# Category: Math

## Closed sets in the lower limit topology.

Would an interval of the form [a,b][a,b] be closed in the lower limit topology Rℓ\mathbb{R}_\ell. Here is why I think it is: Because Rℓ\mathbb{R}_\ell is finer than the standard topology on R\mathbb{R}, then all the basis elements of this standard topology are in the lower limit topology; i.e., the sets …

## Singular value decomposition and inverse of square matrix

I’ve previously touched the subject in this question. There (and subsequently on other places), I’ve learned that if a SVD is applied to a square matrix MM, M=USVTM=USV^T, then the inverse of MM is relatively easy to calculate as M−1=VS−1UTM^{-1}=V S^{-1}U^T. I’ve implemented the SVD algorithm and began to receive …

## Is this function continuous at 0?

Let f(x)=x2yx+yf(x) = \frac{x^2y}{x+y} for (x,y)≠(0,0)(x,y) \neq (0,0); f(0)=0f(0)=0 Is this function continuous at the origin? If I use polar coordinates, I find that it is continuous. But if I try the limit on the Ox axis and the line y=x3−xy=x^3-x, the limits yielded are different. ================= 1 This …

## Proof with restrictions on mod

Let n be a positive integer. Prove that 12+22+32+…+(n−1)2≡0(modn)1^2 +2^2 +3^2 + … +(n-1)^2\equiv 0 \pmod n if and only if n≡±1(mod6)n\equiv \pm 1 \pmod 6 So I figured out that 12+22+32+…+(n−1)21^2 +2^2 +3^2 + … +(n-1)^2 is equivalent to (n−1)(n)(2n−1)6\frac{(n-1)(n)(2n-1)}{6}, but I don’t see why the only restriction has …

## Fixed-point iteration and continuity of parameters

Let XX a compact set and A⊆RA\subseteq \mathbb{R}. Consider a continuous function f:X×A→Xf\colon X\times A\to X and construct a fixed-point iteration as follows xk+1=f(xk,a),x0∈X,a∈A.(⋆) x_{k+1}=f(x_k,a),\quad x_0\in X, a \in A.\quad (\star) My question: If (⋆)(\star) admits a unique fixed point, denoted by Fix(fa)\mathrm{Fix}(f_a), for all a∈Aa\in A, can we conclude …