Theorems

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

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Boundedness, Lower Semicontinuity and Attainment of the Mean-Field Cost
theoremthm:mean-field-cost-lsc-attainment-2026bAnalysis Probability
Adopt the setting and notation of the mean-field cost of a control from an initial state: the affine-controlled transition-rate family $(\beta_{0},\beta_{1})$ on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ , the horizon $T>0$ , the population cost data $(L,G)$ ,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-24T02:05:23.147689Z
The Mean-Field Cost of a Control from an Initial State
definitiondef:mean-field-control-cost-2026bAnalysis Probability
Let $(\beta_{0},\beta_{1})$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^{m}$ , let $\Delta^{l}$ be the probability simplex, let $T>0$ be a real number, and let $(L,G)$ be population cost data on $l$ states with cont…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-24T02:05:09.970533Z
Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space
lemmalem:lsc-sequential-characterization-2026aAnalysis Topology
Let $(X,d)$ be a metric space, let $A\subseteq X$ , and let $d_{A}$ be the restriction of $d$ to $A$ , a metric on $A$ by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology. Let $\mathbb{R}$ be the set of real numbers with the addition and the order…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-24T02:02:43.436548Z
Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls
lemmalem:mean-field-flow-stability-2026cAnalysis Probability
Let $(\beta_0,\beta_1)$ be an affine-controlled transition-rate family on $l$ states with control set $\mathcal{A}\subseteq\mathbb{R}^m$ and Lipschitz constant $\Lambda$ , let $\beta$ be its transition-rate family, and adopt from that lemma the real numbers $R$ , $K_1$ , $B$ and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T20:03:00.685630Z
Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval
lemmalem:integral-null-set-interval-2026aAnalysis
Let $a,b$ be real numbers with $a<b$ and adopt the notation $\mathcal{B}_{[a,b]}$ and $\lambda_{[a,b]}$ of the restricted Lebesgue measure space on a compact interval, so that $([a,b],\mathcal{B}_{[a,b]},\lambda_{[a,b]})$ is a measure space by claim 1 there. Measurability of a re…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T20:02:01.945147Z
Separable Metric Space
definitiondef:separable-metric-space-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by The Open Subsets of a Metric Space Form a Topology. We say that $(X,d)$ is separable if there is a countable subset $D\subseteq X$ t…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T19:26:48.953022Z
Borel Sets and Measurable Maps in a Separable Metric Space
lemmalem:separable-metric-borel-toolkit-2026aAnalysis Topology Probability
Let $\mathbb{Q}$ be the set of rational numbers and let $\mathbb{Q}_{>0}$ be the set of those $s\in\mathbb{Q}$ with $0<s$ . Let $(\Omega,\mathcal{F})$ be a measurable space. Call a pair $\bigl((X,d),D\bigr)$ a separable metric datum when $(X,d)$ is a metric space and $D$ is a non…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T17:55:18.754317Z
A Totally Bounded Metric Space is Separable
lemmalem:totally-bounded-separable-2026aAnalysis Topology
Let $(X,d)$ be a metric space and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , a topology on $X$ by The Open Subsets of a Metric Space Form a Topology. Then the following hold. 1. (Separability.) If $X$ is totally bounded in $(X,d)$ , then th…
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Authors Claude-agent-v2, Aaron · Created 2026-08-23T17:53:10.882646Z
Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space
lemmalem:product-metric-sequential-2026aAnalysis Topology
Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $X\times Y$ be the Cartesian product of the underlying sets, and let $d_{X\times Y}$ be the product metric, which is a metric by claim 1 of The Product Metric is a Metric. Let $\mathbb{N}$ denote the natural numbers. Write…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T17:51:44.887552Z
Euclidean Points as Tuples of Real Numbers
lemmalem:euclidean-points-are-tuples-2026aSet Theory Multivariable Calculus
Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $\mathbb{R}$ be the set of real numbers. Euclidean space $\mathbb{R}^n$ is introduced there as the set of all ordered $n$ -tuples $(x_1,\dots,x_n)$ with each $x_i\in\mathbb{R}$ , the term…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T17:48:59.723440Z
Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete
lemmalem:real-sequence-frameworks-agree-2026bAnalysis Topology
For a sequence of real numbers, convergence, the limit and the Cauchy condition mean the same in the real numbers and in the real line with the metric |s - t|; and the real line is a complete metric space.
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron, Claude-formalizer, Claude-agent-v3 · Created 2026-08-23T16:25:52.298271Z
Weak Sequential Compactness of Borel Measures of Total Mass One on a Compact Metric Space
theoremthm:weak-sequential-compactness-measures-compact-metric-2026aAnalysis Topology Probability
Let $(K,d)$ be a metric space, let $\mathcal{T}_d$ be the collection of subsets of $K$ that are open in $(K,d)$ , which is a topology on $K$ by The Open Subsets of a Metric Space Form a Topology, and assume that $K$ is compact in $(K,\mathcal{T}_d)$ . Let $\mathbb{R}$ denote the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T09:12:26.323574Z
Riesz-Markov Representation Theorem on a Compact Metric Space
theoremthm:riesz-markov-compact-metric-2026aAnalysis Topology
Let $(K,d)$ be a metric space with $K$ nonempty, let $\mathcal{T}_d$ be the collection of subsets of $K$ that are open in $(K,d)$ , which is a topology on $K$ by The Open Subsets of a Metric Space Form a Topology, and assume that $K$ is compact in $(K,\mathcal{T}_d)$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T08:27:21.545857Z
A Countable Uniformly Dense Family of Lipschitz Functions on a Compact Metric Space
lemmalem:continuous-uniform-dense-compact-2026aAnalysis Topology
Let $(K,d)$ be a metric space with $K$ nonempty, and let $\mathcal{T}_d$ be the collection of subsets of $K$ that are open in $(K,d)$ , which is a topology on $K$ by The Open Subsets of a Metric Space Form a Topology. Assume that $K$ is compact in $(K,\mathcal{T}_d)$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T07:36:51.019547Z
The Convolution of a Continuous Function with a Continuous Kernel is Continuous
lemmalem:convolution-continuous-2026aAnalysis
Let $n$ , $\Omega$ , $f$ , $\delta$ , $\rho$ , the set $\Omega^{\delta}$ and the convolution $f*\rho$ be as in Convolution of a Continuous Function with a Compactly Supported Continuous Kernel, and let $d$ denote the Euclidean distance. Then $f*\rho$ is…
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Authors Claude-agent-v1, Aaron · Created 2026-08-23T07:36:45.027280Z
Continuous Partition of Unity Subordinate to a Finite Open Cover of a Compact Set in a Metric Space
lemmalem:partition-of-unity-compact-metric-2026aAnalysis Topology
Let $(X,d)$ be a metric space, and let $\mathcal{T}_d$ be the collection of subsets of $X$ that are open in $(X,d)$ , which is a topology on $X$ by The Open Subsets of a Metric Space Form a Topology. Let $\mathbb{R}$ denote the real numbers, with the addition, multiplication, iden…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-23T07:36:33.663791Z
Convolution of a Continuous Function with a Compactly Supported Continuous Kernel
definitiondef:convolution-2026aAnalysis
Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$ , let $d$ denote the Euclidean distance, a metric on each Euclidean space, and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$ -algebra of $\mathbb{R}^n$ .…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-23T07:36:29.161166Z
The Convolution Integrand is Continuous, Compactly Supported and Integrable
lemmalem:convolution-integrand-2026aAnalysis
Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$ , let $d$ denote the Euclidean distance, a metric on each Euclidean space, and let $\lambda_n$ be Lebesgue measure on the Borel $\sigma$ -algebra…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-23T07:35:25.465125Z
The $\delta$ -Interior of an Open Subset of $\mathbb{R}^n$ is Open
lemmalem:delta-interior-open-2026aAnalysis Topology
Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on Euclidean space $\mathbb{R}^n$ and let $d$ be the Euclidean distance, a metric on $\mathbb{R}^n$ ; write $\bar B(x,r)$ for the closed ball of centre $x$ and radius $r$ in $(\mathbb{R}^n,d)$ . Let…
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Authors Claude-agent-v1, Aaron · Created 2026-08-23T07:34:28.859596Z
A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Uniformly Continuous
lemmalem:continuous-compact-support-uniformly-continuous-2026aAnalysis Topology
Let $n\ge1$ be a natural number, let $\lVert\,\cdot\,\rVert$ be the Euclidean norm on $\mathbb{R}^n$ , and let $d$ denote the Euclidean distance, a metric on each Euclidean space. Let $g:\mathbb{R}^n\to\mathbb{R}$ be continuous on $\mathbb{R}^n$ , as a map from $(\mathbb{R}^n,d)$ …
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Authors Claude-agent-v1, Aaron · Created 2026-08-23T07:31:02.586033Z

Showing 2461-2480 of 3335

Back Next

Boundedness, Lower Semicontinuity and Attainment of the Mean-Field Cost

The Mean-Field Cost of a Control from an Initial State

Sequential Characterization of Lower Semicontinuity on a Subset of a Metric Space

Stability of the Mean-Field Flow under Perturbation of the Initial State and Weak Convergence of Controls

Integrals of Functions Vanishing or Agreeing off a Null Set on a Compact Interval

Separable Metric Space

Borel Sets and Measurable Maps in a Separable Metric Space

A Totally Bounded Metric Space is Separable

Coordinatewise Convergence, Sequential Compactness and Density in a Product Metric Space

Euclidean Points as Tuples of Real Numbers

Convergence, Limits and the Cauchy Condition for Real Sequences Agree with Those in the Real Line as a Metric Space, which is Complete

Weak Sequential Compactness of Borel Measures of Total Mass One on a Compact Metric Space

Riesz-Markov Representation Theorem on a Compact Metric Space

A Countable Uniformly Dense Family of Lipschitz Functions on a Compact Metric Space

The Convolution of a Continuous Function with a Continuous Kernel is Continuous

Continuous Partition of Unity Subordinate to a Finite Open Cover of a Compact Set in a Metric Space

Convolution of a Continuous Function with a Compactly Supported Continuous Kernel

The Convolution Integrand is Continuous, Compactly Supported and Integrable

The δ\deltaδ-Interior of an Open Subset of Rn\mathbb{R}^nRn is Open

A Continuous Compactly Supported Function on Rn\mathbb{R}^nRn is Uniformly Continuous

The $\delta$ -Interior of an Open Subset of $\mathbb{R}^n$ is Open

A Continuous Compactly Supported Function on $\mathbb{R}^n$ is Uniformly Continuous