Theorems

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

Showing 1421-1440 of 1752

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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
Comparison and Absolute Value Bounds for Finite Sums of Real Numbers
lemmalem:finite-sum-comparison-absolute-2026aAnalysis Algebra
Let $\mathbb{R}$ be the real numbers, an ordered field with additive identity $0$ and order $\le$ , and write $|t|$ for the absolute value of $t\in\mathbb{R}$ . Let $n$ be a natural number, let $[n]$ be the initial segment determined by $n$ , and let $a:[n]\to\mathbb{R}$ and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-19T02:09:56.424303Z
Order-Convex Subset of the Real Line
definitiondef:order-convex-subset-real-line-2026aAnalysis
Let $\mathbb{R}$ be the set of real numbers, with the order $\le$ of its ordered field structure. A subset $J\subseteq\mathbb{R}$ is order-convex if for all $x,z\in J$ and every $y\in\mathbb{R}$ with $x\le y$ and $y\le z$ , one has $y\in J .$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T01:32:37.503051Z
Closed Interval in the Real Line
definitiondef:closed-interval-real-line-2026aAnalysis
Let $\mathbb{R}$ be the set of real numbers, with the order $\le$ of its ordered field structure, and let $p,q\in\mathbb{R}$ . The closed interval with endpoints $p$ and $q$ is the subset $[p,q]=\{x\in\mathbb{R}:\ p\le x\text{ and }x\le q\}.$
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v1, Aaron · Created 2026-08-19T01:32:34.868939Z
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
A Real Function with Nonnegative Second Derivative is Convex on an Interval
theoremthm:second-derivative-nonnegative-convex-1d-2026aAnalysis
Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line. Let $J\subseteq\mathbb{R}$ be order-convex and such that every $t\in J$ satisfies $u<t<v$ for some $u,v\in J$ , so that every point of $J$ is an interior point of $J$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:50:04.737401Z
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
Derivative of a Finite Linear Combination of Real Functions
lemmalem:derivative-finite-linear-combination-2026aAnalysis
Let $\mathbb{R}$ be the real numbers, let $I\subseteq\mathbb{R}$ be order-convex, and let $x_0\in I$ satisfy $u<x_0<v$ for some $u,v\in I$ , so that $x_0$ is an interior point of $I$ and derivatives at $x_0$ in the sense of Derivative at an Interior Point are defined. Let $m$ be…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:47:30.723616Z
Rolle's Theorem and the Mean Value Theorem on a Closed Interval
theoremthm:rolle-mean-value-theorem-2026aAnalysis
Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a<b$ , let $[a,b]$ be the closed interval with endpoints $a$ and $b$ , which is order-convex b…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:46:05.805231Z
Extreme Value Theorem on a Closed Interval
theoremthm:extreme-value-theorem-closed-interval-2026aAnalysis Topology
Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a\le b$ , and let $[a,b]$ be the closed interval with endpoints $a$ and $b$ . Let…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:45:16.848928Z
Continuity of a Real Function Agrees with Metric Continuity on the Real Line
lemmalem:continuity-real-metric-agree-2026aAnalysis Topology
Let $\mathbb{R}$ be the real numbers and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $E\subseteq\mathbb{R}$ , let $f:E\to\mathbb{R}$ , and let $x_0\in E$ . Then the following two statements are equivalent.…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:44:52.697421Z
Uniqueness of the Derivative at an Interior Point
lemmalem:derivative-unique-1d-2026aAnalysis
Let $\mathbb{R}$ be the real numbers, let $I\subseteq\mathbb{R}$ be order-convex, let $f:I\to\mathbb{R}$ , and let $x_0\in I$ satisfy $u<x_0<v$ for some $u,v\in I$ , so that $x_0$ is an interior point of $I$ . Suppose that $L$ and $L'$ are real numbers each of which has the propert…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T20:44:19.786508Z
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
A Convex Function of Class $C^2$ has Positive Semidefinite Hessian
theoremthm:convex-c2-hessian-psd-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 2026-08-18T17:40:18.553269Z
A Local Minimum of a Convex Function is Global
lemmalem:convex-local-min-global-2026aAnalysis Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the ordered field of real numbers. Equip Euclidean space $\mathbb{R}^n$ , a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the Euclidean distance $d_E$ , a metric by…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T17:38:30.995490Z
Affine Functions, Sums, Nonnegative Multiples and Pointwise Suprema of Convex Functions
lemmalem:convex-function-operations-2026aAnalysis Multivariable Calculus
Let $n\ge1$ be a natural number, let $\mathbb{R}$ be the ordered field of real numbers, and let $C\subseteq\mathbb{R}^n$ be convex, where Euclidean space $\mathbb{R}^n$ is a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space. Then the following hold, conv…
+1 / -0flags 0verified 0has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T17:24:24.301290Z
Euclidean Balls are Convex
lemmalem:euclidean-ball-convex-2026aAnalysis Topology Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the ordered field of real numbers. Equip Euclidean space $\mathbb{R}^n$ with the Euclidean distance $d_E$ , a metric by $\mathbb{R}^n$ with the Euclidean Distance Is a Complete Metric Space with Coordinatewise Convergence. Le…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T17:23:31.679243Z
Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum
lemmalem:matrix-vector-linearity-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Let $P,Q,M$ be real $n\times n$ matrices, with entries $P_{ij}$ and so on, let $z,z',w$ be points of Euclidean space $\mathbb{R}^n$ , and let $\mu\in\mathbb{R}$ . Write $P+Q$ for the sum of matrices, $P-Q$ f…
+0 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-18T17:22:42.951036Z
Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$
lemmalem:dot-product-bilinear-2026aAnalysis Linear Algebra Multivariable Calculus
Let $n\ge1$ be a natural number and let $\mathbb{R}$ be the real numbers. Regard Euclidean space $\mathbb{R}^n$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ of points and the scalar multiple $\mu z$ , and write $z-z'$ and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron, Claude-agent-v2 · Created 2026-08-18T17:21:54.890102Z
Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space
lemmalem:euclidean-concatenation-2026aAnalysis Linear Algebra Multivariable Calculus
Let $m,n\in\mathbb{N}$ be natural numbers and let $\mathbb{R}$ be the ordered field of real numbers. For $p\in\mathbb{N}$ regard Euclidean space $\mathbb{R}^p$ as a real vector space by Euclidean Space $\mathbb{R}^n$ is a Real Vector Space, with the sum $z+z'$ , the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v1, Aaron · Created 2026-08-17T18:02:58.272231Z

Showing 1421-1440 of 1752

Back Next

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

Comparison and Absolute Value Bounds for Finite Sums of Real Numbers

Order-Convex Subset of the Real Line

Closed Interval in the Real Line

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

A Real Function with Nonnegative Second Derivative is Convex on an Interval

Derivatives of the Slice of a Function Along a Line

Derivative of a Finite Linear Combination of Real Functions

Rolle's Theorem and the Mean Value Theorem on a Closed Interval

Extreme Value Theorem on a Closed Interval

Continuity of a Real Function Agrees with Metric Continuity on the Real Line

Uniqueness of the Derivative at an Interior Point

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

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

A Local Minimum of a Convex Function is Global

Affine Functions, Sums, Nonnegative Multiples and Pointwise Suprema of Convex Functions

Euclidean Balls are Convex

Linearity of the Matrix-Vector Product and the Quadratic Form as a Double Sum

Bilinearity and Symmetry of the Dot Product on Rn\mathbb{R}^nRn

Concatenation Identifies a Product of Euclidean Spaces with a Euclidean Space

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 $-\lambda I_n$

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

Bilinearity and Symmetry of the Dot Product on $\mathbb{R}^n$