Newest Questions
1,698,241 questions
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The surface $x + y + z = \sin xyz$: Can it be visualized?
The surface $$x + y + z = \sin xyz$$ is very hard to visualize:
What can be said about it, geometrically or visually?
Results so far
Start with a similar, but simpler, curve: $$x + y = \sin xy.$$ It ...
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Need help understanding a particular step in the proof of Cauchy's root test.
I am questioning a particular step of the solution presented to the following question:
Cauchy’s root test for convergence states the following: Given a series $\sum_{k=1}^\infty a_k$, define
$$\rho=\...
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geodesic distance in embedded manifolds
I wish to describe a problem encountered in my research and am seeking advice, or just pointers on where to look.
My setting is as follows. Given a scatterplot of data in $\mathbb{R}^D$, we wish to ...
- 173
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$T$ is Hermitian positive iff $T$ is Hermitian and $Sp(T) \in \Bbb{R_+}$
let $H$ be a complex Hilbert space and $T \in L(H)$.
We know $T$ is Hermitian iff $ \langle Tx,x \rangle \in \Bbb{R}$
Now I would like to show $\langle Tx,x \rangle \ge 0$ iff $T$ is Hermitian and $Sp(...
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Type decomposition von Neumann algebra
I'm reading 'Lectures on von Neumann algebras' by Stratila and Zsido. Currently, I'm focused on the following fragment:
In the proof of Theorem 4.17, the decomposition (iii) seems very strange to me.
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Example of non-commutative finite ring without unity having no zero-divisors. And optimal examples.
I am trying to complete the following table with examples, but I cannot find a finite non-commutative ring without unity that is also a ring without zero divisors.
Let $K:=<\bar2>$ in $\mathbb{Z}...
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Continuity and differentiability at the origin of a logarithmic function with a parameter $\alpha$
Let $\alpha > 0$ and $f : \mathbb{R}^{2} \to \mathbb{R}$ be the function defined by
$$
f(x, y) = \begin{cases}
\dfrac{\log(1 + |xy|^\alpha)}{x^2 + y^4}, & \text{if } (x, y) \neq (0, 0), \\
0, ...
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Some simple power series covergence [closed]
I consider a sequence:
1/2 + 1/4 + 1/8 + 1/16 + ...
It obviously converge to 1.
Now,
I am known that every type of power series has a name. Is there a name for this sort of power series:
f(x)=x/2+(x^2)...
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Definition properly infinite projection von Neumann algebra
A projection $e$ in a von Neumann algebra $M$ is called properly infinite if for all central projections $q\in M$, it holds that finiteness of $qe$ implies $qe=0$.
According to this definition, it ...
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Validity of Green's Theorem in a specific case
be C1 and C2 two closed, smooth simple curves that not cross each other, with C1 in the interior of C2. Lets call A the region they enclose. Be D an open set,and A contained in D and consider a field ...
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Instances when deciding convexity of degree-four polynomials is in P
I was reading this paper, which contains a beautiful proof of the NP-hardness of deciding the convexity of quartic polynomials, including homogenous polynomials.
For what class of quartic polynomials ...
1
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1
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About the first fixed point of the $\aleph$ functional
I need help proving the following:
Show that $2^{\aleph_0}$ cannot be the first fixed point of the $\aleph$ functional.
I don't know where to start, I think maybe by proving the cofinality of the ...
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Affine equation of Kummer surface of product of elliptic curves
Let $A=E_1\times E_2$ where $E_1$ and $E_2$ are defined by two cubic polyonomials $f_1(x_1),f_2(x_2)$. Then this paper on page 15 claims that the Kummer surface of $A$ is given by the affine equation $...
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Missing function in composition of two functions
Let $(f\circ g)(x) =x^4+2x^3-3x^2-4x+6$ and $g(x)=x^2+x-1$. Find $f(x)$, it seem to be $f$ will have the formula $f(x)=ax^2+bx+c$. Plugging $g(x)$ in $f(x)$, we get $$ f(x^2+x-1)=a(x^2+x-1)^2+b(x^2+x-...
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Base change of modules and the group of homomorphisms
Let $A,B$ be commutative rings and in particular, $B$ is an $A$-algebra defined by a homomorphism $f:A \to B$. I want to prove that the following three conditions are equivalent.
$B \otimes_A B\cong ...
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