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Let $(R,m,k)$ be a (noetherian) regular local ring of depth=dimension $d$, and let $D$ be a dualizing module for $R$ (say, the injective envelope of $R/m$).
Then is $D_p$ dualizing for $R_p$ for any prime $p$ of $R$ (more generally, if $R$ is Gorenstein and $p$ is a prime such that $R_p$ is also Gorenstein)? If it i... |
I have a certain side ratio of small rectangle, and a side ratio for a large rectangle. How can I write an equation that gives me the number of small rectangles that will make the large rectangle? Is there a formula for that?
Apologies for the title -- it was a last-ditch attempt.
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As opposed to the algebraic completion of $\mathbb{Q}$, which yields the algebraic numbers, we can say that $\mathbb{R}$ is complete in the sense that every non-empty subset of $\mathbb{R}$ bounded by above has a supremum.
So, it isn't algebraically complete, but is it topologically or metrically complete? What would b... |
Let G be a finite group and $\phi$ a homomorphism from G to $\mathbb{C}-\{0\}$ such that $\phi(G)\neq \{1\}$. It is claimed, in a book Im studying, that $\phi(G)=\{\zeta_n^r : 1\leq r \leq n\}$, $n\geq 2$. Prove this.
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In http://mathproofs.blogspot.com/2005/07/mapping-square-to-circle.html, there is a derivation of the mapping from a unit square to a unit circle. Looking up wikipedia tells me the canonical form of an ellipse is:
$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
and the aforementioned article starts with:
$\frac{(x^')^2}{x^2} +... |
I've been studying the axiomatic definition of the real numbers, and there's one thing I'm not entirely sure about.
I think I've understood that the Archimedean axiom is added in order to discard ordered complete fields containing infinitesimals like the hyperreal numbers. Additionally, this property clearly cannot be ... |
Is any transposition a product of simple transpositions? If yes, how can you prove this?
|
As everyone knows, the set of real numbers is uncountable. The most ubiquitous proof of this fact uses Cantor's diagonal argument. However, I was surprised to learn about a gap in my perception of the real numbers:
A computable number is a real number that can be computed to within any desired precision by a finite, te... |
I'm writing a program that multiples a matrix by its transpose, and was trying to find efficiency hacks I could exploit considering that the two matrices being multiplied are related. Any ideas?
|
What are $y^+$ and $y^-$ supposed to represent? $y$ is a vector.
|
If $$\lim_{x \rightarrow a}\frac{f(x)}{g(x)}=0$$ where $a$ is a real number, or could be $ \pm\infty$, and $f,g$ are continuous functions (not necessary polynomials), with $g(a)\neq 0$.
what we can conclude about $f$ and $g$ ?
|
I'm working my way through these notes on stochastic calculus:
The following is taken from section 2.20:
In discrete probability, equivalence classes are measurable. (Proof: for any $\omega^\prime$ not equivalent to $\omega$ in $\mathcal{F}$, there is at least one $B_{\omega^\prime} \in \mathcal{F}$ with $\omega \in B... |
Let $\{A_n\}$ and $\{B_n\}$ be two bases for an $N$-dimensional Hilbert space. Does there exist a unit vector $V$ such that:
$$(V\cdot A_j)\;(A_j\cdot V) = (V\cdot B_j)\;(B_j\cdot V) = 1/N\;\;\; \ \text{for all} \ 1\le j\le N?$$
Notes and application:
That the $\{A_n\}$ and $\{B_n\}$ are bases means that
$$(A_j\cdot... |
Can you give me some hints to prove equality:
$$\sum_{m,n=1}^{\infty} \frac1{(m^2+n^2)^2} =\zeta (2)\ G-\zeta(4)=\frac{\pi^2}{6}\ G-\frac{\pi^4}{90}$$
where $\zeta (t):= \sum\limits_{n=1}^{+\infty} \frac{1}{n^t}$ is the Riemann zeta function and $G := \sum\limits_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} \approx 0.915 9... |
this is homework, but after banging my head against the wall for a day, I feel justified asking for a hint.
Let $X$ a compact Riemann Surface, $D,E \in Div(X)$ divisors on $X$ with $degE\geq 0$.
Show that $dimL(D+E) \leq dimL(D) + deg(E)$.
I think if I can show that for $E = 1x$, $dim(D+1x) \leq DimL(D) + 1$, then this... |
I call a 4-cycle permutation simple if I can write it as $(i,i+1,i+2,i+3)$ so $(2,3,4,5)$ is a simple 4-cycle but $(1,3,4,5)$ is not. I want to write $(1,2,3,5)$ as a product of simple 4-cycles. So this is what I did:
$$
(1,2,3,5)=(1,2)(1,3)(1,5)
$$
but
$$\begin{align}
(1,3)&=(2,3)(1,2)(2,3)\\
(1,5)&=(4,5)(3,4)(2,3... |
Show that the solution vectors of a consistent nonhomogeneous system of m linear equations in n unknowns do not form a subspace of $\mathbb{R}^n$.
I'm not really sure how to go about this problem. I know that I'm suppose to check if the vectors are with the subspace by addition and scalar multiplication but I'm not r... |
Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$.
It is a fact that the eigenvalue problem for the Laplace operator with homogeneous Dirichlet boundary condition, i.e.:
$$\begin{cases} u_{xx}+u_{yy}=-\lambda u, &\text{in } \mathcal... |
This came up in a discussion with Pete L. Clark on this question on complete ordered fields. I argued that every Cauchy sequence in the hyperreal field is eventually constant, hence convergent; he asked whether the same is true for arbitrary Cauchy nets in $\mathbb{R}^*$. I'm not sure how to deduce this either from th... |
I had this question in my mind for a long time but I was not sure if it makes sense to anyone.I would appreciate your valuable thoughts on this question.
How to identify points of discontinuity of a function $f :\mathbb{R} \to \mathbb{R}$ given samples of the function (obtained by using Nyquist sampling or any other... |
$1:$ the problem
Let $f : U \to \Bbb{C}$ be analytic on some open set $U$ that includes the closed unit ball. Define a path $\gamma$ by:
$\gamma(t) = e^{i t}$, -$\pi < t \leq \pi$
I want to show that the following holds:
$\int_\gamma\ e^{\lambda\ f(z)}\ dz =\ \left(\frac{2\pi}{-\lambda\ f''(1)}\right)^{\frac{1}{2}}\ ... |
Let $Y$ be a Hausdorff space, and let $f:S^{n-1} \to Y$ be continuous. Then $D^n \coprod_f Y$ is called the space obtained from $Y$ by attaching an n-cell (denoted $e^n$) via $f$ and is denoted $Y_f$
The charactersitc map $\Phi$ is the composite:
$D^n \hookrightarrow D^n \coprod Y \to D^n \coprod_f Y \to Y_f$ so that $... |
Given the equations,
$$v_1+v_2=a-1$$
$$v_1v_2=b$$
for what ranges of $a$ and $b$, can I be sure to find $0<v_1<1$ and $0<v_2<1$.
Also, for what ranges, can I be sure to find at least one $v_i$ such that $0<v_i<1$
|
Let $a,b$ be positive integers.
When $$k = \frac{a^2 + b^2}{ab+1}$$ is an integer, it is a square.
Proof 1: (Ngô Bảo Châu): Rearrange to get $a^2-akb+b^2-k=0$, as a quadratic in $a$ this has two values: $a$ and $kb - a = (b^2-k)/a$. (The second root is determined in two different ways from the expansion $(x-r_1)(x-r_2... |
What is the relationship between homology and graph theory? Can we form simplicial complexes from a graph $G$ and compute their homology groups? Are there any practical results in looking at the homology of simplicial complexes formed from a graph?
|
I want to know the answer/references for the question on decay of Fourier series coefficients and the differentiability of a function. Does the magitude of fourier series coefficients {$a_k$} of a differentiable function in $L^2 (\mathbb{R})$ (or in any suitable space) decay as fast as or faster than $k^{-1}$. I want t... |
what are the parts or the variables present in the bicorn equation?
|
Given natural numbers $a,b,n$, prove $b\mid a \implies (n^b-1)\mid (n^a-1)$.
I tried the simple method of beginning with $b\mid a \implies \exists k \in \mathbb{N} $ such that $bk=a$ and then raising $n$ to the power of the LHS and the RHS and eventually forming $(n^b)^k-1=n^a-1$. That's obviously not enough. It trie... |
I know that if I have any permutation, this permutation can be written as the product of transpositions. Now the number of these transpositions may be odd or even depending on my permutation. I also know that any transposition can be written as the product of simple transpositions (the pair $(i,i+1)$). So this means th... |
I am trying to decide if $$\sum_{n=1}^\infty \left(\frac{1}{n}-1\right)^{n^2}$$ converges. By the alternating series test, as far as I can see, the series converges. This is also true by the root test. In both cases I assume that $$\left|\left(\frac{1}{n}-1\right)^{n^2}\right| = \left(1-\frac{1}{n}\right)^{n^2}$$ and I... |
Suppose I have an some plane and an identity mapping on the points of the plane. I see that the identity can be expressed as a product of an even number of reflections, since any reflection has itself as its own inverse. But why is it impossible to ever express the identity as the product of an odd number of reflection... |
Let $X$ and $Y$ be topological spaces with a free G-action, where $G$ is group. Suppose $p:X\rightarrow Y$ is a Galois covering which is G-equivariant map.
Then will it be true that the induced map $\tilde{p}:X/G\rightarrow Y/G$ is also
a Galois covering?
|
Consider a 3-dimensional body $B$. The space of all orientations of $B$ in 3-space is the rotation group $SO(3)$, and is often represented in computer science applications as the quaternions with $q$ and $-q$ identified. However, if $B$ is a cube, for example, not all its orientations are distinguishable. From my curso... |
If I have a homomorphism $\phi : R \rightarrow S$ between integral domains, how can I show that if the kernel is non-zero then it is a maximal ideal in R?
|
Hey guys,
I need to prove or refute that once given an eigenvalue t of a matrix AB and B is invertible,
so t is also eigenvalue of A.
I believe it's not true, but sadly beliefs are not enough in math : )
Thank you.
|
I'm trying to find
$$\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}} .$$
I tried couple of methods: Stolz, Squeeze, D'Alambert
Thanks!
Edit: I can't use Stirling.
|
I know that in general the number of ideal classes are not 1, and that there are only 9 imaginary quadratic number fields which are principal ideal domains, i.e. $\mathbb(Q(\sqrt{-m}))$ where m is 1, 2, 3, 7, 11, 19, 43, 67, 163, and that Gauss had conjectured that there are infinitely many real quadratic number fields... |
Is it that the values in each column add up to one? Also, how does this work?
|
How to express this function in closed form without condition verfication and Re and Im functions (only with absolute value function)?
$$f(z)= \begin{cases}
- \operatorname{sgn}(\operatorname{Im}(z)), & \mbox{if } \operatorname{Re}(x)=0 \\
\operatorname{sgn}(\operatorname{Re}(z)), & \mbox{if } \operatorname{Re}... |
I am dealing with a piece of software which calculates temperatures, and corrects these based on measurements (which may be in error) as follows:-
The difference between the corresponding calculated and measured temperatures are found, and arranged in ascending order.
From this set of temperature differences, subsets ... |
Sorry for the vague title... I've proved a number theoretical result for the imaginary quadratic fields (it was already known for the rationals). I think it would be much easier to sell if I could prove it for all quadratic fields. I think to do it will require using some special properties of the field, as proving... |
I am trying to utilize Numpy's fft function, however when I give the function a simple Gaussian function the FFT of that Gaussian function is not a Gaussian, its close but its halved so that each half is at either end of the x axis.
I was told by my professor that this has something to do with how when Fourier transfor... |
Say you have the polynomial
$$ x^4 + 2 + x^{-4} $$
Looking at it, you see you can do
$$\begin{align*}
x^4 + 1 + 1 + x^{-4} & =x^2( x^2 + x^{-2} ) + x^{-2}( x^2 + x^{-2} )\\
&= \left( x^2 + x^{-2} \right)^2.
\end{align*}$$
Another one is
$$\begin{align*}
x^2 + \frac{1}{2} + \frac{1}{16x^2} &=
x^2 + \frac{1}{4} + ... |
Let $A, B$ be positive-definite matrices and $Q$ a unitary matrix, furthermore suppose $A=BQ$.
Prove or disprove: $A=B$.
I'm having a hard time figuring out where to begin. Thanks.
|
Given two lines joining A,B and C, D in 3D how do I figure out if they intersect, where they intersect and the ratio along AB at which the intersection happens?
I can quite hapilly work out the equation for the lines in different forms. I'm guessing that you need to change them to parametric form, equate the equations ... |
It is well known that $x^2-1=0$ has two roots in $\mathbb{C}$, namely $\pm 1$. In general $x^n-1=0$ has exactly $n$ roots in $\mathbb{C}$. But what happens when $n$ is non integer (rational or real or even complex)? For example how many roots does $x^{1.9}-1=0$ have (counting multiplicity)? Intuitively I suspect that f... |
This is related to my first question. In order to get what I don't get, I ll go with something much more specific here.
It is well known that $ZF \vdash Con(ZF) \longrightarrow Con(ZF + AF)$. The way I know to prove it uses :
Lemma 1 : If $T$ and $S$ are two theories in set theory language, $M$ is a class (i.e. a formu... |
Is there any infinite metabelian group or it is naturally defined to be finite? In fact, I am investigating for finding another solvable group around (apart of $\mathbb{Z}_{p^{\infty}}$ or additive group $\mathbb{Z}$).
|
I was reading through a paper, Learning Fast Approximations of Sparse Coding (LeCunn, ICML 2010) and ran across a phrase I'm not terribly familiar with. Apologetically, I quote with very little context (though I'll link to a copy of the paper):
[a function described in the paper] is a coordinate-wise shrinking functi... |
Given $n$ distinct elements, how many Young Tableaux can one make?
|
In my book - Algebraic Number Theory and Fermat's Last Theorem Ian Stewart, David Tall - there is the exercise:
Prove that the ring of integers of $\mathbb{Q}(\zeta_5)$ (the 5'th cyclotomic ring) is Euclidean.
I can prove that the integers of $\mathbb{Q}(\sqrt{-1})$ are Euclidean by a geometric argument but this does... |
Here is a problem a have got in my homework.
Given a set of $X_1, ... X_n \sim F$ i.i.d values find the variance of $T_n = \overline{X}_n^2$ where $\overline{X}_n = \frac{\sum_i{X_i}}{n}$.
I actualy have an answer for this problem: $V(T_n) = \frac{4 \mu^2 \alpha_2}{n} + \frac{4 \mu \alpha_3}{n^2} + \frac{\alpha_4}{n^3}... |
There is the example 4.4.ii in “A First Course in Abstract Algebra” by Rotman.
If $A$ is a symmetric matrix with
coefficients in $k$, define an inner
product by $(v,w)=v^T\cdot A\cdot w$. The reader may prove that
this is an inner product and that it is
non-degenerate iff $A$ is non-singular.
Non-degenerate :... |
I am an economist and I am trying to figure out the stability of this system,
$$\begin{aligned}\frac{\partial x}{\partial t} &= \frac{a}{y} + x b \\ \frac{\partial y}{\partial t} &= C + y c + \frac{d}{x} \end{aligned}$$
I have really no clues about how to deal with the nonlinearities.
Any help?
And how to find the equi... |
I think I might have asked this question before, but I can't find it on the site, so I sincerely apologize if I am making a duplicate. But anyway, I have been working on this proof for several weeks and am stumped.
If $\pi(x)$ is the number of primes less than or equal to $x$, prove that
$$\lim_{x\to\infty}\frac{\pi... |
I'm referring to an answer posted on Math Overflow (see the post by fedja on https://mathoverflow.net/questions/59115/a-set-for-which-it-is-hard-to-determine-whether-or-not-it-is-countable)
The question is whether the set of real numbers $a > 1$ so that for $K > 0$ the distance between $K a^n$ and its nearest integer a... |
I have read some old discussions about this topic and would like to get some up-to-date advice, if possible.
How can I take math notes, write formulas and draw graphs on my pc (win 7), the easiest and quickest way?
Thanks for your consideration.
|
Let me introduce my problem: Let $C \subset \mathbb{C}$ be the unit circle of the complex plane and $Z=\left\{ z_n \right\}_{n \in \mathbb{N}} \subset C$ be a dense subset of the unit circle meaning that $\bar{Z}=C$ or otherwise put that for all $c \in C$ there is a subsequence $\left\{z_{n_{k}}\right\}_{k \in \mathbb{... |
I'm a beginner in mathematics and there is one thing that I've been wondering about recently. The formula for the normal distribution is:
$$f(x)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\displaystyle{\frac{(x-\mu)^2}{2\sigma^2}}},$$
However, what are $e$ and $\pi$ doing there? $\pi$ is about circles and the ratio to its diamet... |
In the course of my research I have come across the following integral:
$\int_{0}^{\infty} e^{- \Lambda \sqrt{(z^2+a)^2+b^2}}\mathrm{d}z$.
This initially looks like it should be solvable by some suitable change of variable which will allow you to get it into a gaussian form. Unfortunately after trying for awhile I can... |
I don't understand the concept of parametrization or how to go about solving this question.
Any help would be appreciated!
|
Recently, I answered to this problem:
Given $a<b\in \mathbb{R}$, find explicitly a bijection $f(x)$ from
$]a,b[$ to $[a,b]$.
using an "iterative construction" (see below the rule).
My question is: is it possible to solve the problem finding a less exotic function?
I mean: I know such a bijection cannot be monotone,... |
I'm trying to prove that $a^n+1$ can only be prime if $n$ is a power of $2$. Is there a general factorization of $a^n+1$?
|
I am currently working on a research project that is trying to show that some numerical integration techniques that do well on separable convex functions will not necessarily do well on non-separable convex functions. I hope to show this using by testing these numerical integration techniques on functions of the form:
... |
(I asked this on Stack Overflow a while ago and didn't get a satisfying answer, so I'm trying again here.)
The Fast Fourier Transform takes O(N log N) operations, while the Fast Wavelet Transform takes O(N). But what, specifically, does the FWT compute?
Although they are often compared, it seems like the FFT and FWT a... |
_EDIT_ I'd like to do this to $d$ digits of precision.
I wonder what the fastest way to get roots of a value on the unit circle is. More specifically, if I have a fraction of naturals, $p/q$, and natural $n$, what is the fastest way to find
$\sqrt[n]{e^{i(2\pi)p/q}}$
I'm considering using lookup tables and such. I gu... |
Suppose $f\in L^{p}(\mathbb{R}^{n}) \cap L^{q}(\mathbb{R}^{n})$. How can I prove that for any $p \lt r \lt q$,
$$
\lVert f \rVert_{r} \leq
(\lVert f \rVert_{p})^{(1/r-1/q)/(1/p-1/q)}
(\lVert f \rVert_{q})^{(1/r-1/q)/(1/p-1/q)}\:?
$$
I tried using Hölder to show that $f$ is in $L^{r}$ but I'm completely lost on how to ... |
I am looking for some insight into a problem:
Consider a group of $T$ persons, and let $a_1, a_2, ..., a_T$ denote the height of these $T$ persons. Suppose that $n$ are selected from this group at random without replacement, and let $X$ denote the sum of the heights of these $n$ persons. I'm supposed to find the mean a... |
I'm trying to figure out how to actually compute maps in the Mayer-Vietoris sequence as it's quite useless unless you actually know some maps. Hatcher computes the homology for the Klein bottle $K$ on page 151. I still can't seem to understand the logic he uses to determine the map
$H_1(A\cap B)\to H_1(A)\oplus H_1(B)$... |
Is it possible to determine (and if so, how) the complex conjugate $\bar{z}$ of $z$, if you don't already know that $z = x + i y$?
I think you can use $\log(z)$ to get the angle, and therefore the ratio of $y$ and $x$. But how do you get $|z|$, the radius? How then to get $r$ (so that $x = {\rm Re}(z) = r \cos(\log(z)... |
In order to show that $\mathbb{R}P^n$ has the cellular structure $e^0 \cup \ldots \cup e^n$ we have the characteristic map, we take $u \in D^n$ and define $$ \Phi(u) = [u_1,\ldots,u_n,\sqrt{1-\| u \| ^2}]$$
What I really want to see is that $\Phi|D^n-S^{n-1}$ maps homeomorphically onto $\mathbb{R}P^n - \mathbb{R}P^{n-1... |
Does the series: $$\sum\limits_{n=2}^{\infty} \frac{\cos(\log{n})}{n \cdot \log{n}}$$ converge or diverge?
I know that $|\cos(\log{n})| \leq 1$, but I really cannot apply it here. Any ideas on how to attack this problem
|
I am reading:
A right cylinder with cross section $\Omega$ is a solid that is formed by translating $\Omega$ along a line, or _axis_, that is perpendicular to it.
So my understanding now is a cylinder is any solid that can be formed by translating a 2D shape straight along an axis.
So extrusion forms a "cylinder", re... |
Every non-decreasing function from R to R is injective?
Prove or provide counter-example if False.
False
Definition of increasing:
for all x and y, x <= y then f(x) <= f(y)
If f is not injective then there exists x1 ≠ x2 such that f(x1) = f(x2)
Example: f(x) = 0
|
That is, let $f:X \rightarrow Y$ be a map of spaces such that $f_*: H_*(X) \rightarrow H_*(Y)$ induces an isomorphism on homology. We get an induced map $\tilde{f}: \Omega X \rightarrow \Omega Y$, where $\Omega X$ is the loop space of $x$. Does $\tilde{f}$ also induce an isomorphism on homology?
|
I want to show the following:
Let $I_{1},...,I_{n}$ be ideals of a commutative ring (with 1) $A$ such that $\bigcap_{i=1}^{n} I_{i}$ is the zero ideal. If each quotient $A/I_{i}$ is a Notherian ring show that $A$ is a Notherian ring.
My work:
Let $f: A \rightarrow A/I_{1} \times A/I_{2} \times .... \times A/I_{n}$ be t... |
A family of pseudometrics defined on a set gives rise to a uniform structure on that set. Moreover (up to uniform equivalence, anyway) every uniform structure arises this way. Let $A$ and $B$ be families of pseudometrics defined on sets $X$ and $Y$ respectively. There ought to be a way to characterize the uniform conti... |
Let $G_1$ and $G_2$ be finite groups, and let $n$ be the number of epimorphisms (i.e. surjective hom.) from $G_1$ to $G_2$.
Then what is the number of epimorphisms from $G_1$ to $G_2$ with distinct kernels? How?
|
I am wondering two different forms of the second mean value theorem for integration. For the one in wikipedia, I also wonder where I can find a proof.
The form I read from another reference is that:
$G:[a,b]\to \mathbb{R}$ is a monotonic function and $\phi : [a, b] \to \mathbb{R}$ is an integrable function, then there ... |
It is known that all prime ideals are irreducible (meaning that they cannot be written as an finite intersection of ideals properly containing them). While for Noetherian rings an irreducible ideal is always primary, the converse fails in general. In a recent problem set I was asked to provide an example of a primary i... |
I am trying to inductively prove that for any string s, the reverse of the reverse of string s is string s.
|
Here I asked how I can write a particular 4-cycle as a product of simple 4-cycles and I understand the solutions given. I now want to prove that every 4-cycle can be written as the product of simple 4-cycles. The only way that I know of to prove this is induction. I have no problem with induction but my problem is that... |
I read this recent question on proving that the identity is never the product of an odd number of reflections. After googling a bit, I found a Lemma in J. Aarts Plane and Solid Geometry, but I didn't fully understand the proof. I'll post the relevant sections, and the google book can be found here.
The Lemma that ans... |
I got this answer from WolframAlpha. Does anyone know how even to approach it to obtain the solution using digamma function. Please don't solve it, just show me in the right direction!
$$
\sum_{k=1}^{n}{1 \over n^{2} - 4\left(k - 1\right)^{2}}
={\pi n\cot\left(\pi n/2\right) - n\Psi\left(n/2\right) + n\Psi\left(3 n/2\... |
I've just read something about calculating sample standard deviation. Some explanations say that it is because of degree of freedom. With a constraint that the mean should be zero, we do actually have only n-1 degree of freedom. Well, this explanation is pretty convincing.
But there is still another question stick in m... |
Given some ring $R$ and two ideals $I$ and $J$ of $R$ such that $I \neq J$, is it possible for $R/I \cong R/J$?
|
Say in $S_4$, I want to write $(1,2,3,4)$ as a product of 3 cycles? How should I do it? In general, how do you decompose larger cycles as the product of smaller cycles (but all having the same cycle length)?
|
Background: In Electrodynamics, the scalar permittivity $\epsilon(\omega)$ relates the Electric displacement field $\vec D$ to the electric field $\vec E$ as $\vec D=\epsilon\vec E$ when assuming a linear, isotropic medium. Causality requires that in time-space the Fourier transform $\tilde\epsilon(t)$ must vanish for ... |
Which books would you recommend, for self-studying homological algebra, to a beginning graduate (or advanced undergraduate) student who has background in ring theory, modules, basic commutative algebra (some of Atiyah & Macdonald's book) and some (basic) field theory?
I would especially like to hear your opinions on th... |
For a given $n \times n$-matrix $A$, and $J\subseteq\{1,...,n\}$ let us denote by $A[J]$ its principal minor formed by the columns and rows with indices from $J$.
If the characteristic polynomial of $A$ is $x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$, then why $$a_k=(-1)^{n-k}\sum_{|J|=n-k}A[J],$$
that is, why is each coeffici... |
Let $T$ be an operator on Hilbert space.
Define $\sigma(T)=\lbrace \lambda\in \mathbb{C} | \lambda I - T~\textrm{is not invertible}\rbrace$.
How can I prove that $\sigma(T^n)=\lbrace \lambda^n|\lambda\in \sigma(T)\rbrace$?
|
This is related to my previous question. The thing was to show :
$ZF \vdash \neg Con(ZF + AF) \longrightarrow \neg Con(ZF)$
If $ZF + AF$ is inconsistant then there is a finite number of $ZF + AF$ axioms, $\psi_1,\dots,\psi_n$ such as $\psi_1 \wedge \dots \wedge \psi_n \longrightarrow \phi \wedge \neg \phi$
We know tha... |
Is it true that "There is a part of $\mathrm{GL}_n(F)$ which is isomorphic to $\mathrm{PSL}_n(F)$, $n \geq 4$"? So since $\mathrm{PSL}_n(F)$ is not solvable (it is simple non-abelian group), the former linear group is simply not solvable.
|
I have the following problem:
Determine the point spectrum and the continuous spectrum of the operator $$(A\psi )(x)=\theta (x)(\cos x)\psi (x)$$ on $L_2(\mathbb R,dx)$, where $\theta(x)=0$ for $x<0$, $\theta(x)=1$ for $x\geq 0$.
Now, for the point spectrum I guess I only have to find a $\psi(x)$ such that $(A\psi )(... |
Are we just trying to prove every theorem or find theories which lead to a lot of creativity or what?
I've already read G. H. Hardy Apology but I didn't get an answer from it.
|
Every surjective function from $\mathbb{R}$ to $\mathbb{R}$ is unbounded.
Every unbounded function from $\mathbb{R}$ to $\mathbb{R}$ is surjective.
Is it possible for either of these statements to be false? I have a feeling there is some counterexample that I am missing but I cannot figure it out.
My understanding is... |
Let $A\subset \mathbb{C}$ be a compact subset.
Since $A$ is compact and metric space, it is separable, say $\overline{\lbrace a_n\rbrace_{n=1}^\infty}=A$.
Let $\mathcal{l}^2(\mathbb{Z})$ be the Hilbert space consisting of $L^2$-summable sequences and $\lbrace e_n\rbrace_{n=1}^\infty$ be the canonical basis of $\mathc... |
I'm trying to show that $\mathbf{W} = \mathrm{span} \{ 1+x, 1-x^2 \}$ where $\mathbf{W} = \{a + b + ax - bx^2 \mid a,b \in\mathbb{R}\}$.
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I was reading the Mathematica Guidebook to Symbolics by Michael Trott, and saw the following expression to evaluate the branch points of a Riemann surface given by $f(x,a)=x^3+x^2+ax-1/2$.
Eliminate[{#==0},D[#,x]==0},x]&[eq=x^3+x^2+ax-1/2]
This gives the result $36a-4a^2+16a^3+19=0$. The roots of this equation are the... |
I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$,
is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter)
the same as the center of mass/gravity of $P$, assuming $P$ is a homogenous solid.
The conditions force quite a bit of symmetry. Perhaps only the Pla... |
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