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8
votes
2answers
7k views

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 ...
8
votes
2answers
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 ...
8
votes
1answer
192 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...
8
votes
1answer
363 views

How to prove that $n\sum_{d\mid n}\frac{|\mu(d)|}{d}=\sum_{d^2\mid n}\mu(d)\sigma\left(\frac{n}{d^2}\right)$?

This is problem 11 part b in chapter 3 of Tom M. Apostol's "Introduction to Analytic Number Theory". A variation on Euler's totient function is defined as $$\varphi_1(n) = n \sum_{d \mid n} \frac{|\mu(...
8
votes
1answer
138 views

What are the conditions for $n^2 \nmid(n-1)!$

Q: What are the conditions for $n^2 \nmid (n-1)!$, given that $2\le n \le 100$ and $n\in \mathbb{N}$? According to me the two conditions must be: 1. $n$ is a prime number (since the factorization of $...
8
votes
1answer
3k views

What is a random measure?

What is a random measure? The Wikipedia article is quite confusing and used to formulate only a random counting measure instead of just the random measure. I'm working on the Dirichlet process and ...
8
votes
1answer
170 views

How to fill up $(0,1)$ with disjoint closed intervals all total measure one

This is a problem which was proposed, but not chosen, in a Mathematics competition for University students not long ago, and its solution is missing: Let $\sum_{n=1}^\infty a_n=1$, where $a_n>0$,...
8
votes
1answer
1k views

for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if ?? CSIR - June $2013$

Question is : for two non zero complex polynomial $p(z),q(z)$ we have $p(z)\overline{q(z)}$ is analytic if and only if $p(z)$ is Constant $p(z)q(z)$ is Constant $q(z)$ is Constant $\overline{p(z)}...
8
votes
1answer
380 views

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.
8
votes
2answers
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 ...
8
votes
1answer
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 ...
8
votes
2answers
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 ...
8
votes
2answers
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 ...
8
votes
2answers
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 ...
8
votes
2answers
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 ...
8
votes
1answer
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 ...
8
votes
2answers
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 ...
8
votes
2answers
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 ...
8
votes
1answer
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, ...
8
votes
2answers
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 ...
8
votes
2answers
432 views

There are 3 points in the spectrum of some self-adjoint element of a non-unital C*-algebra.

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 ...
8
votes
2answers
466 views

If $\displaystyle \lim _{x\to +\infty}y(x)\in \mathbb R$, then $\lim _{x\to +\infty}y'(x)=0$ [duplicate]

Not homework. I need this (or something similar) to solve 4. in this question. Let $y:(a ,+\infty)\to \mathbb R$ be $C^1$. Prove that $$\lim_{x\to +\infty}y(x)=\eta\text{ for some }\eta\in \mathbb R\...
8
votes
1answer
14k views

How to get the roots of a quartic function when given a quadratic factor

We have the function $$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$ $x^2 -x - 6$ is a factor of this function. Find all the roots of the polynomial. So we have $(x-3)(x+2)$, and since it is a quartic we need ...
8
votes
2answers
183 views

Does there exist an infinite number string without any 'refrain'?

Let us consider an infinite or finite number string which consists of $0,1,2$. Then, let us call an adjacent pair of repeating number(s) 'a refrain'. For example, we have three refrains in the ...
8
votes
1answer
607 views

Where to get help with Homotopy type theory?

I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ...
8
votes
1answer
230 views

Subadditivity of positive index of inertia

Denote the number of positive eigenvalues of a Hermitian matrix $H$ by $P_H$. If $A,B$ Hermitian, show that $$P_{A+B}\leq P_A+P_B.$$
8
votes
2answers
168 views

$R$ is a ring with unity, and for each $a \in R$, there exists $x \in R$ such that $a^2x=a$. Show that $ax=xa$.

Let $R$ be a ring with unity. For each $a \in R$, there exists $x \in R$ such that $a^2x=a$. Show that $ax=xa$. I know that $R$ has no nonzero nilpotent elements and $axa=a$. Thus I tried to show ...
8
votes
2answers
246 views

Set theory based on inclusion

There are several axiomatizations of set theory based on inclusion rather than membership. I found only two papers, but they are both in German, and I could not read them even using a disctionary. Can ...
8
votes
1answer
209 views

Need a general formula for $\frac{d^n}{dx^n}\left(f(x)^m\right)$

Let $m,n\in\mathbb{N}$. I need to express the derivative $\displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right)$ in terms of sums/products of the derivatives of the function $f$ itself. Here are results ...
8
votes
3answers
253 views

How can we find the gcd for elements (binomial coefficient)?

$\gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right)$ i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ...
8
votes
3answers
2k views

conditional probability question from sheldon ross

In any given year a male automobile policyholder will make a claim with probability $p_{m}$, and a female policyholder will make a claim with probability $p_{f}$, where $p_{f} \neq p_{m}$. The ...
8
votes
2answers
432 views

show that $a^3+b^3+c^3-3abc\ge2(\frac{b+c}{2}-a)^3$

let $a,b,c\ge 0$,show that: $$a^3+b^3+c^3-3abc\ge2 \left(\dfrac{b+c}{2}-a\right)^3$$ my try: $$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ac)$$ then let $b-a=x,c-a=y$ But following I don't can't ...
8
votes
2answers
693 views

characterization of projective/injective/flat modules via $\operatorname{Hom}$ and $\otimes$

Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is ...
8
votes
1answer
872 views

What are the one-parameter subgroups of GL?

Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms $\lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk$ of algebraic groups) completely classified? The obvious ...
8
votes
4answers
820 views

Prove $\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$

I am trying to show that, for all $x>0:$ $$\frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1}$$ This inequality is closer than I expected. I have tried exponentiating, power series, and have achieved ...
8
votes
3answers
269 views

On simple groups $G$, where $2\mid |G|$, $4\not\mid |G|$

The (old) exam I'm looking at has the following problem: Suppose the order of $G$ is even, but is not divisible by $4$. Prove that $G$ is not simple. A group with $2$ elements is clearly a counter-...
8
votes
2answers
772 views

Is it true that every element of $\mathbb{F}_p$ has an $n$-th root in $\mathbb{F}_{p^n}$?

It is not hard to prove that every element of $\mathbb{F}_p$ has a square root in $\mathbb{F}_{p^2}$: take any $a \in \mathbb{F}_p$ and consider the polynomial $f = X^2 - a$. If $f$ has a root in $\...
8
votes
1answer
1k views

finite subgroups of SO(3)

As is well-known, all finite subgroups of $SO(3)$, except for cyclic and dihedral groups, are isomorphic to one of: $A_4$ $S_4$ $A_5$ The classical proof of this fact uses the geometry of regular ...
8
votes
1answer
193 views

Solve $10x+2x^2+x^3=20$ using only algebra and geometry?

The cubic formula and modern math is not allowed, only algebra, geometry, and the like. Supposedly this problem was given to Fibonacci. Here is the whole paragraph I read: In Flos Fibonacci gives ...
8
votes
2answers
1k views

Sum of two irrational radicals is irrational?

If $a,b,m$ and $n$ are positive integers such that $\sqrt[m]{a}$ and $\sqrt[n]{b}$ are irrational numbers, how can we prove that the sum $\sqrt[m]{a}+\sqrt[n]{b}$ is also irrational?
8
votes
1answer
204 views

Closed form of $\operatorname{Li}_2(\varphi)$ and $\operatorname{Li}_2(\varphi-1)$

I am trying to calculate the dilogarithm of the golden ratio and its conjugate $\Phi = \varphi-1$. Eg the solutions of the equation $u^2 - u = 1$. From Wikipdia one has the following \begin{align*}...
8
votes
3answers
213 views

The Keys problem

Another challenge: A calculator has two special keys: A key transforms a number x in the number 2x. B key transforms a number x in the number 2x - 1. Is it true that if you start with any positive ...
8
votes
1answer
498 views

A Conjecture about Maximal Prime Gaps

As it is well known that prime number is $2,3,5\cdots \cdots$, thus all these prime number are denoted by$p_{1},p_{2},\cdots \cdots ,p_{n}\cdots \cdots$. The prime maximal gap $\max_{p_{n+1}\leqslant ...
8
votes
1answer
604 views

are elementary symmetric polynomials concave on probability distributions?

Let $S_{n,k}=\sum_{S\subset[n],|S|=k}\prod_{i\in S} x_i$ be the elementary symmetric polynomial of degree $k$ on $n$ variables. Consider this polynomial as a function, in particular a function on ...
8
votes
2answers
307 views

Primes inert in quadratic field of class number one

Let $K = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic field of class number one (i.e. every ideal in $\mathcal{O}_K$ is principal, i.e. $\mathcal{O}_K$ is a principal ideal domain). Let $d_K$ be ...
8
votes
4answers
22k views

Archimedes' derivation of the spherical cap area formula

Archimedes derived a formula for the area of a spherical cap. so Archimedes says that the curved surface area of a spherical cap is equal to the area of a circle with radius equal to the distance ...
8
votes
1answer
263 views

If $m^*(E)=\infty$, then $E=\bigcup_{k=1}^{\infty}E_k$, $E_k$ measurable and $m^*(E_k)<+\infty$

Reading Royden's fourth edition of Real Analysis. I'm working with outer measure defined as $$m^*(E)=\inf\left\{\sum_{n=1}^\infty l(I_n):\,E\subset \bigcup_{n=1}^\infty I_n\right\},$$ where each $...
8
votes
3answers
3k views

What are discrete and fast Fourier transform intuitively?

I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them.
8
votes
1answer
295 views

Finite Series - reciprocals of sines

Find the sum of the finite series $$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$ This problem was asked in a test in my school. The answer seems to be $\dfrac{\cos1^{\circ}}{\sin^...
8
votes
2answers
6k views

Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$

Be a continuous function $f:[0,1] \rightarrow [0,1]$. Show that there is a $\zeta \in [0,1]$ with $f(\zeta)=\zeta$ ($\zeta$ is called fixed point). Consider the function $g:[0,1] \rightarrow [-1,1]$,...

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