# All Questions

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### Understanding the syntax for derivatives - dy/dx

I'm new to calculus, and I'm trying to understand the syntax of derivatives: $$\frac {dy}{dx}$$ At a glance it implies some kind of division and some variable "d" has entered the picture. Does this ...
201 views

### How find this function equation $(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that satisfying the function equation $$(f(x))^2-(f(y))^2=f(x+y)\cdot f(x-y)$$ By the way :I have see this problem( is ...
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### Uniquely complemented lattice that is non-modular

I'm looking for an explicit example of a uniquely complemented lattice that is non-modular, since neither of the two non-modular lattices described here at wikipedia have this property. Thanks.
5k views

### Any two norms on finite dimensional space are equivalent

Any two norms on a finite dimensional linear space are equivalent. Suppose not, and that $||\cdot||$ is a norm such that for any other norm $||\cdot||'$ and any constant $C$, $C||x||'<||x||$ for ...
2k views

### How to prove that a vector bundle is trivial iff there are n global sections that form a basis on each fiber?

I can prove the only if part. My attempt to prove if part is the following: Given $n$ global sections $s_1, s_2, ..., s_n$ of a vector bundle $E$ on a smooth manifold $M$ such that they form a basis ...
360 views

### Multiplicative nature of the separability degree

In what follows, let $E / F$ be an algebraic extension, $h(x),f(x)\in F[x]$ polynomials, $h(x)$ irreducible. Definitions. We say $h(x)$ is separable if it has not repeated factors. We say $f(x)$ is ...
249 views

### What is going on in this degree 8 number field that fails to be a quaternion extension of $\mathbb{Q}$?

This is a soft but very mathematically hands-on question. Hopefully it will be interesting to more than just me. Thanks in advance for your help in thinking clearly about what follows. I have been ...
701 views

### If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?

This appears on problem 1 of chapter 1 in Stein & Shakarchi's Real Analysis: Given an irrational $x$, one can show (using the pigeon-hole principle, for example) that there are infinitely many ...
4k views

### Recovering eigenvectors from SVD

I am dealing with a problem similar to principal component analysis. Aka, I have a matrix and i want to recover the 'most efficient basis' to exaplin the matrix variability. With a square matrix these ...
1k views

### What's the best way to teach oneself both Category Theory & Model Theory?

I've done a bit of reading around both Category Theory & Model Theory (CT & MT) as a novice in each field. I'm interested in how they might combine, particularly when applied to Algebra. [So ...
2k views

### Proof that $\sin(x)$ don't have limit to infinity

I just used the Heine's definition. Let $\alpha,\delta \in \mathbb{R}$ such that $\sin(\alpha)=a$ and $\sin(\delta)=b$. Let $(u_{n})=\alpha+2\pi n$ and $(v_{n})=\delta+2\pi n$ and $f(x)=\sin(x)$. So ...
696 views

### Does every Lie algebra come from commutator of some associative product operation?

Suppose $\mathfrak{g}$ is an Lie algebra. Is it possible to define an associative product operation $\star$ on $\mathfrak{g}$ such that $[A,B]=A\star B - B \star A$ ? If it is not possible to do so ...
515 views

### How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
363 views

### How to find value of $x+y+z+u+v+w$

let $x,y,z,u,v,w$ be positive integer numbers,and such $$1949(xyzuvw+xyzu+xyzw+xyvw+xuvw+zuvw+xy+xu+xw+zu+zw+vw+1)=2004(yzvw+yzu+yzw+uvw+y+u+w)$$ Find this value of $$x+y+z+u+v+w=?$$ My try: maybe ...
Let $A$ be a non-unital C*-algebra. I would like to know a simple way to show that $A$ contains a self-adjoint element whose spectrum has at least $3$ elements. Note that the spectrum of an ...