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The protocol combines encrypted ballots so that only the totals for each candidate are revealed, not the individual voters' choices. All encrypted ballots can be combined using a mathematical protocol that preserves the sum without decrypting the individual components.

Conditions: All ballots are encrypted before submission.; Aggregation is performed on encrypted data.; Only final totals are revealed.

The partial sum $S_n$ is the finite sum of the first $n$ terms, whereas the infinite series $S$ is defined as the limit of $S_n$ as $n \to \infty$. While $S_n$ forms a sequence of values, $S$ represents the single limiting value (or divergence) that this sequence approaches.

Conditions: $S_n$ denotes the sum of the first $n$ terms.; $S$ denotes the infinite series $\sum_{n=1}^{\infty} a_n$.; The limit is taken as $n \to \infty$.

Risk-neutral probabilities are defined as the specific set of probabilities chosen precisely to make the discounted price process of any asset a martingale. They are not derived from historical data or investor beliefs, but from the mathematical requirement that discounted prices behave like fair games.

Conditions: Operating within a risk-neutral valuation framework.; Prices are discounted at the risk-free rate.

The degree of diversification, denoted as $d_j$, is defined as $1 - e_j$. It represents the amount of useful information or distinguishing power provided by a specific criterion among the alternatives.

Conditions: The entropy value $e_j$ has already been calculated for the criterion.; The degree of diversification is always a positive real number in this context.

The video concludes $X$ is compact because it demonstrates that an open cover $C$ of the finite space $X$ has a finite subcover $C_1$. The speaker applies the definition of compactness (every open cover has a finite subcover) to this finite setting.

Conditions: $X$ is finite.; Compactness is defined by the requirement that every open cover of $X$ has a finite subcover.

The steel specimen used in the experiment has a diameter of 8 mm and a length of 40 mm, denoted as $\Phi 8 \times 40\ \text{mm}$.

Conditions: The specimen is a cylindrical steel bar.; Surface oil is removed before exposure.

The squaring function $f(x)=x^2$ maps distinct real inputs, such as opposites like $2$ and $-2$, to the same output ($4$). This behavior is permissible under the definition of a function, which requires exactly one output per input but allows multiple inputs to share an output. Consequently, $f(x)=x^2$ is not injective (one-to-one), though it remains a valid function.

Conditions: The function is $f(x)=x^2$.; Inputs are distinct real numbers.

Matrix addition requires matching dimensions so that every entry in one matrix has exactly one corresponding partner in the other matrix. Without identical row and column counts, some positions would lack a pair to add, making the operation undefined under standard rules.

Conditions: Standard definition of matrix addition applies.; Matrices represent rectangular arrays of numbers.

Mathematical modeling is the process of translating a real-world problem into a mathematical model by making specific assumptions and approximations while still capturing the essence of the original problem.

Conditions: Must start with a real-world problem; Must result in a mathematical model; Requires making assumptions and approximations

A metric space is formally defined as an ordered pair $(A, d)$, where $A$ is a non-empty set and $d$ is a distance function mapping pairs of elements from $A$ to the real numbers $\mathbb{R}$. The function $d$ must satisfy four specific properties for all elements in $A$.

Conditions: $A$ is a non-empty set.; $d$ is a function $d: A \times A \rightarrow \mathbb{R}$.; The four properties of the distance function must hold.

The Pythagorean theorem can be used because the perpendicular component displacements of a vector in an orthonormal Cartesian plane form the two legs of a right triangle, with the vector itself acting as the hypotenuse. The theorem relates the lengths of the legs to the length of the hypotenuse, allowing the calculation of the vector's magnitude (length).

Conditions: The coordinate axes are perpendicular (orthonormal).; The vector is drawn from the origin to its terminal point.; The magnitude is the Euclidean length of the vector.

The example concludes this because the conditional expectation of the discounted time-3 stock price, given information at time 2, equals the discounted time-2 stock price. Specifically, $\mathbb{E}_2\left[\frac{S_3}{(1+r)^3}\right] = 82.64$ and $\frac{S_2}{(1+r)^2} = 82.64$, satisfying the martingale condition $M_n = \mathbb{E}_n[M_{n+1}]$.

Conditions: The expectation is taken under the risk-neutral probability $p = 2/3$.; Discounting uses the interest rate $r = 0.1$.