$\left\lfloor...
$$\left\lfloor \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+\frac{1}{\sqrt{4}}+...+\frac{1}{\sqrt{1024}}\right\rfloor =?$$ I can't find $$? \leq...
View ArticleWhat exactly is a number? [duplicate]
We've just been learning about complex numbers in class, and I don't really see why they're called numbers.Originally, a number used to be a means of counting (natural numbers).Then we extend these...
View ArticleAnother proof that dividing by $0$ does not exist -- is it right?
Ok I am in grade 9 and I am maybe too young for this.But I thought about this, why dividing by $0$ is impossible.Dividing by $0$ is possible would mean $1/0$ is possible, which would mean $0$ has a...
View ArticleAre there mathematical concepts that exist in dimension $4$, but not in...
Are there mathematical concepts that exist in the fourth dimension, but not in the third dimension? Of course, mathematical concepts include geometrical concepts, but I don't mean to say geometrical...
View ArticleProperties of "derivative of a recurrence"
Long-story-short, I'm required to come up with an unconventional concept for a presentation, and thought of the following:An operator $D$ such that for any recurrence relation, we get that$$D(x_n) = n...
View ArticleHow far away is $\operatorname{SL}_2(3)$ from being quasisimple?
Definition: A group $G$ is quasisimple if it is perfect (i.e., $G=[G,G]$, its derived subgroup) and $\operatorname{Inn}(G)$ is simple.NB: I know that $\operatorname{Inn}(G)\cong G/Z(G)$.The...
View ArticleAre covariance and contravariance asymmetric?
When I first learned some category theory, I was under the impression that covariant and contravariant phenomena were formally dual, and therefore had the same "value." However, as time has passed I...
View ArticleWhy the construction of ring of sections of a bundle?
I am reading page 78, Theorem 4.2.15 of the book Galois Theories, by Francis Borceus and George Janelidze. He stated the following definitions and results.For a commutative ring $A$ with identity, a...
View ArticleHow to find largest number of collinear points from a set of coordinates
Suppose we have a set of $n$-coordinates i.e. $(x_1,y_1),(x_2,y_2),\cdots ,(x_n,y_n)$ , I have the following question:$(1)$ How does one pick out the collinear sets of coordinates? (preferably the...
View ArticleChallenging Integrals for High School Students
I am now in my last year of high school. We have covered all the techniques useful for indefinite integration that are included in our Maths and Further Maths courses. This includes:Integration by...
View ArticleMathematical aphorisms: theorems captured in everyday analogies [closed]
What are some popular mathematical aphorisms or sayings that capture the intuition behind a theorem?Statements such asA drunk man will find his way home, but a drunk bird may get lost forever help...
View ArticleOn the near-integer $163/\ln(163)$
This question, concerning the approximation $\frac{163}{\ln(163)}\approx 2^5$, was posted on MO 5 years ago: Why Is 163/ln(163) a Near-Integer?.It was concluded that it had nothing to do with 163 being...
View ArticleWhat other tricks and techniques can I use in integration?
So far, I know and can use a reasonable number of 'tricks' or techniques when I solve integrals. Below are the tricks/techniques that I know for indefinite and definite integrals separately.Indefinite...
View Articlebecoming a better speaker in mathematical presentations [closed]
If this question is not relevant to the site. I apologize and I will remove my question. This is a soft question:What would you recommend for becoming a better speaker in mathematical presentations?I...
View ArticleIntersection Points of Sine Functions with Incommensurate Periods
I’ve been thinking about the intersection points of two sine functions with irrationally related time periods:$f(x) = \sin(2x) \quad \text{and} \quad g(x) = \sin\left(\frac{2\pi...
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notation for quantifiers (brackets, commas, colons, etc)
The following are variations in expressing "there exists a natural number $n$, such that $2T = n \cdot (n+1)$":$\exists n\in \mathbb {N},\; 2T = n \cdot (n+1)$$\exists n\in \mathbb {N}\; [2T = n \cdot...
View ArticlePrerequisites for learning Symplectic Geometry up to Fukaya Categories
I want to create a far-reaching plan for self-study of Symplectic Geometry with a view toward Homological Mirror Symmetry. In particular, I would like to prepare myself for being able to read Denis...
View ArticleStruggling with the book Challenge and thrills of pre college mathematics
I am a highschool student and I am preparing for IOQM (which is the first level for IMO qualification in india, the exam is objective type), I got the book"Challenge and Thrills of Pre-College...
View ArticleQuestion about fields medal [closed]
Continuing Are there any well known mathematicians who published very little? my question is fields medalist with least number of papers. Is there any more probability of getting fields medal in...
View ArticleConvention about sub-$\sigma$-algebras
I will illustrate my question with the following example. Let $\mathcal{B}({\mathbb{R}})$ and $\mathcal{B}({[0, \infty)})$ be the Borel $\sigma$-algebras on $\mathbb{R}$ and $[0, \infty)$ with respect...
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