Theorems

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

Showing 81-100 of 199

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Derivatives Along a Segment for $C^1$ Functions on a Euclidean Open Set
lemmalem:segment-derivative-c1-2026aMultivariable Calculus
Let $n$ be a natural number, let $W$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line. Let $f:W\to\mathbb{R}$ be of class $C^1$ on $W$ (via clause 3 there, $f$ being real-valued), wit…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-08-26T23:42:13.180684Z
$C^k$ Maps on a Euclidean Open Set
definitiondef:ck-map-euclidean-2026aMultivariable Calculus
Let $n,m$ be natural numbers, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers, and let $f=(f_1,\dots,f_m):U\to\mathbb{R}^m$ , with coordinate functions $f_j:U\to\mathbb{R}$ . For a natural number $k$ , we define what it means for $f$ …
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-26T23:39:48.174595Z
Partial Derivative on a Euclidean Open Set
definitiondef:partial-derivative-euclidean-2026aMultivariable Calculus
Let $n$ be a natural number, let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ , let $\mathbb{R}$ be the real numbers with absolute value $|\cdot|$ , let $f:U\to\mathbb{R}$ , let $a=(a_1,\dots,a_n)\in U$ , and let $i\in\{1,\dots,n\}$ . We say that the…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-08-26T23:39:37.706348Z
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
A Convex Function is Continuous near an Interior Point
corollarycor:convex-function-continuous-near-interior-point-2026aAnalysis Multivariable Calculus
Let $n$ , $[n]$ , $\mathbb{R}$ , $\mathbb{R}^{n}$ , the Euclidean norm $\lVert\cdot\rVert$ , the Euclidean metric $d_{E}$ and the closed balls $\bar{B}_{d_{E}}(x,s)$ of Closed Ball in a Metric Space be as in A Convex Function is Lipschitz on a Ball around an Interior Point. Regard…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-20T02:17:42.205853Z
A Convex Function is Lipschitz on a Ball around an Interior Point
theoremthm:convex-function-lipschitz-near-interior-point-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $[n]$ be the initial segment determined by $n$ , and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, with the absolute value written $|\cdot|$ . On Euclidean space…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-20T02:16:30.039490Z
Increment Bound for a Convex Function through an Extended Point
lemmalem:convex-function-increment-bound-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $C$ be a convex subset of Euclidean space $\mathbb{R}^{n}$ , a real vector space by…
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-20T02:16:04.235003Z
A Convex Function Bounded Above on a Set Symmetric about a Point is Bounded Below on It
lemmalem:convex-function-bounded-below-reflection-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, let $C$ be a convex subset of Euclidean space $\mathbb{R}^{n}$ , a real vector space by…
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-20T02:15:39.692906Z
A Convex Function is Bounded Above near a Point by its Values at Coordinate Neighbours
lemmalem:convex-function-bounded-above-crosspolytope-2026aAnalysis Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $[n]$ be the initial segment determined by $n$ , and let $\mathbb{R}$ be the set of real numbers with the operations and the order $\le$ of its ordered field structure, with the absolute value written $|\cdot|$ . Let $C$ be a convex su…
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-20T02:05:07.387443Z
Jensen's Inequality for Finite Convex Combinations
theoremthm:jensen-inequality-finite-2026aAnalysis Multivariable Calculus
Let $n$ and $N$ be natural numbers with $1\le n$ and $1\le N$ , let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and let $[N]$ be the initial segment determined by $N$ . Let $C$ be a convex subset of Euclidean space $\mathbb{R}^n$ , regarded…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:12:24.879393Z
Properties of the Norm of a Symmetric Real Matrix
lemmalem:symmetric-matrix-norm-properties-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number with $1\le n$ and let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure and the absolute value $|\cdot|$ . Let $A$ and $B$ be symmetric real $n\times n$ matrices and let $\mu,\lambda\in\mathbb{R}$ . Write $I_n$ for the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:12:19.168682Z
Small Cases, Reduction, and Membership for Convex Combinations
lemmalem:convex-combination-properties-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n$ and $N$ be natural numbers with $1\le n$ and $1\le N$ , let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$ . Regard Euclidean space $\mathbb{R}^n$ as a rea…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:11:55.235742Z
A Closed Euclidean Ball is Convex and Compact
lemmalem:closed-euclidean-ball-convex-compact-2026aAnalysis Topology Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, let $x$ be a point of Euclidean space $\mathbb{R}^n$ , regarded as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:11:44.249606Z
Testing a Block Semidefinite Inequality on the Diagonal
lemmalem:block-order-implies-matrix-order-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $\mathbb{R}$ be the real numbers, let $\alpha\in\mathbb{R}$ , and let $X$ and $Y$ be symmetric real $n\times n$ matrices. Write $I_n$ for the identity matrix of size $n$ , write $0_n$ for the real $n\times n$ matrix all of whose entrie…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:11:30.882818Z
The Quadratic Form of a Real Square Matrix is Bounded on the Closed Unit Ball
lemmalem:matrix-quadratic-form-bounded-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n$ be a natural number with $1\le n$ , let $[n]$ be the initial segment determined by $n$ , and let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure and the absolute value $|\cdot|$ . Let $A$ be a real $n\times n$ matrix, with entries…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:10:47.443139Z
Action and Quadratic Form of a Block Matrix
lemmalem:block-matrix-quadratic-form-2026aAnalysis Linear Algebra Multivariable Calculus
Let $m$ and $n$ be natural numbers with $1\le m$ and $1\le n$ , let $\mathbb{R}$ be the real numbers, and for a natural number $p$ let $[p]$ be the initial segment determined by $p$ . Let $A$ , $B$ , $C$ and $D$ be real matrices of sizes $m\times m$ , $m\times n$ , $n\times m$ and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:10:31.208216Z
Convex Combination of Finitely Many Points of $\mathbb{R}^n$
definitiondef:convex-combination-rn-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n$ and $N$ be natural numbers with $1\le n$ and $1\le N$ , let $\mathbb{R}$ be the real numbers with the order $\le$ of its ordered field structure, and let $[N]$ and $[n]$ be the initial segments determined by $N$ and by $n$ . Let $x:[N]\to\mathbb{R}^n$ be a family of points…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T02:10:10.857337Z
A Function of Class $C^2$ with Positive Semidefinite Hessian is Convex
theoremthm:hessian-psd-implies-convex-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $U\subseteq\mathbb{R}^n$ be an open and convex subset of Euclidean space $\mathbb{R}^n$ , a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, and let $f:U\to\mathbb{R}$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:50:59.457957Z
Derivatives of the Slice of a Function Along a Line
lemmalem:line-slice-derivative-2026bAnalysis Multivariable Calculus
Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the real numbers, and let $U$ be an open subset of Euclidean space $\mathbb{R}^n$ . Write $z\cdot z'$ for the dot product of points of $\mathbb{R}^n$ and $Av$ for the matrix-vector product. Let $f:U\to\mathbb{R}$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:48:29.706187Z
A Semiconvex Function of Class $C^2$ has Hessian Bounded Below by $-\lambda I_n$
corollarycor:semiconvex-c2-hessian-bound-2026aAnalysis PDE Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $U\subseteq\mathbb{R}^n$ be an open and convex subset of Euclidean space $\mathbb{R}^n$ , a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, let $\lambda\in\mathbb{R}$ satisfy…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T17:52:45.222413Z

Showing 81-100 of 199

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Derivatives Along a Segment for C1C^1C1 Functions on a Euclidean Open Set

CkC^kCk Maps on a Euclidean Open Set

Partial Derivative on a Euclidean Open Set

Euclidean Points as Tuples of Real Numbers

A Convex Function is Continuous near an Interior Point

A Convex Function is Lipschitz on a Ball around an Interior Point

Increment Bound for a Convex Function through an Extended Point

A Convex Function Bounded Above on a Set Symmetric about a Point is Bounded Below on It

A Convex Function is Bounded Above near a Point by its Values at Coordinate Neighbours

Jensen's Inequality for Finite Convex Combinations

Properties of the Norm of a Symmetric Real Matrix

Small Cases, Reduction, and Membership for Convex Combinations

A Closed Euclidean Ball is Convex and Compact

Testing a Block Semidefinite Inequality on the Diagonal

The Quadratic Form of a Real Square Matrix is Bounded on the Closed Unit Ball

Action and Quadratic Form of a Block Matrix

Convex Combination of Finitely Many Points of Rn\mathbb{R}^nRn

A Function of Class C2C^2C2 with Positive Semidefinite Hessian is Convex

Derivatives of the Slice of a Function Along a Line

A Semiconvex Function of Class C2C^2C2 has Hessian Bounded Below by −λIn-\lambda I_n−λIn​

Derivatives Along a Segment for $C^1$ Functions on a Euclidean Open Set

$C^k$ Maps on a Euclidean Open Set

Convex Combination of Finitely Many Points of $\mathbb{R}^n$

A Function of Class $C^2$ with Positive Semidefinite Hessian is Convex

A Semiconvex Function of Class $C^2$ has Hessian Bounded Below by $-\lambda I_n$