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It is known that for $k+1$ runners, one can assume the following WLOG: $\gcd(v_1,...,v_k)=1$ $(k+1)|v_1$ $V$ is neither the set of arithmetic/geometric progression nor prime numbers. … Although it wasn't stated in any paper, it is trivially verifiable that one can also assume the following WLOG: $\forall r\in\mathbb{Z}$ s.t. $2\leq r\leq k+1$, $\exists v_i\in V$ s.t. $r|v_i$. …

Consider the following: fix a function $\bar{b} : \mathbf{R}_+ \to [0, \infty]$, and define \begin{align} \mathcal{S} \left( \bar{b} \right) := \left\{ b : \mathbf{R}_+ \to [0, \infty] \, \text{s.t.} …

A start for (A): we can ask the same question for the closure of the torsion subgroup of G (a subgroup since G is abelian), so WLOG we can assume the torsion subgroup is dense in G. …

For a hyperelliptic curve $C$ of genus $g$ (over an algebraically closed field of characteristic not two) what is the smallest $d$ for which $C$ can be embedded in some $\mathbb{P}^n$ (I guess $n=3$ wlog …

As there is a weakly converging sub-sequence, we can WLOG assume that $\{\|u_n\|^2u_n\}$ converges weakly to $u_0\in H$. Is it right that $u_0=\|u\|^2u$ ? …

Since having type/cotype is a local property I guess we can assume wlog that $Y$ and $Z$ are finite dimensional, so that we can get rid of the closure but even then I don't know how to proceed. …

(WLOG null-pad to have $m=n$ and sort descendingly.) It's natural to ask when the value is an integer. …

In other words, wlog can one assume that the coproduct is the * of the product? …

So, wlog $D_1$ is not an elliptic curve. But then $D_1$ is ample (by Nakai-Moishezon). And this implies that $D_1 +\ldots +D_n$ is also ample. I couldn't figure it out for abelian threefolds. …

Now, if $V$ has a unique global minimiser at (wlog) $V \left(\mathbf{0}\right) = 0$, one can usually argue that as $\beta \to \infty$, $P_\beta$ converges in law to a delta measure at $\mathbf{0}$, maybe … My situation is that $V$ takes its minimum value (again, wlog taken to be $0$) along a codimension-1 submanifold, i.e. along $$\mathcal{F} = \{ x \in (-1, 1)^d : V(x) = 0 \}.$$ Now, I would like to reason …

WLOG, we assume $\mu (A_1)=0$. Because $(\Omega, \cA_1, \mu)$ is complete, $C \in \cA_1 \subset \cB$. Let's prove that $\Sigma$ is a $\lambda$-system. … WLOG, we assume $\mu (A_2) >0$. Then $\mu (A_2) = \mu (A_1) >0$. …

When I was trying to solve the problem I assumed WLOG, due to symmetry, that the center of $C_3$ is placed at some point along the x-axis $(d,0)$ and found $d$ to satisfy $2(R_3^2 + d^2) = R_1^2 + R_2 …

Update: for $P$ above $2$, (wlog) $ord_P(a) = 0$ and $ord_P(b)=1$ there is an algorithm that outputs a solution (Algorithm 6.2) …

I have a zero mean multivariate normal probability distribution where WLOG each marginal variance is unity and all pairwise correlation coefficient are equal and positive. …

\end{equation} Indeed, if at least two of the three numbers are equal, then the inequality holds (with equality) (thus we may assume wlog $p_{1}<p_{2}<p_{3}$). …