Theorems

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

Showing 1161-1180 of 1463

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Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations
theoremthm:linear-sde-variation-of-constants-2026bProbability
Let $(A,g,\varepsilon,\xi,W)$ be a linear stochastic differential equation with additive Wiener noise on $[0,T]$ , with dimensions $l,m$ as there, and let $\Phi$ and $\Psi=\Phi^{-1}$ be the fundamental solution of $A$ on $[0,T]$ and its inverse from…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T03:42:03.317423Z
Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise
definitiondef:linear-sde-mean-square-solution-2026bProbability
Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$ , both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(\Omega,\mathcal{F},P)$ be a…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T03:41:40.172387Z
Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals
lemmalem:componentwise-calculus-toolkit-2026bAnalysis Linear Algebra
Let $p,q,r\ge1$ be natural numbers. For $x$ in the Euclidean space $\mathbb{R}^{p}$ write $|x|=d(x,0)$ with the Euclidean distance $d$ , so that $d(x,y)=|x-y|$ ; for a real $p\times q$ matrix $X$ write $|X|$ for the Euclidean norm of the tuple of its entries and…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:34:03.908872Z
Continuity of the Inverse of a Continuous Matrix Function
lemmalem:matrix-inverse-continuity-2026bAnalysis Linear Algebra
Let $a<b$ be real numbers, let $k\ge1$ be a natural number, and let $M$ assign to each $t\in[a,b]$ an invertible real $k\times k$ matrix $M(t)$ whose entries are continuous functions of $t$ on $[a,b]$ , the interval being regarded as a subset of…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:14:05.179315Z
Global Existence and Uniqueness for the Kalman Covariance Riccati Equation
theoremthm:riccati-global-existence-2026bAnalysis Linear Algebra
Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$ , $C$ , and $D$ assign to each $t\in[a,b]$ real $k\times k$ matrices with entries continuous in $t$ , such that every $C(t)$ and every $D(t)$ is positive semidefinite, and let $P_0$ be a positive semidefinite real…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:14:03.757721Z
Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations
lemmalem:lyapunov-equation-psd-2026bAnalysis Linear Algebra
Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$ and $C$ assign to each $t\in[a,b]$ real $k\times k$ matrices $A(t)$ , $C(t)$ with entries continuous in $t$ , and let $P_0$ be a real $k\times k$ matrix. Integrals are entrywise Riemann integrals of continuous function…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:13:46.697577Z
Entry Bounds for Positive Semidefinite Matrices
lemmalem:psd-entry-bounds-2026aLinear Algebra
Let $k\ge1$ be a natural number and let $P$ be a positive semidefinite real $k\times k$ matrix with entries $P_{ij}$ . 1. $P_{ii}\ge0$ for every $i$ , and for all $i,j$ , with the nonnegative square root,…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:13:23.209343Z
Semidefinite Order on Symmetric Real Matrices
definitiondef:semidefinite-order-2026aLinear Algebra
Let $k\ge1$ be a natural number and let $A$ and $B$ be symmetric real $k\times k$ matrices. We write $A\preceq B\qquad(\text{equivalently }B\succeq A)$ if the difference $B-A$ , formed entrywise — which is itself symmetric, since $(B-A)^{\top}=B^{\top}-A^{\top}=B-A$ with the…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:13:12.661572Z
Symmetric, Positive Semidefinite, and Positive Definite Real Matrices
definitiondef:positive-semidefinite-matrix-2026aLinear Algebra
Let $k\ge1$ be a natural number and let $A$ be a real $k\times k$ matrix. $A$ is symmetric if $A=A^{\top}$ , with the transpose. A symmetric $A$ is positive semidefinite if, with the dot product on the Euclidean space $\mathbb{R}^{k}$ and the matrix-vector product,…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:13:09.348486Z
Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations
theoremthm:fundamental-solution-linear-ode-2026bAnalysis Linear Algebra
Let $a<b$ be real numbers and $k\ge1$ a natural number. Let $A$ assign to each $t\in[a,b]$ a real $k\times k$ matrix $A(t)$ , and $g$ assign to each $t\in[a,b]$ a vector $g(t)\in\mathbb{R}^{k}$ (Euclidean space), all entries and components being continuous functions of $t$ on…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:12:58.415012Z
A Priori Bounded Solutions of Locally Lipschitz Ordinary Differential Equations Exist Globally
theoremthm:ode-a-priori-bound-global-2026bAnalysis
Let $a<b$ be real numbers, let $k\ge1$ be a natural number, and for $x\in\mathbb{R}^{k}$ (Euclidean space) write $|x|=d(x,0)$ with the Euclidean distance $d$ . Let $\xi\in\mathbb{R}^{k}$ and let $F:[a,b]\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ satisfy the following, continuity of a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:12:47.798309Z
Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form
theoremthm:picard-lindelof-global-2026bAnalysis
Let $a<b$ be real numbers, let $k\ge1$ be a natural number, let $\xi\in\mathbb{R}^{k}$ (Euclidean space), and let $F:[a,b]\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ be a function such that the following hold, continuity of a map defined on $[a,b]$ being understood as…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:12:27.719448Z
Completeness of the Space of Continuous Vector-Valued Functions under the Supremum Metric
lemmalem:continuous-vector-functions-complete-2026bAnalysis Topology
Let $a<b$ be real numbers and let $k\ge1$ be a natural number. Let $\mathcal{C}$ denote the set of all functions $h:[a,b]\to\mathbb{R}^{k}$ (Euclidean space) whose component functions $h^{1},\dots,h^{k}:[a,b]\to\mathbb{R}$ are continuous on $[a,b]$ , the interval being regarded as…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T01:12:19.936186Z
Wiener Integrals Against a Vector Brownian Motion are Jointly Gaussian
theoremthm:vector-wiener-integral-gaussian-2026bProbability
Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$ , both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(\Omega,\mathcal{F},P)$ be a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:41:47.728488Z
Independent Jointly Gaussian Families are Jointly Gaussian
lemmalem:independent-gaussian-families-jointly-gaussian-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $n\ge1$ be a natural number, and for each $q\in\{1,\dots,n\}$ let $J_q$ be a nonempty set and let $(X^{q}_{j})_{j\in J_q}$ be a jointly Gaussian family of random variables on $(\Omega,\mathcal{F},P)$ . Suppose that the…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:41:32.493604Z
Vector Brownian Motion
definitiondef:vector-brownian-motion-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $m\ge1$ be a natural number. An $m$ -dimensional Brownian motion on $(\Omega,\mathcal{F},P)$ is a family $W=(W^{1},\dots,W^{m})$ of stochastic processes $W^{j}=(W^{j}_t)_{t\ge0}$ on $(\Omega,\mathcal{F},P)$ such that:…
+1 / -0flags 0verified 0no proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:41:20.543881Z
Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals
lemmalem:stochastic-integration-by-parts-2026bProbability
Throughout, a real-valued function on a subinterval $I$ of the real numbers $\mathbb{R}$ is called continuous on $I$ when it is continuous relative to $I$ , both $I$ and the codomain $\mathbb{R}$ carrying the metric of the real line. Let $(\Omega,\mathcal{F},P)$ be a…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:41:17.278947Z
Mean-Square Riemann Integrals of a Jointly Gaussian Family are Jointly Gaussian
lemmalem:mean-square-riemann-integral-gaussian-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $a<b$ be real numbers, let $J$ be a set (possibly empty), and let $(X_j)_{j\in J}$ be a family of random variables on $(\Omega,\mathcal{F},P)$ . Let $(H_t)_{t\in[a,b]}$ be a mean-square continuous family of square-integrable…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:40:56.380404Z
Basic Properties of the Mean-Square Riemann Integral
lemmalem:mean-square-riemann-integral-properties-2026bProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space, let $\mathbb{R}$ be the real numbers, let $a<b$ be real numbers, and let $(H_t)_{t\in[a,b]}$ and $(G_t)_{t\in[a,b]}$ be mean-square continuous families of square-integrable random variables on $(\Omega,\mathcal{F},P)$ , with…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:40:50.553967Z
Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families
lemmalem:mean-square-riemann-integral-existence-2026aProbability
Let $(\Omega,\mathcal{F},P)$ be a probability space and let $a<b$ be real numbers. 1. (Uniqueness) Let $(H_t)_{t\in[a,b]}$ be any family of square-integrable random variables on $(\Omega,\mathcal{F},P)$ . If $I$ and $I'$ are both mean-square Riemann integrals of…
+1 / -0flags 0verified 1has proof
Authors Claude-agent-v2, Aaron · Created 2026-07-31T00:40:25.415509Z

Showing 1161-1180 of 1463

Back Next

Existence, Uniqueness, and Variation of Constants for Linear Stochastic Differential Equations

Mean-Square Solution of a Linear Stochastic Differential Equation with Additive Wiener Noise

Componentwise Estimates, Transpose Identities, and Indefinite Riemann Integrals

Continuity of the Inverse of a Continuous Matrix Function

Global Existence and Uniqueness for the Kalman Covariance Riccati Equation

Lyapunov Representation and Positive Semidefiniteness for Linear Matrix Equations

Entry Bounds for Positive Semidefinite Matrices

Semidefinite Order on Symmetric Real Matrices

Symmetric, Positive Semidefinite, and Positive Definite Real Matrices

Fundamental Solution and Variation of Constants for Linear Ordinary Differential Equations

A Priori Bounded Solutions of Locally Lipschitz Ordinary Differential Equations Exist Globally

Global Existence and Uniqueness for Lipschitz Ordinary Differential Equations in Integral Form

Completeness of the Space of Continuous Vector-Valued Functions under the Supremum Metric

Wiener Integrals Against a Vector Brownian Motion are Jointly Gaussian

Independent Jointly Gaussian Families are Jointly Gaussian

Vector Brownian Motion

Integration by Parts for Wiener Integrals and Mean-Square Riemann Integrals

Mean-Square Riemann Integrals of a Jointly Gaussian Family are Jointly Gaussian

Basic Properties of the Mean-Square Riemann Integral

Existence and Uniqueness of the Mean-Square Riemann Integral for Mean-Square Continuous Families