Theorems

A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.

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Joint Measurability of the State and Control of the Controlled N-Agent Dynamics
lemmalem:n-agent-joint-measurability-2026bProbability
Adopt the setting of the controlled $N$ -agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$ , and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ : a transition-rate family $\beta$ with control set…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:53:42.616939Z
Regularity and Derivative Bounds of the Extended Aggregate State Drift
lemmalem:extended-drift-regularity-2026bProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ , let $(U,V,\bar{\beta})$ be a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:53:32.139939Z
Twice Continuously Differentiable Extension of Population Cost Data
definitiondef:c2-population-cost-extension-2026cProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $(L,G)$ be population cost data on $l$ states with control dimension $m$ , let $\Delta^l\subset\mathbb{R}^l$ be the probability simplex, and let $K\ge0$ be a real number. Points of $\mathbb{R}^l\times\mathbb{R}^m$ ar…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:51:22.945434Z
Twice Continuously Differentiable Extension of a Transition-Rate Family
definitiondef:c2-transition-rate-extension-2026cProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ , let $\Delta^l\subset\mathbb{R}^l$ be the…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:51:09.565569Z
Fluctuation Processes of the Controlled N-Agent Dynamics
definitiondef:n-agent-fluctuation-processes-2026cProbability
Adopt the setting of the controlled $N$ -agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$ , and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ : a transition-rate family $\beta$ on $l$ states with contr…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:43:16.169976Z
The Mean-Field Cost Functional
definitiondef:mean-field-cost-2026cProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , let $B$ be a nonnegative real number, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ , le…
+0 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:43:08.810002Z
Mean-Field Trajectory Pair
definitiondef:mean-field-trajectory-pair-2026cProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , let $B$ be a nonnegative real number, let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ , le…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:43:02.874076Z
Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law
lemmalem:poisson-interarrival-exponential-2026aProbability
Let $Y=(Y_u)_{u\ge0}$ be a homogeneous Poisson process with rate $1$ on a probability space $(\Omega,\mathcal{F},P)$ , all of whose paths are counting paths, and let $\tau_k=\tau_k(Y)$ denote the $k$ -th jump time for each natural number $k\ge1$ . (a) Almost surely: every $\tau_k$ …
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:36:43.122705Z
The N-Agent Cost Functional
definitiondef:n-agent-cost-2026bProbability
Let $N$ , $l$ , $\tilde{l}$ , $m$ be natural numbers with $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ , $m\ge1$ , and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ . Adopt the setting of the controlled $N$ -agent dynamics: a transition-rate family $\beta$ on $l$ states…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:31:58.998510Z
Martingale Decomposition of the Empirical State Measure and the Observation Process
theoremthm:n-agent-martingale-decomposition-2026cProbability
Adopt the setting of the controlled $N$ -agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$ , and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ : a transition-rate family $\beta$ with control set…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:31:52.164354Z
Compensated Counters of the Controlled N-Agent Dynamics are Square-Integrable Martingales
lemmalem:n-agent-compensated-martingales-2026bProbability
Adopt the setting of the controlled $N$ -agent dynamics with $N$ agents, $l$ states, $\tilde{l}$ observation channels, and control dimension $m$ , and let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ : a transition-rate family $\beta$ with control set…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:31:27.413973Z
Fresh-Start Property of the Controlled N-Agent Dynamics
lemmalem:n-agent-fresh-start-2026bProbability
Adopt the setting of the controlled $N$ -agent dynamics, with control dimension $m$ and with $\mathcal{A}$ a nonempty subset of Euclidean space $\mathbb{R}^m$ : a transition-rate family $\beta$ with control set $\mathcal{A}$ and rate bound $B$ , an observation-rate family…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:31:15.679715Z
Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics
theoremthm:n-agent-dynamics-existence-2026bProbability
Let $N$ , $l$ , $\tilde{l}$ , $m$ be natural numbers with $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ , $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , let $B$ and $\tilde{B}$ be nonnegative real numbers, let $\beta$ be a transition-rate family on $l$ states…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:29:54.200435Z
Aggregate Fluctuation Covariance
definitiondef:aggregate-fluctuation-covariance-2026bProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , and let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ . The aggregate fluctuation covariance…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:28:15.599617Z
Aggregate Observation Drift
definitiondef:aggregate-observation-drift-2026aProbability
Let $l$ and $\tilde{l}$ be natural numbers with $l\ge2$ and $\tilde{l}\ge1$ , and let $\tilde{\beta}$ be an observation-rate family on $l$ states with $\tilde{l}$ observation channels and rate bound $\tilde{B}$ . The aggregate observation drift of $\tilde{\beta}$ is the function…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:28:07.962675Z
Aggregate State Drift
definitiondef:aggregate-state-drift-2026bProbability
Let $l$ and $m$ be natural numbers with $l\ge2$ and $m\ge1$ , let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ , and let $\beta$ be a transition-rate family on $l$ states with control set $\mathcal{A}$ and rate bound $B$ . The aggregate state drift of $\beta$ …
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:28:03.189941Z
Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable
lemmalem:continuous-composition-measurable-2026aAnalysis Probability
Let $(X,\mathcal{A})$ be a measurable space, let $d\ge 1$ be a natural number, and let $E$ be a nonempty subset of Euclidean space $\mathbb{R}^d$ . Let $g:E\to\mathbb{R}$ be sequentially continuous on $E$ : whenever $(x_n)_{n\in\mathbb{N}}$ is a sequence in $E$ and $x\in E$ with th…
+0 / -0flags 0verified 0has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:27:58.070265Z
Solution of the Controlled N-Agent Dynamics
definitiondef:n-agent-controlled-dynamics-2026bProbability
Let $N$ , $l$ , $\tilde{l}$ , $m$ be natural numbers with $N\ge1$ , $l\ge2$ , $\tilde{l}\ge1$ , $m\ge1$ . Let $\mathcal{A}$ be a nonempty subset of Euclidean space $\mathbb{R}^m$ . Fix a transition-rate family $\beta$ on $l$ states with control set $\mathcal{A}$ and rate bound $B$ , an…
+0 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:27:07.711078Z
Observation-Driven Control Policy
definitiondef:observation-driven-control-policy-2026aProbability
Let $m$ and $\tilde{l}$ be natural numbers with $m\ge 1$ and $\tilde{l}\ge 1$ , and let $T>0$ be a real number, called the horizon. For each natural number $k\ge 1$ define the record space…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:26:30.233768Z
N-Agent Driving System
definitiondef:n-agent-driving-system-2026aProbability
Let $N$ , $l$ , and $\tilde{l}$ be natural numbers with $N\ge 1$ , $l\ge 2$ , and $\tilde{l}\ge 1$ . An $N$ -agent driving system with $l$ states and $\tilde{l}$ observation channels is a probability space $(\Omega,\mathcal{F},P)$ together with the following data.…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-01T04:24:56.638767Z

Showing 921-940 of 1090

Back Next

Joint Measurability of the State and Control of the Controlled N-Agent Dynamics

Regularity and Derivative Bounds of the Extended Aggregate State Drift

Twice Continuously Differentiable Extension of Population Cost Data

Twice Continuously Differentiable Extension of a Transition-Rate Family

Fluctuation Processes of the Controlled N-Agent Dynamics

The Mean-Field Cost Functional

Mean-Field Trajectory Pair

Jump Times of the Homogeneous Poisson Process: Finiteness and Exponential Interarrival Law

The N-Agent Cost Functional

Martingale Decomposition of the Empirical State Measure and the Observation Process

Compensated Counters of the Controlled N-Agent Dynamics are Square-Integrable Martingales

Fresh-Start Property of the Controlled N-Agent Dynamics

Existence, Uniqueness, and Regularity for the Controlled N-Agent Dynamics

Aggregate Fluctuation Covariance

Aggregate Observation Drift

Aggregate State Drift

Sequentially Continuous Functions of Measurable Euclidean Maps are Measurable

Solution of the Controlled N-Agent Dynamics

Observation-Driven Control Policy

N-Agent Driving System