Theorems

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

Showing 941-960 of 1428

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Continuous Real-Valued Functions on a Compact Interval are Bounded
lemmalem:continuous-compact-interval-bounded-2026bAnalysis
Let $a$ and $b$ be real numbers with $a<b$ , and let $g:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ , the interval being regarded as a subset of the real line with the absolute value metric and $\mathbb{R}$ carrying the same metric. Then there is a real number $C\ge0$ such that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-07T18:51:23.166299Z
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
Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions
theoremthm:spectral-theorem-self-adjoint-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ ; write $n=\dim V$ for its dimension and let $[n]$ be the initial segment determined by $n$ . Let $T$ be a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T04:00:18.118528Z
A Self-Adjoint Operator on a Finite-Dimensional Space has a Unit Eigenvector
theoremthm:self-adjoint-eigenvalue-existence-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ . Let $T$ be a linear operator on $V$ that is self-adjoint. Then there are a unit vector $x_{0}\in V$ and a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T03:58:10.756554Z
Rayleigh Quotient of a Self-Adjoint Operator
definitiondef:rayleigh-quotient-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $T$ be a linear operator on $V$ that is self-adjoint, and let $S$ be the set of unit vectors of $V$ . The Rayleigh quotient of $T$ is the map $R_{T}:S\to\mathbb{R}$ sending each $x\in S$ to…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:36:02.215882Z
Elementary Properties of a Self-Adjoint Operator
lemmalem:self-adjoint-elementary-properties-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and let $T$ be a linear operator on $V$ that is self-adjoint. Then the following hold. 1. (Real values on the diagonal) For every $x\in V$ the complex number…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:36:01.059876Z
Eigenvector with Eigenvalue
definitiondef:eigenvector-eigenvalue-2026aAlgebra Linear Algebra
Let $V$ be a complex vector space with zero vector $0_{V}$ , let $T$ be a linear operator on $V$ , let $\lambda$ be a complex number, and let $x\in V$ . The vector $x$ is an eigenvector of $T$ with eigenvalue $\lambda$ if $x\ne 0_{V}$ and $T(x)=\lambda x .$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:35:49.959830Z
Elementary Arithmetic in an Ordered Field
lemmalem:ordered-field-arithmetic-2026aAnalysis Algebra
Let $F$ be an ordered field, with the additive identity $0$ , multiplicative identity $1$ , additive inverses $-x$ and multiplicative inverses $x^{-1}$ of a field, and with its order $\le$ ; write $x-y=x+(-y)$ . Let $a,b,x,y\in F$ . Then the following hold.…
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:35:46.545206Z
The Orthogonal Complement of a Unit Vector
lemmalem:orthogonal-complement-unit-vector-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ ; write $n=\dim V$ for its dimension. Let $u\in V$ be a unit vector, let $\tilde{u}\in V^{1}$ be the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:23:56.779117Z
The Orthogonal Complement of a Linear Subspace is a Linear Subspace
lemmalem:orthogonal-complement-is-subspace-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $W$ be a linear subspace of $V$ . Then the orthogonal complement $W^{\perp}$ is a linear subspace of $V$ .
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:23:45.795909Z
Orthogonal Complement of a Linear Subspace
definitiondef:orthogonal-complement-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space and let $W$ be a linear subspace of $V$ . The orthogonal complement of $W$ is the set $W^{\perp}=\bigl\{x\in V\ :\ \langle w,x\rangle=0\ \text{ for every }w\in W\bigr\},$ where $0$ is the zero…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:22:26.614020Z
Dimension of a Finite-Dimensional Complex Inner Product Space
definitiondef:dimension-inner-product-space-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ . The dimension of $V$ , written $\dim V$ , is the natural number $n$ for which there is an $n$ -tuple in $V$ t…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:22:23.816875Z
Orthonormal Bases and Basis Size in a Finite-Dimensional Inner Product Space
lemmalem:inner-product-space-basis-size-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space with zero vector $0_{V}$ , and suppose that $V$ is finite-dimensional and $V\ne\{0_{V}\}$ . Then the following hold. 1. (Existence) There are a natural number $n$ and an $n$ -tuple $e\in V^{n}$ that…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:21:59.411303Z
Gram-Schmidt Orthonormalisation
theoremthm:gram-schmidt-2026aAnalysis Linear Algebra
Let $V$ together with $\langle\cdot,\cdot\rangle$ be a complex inner product space, let $m$ be a natural number, and let $b\in V^{m}$ be an $m$ -tuple in $V$ that is linearly independent. For $k\in[m]$ , with $[k]$ the initial segment determined by $k$ , write $x|_{[k]}$ for the res…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:14:44.632164Z
A Finite Spanning Family Contains a Basis
lemmalem:spanning-family-contains-basis-2026aAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$ , and suppose $V\ne\{0_{V}\}$ . Let $n$ be a natural number and let $v\in V^{n}$ be an $n$ -tuple in $V$ that spans $V$ . Then there are a natural number $r$ with $r\le n$ , in the order on $\mathbb{N}$ ,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T01:14:17.189462Z
Elementary Properties of Linear Independence
lemmalem:linear-independence-elementary-2026aAlgebra Linear Algebra
Let $K$ be a field, let $V$ be a vector space over $K$ with zero vector $0_{V}$ , let $n$ be a natural number with the order relations $<$ and $\le$ , and let $v\in V^{n}$ be an $n$ -tuple in $V$ . For $j\in[n]$ , with $[j]$ the initial segment determined by $j$ , write…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-04T00:54:42.140133Z

Showing 941-960 of 1428

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Continuous Real-Valued Functions on a Compact Interval are Bounded

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

Spectral Theorem for a Self-Adjoint Operator in Finite Dimensions

A Self-Adjoint Operator on a Finite-Dimensional Space has a Unit Eigenvector

Rayleigh Quotient of a Self-Adjoint Operator

Elementary Properties of a Self-Adjoint Operator

Eigenvector with Eigenvalue

Elementary Arithmetic in an Ordered Field

The Orthogonal Complement of a Unit Vector

The Orthogonal Complement of a Linear Subspace is a Linear Subspace

Orthogonal Complement of a Linear Subspace

Dimension of a Finite-Dimensional Complex Inner Product Space

Orthonormal Bases and Basis Size in a Finite-Dimensional Inner Product Space

Gram-Schmidt Orthonormalisation

A Finite Spanning Family Contains a Basis

Elementary Properties of Linear Independence