The Existence And Uniqueness Theorem

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Math 361S Lecture Notes Numerical solution of ODEs

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The following is a fundamental theorem about existence and uniqueness for ODE’s. Theorem 2.1. If f: [a;b] Rd!Rd is continuously di erentiable, then in a neigh-borhood [a;a+ ") around a, the solution to (2.1a){(2.1b) exists and is unique. Note that the solution may not exist for all t2[a;b] because the solution may diverge.

Theorem, Existence, Uniqueness, Existence and uniqueness

System of First Order Diﬀerential Equations

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A(t)x(t)+b(t): The following theorem gives existence and uniqueness of solutions, Theorem 1.1. If the vector-valued functions A(t) and b(t) are con-tinuous over an open interval I contains t0; then the initial value prob-lem ‰ x0(t) = A(t)x(t)+b(t) x(t0) = x0 has an unique vector-values solution x(t) that is deﬁned on entire in-

First, System, Order, Equations, Diﬀerential, Theorem, Existence, System of first order diﬀerential equations, Uniqueness, Existence and uniqueness

7. Some irreducible polynomials

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existence of an element of order 4 in (Z =p) . Thus, x2 + 1 is irreducible in such k[x]. ... Dirichlet’s theorem on primes in arithmetic progressions assures that there are in nitely many such. The presence of ... For uniqueness, suppose R(x) were another polynomial of degree <ntaking the same values at ndistinct points

Theorem, Irreducible, Existence, Uniqueness

Analytic Solutions of Partial Di erential Equations

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Existence and uniqueness of solutions SolvingPDEsanalytically isgenerallybasedon ndingachange ofvariableto transform the equation into something soluble or on nding an integral form of the solution. First order PDEs a @u @x +b @u @y = c: Linear equations: change coordinate using (x;y), de ned by the characteristic equation dy dx = b a;

Solutions, Equations, Partial, Analytics, Erential, Existence, Uniqueness, Existence and uniqueness, Analytic solutions of partial di erential equations

Numerical Analysis

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16.2 Existence and uniqueness of solutions 258 16.3 Basic discretization methods 262 16.4 Convergence of discretization methods 266 16.5 More reading 269 16.6 Exercises 269 16.7 Solutions 271 Chapter 17. Higher-order ODE Discretization Methods 275 17.1 Higher-order discretization 276 17.2 Convergence conditions 281 17.3 Backward ...

Analysis, Numerical, Numerical analysis, Existence, Uniqueness, Existence and uniqueness

The Moore-Penrose Pseudoinverse (Math 33A: Laub)

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Theorem: (Existence) The linear system Ax = b; A 2 IRm£n; b 2 IRm (3) has a solution if and only if R(b) µ R(A); equivalently, there is a solution to these m equations in n unknowns if and only if AA+b = b. Proof: The subspace inclusion criterion follows essentially from the deﬂnition of the range of a matrix. The matrix criterion is from ...

Theorem, Existence

Microeconomic Theory

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Lecture Notes 1 Microeconomic Theory Guoqiang TIAN Department of Economics Texas A&M University College Station, Texas 77843 (gtian@tamu.edu) August, 2002/Revised: February 2013

Theory, Microeconomics, Microeconomic theory