Wacky Math Formula Test

Testing the appearance of math formulas in search of bugs.

BREAK

\(\frac{f_{c} + 1}{2}\)

\(\mathcal{D}\)

\(\mathcal{O}(n)\)

\(\mathcal{S}\)

\(\mathcal{1/3}\)

\(2f_{c} + 1\)

\(f_{c} + 1\)

\(r + 1\)

\(f_{c}\)

\(2f_{d} + 1\)

\(f_{d} + 1\)

\(f_{d}\)

\(b_{1}\)

\(2f_{t} + 1\)

\(2f + 1\)

\(f_{t} + 1\)

\(f_{t}\)

\(3f_{t} + 1\)

\(\frac{1}{3}\)

\(S_{r}\)

\(W_{h}\)

\(C_{h}\)

\(\sigma\)

\(\sigma_{h}\)

\(\mathcal{x}\)

\(\mathcal{y}\)

\(\mathcal{x} – \mathcal{y} = 0\)

\(\mathcal{D}_{r}\)

\(\mathcal{x} – \mathcal{yz} \leq \mathcal{D}_{r}\)

(\mathcal{x} – \mathcal{yz} > \mathcal{D}_{r})

$$V_{sphere} = \frac{4}{3}\pi r^3$$

A DORA-CC protocol among \(n\) nodes \(p_1,p_2,\dots,p_n\) with each node having inputs \(v_i\), for a given agreement distance \(\mathcal{D}\), guarantees that:

1. [Termination] : All honest nodes eventually agree on some value.
2. [Agreement] : The output value \(\mathcal{S}\) for all the nodes is the same.
3. [Validity] : \(H_{min} – \mathcal{D} \leq \mathcal{S} \leq H_{max}\), where \(H_{min}\) and \(H_{max}\) denote the minimum and maximum values from honest nodes.

Given an agreement distance \(\mathcal{D}\), we say that two values \(v_1\) and \(v_2\) agree with each other, if \(|v_1 – v_2| \leq \mathcal{D}\). That is, if two values differ at most by the agreement distance, then they are said to agree with each other.

A set of values \(CC\) is said to form a {\em coherent cluster}, if \(\forall v_1, v_2 \in CC: |v_1 – v_2| \leq \mathcal{D}\). In other words, a coherent cluster is a set of values where all the values in that set agrees amongst themselves.

Let \(S_r\) denote the \(S\)-value of round \(r\). A circuit-breaker function \(\frac{|S_r – S_{r-1}|}{S_{r-1}} \geq thr\) triggers and breaks the circuit (or halts the trade) when \(S_r\) deviates from \(S_{r-1}\) by more than some percentage threshold defined by \(thr\).