition:
A group $G$ acting on a set $\Omega$ is said to be “Sharply m-Transitive” iff $$\forall (a_1,a_2…,a_m) , (b_1,b_2…,b_m) \in \Omega^{m};\ ∃! g \in G , a_i^g=b_i, 1\leq i\leq m$$
Wh... |
H: Division by $2p+1$
Can $\left\lfloor{\dfrac{x}{2p+1}} \right\rfloor$ be expressed in terms of $\left\lfloor{\dfrac{x}{p}} \right\rfloor$ for prime $p$?
How to divide by $2p+1$ by only using division by $p$?
EDIT:
The above formulation is wrong. I meant "expressed in terms" in a sense broader that "a function that ... |
H: Group presentation for semidirect products
If $G$ and $H$ are groups with presentations $G=\langle X|R \rangle$ and $H=\langle Y| S \rangle$, then of course $G \times H$ has presentation $\langle X,Y | xy=yx \ \forall x \in X \ \text{and} \ y \in Y, R,S \rangle$. Given two group presentations $G=\langle X|R \rang... |
H: Diagonal Lemma justification
Given the diagonal lema stated as above:
Diagonal Lema. Let $\mathfrak{T}$ be a theory wich is capable of representing the primitive recursive functions, and a codification schema for formulas in $\mathfrak{T}$ such that $\ulcorner \phi \urcorner$ is the codification of $\phi$. For all ... |
H: Lower bound for $\|A-B\|$ when $\operatorname{rank}(A)\neq \operatorname{rank}(B)$, both $A$ and $B$ are idempotent
Let's first focus on $k$-by-$k$ matrices. We know that rank is a continuous function for idempotent matrices, so when we have, say, $\operatorname{rank}(A)>\operatorname{rank}(B)+1$, the two matrices ... |
H: Countably Compact vs Compact vs Finite Intersection Property
There is this exercise: Show that countable compactness is equivalent to the following condition. If ${C_n}$ is a countable collection of closed sets in S satisfying the finite intersection hypothesis, then $\bigcap_{i=1}^\infty C_i$ is nonempty.
Definit... |
H: Changing order of summation
I would like to rewrite the sum
$$\sum_{i=1}^K \sum_{l=-\infty}^\infty \sum_{j=-\infty}^\infty f(i+lK;j-l)$$
In the form
$$ \dots\sum_{s=-\infty}^\infty \sum_{w=-\infty}^\infty f(s,w)$$
where $s=i+lK$, $w=j-l $. How do I do it?
AI: I think that the sum is exactly $$\sum_{s=-\infty}^\inf... |
H: Why is the matrix representing a non-degenerate sesquilinear form invertible?
Let's consider a finite-dimensional vector space $E$ on the field $\mathbb{K}$ (where $\mathbb{K}=\mathbb{C} \ \text{or}\ \mathbb{R}$) and a sesquilinear (or bilinear if $\mathbb{K}=\mathbb{R}$) form $q:E\times E \rightarrow \mathbb{K}$.
... |
H: Primitive element of $\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q}$
Is there a clever way to determine a primitive element of the finite extension
$$F=\mathbb{Q}(\sqrt{2}+i,\sqrt{3}-i)/\mathbb{Q} \text{ ?}$$
On simpler examples, I've been able to find one by determining all field morphisms $\sigma: F\to\mathbb{C}$ ... |
H: Probability a coin comes up heads more often than tails
I am told that a fair coin is flipped $2n$ times and I have to find the probability that it comes up heads more often that it comes up tails.
Please, how do I find the required probability?
AI: Note that we have $$P(\text{# Heads} > \text{#Tails}) + P(\text... |
H: Calculus, Problem.
$$\large f\left(x\right) = \int\limits_{\cos x}^{\sin x} e^{t^2+xt}dt.$$Compute $f'\left(0\right)$.
I can't get it right -sigh- :/
AI: If we have $$f(x) = \int_{a(x)}^{b(x)} g(t,x) dt,$$ then for "nice enough" $g(t,x)$ $$f'(x) = \int_{a(x)}^{b(x)} \dfrac{\partial g(t,x)}{\partial x} dt + g(b(x)... |
H: Eigenvectors of $P^{-1}AP$
Let $A\in M_{n}(\mathbb{C})$ and assume that $A$ is diagonalizable,
let $P\in M_{n}(\mathbb{C})$ be an invertible matrix.
My question is what are the eigenvectors of $P^{-1}AP$ ?
I think it's probably something like $P$(eigenvectors of $A)$ , but I don't remember...
I appriciate any help.... |
H: Understanding torsion from a presentation
Let $F_2 = \langle a,b \rangle$ be the free group on two generators, and for each word $w \in F_2$, let $G(w) = \langle a, b \ | \ w \rangle$. Is the following statement true?
$G(w)$ is torsion-free if and only if for all $k \geq 2$ and for all $v \in F_2$, $w \neq v^k$
In ... |
H: Minimize $\| ACE \|$ by geometrical means
I have the following figure
Where $AB=10$m, $BD=12$m and $DE=12$m. The point C can slide
along the segment BD. Now the problem is to minimize the distance from A to D
going along the dashed line. The problem can be solved using simple analysis and differentiation. Let $B... |
H: If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.
Let $R$ be a UFD and $P$ a prime ideal. Here we are defining a UFD with primes and not irreducibles.
Is the following true and what is the justification?
If $a$ in $R$ is prime, then $(a+P)$ is prime in $R/P$.
AI: The bijection between ideals of $\,R/I\,$ an... |
H: Does commutativity imply Associativity?
Does commutativity imply associativity? I'm asking this because I was trying to think of structures that are commutative but non-associative but couldn't come up with any. Are there any such examples?
NOTE: I wasn't sure how to tag this so feel free to retag it.
AI: Consider ... |
H: For any sequence in $L^2$ there is a function in $L^2$ s.t. is not orthogonal to any point of the sequence
How to prove that for any sequence $(f_n) \subset L^2[0,1]\setminus \{0\}$ there is a function $g \in L^2[0,1]$ such that
$$\int f_n g dx \neq0\ \forall n\geq 1?$$
I tried to use a weak limit of $sign(f_n)$ ... |
H: How to get $ \cot(\theta/2)$ from $ \frac {\sin \theta} {1 - \cos \theta} $?
According to wolfram alpha, $\dfrac {\sin \theta} {1 - \cos \theta} = \cot \left(\dfrac{\theta}{2} \right)$.
But how would you get to $\cot \left(\dfrac{\theta}{2} \right)$ if you're given $\dfrac {\sin \theta} {1 - \cos \theta}$?
AI: All ... |
H: Intersection of compositum of fields with another field
Let $F_1$, $F_2$ and $K$ be fields of characteristic $0$ such that $F_1 \cap K = F_2 \cap K = M$, the extensions $F_i / (F_i \cap K)$ are Galois, and $[F_1 \cap F_2 : M ]$ is finite. Then is $[F_1 F_2 \cap K : M]$ finite?
AI: No. First, the extension $\mathbb{... |
H: What should a PDE/analysis enthusiast know?
What are the cool things someone who likes PDE and functional analysis should know and learn about? What do you think are the fundamentals and the next steps? I was thinking it would be good to know how to show existence or even to know where to start to show existence of... |
H: Is my proof that $(p \wedge \neg p) \Rightarrow q$ correct?
I was asked by a professor a while ago to prove $(p \wedge \neg p)$ implies $q$. Whether through laziness or cleverness, I came up with the following proof:
$p \wedge \neg p$ (by assumption).
Assume by way of contradiction $\neg q$.
$p \wedge \neg p$, th... |
H: "Best practice" for finding the language of a formal grammar
If I've been given a formal grammar like
$$
\begin{eqnarray}
S & \Rightarrow & \lambda & | & 0A & | & 1B \\
A & \Rightarrow & 1S & | & 0AA \\
B & \Rightarrow & S & | & 0S & | & 1BB
\end{eqnarray}
$$
what is "the best" (or just a "good") way to find th... |
H: Null space of a matrix
I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video05.html (about 0:38:10) related to convex optimization and for optimization it had a certain affine function equality constraint like
$$Ax=b$$
The lecturer then obtained the equivalent optimization problem removi... |
H: Find $n$ in $n \log_2 n = c$
I'm trying to find the value for $n$ in the following equation.
$$n \log_2 n = c$$
what is $n$?
thanks,
Tim
AI: There is no closed solution formula for such equations. You will have to find the solution numerically -- that is, by trial and error, bisection, Newton iteration or the like.... |
H: Embedding of standard model of arithmetic to PA-model
I am working on the following problem:
Let $ S_{Arithmetic} = \{+, *, 0, 1\}, \mathfrak{M} $ a model for PA (first-order peano axioms) }, and $ \mathbb{N} = (\mathbb{N},+ ^{\mathbb{N}}, *^{\mathbb{N}},0^{\mathbb{N}},1^{\mathbb{N}} )$.
Construct an embedding $f ... |
H: finding a solution of a heat equation
Let $f\in C^0(\mathbb R^n)\cap L^\infty(\mathbb R^n)$ and $\alpha\in\mathbb R$.
How can you find a solution $u$ of $$\begin{cases}\frac{\partial u}{\partial t}(x,t)-\Delta u(x,t)&=&-\alpha\cdot u(x,t) &\text{ in }]0,\infty[\times\mathbb R^n\\ \space\space\space\space\space\spac... |
H: Doubt on displacement of a parabola
Find the equation which is an displacement of $x² - 3x + 4$ and passes though point $(-3, 3)$ and $(2, 8)$
I've already mounted an simple system of equations which looks obvious
$$\begin{align}
8 &= 4a + 2b + c \\
3 &= 9a - 3b + c
\end{align}$$
but I'm stuck here, does someone k... |
H: Famous papers in algebraic geometry
I'm reading the Mathoverflow thread "Do you read the masters?", and it seems the answer is a partial "yes".
Some "masters" are mentioned, for example Riemann and Zariski. In particular, a paper by Zariski is mentioned, but not its title nor where it was published, so I have been ... |
H: What consitutes an exponential function?
I was recently having a discussion with someone, and we found that we could not agree on what an exponential function is, and thus we could not agree on what exponential growth is.
Wikipedia claims it is $e^x$, whereas I thought it was $k^x$, where k could be any unchanging... |
H: How do I differentiate this? (logarithm & chain)
I keep getting wrong results when trying to differentiate this:
${\partial \over \partial x} \ln{(x - \sqrt{x^2+a^2})}$
Thanks for hints!
AI: There may be a typo in the question, since the thing inside the logarithm is $\le 0$. So we solve a different problem, findin... |
H: Thompson's Conjecture
I have heard that the following is a conjecture due to Thompson:
The number of maximal subgroups of a (finite) group $G$ does not exceed the order $|G|$ of the group.
My question is: did Thompson really conjecture this? If so, is there any literature on the subject?
AI: I've seen this conjectu... |
H: Formal language homework problem - extend(L)
This is my attempt to solve an exercise from a formal languagues class.
Consider the following definition:
extend(L) = { w $\in$ $\Sigma^*$ | $\exists$ x $\in$ L, y$\in$ $\Sigma^*$ . w = xy }
In words: extend(L) is the set of all strings with some prefix in L;
1- Are ... |
H: Finding the recurrence relation
This is actually a very simple problem, but I am going to type everything out in case I really overlooked something
I am trying to find a recurrence relation in the series solution I got.
The ODE is $y'' + xy'- y = 0$.
$\begin{align}
y'' + xy' - y & = \sum_{n=2}^{\infty} n(n - 1)a_... |
H: Recommend a space to analyze the bearing of the vector between any two points
In Euclidean space, given any two points, the vector connecting them can be characterized by length (distance) and direction (bearing). Now I am only interested in the bearing part. And I found it is inconvenient to analyze the bearing pr... |
H: Statistics: How to measure how accurately probabilities are reported?
If you roll a six sided die a bunch of times, and count how many times the number 1 shows up, you'd expect it to show up about 1/6 of the time.
Now if you roll this die 1000 times, and the number 1 shows up 600 times, you'd know something is amis... |
H: What is the thing inside a sum called?
You know how the "thing" inside an integral, we call that an integrand. Does any know what the $a_n$ in a typical $\sum a_n$ is called? Or do we only have names if it is an infinite series? I could've sworn there is a name or there really should be a name
AI: Any element of a ... |
H: On the space of ultrafilters on $N$
I meet the space $X$ of ultrafilters on $N$ with the topology generated by sets of the form $\{p\}\cup A$ where $A\in p \in X$. I can't understand the definition of the topology.
Is the points in $N$ are all discrete? Could someone help me to understand this space? Any help will... |
H: The smallest possible value of $x^2 + 4xy + 5y^2 - 4x - 6y + 7$
I have been trying to find the smallest possible value of
$x^2 + 4xy + 5y^2 - 4x - 6y + 7$, but I do not seem to have been heading in any direction which is going to give me an answer I feel certain is correct. Any hints on how to algebraically appro... |
H: When does an "infinite polynomial" make sense?
Suppose I pick a collection $A \subset \mathbb{C}$ of points in the complex plane and attempt to construct a "polynomial" with those roots via,
$$f(z):=\Pi_{\alpha \in A} (z-\alpha).$$
If $A$ is finite, we get a polynomial.
If $A=\{n\pi:n \in \mathbb{Z}\}$, according ... |
H: Does localization preserves dimension?
Does localization preserves dimension?
Here's the problem:
Let $C=V(y-x^3)$ and $D=V(y)$ curves in $\mathbb{A}^{2}$. I want to compute the intersection multiplicity of $C$ and $D$ at the origin.
Let $R=k[x,y]/(y-x^3,y)$. The intersection multiplicity is by definition the dimen... |
H: $G_{1}/N_{1} \cong G_{2}/N_{2}$ and $N_{1} \cong N_{2} \Rightarrow G_{1} \cong G_{2}$?
1) Suppose $G_{1}$ and $G_{2}$ are groups with respective normal subgroups $N_{1}$ and $N_{2}$. Suppose $G_{1}/N_{1} \cong G_{2}/N_{2}$ and $N_{1} \cong N_{2}$. Does this imply that $G_{1} \cong G_{2}$?
2) Suppose $G/N \cong H$ a... |
H: Is the diagonalization of A Invertable?
There is a theorem which says that given a diagonalizable matrix $A$ such that $P^{-1}AP=D$ if $D$ is invertible then A is invertible.
I suspect that the other direction isn't true, but I can't think of a counter example.
AI: You know that your matrix $P$ is invertible. Now w... |
H: Topology for beginners
Possible Duplicate:
best book for topology?
Please Suggest some good books on Topology and Functional Analysis.
It would be good if somebody can post links of video lectures related to these.
Thanks in advance!!
AI: I really liked Topology by Munkres. It covers a lot of general topology.
F... |
H: Simple Property of GCD and Modular Arithmetic
I'm stuck on proving a rather elementary property, as I'm not really sure how to start off the approach. Suppose $g^a\equiv 1$ mod $m$ and $g^b\equiv 1$ mod $m$. Does this imply that $g^{\gcd(a,b)}\equiv 1$ mod $m$?
Here's my attempt:
By definition, we know that $m\mid ... |
H: Proving Linear Independence of Sequences
Good morning,
my question is about proving the linear independence of sequences. In the theory of linear difference equations, one needs a fact, that for all distinct $\lambda_1,\ldots,\lambda_m \in \mathbb{C}$ and all $k_1,\ldots,k_m \in \mathbb{N}$, the functions (sequence... |
H: Do there exist infinitely many primes $p$ such that $a^{p-1}\equiv 1$ $\text{mod } p^2$ for fixed a?
I noticed that Hardy and Wright in their "An Introduction to Theory of Numbers"(sixth edition) have asked the following:
Is it ever true that $$2^{p-1}\equiv 1 \bmod p^2 \tag{*}\;\;\;?$$
They have pointed out tha... |
H: A $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$
Сould any one help me how to show $C^{\infty}$ function from $\mathbb{R}^2$ to $\mathbb{R}$ can not be injective?
AI: $\mathbb{R}^2\setminus\{x\}$ is connected for any $x\in\mathbb{R}^2$.
Only continuity is required for the argument. |
H: if $G$ is a finite group of an odd order, then the product of all the elements of $G$ is in $G' $
Let $G$ be a finite group of an odd order, and let $x$ be the product of all the elements of $G$ in some order. Prove that $x \in G' $
My proof:
(1) If $G$ is abelian then it is very simple to prove.
(2) If $G$ is not ... |
H: How do I prove exchangeable modularity?
How do I prove that, considering all numbers natural, and p and i relatively prime,
$mp+n \not \equiv 0 \pmod i$
is the same as
$m-x \not \equiv 0 \pmod i$
considering x a natural number and the solution of
$xp+n \equiv 0 \pmod i$
?
AI: Since $p$ and $i$ are relatively prime,... |
H: a question on Pixley-Roy topology
Let $X$ be a $T_1$ space and let $F[X]$ be $\{x\subset X:\text{is finite}\}$ with Pixley-Roy topology.
If $X$ is not discrete, how to prove $F[X]$ is not a Baire space?
Thanks ahead:)
Definition of Pixley-Roy topology: Basic neighborhoods of $F\in F[X]$ are the sets
$$[F,V]=\{H\in... |
H: Ways to compute the limit of $\sum_{k=1}^\infty \frac{k^{n-1}}{(k+1)^n}$ as $n\to\infty$?
Consider the sum
$$S(n) = \frac{1^{n-1}}{2^n} + \frac{2^{n-1}}{3^n} + \frac{3^{n-1}}{4^n} + \cdots \infty = \sum_{k=1}^\infty \frac{k^{n-1}}{(k+1)^n}$$
How do I find the value of $\lim_{n\to\infty}S(n)$?
I am guessing it wou... |
H: A problem in combinatorics
8 subjects need to be given to 4 students. In how many ways can it be done so that the third student gets an odd number of subjects.
I tried combination with repetition $9 \choose 7$+$7 \choose 5$+$5 \choose 3$+$3 \choose 1$
but i'm not quite sure this is the right way to solve the proble... |
H: question on complete vector fields
Could any one help me to solve these two problems on vector fields
Any $\mathbb{C}^{\infty}$ vector field on a compact manifold is complete.
Is every vector field on $\mathbb{R}$ complete?
AI: The 1st is true, not the 2nd. The flow associated to $\dot{x}=x^2$ is given by $\phi_t(... |
H: how many distinct sets can be formed
QUESTION (Edited to make it more readable)
If A and B are two different nonempty sets, how many distinct sets can be formed with these sets using as many unions,intersections,complements and parentheses as desired.
EDIT: My question was actually about generalising the homework q... |
H: Solve the equation: $e^x=mx^2$
I need to find out the maximum possible number of real roots of the equation:
$$e^x=mx^2$$
where m is a real parameter.
I'm interested in some easy approaches. Moreover, is it possible to solve it without using derivatives at all? Thanks.
AI: Provided $m > 0$ there is always a negativ... |
H: Lifting additive characters
Let $K$ a finite extension of $\mathbb{Q}_p$ ($p$ prime different from 2) and let $G_K$ the absolute Galois group of $K$.
Let $\bar{u} : G_K \longrightarrow \mathbb{F}_p$ a continuous additive character. Is it always possible to lift $\bar{u}$ to an additive character $u : G_K \longrig... |
H: Boundedness of the solution of the heat equation
The general solution of the heat equation
$$\left\{\begin{array}{rcl}
\partial_tu-\Delta u &=& 0\\
u(x,0)&=& f
\end{array} \right.$$
is given by $$u(x,t)=\int\limits_{\mathbb R^n}\Phi(x-y,t)f(y)\mathrm dy$$
with the fundamental solution $\Phi$ (wikipedia).
So why is ... |
H: A good commutative algebra book
Possible Duplicate:
Reference request: introduction to commutative algebra
I'm looking for a good book on commutative algebra covering most of (but not limited to) :
Basic Galois theory and Module algebra
Primary decomposition of ideals
Zariski topology
Nullstellensatz, Hauptidea... |
H: If $R_2$ is an $R_1$-algebra, then is $R_2 \otimes_{R_1} M$ an $R_2$-module?
If we have a ring homomorphism $f\colon R_{1}\rightarrow R_{2}$, and if $M$ is an $R_{1}$-module, my question is: Can we show that the $R_{1}$-module $R_{2}\otimes_{R_{1}}M$ is somehow also an $R_{2}$-module?
AI: Yes. This is called extend... |
H: Square with variable $x$ inside
I am learning about how to calculate the length of a path with integration.
The equation is:
$$\sqrt{1+\Big (\frac{dy}{dx} \Big)^2} $$
So I have to integrate it between $a$ and $b$. In my book I have an example, I understand it but don't know how he solved this:
$$\sqrt{\frac{1}{4}... |
H: What is this automorphism-related subgroup?
Let $G$ be a group, let $H\leq G$ and let $\phi\in\operatorname{Aut}(G)$. Then what "is" the subgroup $K=\langle h^{-1}(h\phi): h\in H\rangle$?
Does it, or its normal closure, have a name? Does it have any interesting properties?
Stuff seems to get interesting if you assu... |
H: Hyperplane in projective space
Let $P_0,P_1,\ldots,P_r$ be distinct points in $\mathbb{P}^n$. Why there is a hyperplane $H$ in $\mathbb{P}^n$ passing through $P_0$ but not through any of $P_1,\ldots,P_r$?
AI: By projective duality, your question is equivalent to asking why, given a finite collection of distinct hyp... |
H: Point belonging to plane
In $\Bbb R^4$, I have a plane (given by its cartesian equation) and a point (given by its coordinates).
How can I check if it belongs to the plane?
AI: Some related cases:
If the equation of a hyperplane is in the form $a_1x_1 + a_2x_2 +
a_3x_3 + a_4x_4 = b$, to check whether a point $(... |
H: What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$
I would like to evaluate the sum
$$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$
Here is my attempt:
Letting
$$f(z)=\frac{1}{az^2+bz+c}$$
The poles of $f(z)$ are located at
$$z_0 = \frac{-b+\sqrt{b^2-4ac}}{2a}$$
and
$$z_1 = \frac{-b-\sqrt{b^2-4... |
H: Prove that if $n \in \mathbb{N}$, $n\ge 1$
As the title says. I encounter this problem in Bernd Schroeder's book of "Mathematical Analysis: A Concise Introduction", p.15. It essentially characterizes natural number from the axioms regarding real number, i.e. the axioms of addition and multiplication of $\mathbb{R}$... |
H: Evaluating $\int_0^{\sqrt{3}}{\frac{\sqrt{1+x^2}}{x}}\,dx$
Could someone please show me how to evaluate this integral (maybe doing all the steps)?
$$\int_0^{\sqrt{3}}{\frac{\sqrt{1+x^2}}{x}}\,dx$$
I prefer if you avoid to follow the same method used by WolframAlpha (with $\csc$, $\sec$ ecc).
This is what I tried 't... |
H: Limit of exponentials
Why is $n^n (n+m)^{-{\left(n+m\over 2\right)}}(n-m)^{-{\left(n-m\over 2\right)}}$ asymptotically equal to $\exp\left(-{m^2\over 2n}\right)$ as $n,m\to \infty$?
AI: By Stirling's approximation we have $$ \binom{2n}{n+m}= \frac{(2n)!}{(n+m)!(n-m)!} \sim \frac{\sqrt{2\pi n} (2n/e)^{2n}}{\sqrt{2\p... |
H: Variational distance basic properties
The variational distance between two probability distributions $X$ and $Y$ taking values on the same alphabet $\mathcal A$ is defined as
\begin{equation}
\delta (X,Y)=1/2\sum_{a\in A} |p_X(a)-p_Y(a)|$
\end{equation}
There are two very basic claims with regard to the variation... |
H: How to get the angle in the right triangle?
I have two coordinates which represent the mouse position with respect to the center of the screen ([0, 0] meaning the center, y increases downwards).
So, [0, 0] is one corner of the triangle, and mousePos is another. Now, the position of the mouse should determine the di... |
H: Math books pointing towards Probability Theory
I work as a professional composer, and I also program most of my own software. I failed every year of math in high school.
I am studying Bayesian Probabilities in reference to music, and while I understand most of what is being said I can't help but feel progress woul... |
H: Mlodinow. The Drunkard's Walk. An example from the book.
This excerpt is from Leonard Mlodinow's book The Drunkard's Walk:
And
although Fortune is fair in potentialities, she is not fair in outcomes.
That means that if each of 10 Hollywood executives tosses 10 coins,
although each has an equal chance of bein... |
H: Product of spheres embeds in Euclidean space of 1 dimension higher
This problem was given to me by a friend:
Prove that $\Pi_{i=1}^m \mathbb{S}^{n_i}$ can be smoothly embedded in a Euclidean space of dimension $1+\sum_{i=1}^m n_i$.
The solution is apparently fairly simple, but I am having trouble getting a star... |
H: Is this map of domains a Jordan homomorphism?
Let $\phi\colon D\to D'$ be a map of division rings, such that $\phi$ is a
homomorphism of the additive groups, respects unity, and if $a\neq 0$, $\phi(a)\neq 0$, and $\phi(a)^{-1}=\phi(a^{-1})$. It's a theorem of L.K. Hua that $\phi$ is either a homomorphism of anti-h... |
H: What are some (or even one) interesting examples of (non-group) semigroups?
I'm going to give a lecture on Alon and Schieber's Tech Report on computing semigroup products (Optimal Preprocessing for Answering On-Line Product Queries). Basically, given a list of elements $a_1,\ldots,a_n$ and a bound $k$, they show h... |
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