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It is well known that any finite group can be embedded in Symmetric group $S_n$, $GL(n,q)$ ($q=p^m$) for some $m,n,q\in \mathbb{N}$. Can we embed any finite group in $A_n$, or $SL(n,q)$ for some $n,q\in \mathbb{N}$?
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I have a question to ask down below, that I have been having some trouble with and would like some help and clarification on.
Suppose A is an $n \times n$ matrix with (not necessarily distinct) eigenvalues $\lambda_{1}, \lambda_{2}, \ldots, \lambda_{n}$. Can it be shown that:
(a) The sum of the main diagonal entrie... |
I am trying to understand if there is an error in this question, or if the answer is trivial.
Define a topology on $[0,\infty)$ with open sets
$(a,\infty), a \in (0,\infty)$
$[0,\infty)$ and $\emptyset$.
Show $[0,\infty)$ is compact in this topology.
Since we can only build an open cover out of open sets, surely we c... |
I have come across this question in a course about Cayley graphs recently. I don't really have a clue how to answer this question. Here it is.
Consider two groups $A,B$. Let $K< A*B$ be the kernel of the homomorphism $\phi: A * B \to A \times B$ that extends the inclusions $A \hookrightarrow A \times B$ and $B \hookrig... |
Let K be an algebraically closed field of arbitrary characteristic. Then there is a Grassmannian of subspaces of fixed dimension say m of n-dimentional space and it is a projective variety. If you have an algebra A over field K you can vew say left modules over this algebra and for a given module of dimension n you can... |
Are there two matrices $A_{m\times n}$, and $B_{n\times m}$ such that $A.B=I_{m}$ and $B.A=I_{n}$ (here $m\neq n$)
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Why are Artinian rings of Krull dimension 0?
As in the example of $\mathbb{Z}/(6)$, the ideal $\mathbb{Z}/(2)$ is prime, I think. So, Artinian rings may contain prime ideals. But why does not the primes ideal contain other prime ideals properly?
Another question is that, when a ring is of Krull dimension 0, is it neces... |
The quaternion group is non-abelian p-group with every proper subgroup cyclic. In general, if $G$ is a non-abelian $p$-group with every proper subgroup cyclic, is it necessarily generalized quaternion?
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Free modules are projective, and projective modules are direct summands of free modules.
Are there examples of projective modules that are not free?
(I know this is not possible for modules of fields.)
Free modules are torsion-free. But is the inverse true?
Are there examples of torsion-free modules that are not ... |
I had been working on this problem here below, but seem to not know a precise and clean way to show the proof to the question below. I had about a few ways of doing it, but the statements/operations were pretty loosely used. The problem is as follows:
Suppose $\mathbf{A}$ is an $n \times n$ matrix with (not necessarily... |
If $G$ is a finite group, then it is well known that there are finitely many inequivalent irreducible representations of $G$ over $\mathbb{C}$; moreover the sum of squares of dimensions of the representations is equal to $|G|$. Also, the dimension of representation divides $|G|$.
If we consider representations over $\... |
My Maths notes don't explain this, but is there a saddle point or a local maximum?
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How the the monodromy group and deck transformation group of a covering $p\colon Y\rightarrow X$ are related in the cases when covering is branched and covering is unbranched?
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I'm trying to derive a result for the number of possible combinations of r objects from n, when we have unlimited numbers of objects to select from. (Wikipedia suggests that these are called multi-combinations). My current line of reasoning is like this:
Suppose we can choose 4 letters from an unlimited set containing ... |
Fourier series of function f:
$$f(x)=\sum_{s=-\infty}^{\infty}f_{s}\exp(2\pi isx)$$
Suppose $f_{s}\sim\frac{1}{s^{p}}$. What can we say about $f(x)$? Can we find some bounds for $f(x)$ like $f(x)<c\sum_{s}\frac{\exp(-2\pi isx)}{s^{p}}$?
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When trying to reproduce the plot of Tupper's self-referential formula (original paper is available here) with Mathematica I have faced unexpected difficulties.
First of all, the algorithm does not give the image shown and I was forced to find the right way by myself (code for Mathematica):
k = 96093937991895888497167... |
im doing a question that involves a shape with 8 faces, 10 vertices and 16 edges. Can anyone enlighten me as to what this shape is called?
Many Thanks
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How can I prove by definition (with $\epsilon$ and $N$) that, for $c>0$,
$$\lim_{n \to \infty}\ \sqrt[n]{c} = 1 ?$$
Thanks.
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I noticed that on a road, if the number of cars reaching a certain critical stage, all the cars would slow down immediately.
Is there any theory/model that connects the number of cars $N$ with the average car speed $V$, and maybe with road width $R$ ( and other variables)?
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If I know the following parameters how to find h co-ordinate of ellipse center
1.Circle : center (0,0), radius = r
2.Ellipse : center (h,k), semi-major axis = a and semi-minor axis = b
3.I know that ellipse is tangent to circle.
If I know all parameters, except h , what is the formula /equation find h ?
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This question requires finding the Cartesian equation for the locus:
$|z-1| = 2|z+1|$
that is, where the modulus of $z -1$ is twice the modulus of $z+1$
I've solved this problem algebraically (by letting $z=x+iy$) as follows:
$\sqrt{(x-1)^2 + y^2} = 2\sqrt{(x+1)^2 + y^2}$
$(x-1)^2 + y^2 = 4\big((x+1)^2 + y^2\big)$
$... |
Suppose I have $G = \mathbb{Z}_{p^m} \times \mathbb{Z}_{p^n}$ for $m, n$ distinct natural numbers and $p$ a prime. Is there a combinatorial way to determine the number of automorphisms of $G$?
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I was reading something today that was talking in terms of 10%, 100% and 1000% faster. I assumed that 10% faster means it takes 10% less time (60 seconds down to 54 seconds).
If that is correct wouldn't 100% faster mean 0 time and 1000% mean traveling back in time?
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Could someone walk me through this derivative?
Thanks.
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I'm intested in learning how to get a sum in difference calculus. I'm willing to settle, at least for now, in learning how to integrate the following integral in calculus:
$\int(a^x(1-a^x)(1-a^x))dx$
The sum that I'm really interested in learning how to get, hopefully using difference calculus, is:
$\sum_x{a^x(1-a^x)(... |
Let $k$ be a commutative ring, and $A$ be a $k$-module. Does $\mathrm{End}(A) \cong A \otimes_k A^*$ hold in general? It holds for finite dimensional vector spaces, but I'm not sure how to construct an element of $A \otimes_k A^*$ that corresponds to $\mathrm{id}_A$ in general, without picking a basis.
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Here is a formula: $\text{angle}=\arctan(dy/dx)$.
I can find an angle with my calculator for any value except $dx=0$.
My question is: is there no angle or, is there something that says when $dx=0$ the angle is found differently?
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Using only the delta definition of a limit, how can we prove that the sequence $\{a_n\}$, where $a_n = \sin n$, as $n$ tends to infinity does not have a limit?
Thanks!
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Given $X$ and $Y = a+bX$, I have to prove that:
If $b \lt 0$, then $\rho = -1$. If $b \gt 0$, then $\rho = 1$.
I've gotten to the point where I have:
$$ \rho = \frac{b \cdot \sigma_x }{ \sqrt{\sigma_y^2} }$$
I need to find why $\sigma_y^2 = b^2 \sigma_x^2$.
Can anyone please explain to me why that is the case?
I tri... |
I am working on a project regarding the number of zeros of a harmonic polynomial and am stuck with the proof of this: The zero set of the harmonic polynomial h(z)= $z^n$ - $\bar{z}^n$ consists of n equally spaced lines through the origin.
Any suggestion is appreciated. Thanks.
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I am trying to integrate $x^3(x^2+1)^5$ and I believe the starting point is to utilize long division to break up the polynomial, however I am unsure how to divide this out. Could someone give me what exactly I'm suppose to be dividing?
Edit: Well the reason I ask is if you attempt to use wolframalpha.com, they state t... |
It seems to be very simple question, but I'm afriad that I am missing something.
Ok, so you have a basket with 1 ball- which can be black or white with an equal proability.
Then you insert to the basket white ball, and by coincidental selection you take out white ball out of the basket.
What is the probability that the... |
In the polynomial algebra $\mathbb{C}[X_1, X_2,\ldots, X_n]$, we define a set of symmetric polynomials as follows $h_i(X_k, X_{k+1}, \ldots, X_n)$ = sum of all monomials of total degree $i$ in the set of variables $X_k, X_{k+1}, \ldots X_n$; $\forall i,k = 1,2, \ldots n$. Let $\sigma_i$ be the elementary symmetric poly... |
Let's define the primarithm function, $pog : \mathbb{N} \rightarrow \mathbb{N}$, where $pog(n)$ is the largest number of distinct primes that can divide a natural number $k$, $k \leq n$.
Does this function have a real name and, more importantly, is there a good estimating function that approaches $pog(n)$ as $n$ gets l... |
Let $\mathbb{P}_{\mathbb{C}}$ be the set of Gaussian primes and $\mathbb{P}_{\mathbb{N}}$ the set of primes in $\mathbb{N}$.
Let $\pi_{\mathbf{C}}(\sqrt{n})$ be the number of Gaussian primes with norm $\leq \sqrt{n}$ and $\pi_{\mathbf{N}}(n)$ be, as usual, the number of primes $\leq n$ in $\mathbb{N}$. Recall that nor... |
Let $1<p<\infty$, and define an isometry of normed linear spaces to be a norm-preserving surjection. Then all isometries from $\ell^p_n(\mathbb{R})$ to itself are given by linear transformations $T$ such that Mat$(T)$ is product of a permutation matrix and a diagonal matrix with $1$'s and $-1$'s along the diagonal. How... |
A linear system of equations of the form
$\begin{cases} ux+(u+d)y=u+2d\\
ax+(a+i)y=a+2i\end{cases}$
Will always have the solution $x=-1, y=2$ (easily proven through matrix algebra). How can I prove this by induction?
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How can one tell that a given sequence is perodic modulo m?
For example its easy to see that the sequence $1^1, 2^2, 3^3, \dots$ is periodic modulo 10. But how can we prove this?
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How can I prove that a 3x3 system of linear equations of the form:
$\begin{pmatrix}
a&a+b&a+2b\\
c&c+d&c+2d\\
e&e+f&e+2f
\end{pmatrix}
\begin{pmatrix}
x\\ y\\ z
\end{pmatrix}
=\begin{pmatrix}
a+3b\\
c+3d\\
e+3f
\end{pmatrix}$
for $a,b,c,d,e,f \in \mathbb Z$ will always have infinite solutions and will inter... |
I have some functionals that have lots of parameters, but all are volume integrals (not really important...)
It's a variational function I'm minimizing and I'm only keeping the real solution (because it's my variational wavefunction and I demand only a real solution :) ).
Solve[D[Subscript[E, HO][\[CapitalPsi]] + Subsc... |
How can I find a Chevalley basis of a type $B_2$ when the related Lie algebra is defined as a linear Lie algebra of elements of the form $$x= \begin{pmatrix} 0 & b_1 & b_2 \\ c_1 & m & n \\ c_2 & p & q \end{pmatrix},$$ where $c_1=-b_2^t$, $c_2=-b_1^t$, $q=-m^t$, $n^t=-n$, $p^t=-p$?
When trying to find such a base, the ... |
Let $A$ and $B$ be $n \times n$ matrices. Strassen's algorithm for multiplication works on a partitioning of $A$ and $B$ into $2^2$ submatrices each. Is there any generalization of Strassen's algorithm where the partitioning is done into, say, $k^2$ submatrices?
Note: What I mean by a $3^2$ partitioning of $A$ is:
$$A ... |
I was trying to solve the following exercise:
Suppose that the sequence of independent events $\{A_i\}$ satisfies $$\phi(n) = \sum_{i=1}^{n} P(A_i) \rightarrow \infty \text{ as } n \rightarrow \infty$$
and let $\tau_k :=\min\{n | \sum_{i=1}^{n} 1(A_i) = k\}$. By applying Doob's stopping time theorem to an appropria... |
So the question asks me to prove that an entire function with positive real parts is constant, and I was thinking that this might somehow be related to showing an entire bounded function is constant (Liouville's theorem), but are there any other theorems that might help me prove this fact?
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For an arbitrary ring $R$ and a positive integer $n >1$, are the category of $R$-modules and the category of $M_n(R)$-modules isomorphic?
Here, $M_n(R)$ denotes the $n\times n$ matrices over the ring $R$.
I know these two categories are equivalent, and I guess they are not necessarily isomorphic, but I don't know how t... |
I would like to know a good program for drawing graphs and analyzing them (finding Eulerian circuits, Hamilton cycles, etc.). I would also like to export the drawing to Word.
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From Richard Stanley's Enumerative Combinatorics,
2(j) How many sequences $(a_1, a_2, \ldots, a_{12})$ are there consisting of four 0's and eight 1's, if no two consecutive terms are both 0's?
I can brute force this, but Stanley provides the (mysterious) reasoning ${8 + 1 \choose 4} = 126$.
Any elegant argument is appr... |
Can someone show that the collection of all simple functions is a function space? Also, is the collection an algebra?
Thank you very much for your help...
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Exercise 2.27 in Boyd and Vanderberghe:
Suppose the set C is closed, has nonempty interior, and has a supporting hyperplane at every point in its boundary. Show that C is convex.
Seems to me one approach is to prove that the intersection of all the supporting hyperplanes is exactly C. Clearly this intersection contains... |
Let $0<x_1<x_2<...<x_n<1$. Let us consider the matrix M defined such that:
$$M_{i,j} = \min(x_i, x_j)^2 \left( 3 \max(x_i, x_j) - \min(x_i, x_j) \right)$$
I believe the determinant of M is different from zero. Any idea on how to prove it (if true)?
Of course, any reason why M would be invertible would suit me as well, ... |
$$.670=\frac{x}{(1-x)(2-x)}.$$
Need to change into the quadratic equation.
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The following is an identity concerning the Bessel functions of the first kind $J_n(x)$ for integers $n$ and $m$:
$$\sum_{n=-\infty}^\infty J_n(x) J_{n+m}(x) = \delta(m)$$
where $\delta(x)$ is the Kronecker delta function.
This can be derived from the Jacobi-Anger identity, but is there a simpler way to derive it, for ... |
If I have a know point cloud before and after a transformation (it could squish,
deform, rotate, etc.) and a known center of the point cloud before and after a transformation I'd like to see how a vector is transformed with that system.
I'd like to have each point in the point cloud contribute to the vectors transform... |
Suppose that $T: \mathbb{R}^m \to \mathbb{R}^n$ is a linear transformation and that $v_1, v_2, \ldots, v_p$ are vectors in $\mathbb{R}^m$.
If $W = \{T(v_1), T(v_2), \ldots, T(v_p)\}$ is linearly independent in $\mathbb{R}^n$, does it follow that $S = \{v_1, v_2, \ldots, v_p\}$ is linearly independent in $\mathbb{R}^m$?... |
For complex numbers $a_{n}, b_{n}$, what would be the next step to simplify this expression:
$$\left \|\sum_{n=1}^\infty a_{n} b_{n}\right \|^{2}=\left \langle \sum_{n=1}^\infty a_{n}b_{n},\sum_{m=1}^\infty a_{m}b_{m}\right \rangle$$
Is it $=\sum_{n=1}^\infty \sum_{m=1}^\infty |a_{n}|^{2}|b_{m}|^{2}$ !?
where $|a_{n}|... |
Suppose that we have a sequence of finite sets $A_1, A_2, \ldots$, which partition $\mathbb{N}$. I am making no other assumptions on the $A_n$ - i.e. there could be any amount of interleaving between them. Now suppose we have $S\subset\mathbb{N}$. If $\lim_{n\rightarrow\infty} \frac{|S\cap A_n|}{|A_n|}=0$, does it f... |
Suppose you have a number field $L$, and a non-zero ideal $I$ of the ring of integers $O$ of $L$.
Question part A: Is there prime ideal $\mathcal{P} \subseteq O$ in the ideal class of $I$ such that $p =\mathcal{P} \cap \mathbb{Z}$ splits completely in $O$?
If the extension $L/\mathbb{Q}$ is Galois I think one can sh... |
I am trying to understand the proof in Carmichaels book Diophantine Analysis but I have got stuck at one point in the proof where $w_1$ and $w_2$ are introduced.
The theorem it is proving is that the system of diophantine equations:
$$x^2 + y^2 = z^2$$
$$y^2 + z^2 = t^2$$
cannot simultaneously be satisfied.
The syst... |
I am having trouble showing the following and i was wondering if anyone can point out my mistake.
let f be a 2pi periodic function and
$$f_m(t)=f(mt)$$
show the n-th fourier coefficient
$$\hat{f_m}(n)=\hat{f}(n/m)$$ if m divides n
$$\hat{f_m}(n)=0$$ if m does not divides n
my attempt:
$$\hat{f_m}(n)=\frac{1}{2\pi}\in... |
I've seen the integral $\displaystyle \int_0^1\ln(1-x^2)\;{dx}$ on a thread in this forum and I tried to calculate it by using power series. I wrote the integral as a sum then again as an integral. Here is my calculation:
$$\displaystyle \begin{aligned} \int_0^1 \ln\left(1-x^2\right)\;{\mathrm dx} & = -\int_0^1\sum_{k... |
This was a mathematical induction question proposed in a textbook, and I've exhausted multiple approaches (proving RHS - LHS > 0, splitting the fraction, fractional exponents, etc.)
The geometric mean of $n$ positive numbers $a_1, a_2,\ldots,a_n$ is $\sqrt[n]{a_1a_2 \ldots a_n}$ and their arithmetic mean is $\frac{a_1... |
I want to show through Boolean algebra that the statement
$$\neg Q\Rightarrow (R\Rightarrow \neg (P\land Q))$$
is equivalent to the tautology $Q\lor(\neg Q)$?
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Let I(x) be a function such that I(x)=0 if x$\leq$0, and I(x)=1 if x$>0$. Let $c_{n}$ be a sequence of numbers such that $\sum$ c$_n$ converges absolutely. Let $x_n$ be some sequence of distinct real numbers.
Let $f_n(x)=\sum_{i=1}^{n} c_i I(x-x_i)$ and $f(x)=\sum_{i=1}^{\infty}c_iI(x-x_i)$.
I'm kind of stumped as t... |
Textbooks usually state "it is not hard to check that in modular arithmetic the usual associative, commutative and distributive properties continue to apply". Is there a way other than tedious proof by case analysis? For example, perhaps it is more convincing to notice that, say Z/13, is cyclic group over addition and ... |
Prove that $x^3 \equiv a \pmod{p}$ has a solution where $p \equiv 2 \pmod{3}$?
How can I prove a congruence equation has a solution? I tried to link Fermat's little theorem with this problem, but I couldn't find a way to solve it.
My attempt was:
$$x^3 \equiv 1 \pmod{2}$$
$$x^3 \equiv a \pmod{p}$$
If $p \equiv 2 \pmo... |
Explain why the following are true for arbitrary statements P and Q:
If P is a tautology then (P or Q) ≡ P
|
Given a topological space $X$ and a finite group $G$, can we construct a covering space $Y$ of $X$, such that the full covering transformation group is isomorphic to $G$. (We may assume some conditions, if required, such as connectedness, path-connectedness etc.)
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Is $\mathbb Q(\pi,i\pi):\mathbb Q$ a simple extension?
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How does $W$ behave near $+\infty$ compared to $\log$? In particular, I'm interested in the asymptotic expansion of
$$\frac{W(x)}{\ln(x)}$$
near $\infty$ (but along the positive real line, if that matters). Clearly $W(x)\sim\ln(x)$ as $x$ increases, and the next term looks hyperbolic.
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What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties.
Perhaps this question could be put more generally as: what's the difference between a vector space and a field?
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Let $t \geq 1$ and $\pi$ be an integer partition of $t$. Then the number of set partitions $Q$ of $\{1,2,\ldots,t\}$ for which the multiset $\{|q|:q \in Q\}=\pi$ is given by \[\frac{t!}{\prod_{i \geq 1} \big(i!^{s_i(\pi)} s_i(\pi)!\big)},\] where $s_i(\pi)$ denotes the number of parts $i$ in $\pi$.
Question: Is there... |
Our book gives this problem:
Find the $\mathcal{B}$-matrix for the transformation $\vec{x} \rightarrow A\vec{x}$ when the basis $\mathcal{B} = \{ \vec{b}_1, \vec{b}_2 \}$, where $A = \left[\begin{array}{cc}
3 & 4 \\
-1 & -1 \\
\end{array} \right]$, $\vec{b}_1 = \left[\begin{array}{cc}
2 \\
-1 \\
\end{array} \rig... |
Let $A_i=K_1 \times K_2 \times K_3 \times \cdots \times K_{i-1}$ and $B_i=(K_1+1) \times (K_2+1) \times (K_3+1) \times \cdots \times (K_{i-1}+1)$, where $K_{i-1}=(n-1)2^{n-(i-1)}-1$, $i=2,3,\cdots$. Find $a_i=\frac{A_i}{B_i}$ and $\sum_{i=1}^n a_i$.
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I have a question about how well one can choose a partition of unity.
So suppose $(M,g)$ is an open Riemannian manifold. I would like to have the following statement to be true:
"There exists an open cover $\{U_n\}$ of $M$ which is uniformly locally finite (i.e. each point of $p \in M$ lies in at most $q$ sets) and uni... |
Can anyone check my proof and let me know if I made some mistakes, please?
Let $R$ be an integral domain, let $K$ be its field of fraction Let $M=R^{\oplus A}$ be a free module on $R$ with
basis $A$. View $M$ as a subset of $K^{\oplus A}$. Let $S$ be a subset of M.
Q : Show that if $S$ is linearly independent on $R$,... |
Is it possible to prove that for all kinds of simple polygon, regardless of whether it is convex or concave and with no opening, the centroid of the polygon must ( or may not) lie inside the polygon?
The wiki link above gives example of polygon which has the centroid lying outside the polygon:
A non-convex object migh... |
I have a question about complex analysis. I am trying to get some intuition about residues and integration.
Let $c(t) = e^{2\pi it}$ be the unit circle of the complex plane.
I consider the following functions :
$f(z) = z$
$g(z) = 1/z$
Integrating $f$ around the $c(t)$ circle yields $0$ because f has no pole inside the ... |
Suppose B represents the matrix of orthogonal (perpendicular) projection of $\mathbb{R}^{3}$ onto the plane $x_{2} = x_{1}$. Compute the eigenvalues and eigenvectors of B and explain their geometric meaning.
What I have I come up so far as an attempt to deduce this question down is, for instance. If we pick an arbitrar... |
I'm wondering does anyone know of a scientific calculator with a mod function?
In C# this is shown as follows (just in case there are any other mods that a mathematical non-savant such as myself wouldn't be aware of)
5 % 4 = 1
I've looked through the most common ones on the web but the function is either not in the lis... |
In this question the term rapidly decaying sequence is used. What is the definition of a rapidly decaying sequence.(in cases of terms being purely real or complex.) How to prove that the sequence of Fourier series coefficients of a compactly supported, real valued, smooth function is rapidly decaying ? Please suggest a... |
A rather tight lower bound $c(n)$ of the number of unlabeled digraphs of order $n$ (loops allowed) seems to be
$$c(n) = 2^{n^2}/n!$$
because there are $2^{n^2}$ labeled graphs, almost all of them are rigid, thus division by $n!$ is almost always "correct", but yields a slightly too small result, because it is sometimes... |
According to my math book, in order to find the intersecting line between two planes we need to:
Find the vector product of the direction normals of the two planes
Write the equations of the planes in Cartesian form.
Assume that $z=0$ since the line has to intersect this plane.
Solve simultaneously for a point on the ... |
Q1.
$$\sum\limits_{j=0}^{N-1} \sum\limits_{i=0}^{N-1} \sum\limits_{y=0}^{j} \sum\limits_{x=0}^{i} a(x)a(y)b(|x-y|) $$
Given that $b(0) = 0$, Find the coefficient of $b(k)$, $0\le k \le N-1$ in the above expansion.
Q2. $$\sum\limits_{j=0}^{N-1} \sum\limits_{i=0}^{N-1} \sum\limits_{x=0}^{\min(i,j)} a(x) .$$
Find the coe... |
In "On Fusion Categories" by Etingof, Nikshych and Ostrik Proposition 2.1 there is used the following characterisation of a right dual in a rigid monoidal category (let us restrict to strict monoidal categories for simplicity):
Let $V$ be a simple object in a rigid monoidal category. A simple object $Y$ is the left dua... |
). Recently, I asked you how to find the "Intersection of 2 Lines in 2D" and the answers revolved around the Determinants ( http://en.wikipedia.org/wiki/Line-line_intersection ) or Systems ( Intersection Of Two Lines In 2D ).
Now, based on what I learned here from Isaac about how to get the General Form Linear Equation... |
$x + y + z = 4$
$x^2 + y^2 + z^2 = 4$
$x^3 + y^3 + z^3 = 4$
Any ideas on how to solve for $(x,y,z)$ satisfying the three simultaneous equations, provided there can be both real and complex solutions?
|
Even though there are uncountably many subsets of $\mathbb{N}$ there are only countably many isomorphism classes of countably infinite - or countable, for short - models of the empty theory (with no axioms) over one unary relation.
How many isomorphism classes of
countable models of the empty theory
over one binar... |
Please suggest a good reference for an introduction to Kernel Density Estimation. I am an electrical engineer, and I want some basic introduction so that I can start using it and also understand its working. I have basic knowledge of probability and random variables as taught in engineering courses.
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What is the volume of spheres in higher dimensions?
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I am reading Kai Lai Chung's A Course in Probability Theory. I understood the concept of vague convergence of a sequence of probability measures in the book. But it seems to me the book uses the concept of vague convergence of a sequence of distribution functions without defining it first. But I could miss the definiti... |
Let $A$ be a subspace of $\mathbb{R}^{n-1}$ with $n \geq 4$. And if we know $\pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+1} -A)$, then can we say that $\pi_1(\mathbb{R}^n -A) \cong \pi_1(\mathbb{R}^{n+k} -A)$ for all $k \geq 1, k \in \mathbb{Z}$ ? I have tried to prove it or find an counter example, but failed.... |
As stated in the title, I need a decision procedure for the problem of regular expression equivalence with not. Wikipedia states the problem is in the NONELEMENTARY complexity class. All I really need is link to the paper that describes the problem and its solution. This is NOT a homework problem. I need the proced... |
I would like to like to be able to find the point p3 given point p1 and p2. The length between point p1 and p2 is set to L and the length between point p2 and p3 is set to L. The angle between point p1, p2 and p3 is set to theta. I'm pretty sure the dot product will play a roll but I'm not sure how. Any help would be ... |
This is one of Do Carmo's excersices and I got it as homework. Part (a) is easy and I include it here for the sake of completness. But I am entirely lost on part (b).
A function $g:\mathbb{R} \rightarrow \mathbb{R}$ given by $g(t)=yt+x$, $t,x,y \in \mathbb{R}, y>0$ is called an affine proper function. The subset of all... |
Got this as homework and I don't know how to tackle this. Help please!
Prove that the isometries of $\mathbb{S}^{n} \subset \mathbb{R}^{n+1}$, with the induced metric, are restrictions to $\mathbb{S}^{n}$ of the linear orthogonal transformations.
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I am having a little trouble understanding the notion of non-degeneracy for bilinear maps.
A bilinear map $\psi : V \times V \mapsto \mathbb{R}$ always has a matrix representation, so $\psi(x,y)=x^TAy$.
A bilinear map is non-degenerate if and only if $A$ is full rank. This is also equivalent to stating that $\psi(x,y)... |
I'm trying to figure out how to compute a particular integral using Lebesgue integration.
For a number $a$, define $f(x) = x^a$ for $0 \lt x \leq 1$, and $f(0) = 0$. Compute $\int_0^1 f$.
Here is what I have so far:
$$\int_0^1 f = \int_{(0,1]}f + 0 = (x^a)*m((0,1]) = x^a$$
I'm not sure if I'm doing this correct. I'd ... |
Let $X$ be a Banach space, and let $X^*$ denote its continuous dual space.
Under the weak* topology, do compactness and sequential compactness coincide?
That is, is a subset of $X^*$ weakly* compact if and only if it is weakly* sequentially compact? Does one imply the other?
Perhaps I should make this next one a se... |
Show that if a 3-D surface is tangent to a plane along a curve, then the points of this curve are either parabolic or planar.
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I see an equation like this:
$$y\frac{\textrm{d}y}{\textrm{d}x} = e^x$$
and solve it by "separating variables" like this:
$$y\textrm{d}y = e^x\textrm{d}x$$
$$\int y\textrm{d}y = \int e^x\textrm{d}x$$
$$y^2/2 = e^x + c$$
What am I doing when I solve an equation this way? Because $\textrm{d}y/\textrm{d}x$ actually mean... |
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