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Let $f$ be a complex valued function, defined in a punctured neighbourhood of $z=0$. In other words, it is defined in $U-\{0\}$ for some open connected $U$. We say that $f$ is holomorphic at a point, if there is an open neighbourhood of this point in which $f$ is holomorphic. Now, make the following assumption: For any open set $V$ in $U-\{0\}$ such that $0\in \partial V$, there is at least one point at which $f$ is holomorphic. Can we say that $f$ is holomorphic in a punctured neighbourhood of $z=0$?

This may be a basic question. Let $d$ be a positive integer, and let $B=B(0,r)$ be the open Euclidean ball centered at the origin with radius $r$. We endow $B$ with the Euclidean metric $d$. Let $\hat{B}=B\cup\{\infty\}$ be the one-point compactification of $B$. I know that $\hat{B}$ is metrizable for some metric $\rho$. Can we construct a metric $\eta$ on $\hat{B}$ with the following conditions? The topologies induced by $\rho$ and $\eta$ are equivalent. For $x \in B$ and $\varepsilon \in (0,r)$, we have $\eta(x,\infty)\ge r-\varepsilon \Longrightarrow d(x,0)<r-\varepsilon$. In the first place, I do not know a simple formula for $\rho$. Thanks.

I am having difficulty with a part of Jonathan Wise’s oft-cited element-free proof of the snake lemma in an abelian category. At a high-level, he takes the snake-lemma starting-diagram, then applies a procedure where he replaces objects with quotients, or with kernels, successively, until he forms a snake of isomorphisms. The part I am unable to understand is how he is justified in replacing objects with (certain) kernels. Recall the context of the snake lemma, in the notation of Wise’s notes: He uses the following notation in his proof: $$B/A= \text{coker}(A\rightarrow B)$$ $$B : A= \ker(A\rightarrow B)$$ One of the key lemmas he seems to rely on on is the following: On its face, the lemma says that in an exact sequence with five elements, if the second item maps has a map from some $X,$ you can replace items two and three with their quotient by $X.$ Judging with how he continues the proof of the snake lemma, I think he is rather arguing that we can replace any two sequential objects in an exact sequence by their quotient by such an $X.$ However, this (seemingly?) says nothing about kernels! Then in the proof of the snake lemma itself, he claims that since every object in the diagram has a map into $L’’$ (true), you can replace all the objects $X$ in the diagram by $\ker(X\rightarrow L’’)$. This is where he loses me: He continues to do this and similar until he gets a snaking series of isomorphisms. My questions are firstly, has he justified this procedure and I am not picking up on it? And secondly, if he hasn’t justified it, what is the justification? My guesses are that this may be a reflection of some more general principle that exactness is preserved under taking kernels and quotients in an abelian category, so we’re free to keep “peeling off” unnecessary parts and still preserve the homological data. But what are the rules for doing that, aside from the setting of Lemma 4, which I follow. I have a vague notion that since kernels and cokernels are dual to each other, and abelian categories are self-dual, that maybe Jonathan Wise’s Lemma 4 somehow says something about taking kernels?

enter preformatted text here Let C ⊆ ℝⁿ be a convex set, and let φ : C → ℝ be a convex function. Define the function f : C → ℝ by f(x) = (φ(x))². Is f a convex function?

This is more of a procedural question than a request to help solve a problem, and I hope it'snot considered a foolish question. I've seen a lot of debate recently about the order of operations and multiplication by juxtaposition, and I'm hoping to settle this, even if just for my own knowledge. If a professor of mathematics weighed in, I'd appreciate it, because I'd love to know how you teach it to your students. So, is the consensus among professionals that juxtaposition has a tighter binding than multiplication or division in the order of operations? For example, would 9×6+9(5+3) = 189 Or 9×6+9(5+3) = 126 I am a firm believer in the answer being 189, that juxtaposition does not take priority over multiplication or division in the order of operations, probably for the most part because it doesn't take priority in any programming language that I know, and almost every calculator besides Desmos ignores it as well. However, if I am told that this is not the consensus among math professionals, I'm more than capable of swallowing my pride and admitting I am wrong. Thank you in advance for taking the time to read this.