Articles of reference request

Voronoi diagram with different metric functions

Given a metric space $(X,d)$ and finite number of points $(x_i)_{i=1}^n$ the Voronoi diagram (or the Dirichlet cell) $C_i$ is given by $$ C_i = \{x\in X:d(x,x_i)<\min\limits_{j\neq i}d(x,x_j)\}. $$ I have two question: Does anybody know if it was considered the case with different distance function, namely $$ C'_i = \{x\in X:d_i(x,x_i)<\min\limits_{j\neq i}d_j(x,x_j)\}. $$ where […]

Tutorials on LDPC error correction codes

Please consider this as soft question. Recently, I have been studying channel coding and in particular error correction codes. I am looking for best tutorial (easy to understand) on LDPC error correction codes. Including the decoding of such codes too. I am looking forward for your suggestions.

Runge's Theorem for meromrophic functions

Is there a name for this extension of Runge’s theorem? Theorem: Let $K\subset\mathbb{C}$ be compact, and let $A\subset K^c$ be a set which intersects each component of $K^c$. Let $f$ be meromorphic on an open set $U$ containing $K$. Let $A_f$ denote the set of poles of $f$ in $K$. Then there is a sequence […]

Limiting set theory using symmetry

If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For example, the set of all sets must be excluded. I recall that I came across a paper in Arxiv some time ago which explained […]

Exponential Law for based spaces

I realize most people work in “convenient categories” where this is not an issue. In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with the compact-open topology): $$F(X\times Y,Z)\cong F(X,F(Y,Z))$$ when $X$ is Hausdorff and $Y$ is locally compact Hausdorff. There is also supposed […]

Describe invariant polynomials under action of commutative group of order eight.

I believe the question below should be fairly standard in invariant theory ; I hope someone more familiar with it than me can explain a bit more or point to a reference. Let $F$ be polynomial field $F={\mathbb R}(X_1,X_2,X_3, \ldots ,X_8)$. Denote by $S$ the group of permutations on $\lbrace X_1,X_2,X_3, \ldots ,X_8\rbrace$, which acts […]

Has the 3×3 magic square of all squares entries been solved?

It is my understanding that it has not yet been determined if it is possible to construct a $3$x$3$ magic square where all the entries are squares of integers. Is this correct? Has any published work been done on this problem?

Suggested measure theory books for certain exercises

I was wondering if anyone knows books with difficult exercises of the theorems of monotone and dominated convergence and if the motto of Fatou possible. I use Bartle but it does not have many exercises demonstrations. Thank you very much. regards

Construction of Hadamard Matrices of Order $n!$

I’m trying to get a hand on Hadamard matrices of order $n!$, with $n>3$. Payley’s construction says that there is a Hadamard matrix for $q+1$, with $q$ being a prime power. Since $$ n!-1 \bmod 4 = 3 $$ construction 1 has to be chosen: If $q$ is congruent to $3 (\bmod 4)$ [and $Q$ […]

Reference request: algebraic methods in geometry

I am a (soon to be) third year undergraduate who has just finished courses in linear and abstract algebra. While I enjoyed the study of algebraic structures in their own right, my favorite part of the courses were the applications of the algebraic machinery developed to geometric problems (i.e. the connection between Galois theory and […]