# attaching maps in CW complexes

Suppose I have a finite CW complex $X$ with $p$-skeleton $X^{(p)}$.Suppose that the map $\theta = q_e \circ \varphi_f \colon S^p \to X^{(p)} \to S^p$ is surjective.

Suppose I have a finite CW complex $X$ with $p$-skeleton $X^{(p)}$.Suppose that the map $\theta = q_e \circ \varphi_f \colon S^p \to X^{(p)} \to S^p$ is surjective.

Suppose we have a joint distribution $P(D,X,L) = P(D|X)P(X|L)P(L)$.Here, D and L are discrete but X is a continuous random variable.

In machine learning or statistics, suppose that we have a set of data x, and we assume there are hidden factors z, then we may be interested in p(z|x), so called inference in machine learning.Usually, we estimate it by p(z|x) = p(z,x)/p(x), i.

Suppose $Z \sim Bern(p(X))$, where $p(X)$ is a logistic function of $X$.Then let $Y = \beta X$ where $\beta$ is a constant.

Suppose that $X \sim N(\mu, \sigma^2)$ and that

${P(X \le 5) = 0.8}$ and $P(X\ge 0) = 0.

a) ∀x(A → P(x)) ≡ A → ∀xP(x)

b) ∃x(A → P(x)) ≡ A → ∃xP(x)

My Solution

a) Suppose A is false.Then A -> P(x) is trivially true because if hypothesis is false then conditional statement is trivially true.

Suppose P(S) = x, P(S') = 1-x then for Alice:

P(C) = P(C|S)P(S) + P(C|S')P(S') = 0.43

P(C) = P(C|S)P(S) + P(C|S')P(S') = 0.

Suppose that $P(x)$ is the statement “$x2 +2x = 1$”.If the domain is the set of integers,

what are the truth values of $\forall x P(x)$ and $\exists x P(x)$?

Suppose P is an abstract mathematical problem, an element $x$ is a solution of P if P(x) is true.The uniqueness is defined as P is unique determined if any two solution of P is equal.

Let $P(x) \in \mathbb{R}[x]$ be polynomial with all real roots and has the property that $P(a)=0 \Rightarrow P(a+1)=1$ for all $a \in \mathbb{R}$.I think this problem statement is not true because if we suppose that $P(x)=x$ then $P(0)=0 \Rightarrow P(1)=1$, $\;P(x)$ has no repeated root.

Suppose that $T$ has characteristic $p>0,$ and that $x \mapsto x^p$ is not an automorphism.Since a ring homomorphism from a field into itself is injective or zero, then $x \mapsto x^p$ is not surjective.

Suppose we have arbitrary predicates are $P$ and $Q$.Consider:

$(x=0)$ as $P(x)$ and $(x > 0)$ as $Q(x)$.

Suppose $f\in L^p[0,\infty)$ for some $p>0$.Show that $\lim_{x\to\infty}f(x)=0$.

Content: suppose that $\forall x~(P(x) \wedge Q(x))$ is true.Because $P(a)$ is true and $Q(a)$ is true for every element in the domain, we can conclude that $\forall x~P(x)$ and $\forall x~Q(x)$ are both true.

Suppose $f(x)$ is p-th integrable, i.Let $A_N = \{x\in A \, : \, |f(x)|>N\}$.

Suppose I have X ∼ Bin(1,p).I know that X+X ∼ Bin(1+1,p), which would lead me to the (probably incorrect) assertion that X^2 ∼ Bin(1*1,p).

Suppose $p(x,y)$ is a homogeneous polynomial over $\mathbb{C}$, say, and

$p(x,\lambda x) = 0$ for some $\lambda \in \mathbb{C}.$ Then why exactly must $y- \lambda x$ divide $p(x,y)$?

Suppose $A $ and $X$ are random variables and suppose we have $VaR_u(AX)=5$.Furthermore, suppose we are interested in the following probability $P(AX>VaR_u(AX)|A=a)$.

Suppose that $X$ is a Multinomial($n, \textbf{p}$) r.v.