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Feasibility of finite reaction offsets

Statement

Let $A$ be a finite set of channel labels and $r \in N$ . For each $a \in A$ , take $ν_{a} \in R^{r}$ and $d_{a} \in R$ as independently specified real input data; this theorem does not define channel input/output compounds, a calibration map, or a map producing $d_{a}$ . In an application, separate model data may specify vectors $ν_{a}^{i n}, ν_{a}^{o u t} \in R^{r}$ and set $ν_{a} : = ν_{a}^{o u t} - ν_{a}^{i n}$ , while a separately supplied real number $Δ σ_{a}$ may be designated as a channel's calibrated entropy change and $d_{a} \leq Δ σ_{a}$ as its lower bound. These are additional application premises, not conclusions or inputs derived here. Use the dot product and coordinatewise finite vector sums defined in Finite inequality alternative by Fourier–Motzkin elimination: for $u, v \in R^{r}$ , $u \cdot v : = \sum_{k < r} u (k) v (k)$ , and $(\sum_{a \in A} λ_{a} ν_{a}) (k) : = \sum_{a \in A} λ_{a} ν_{a} (k)$ . A multiplier $λ \in R^{A}$ is a function $A \to R$ . There exists an offset vector $b \in R^{r}$ satisfying

$b \cdot ν_{a} + d_{a} \geq 0 (a \in A)$

if and only if every $λ \in R^{A}$ with $λ_{a} \geq 0$ and $\sum_{a \in A} λ_{a} ν_{a} = 0$ satisfies $\sum_{a \in A} λ_{a} d_{a} \geq 0$ . If a separate model supplies for each $a$ a real number $Δ σ_{a}$ , designated as its transition entropy change, and assumes $Δ σ_{a} \geq d_{a}$ , such a vector guarantees $Δ σ_{a} + b \cdot ν_{a} \geq 0$ for those channels. The criterion is exact for the finite lower-bound inequalities; it does not claim necessity for actual transitions when $d_{a}$ is only a loose bound.

Facts & Assumptions

[A1]

The finite vectors $ν_{a}$ and reals $d_{a}$ are independent input data. No relation between channel input/output compounds, common-meter values, calibrated changes, and $d_{a}$ is assumed or constructed here; the transition conclusion is conditional on separate real values $Δ σ_{a}$ satisfying $Δ σ_{a} \geq d_{a}$ .

[F1]

Finite inequality alternative by Fourier–Motzkin elimination: the stated dot product is a finite coordinate sum and vector sums are coordinatewise; the finite row system $c_{j} \cdot b \geq h_{j}$ is feasible exactly when every nonnegative zero-left-side multiplier has nonpositive combined right side.

[F2]

Finite commutative-monoid sums are invariant under bijective reindexing, split over disjoint unions, and satisfy the finite Fubini rule: every finite row-label set admits an enumeration by a natural-number index set; reindexing by a bijection preserves the finite vector and scalar sums.

Proof

Given: [A1], [F1], and [F2].

1.1A1F1F2algebra

By [F2], choose a natural number $m$ and a bijection $ϕ : m \to A$ ; when $A = \emptyset$ , take $m = 0$ and the empty bijection. Reindex the data as $c_{j} : = ν_{ϕ (j)}$ and $h_{j} : = - d_{ϕ (j)}$ for $j < m$ . The row system $c_{j} \cdot b \geq h_{j}$ is exactly the required offset system. By [F1], it is feasible exactly when every nonnegative multiplier family $μ \in R^{m}$ with $\sum_{j < m} μ_{j} c_{j} = 0$ has $\sum_{j < m} μ_{j} h_{j} \leq 0$ . The bijection transports such a family to $λ_{a} : = μ_{ϕ^{- 1} (a)}$ on $A$ , preserving nonnegativity and both finite sums by [F2]. Hence this condition is precisely $\sum_{a \in A} λ_{a} ν_{a} = 0 \Rightarrow \sum_{a \in A} λ_{a} d_{a} \geq 0$ . For $A = \emptyset$ , $b = 0$ is feasible and the multiplier condition is vacuous. If $r = 0$ and $A$ is nonempty, every $ν_{a} = 0$ ; testing the coordinate multiplier for each channel makes the criterion equivalent to $d_{a} \geq 0$ for every $a$ , and then the unique empty vector $b$ satisfies all rows.

2.1A1step 1.1algebra∎

For any channel with actual change $Δ σ_{a} \geq d_{a}$ , add $b \cdot ν_{a}$ to both sides of that bound and use the solved row inequality $b \cdot ν_{a} \geq - d_{a}$ . Then $Δ σ_{a} + b \cdot ν_{a} \geq d_{a} + b \cdot ν_{a} \geq 0$ . This is a sufficient monotonicity conclusion for the stated channels.

Remarks

For $r = 0$ and nonempty $A$ , the criterion reduces to $d_{a} \geq 0$ for every channel. Strict increase, infinitely many channels, and lower bounds equal to actual changes are separate conditions and are not inferred here.

Feasibility of finite reaction offsets

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources