How to determine the matrix?

Definition of S\mathbf{S} S(a,b,c)≡(a+b+c)2−2(a2+b2+c2)−4abcS(a, b, c) \equiv (a + b + c)^2 – 2(a^2 + b^2 + c^2) – 4abc So if I expand this I get S(a,b,c)=2ab+2ac+2bc−a2−b2−c2−4abcS(a, b, c) = 2ab + 2ac + 2bc – a^2 – b^2 – c^2 – 4abc =4(1−a)(1−b)(1−c)−(a+b+c−2)2 = 4(1 – a)(1 – …

Complex sequence (zi)(z_i) such that ∑izki\sum_i z_i^k converges and ∑i|zi|k\sum_i \vert z_i \vert^k diverges for all kk

How to find a complex sequence (zi)(z_i) such that ∑izki converges for all k∈N\sum_i z_i^k \text{ converges for all } k \in \mathbb N but ∑i|zi|k diverges for all k∈N?\sum_i \vert z_i \vert^k \text{ diverges for all } k \in \mathbb N ? =================      can you find an example that works for …

upper division linear algebra [duplicate]

This question already has an answer here: Why do we call a vector space in terms of vector space over some field? 4 answers I really need help understanding this sentence right here: “Every vector space is regarded as a vector space over a given field, which is denoted by …

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 …

Combinaison problem with exponential solutions

I have the following problem which I can’t find how to solve: There is a pool with R row and T tiles by row. The problem happen in a night. During the first day, there are (R+T)/2 algae. They may be anywhere. During the night, there is proliferation: each tile …