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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 ...
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 ...
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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.
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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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 ...
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 ...
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, ...
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 ...
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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 ...
2answers
466 views

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 $$\lim_{x\to +\infty}y(x)=\eta\text{ for some }\eta\in \mathbb R\... 1answer 14k views ### How to get the roots of a quartic function when given a quadratic factor We have the function$$x^4 + 4x^3 - 17x^2 -24x + 36 = 0.$$x^2 -x - 6 is a factor of this function. Find all the roots of the polynomial. So we have (x-3)(x+2), and since it is a quartic we need ... 2answers 183 views ### Does there exist an infinite number string without any 'refrain'? Let us consider an infinite or finite number string which consists of 0,1,2. Then, let us call an adjacent pair of repeating number(s) 'a refrain'. For example, we have three refrains in the ... 1answer 607 views ### Where to get help with Homotopy type theory? I'm trying to understand the Homotopy Type Theory book. I find myself completely lost in Chapter 2, especially when it starts using higher groupoids. What is the recommended background for this book? ... 1answer 230 views ### Subadditivity of positive index of inertia Denote the number of positive eigenvalues of a Hermitian matrix H by P_H. If A,B Hermitian, show that$$P_{A+B}\leq P_A+P_B.$$2answers 168 views ### R is a ring with unity, and for each a \in R, there exists x \in R such that a^2x=a. Show that ax=xa. Let R be a ring with unity. For each a \in R, there exists x \in R such that a^2x=a. Show that ax=xa. I know that R has no nonzero nilpotent elements and axa=a. Thus I tried to show ... 2answers 246 views ### Set theory based on inclusion There are several axiomatizations of set theory based on inclusion rather than membership. I found only two papers, but they are both in German, and I could not read them even using a disctionary. Can ... 1answer 209 views ### Need a general formula for \frac{d^n}{dx^n}\left(f(x)^m\right) Let m,n\in\mathbb{N}. I need to express the derivative \displaystyle\frac{d^n}{dx^n}\left(f(x)^m\right) in terms of sums/products of the derivatives of the function f itself. Here are results ... 3answers 253 views ### How can we find the gcd for elements (binomial coefficient)? \gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right) i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ... 3answers 2k views ### conditional probability question from sheldon ross In any given year a male automobile policyholder will make a claim with probability p_{m}, and a female policyholder will make a claim with probability p_{f}, where p_{f} \neq p_{m}. The ... 2answers 432 views ### show that a^3+b^3+c^3-3abc\ge2(\frac{b+c}{2}-a)^3 let a,b,c\ge 0,show that:$$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)$$then let b-a=x,c-a=y But following I don't can't ... 2answers 693 views ### characterization of projective/injective/flat modules via \operatorname{Hom} and \otimes Let R be a commutative unital ring and M an R-module. Then M is projective iff \operatorname{Hom}(M,-) is exact, injective iff \operatorname{Hom}(-,M) is exact, and flat iff M\otimes- is ... 1answer 872 views ### What are the one-parameter subgroups of GL? Are the multiplicative one-parameter subgroups of the general linear group (i.e., morphisms \lambda:\Bbbk^\times\to\mathrm{GL}_n\Bbbk of algebraic groups) completely classified? The obvious ... 4answers 820 views ### Prove \frac{1}{\sqrt{x}}\geq \frac{\ln x}{x-1} I am trying to show that, for all x>0:$$\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 ... 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-... 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 \... 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 ... 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 ... 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? 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*}... 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 ... 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 byp_{1},p_{2},\cdots \cdots ,p_{n}\cdots \cdots. The prime maximal gap \max_{p_{n+1}\leqslant ... 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 ... 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 ... 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 ... 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 asm^*(E)=\inf\left\{\sum_{n=1}^\infty l(I_n):\,E\subset \bigcup_{n=1}^\infty I_n\right\},$$where each ... 3answers 3k views ### What are discrete and fast Fourier transform intuitively? I have done both of these in my math courses, but without understanding what they actually are intuitively. I would be very much grateful if you could give me an intuitive explanation of them. 1answer 295 views ### Finite Series - reciprocals of sines Find the sum of the finite series$$\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 $\dfrac{\cos1^{\circ}}{\sin^... 2answers 6k views ### Show that any continuous$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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