could anyone provide me with Emmy Noether’s definition of right and left ideals? The German original and references would be welcome. I am assuming she was the one who first coined those two kinds of one-sided ideals, though the term ideal was already in use in Dedekind. But maybe my assumption is wrong. Feel free […]

I’m taking a course in continuum mechanics. Our book is Continuum Mechanics by Reddy, a Cambridge edition. In the second chapter he introduces tensors and defines them to be polyadics. He is specifically concerned with dyadic which are tensors of rank (2,0). In his presentation of dyadic, the notation is introduced, then some properties are […]

For recreational purpose, i haven’t seen a interesting elemetary probability question quite a while. Is there any surprising elementary probability problem that result in surprising solution like the Monte Hall problem? Please give a few examples.

I was given the following question regarding hyperbolic geometry: In the hyperbolic geometry in the upper half plane, construct two lines through the point $(3,1)$ that are parallel to the line $x=7$. How do I go about doing this? I am very new to non-Euclidean geometry. Thank you.

Cauchy defined infinitesimal as a variable or a function tending to zero, or as a null sequence. While I found the definition is not so popular and nearly discarded in math according to the following statement. (1). Infinitesimal entry in Wikipedia: Some older textbooks use the term “infinitesimal” to refer to a variable or a […]

I am a postgraduate student of mathematics from Slovenia (central Europe) with quite some experience in mathematics. While answering questions on this site, I often encounter the function $\sec(x)$ which is, as I understand, defined as $\sec(x) = \frac1{\cos x}$. During my studies, I never encountered this function. I am wondering two things: How widespread […]

Two related elementary facts in group theory are sometimes called Poincaré’s theorems. If $H\lneq G$ and $[G:H]<\infty$, then there is $N\leq H$, $N\lhd G$ such that $[G:N]<\infty$. The intersection of a finite number of subgroups of finite index is of finite index. Did he prove both? Could you please give me references to the paper(s) […]

It was known in the 17th century that the function $$ t \mapsto \int_{1}^{t} \frac{dx}{x} $$ is logarithmic: a geometric sequence in the domain produces an arithmetic sequence in the codomain. This is. of course, easy to prove with the fundamental theorem of calculus. But is there a simpler, perhaps geometric, way of proving this?

As far as we know, Euler was the first to prove $$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{k=1}^\infty \left( \frac{1}{z-k} + \frac{1}{z+k} \right).$$ I’ve seen several modern proofs of it and they all seem to rely either on the Herglotz trick or on the residue theorem. I recon Euler had neither nor at his […]

Who invented or used very first the double lined symbols $\mathbb{R},\mathbb{Q},\mathbb{N}$ etc. to represent the real number system, rational number system, natural number system respectively?

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