Articles of definition

Complex Functions: Integrability

Let $\Omega$ be a measure space with measure $\lambda$. Denote the space of simple functions by: $$\mathcal{S}:=\{s:\Omega\to\mathbb{C}:s=\sum_{k=1}^{K<\infty}s_k\chi_{A_k:\lambda(A_k)<\infty}\}$$ Denote the positive and negative part of the real and imaginary part by: $$f=\Re_+f-\Re_-f+i\Im_+f-i\Im_-f=:\sum_{\alpha=0\ldots3}i^\alpha f_\alpha$$ Define for positive functions: $$\int fd\lambda:=\sup_{s\in\mathcal{S}:s\leq f}\int sd\lambda\quad(f\geq 0)$$ and for complex functions: $$\int fd\lambda:=\sum_{\alpha=1\ldots3}i^\alpha\int f_\alpha d\alpha$$ as long as all terms of […]

Meaning of variables and applications in lambda calculus

The wikipedia definition of lambda terms is: The following three rules give an inductive definition that can be applied to build all syntactically valid lambda terms: a variable, $x$, is itself a valid lambda term if $t$ is a lambda term, and $x$ is a variable, then $(\lambda x.t)$ is a lambda term (called a […]

What is the most accurate definition of the hyperboloid model of hyperbolic geometry?

For simplicity, let’s focus on the two-dimensional case (the hyperbolic plane in 3-space). I have seen the hyperboloid model defined as variations on the following i) The positive sheet of a two-sheeted hyperboloid (i.e. points that fulfill the equations $x^2 – y^2 – z^2 = 1, x>0$). ii) This is in Minkowski space. iii) The […]

Good way to describe “converging parallel lines”?

Not sure if this question is on topic. I am looking for a nice correct and succinct way to describe “2 lines are limiting parallel ” that is: Understandable for newbies to hyperbolic geometry. Geometrically correct not mentions “ideal points”. (because they don’t really exist. “going in the same direction” is also not allowable, lines […]

Understanding definition of tensor product

The definition I have of a tensor product of vector finite dimensional vector spaces $V,W$ over a field $F$ is as follows: Let $v_1, …, v_m$ be a basis for $V$ and let $w_1,…,w_n$ be a basis for $W$. We define $V \otimes W$ to be the set of formal linear combinations of the mn […]

Is the “binary operation” in the definition of a group always deterministic?

The introduction to group theory that I’m reading requires that the actions of a group are “deterministic”; but the formal definition given makes no mention of this property: A set G is a group if the following criteria are satisfied. There is a binary operation $\cdot$ on $G$. That operation is associative… There is an […]

What needs to be linear for the problem to be considered linear?

Harry Altman presented an excellent question in a comment here: What needs to be linear for the problem to be considered linear? So is it enough to a have linear objective function or other requirements for example to the smoothness?

Definition of local maxima, local minima

Wikipedia says that: A real-valued function $f$ defined on a real line is said to have a local (or relative) maximum point at the point $x^*$, if there exists some $\varepsilon > 0$ such that $f(x^*) \ge f(x)$ whenever $\lvert x − x^*\rvert < \varepsilon$. The value of the function at this point is called […]

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that belongs to $\mathsf{Rel}^{op}\left(B,A\right)=hom_{\mathsf{Rel}^{op}}\left(B,A\right)$? As a triple $\left(A,R,B\right)$ or as a triple $\left(B,R,A\right)$ where in both cases $R\subseteq A\times B$? ACC tells me: “…Thus $\mathcal{A}$ […]

If the empty set is a subset of every set, why isn't $\{\emptyset,\{a\}\}=\{\{a\}\}$?

If the empty set is a subset of every set, why it isn’t written with the elements of a set? like so $\{1,2,3,\emptyset\}$ Or why isn’t $\{\emptyset,\{a\}\}=\{\{a\}\}$? I know one has two elements and the other has 1 but since the empty set is a subset of both, then why it isn’t being mentioned explicitly […]