# Memoirs on Differential Equations and Mathematical Physics

Save this PDF as:

Μέγεθος: px
Εμφάνιση ξεκινά από τη σελίδα:

## Transcript

1 Memoirs on Differential Equations and Mathematical Physics Volume 31, 2004, T. Tadumadze and L. Alkhazishvili FORMULAS OF VARIATION OF SOLUTION FOR NON-LINEAR CONTROLLED DELAY DIFFERENTIAL EQUATIONS WITH CONTINUOUS INITIAL CONDITION

2 Abstract. Formulas of variation of solution for controlled differential equations with variable delays are proved. The continuous initial condition means that at the initial moment the values of the initial function and the trajectory coincide Mathematics Subject Classification. 34K99. Key words and phrases: Delay differential equation, variation formulas of solution.! "\$# % & % ' "(!& "! )"(*\$ # % *(, & - & *( & % *\$!"!. / %. % & & 0 )

3 THE FORMULAS OF VARIATION OF SOLUTION 85 Introduction In the present work the formulas of variation of solution for controlled differential equations with variable delays are proved. These formulas play an important role in proving necessary conditions of optimality for optimal problems. The formulas of variation of solution for various classes of delay differential equations are given in [1] [4]. 1. Formulation of Main Results Let J = [a, b] be a finite interval; O R n, G R r be open sets. Let the function f : J O s G R n satisfy the following conditions: for almost all t J, the function f(t, ) : O s G R n is continuously differentiable; for any (x 1,..., x s, u) O s G, the functions f(t, x 1,..., x s, u), f xi (t, x 1,..., x s, u), i = 1,..., s, f u (t, x 1,..., x s, u) are measurable on J ; for arbitrary compacts K O, M G there exists a function m K,M ( ) L(J, R ), R = [0, ), such that for any (x 1,..., x s, u) K s M and for almost all t J, the following inequality is fulfilled f(t, x 1,..., x s, u) f xi ( ) f u ( ) m K,M (t). Let the scalar functions τ i (t), i = 1,..., s, t R, be absolutely continuous and satisfy the conditions: τ i (t) t, τ i (t) > 0, i = 1,..., s. Let Φ be the set of piecewise continuous functions ϕ : J 1 = [τ, b] O with a finite number of discontinuity points of the first kind, satisfying the conditions cl ϕ(j 1 ) O, τ = min{τ 1 (a),..., τ s (a)}, ϕ = sup{ ϕ(t), t J 1 }; Ω be the set of measurable functions u : J G satisfying the condition: cl{u(t) : t J } is a compact lying in G, u = sup{ u(t) : t J }. To every element = (, ϕ, u) A = J Φ Ω, let us correspond the differential equation with the continuous initial condition ẋ(t) = f(t, x(τ 1 (t)),..., x(τ s (t)), u(t)) (1.1) x(t) = ϕ(t), t [τ, ]. (1.2) Definition 1.1. Let = (, ϕ, u) A, < b. A function x(t) = x(t; ) O, t [τ, t 1 ], t 1 (, b], is said to be a solution of the equation (1.1) with the initial condition (1.2), or a solution corresponding to the element A, defined on the interval [τ, t 1 ], if on the interval [τ, ] the function x(t) satisfies the condition (1.2), while on the interval [, t 1 ] it is absolutely continuous and almost everywhere satisfies the equation (1.1). Let us introduce the set of variation V = {δ = (δ, δϕ, δu) : δ α = const, δϕ = k λ i δϕ i, λ i α, i = 1,..., k, δu α, δu Ω ũ}, (1.3)

4 86 T. Tadumadze and L. Alkhazishvili where δϕ i Φ ϕ, i = 1,..., k, ϕ Φ, ũ Ω are fixed functions, α > 0 is a fixed number. Let x(t) be a solution corresponding to the element = (, ϕ, ũ) A, defined on the interval [τ, t 1 ], t i (a, b), i = 0, 1. There exist numbers ε 1 > 0, δ 1 > 0 such that for an arbitrary (ε, δ ) [0, ε 1 ] V, to the element εδ A there corresponds a solution x(t; εδ ) defined on [τ, t 1 δ 1 ] J 1 (see Lemma 2.2). Due to uniqueness, the solution x(t, ) is a continuation of the solution x(t) on the interval [τ, t 1 δ 1 ]. Therefore the solution x(t) in the sequel is assumed to be defined on the interval [τ, t 1 δ 1 ]. Let us define the increment of the solution x(t) = x(t; ) x(t) = x(t; εδ ) = x(t; εδ ) x(t), (t, ε, δ ) [τ, t 1 δ 1 ] [0, ε 1 ] V. (1.4) Theorem 1.1. Let the function ϕ(t) be absolutely continuous in some left semi-neighborhood of the point and let there exist the finite limits lim ω ω 0 ϕ = ϕ( ), f[ω] = f, ω = (t, x 1,..., x s ) (a, ] O s, f[ω] = f(ω, ũ(t)), ω 0 = (, ϕ(τ 1 ( )),..., ϕ(τ s ( ))). Then there exist numbers ε 2 > 0, δ 2 > 0 such that for an arbitrary (t, ε, δ ) [, t 1 δ 2 ] [0, ε 2 ] V, V = {δ V : δ 0}, the formula is valid, where x(t; εδ ) = εδx(t; δ ) o(t; εδ ) 1 (1.5) δx(t; δ ) = Y ( ; t) [ ( ϕ f )δ δϕ( ) ] β(t; δ ), (1.6) β(t; δ )= τ i() Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ)dξ Y (ξ; t) f u [ξ]δu(ξ)dξ, γ i (t) = τi 1 (t), fxi [t] = f xi (t, x(τ 1 (t),..., x(τ s (t)), Y (ξ; t) is the matrix-function satisfying the equation Y (ξ; t) ξ = Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ), ξ [, t], (1.7) 1 Here and in the sequel, the values (scalar or vector) which have the corresponding order of smallness uniformly for (t, δ ), will be denoted by O(t; εδ ), o(t; εδ ).

5 THE FORMULAS OF VARIATION OF SOLUTION 87 and the condition Y (ξ; t) = { I, ξ = t, Θ, ξ > t. Here I is the identity matrix, Θ is the zero matrix. (1.8) Theorem 1.2. Let the function ϕ(t) be absolutely continuous in some right semi-neighborhood of the point and let there exist the finite limits lim ω ω 0 ϕ = ϕ( ), f[ω] = f, ω = (t, x 1,..., x s ) [, b) O s, ω 0 = (, ϕ(τ 1 ( )),..., ϕ(τ s ( ))). Then for any s 0 (, t 1 ) there exist numbers ε 2 > 0, δ 2 > 0 such that for an arbitrary (t, ε, δ ) [s 0, t 1 δ 2 ] [0, ε 2 ] V, V = {δ V : δ 0}, the formula (1.5) is valid, where δx(t; δ ) has the form δx(t; δ ) = Y ( ; t) [ ( ϕ f )δ δϕ( ) ] β(t; δ ). (1.9) The following theorem follows from Theorems 1.1 and 1.2. Theorem 1.3. Let the conditions of Theorems 1.1 and 1.2 be fulfilled. Moreover, let ϕ f = ϕ f = f 0, and the functions δϕ i (t), i = 1,..., k be continuous at the point (see (1.3)). Then for any s 0 (, t 1 ) there exist numbers ε 2 > 0, δ 2 > 0 such that for an arbitrary (t, ε, δ ) [s 0, t 1 δ 2 ] [0, ε 2 ] V the formula (1.5) is valid, where δx(t; εδ ) has the form δx(t; δ ) = Y (, t) [ f 0 δ δϕ( ) ] β(t; δ ). 2. Auxiliary Lemmas To every element = (, ϕ, u) A, let us correspond the functionaldifferential equation ẏ(t) = f(, ϕ, u, y)(t) = = f ( t, h(, ϕ, y)(τ 1 (t)),..., h(, ϕ, y)(τ s (t)), u(t) ) (2.1) with the initial condition y( ) = ϕ( ), (2.2) where the operator h( ) is defined by the formula { ϕ(t), t [τ, ], h(, ϕ, y)(t) = y(t), t [, b]. (2.3)

6 88 T. Tadumadze and L. Alkhazishvili Definition 2.1. Let = (, ϕ, u) A. An absolutely continuous function y(t) = y(t; ) O, t [r 1, r 2 ] J, is said to be a solution of the equation (2) with the initial condition (2.2), or a solution corresponding to the element A, defined on the interval [r 1, r 2 ], if [r 1, r 2 ], y( ) = ϕ( ) and the function y(t) satisfies the equation (2) almost everywhere on [r 1, r 2 ]. Remark 2.1. Let y(t; ), t [r 1, r 2 ], A, be a solution of the equation (2) with the initial condition (2.2). Then the function x(t; ) = h(, ϕ, y( ; ))(t), t [r 1, r 2 ], (2.4) is a solution of the equation (1.1) with the initial condition (1.2) (see Definition 1.1, (2.3)). Lemma 2.1. Let ỹ(t), t [r 1, r 2 ] (a, b), be a solution corresponding to the element A; let K 1 O be a compact which contains some neighborhood of the set cl ϕ(j 1 ) ỹ([r 1, r 2 ]) and let M 1 G be a compact which contains some neighborhood of the set cl ũ(j ). Then there exist numbers ε 1 > 0, δ 1 > 0 such that for an arbitrary (ε, δ ) [0, ε 1 ] V, to the element εδ A there corresponds a solution y(t; εδ ) defined on [r 1 δ 1, r 2 δ 1 ] J. Moreover, ϕ(t) = ϕ(t) εδϕ(t) K 1, t J 1, u(t) = ũ(t) εδu(t) M 1, t J, y(t; εδ ) K 1, t [r 1 δ 1, r 2 δ 1 ], lim y(t; εδ ) = y(t; ) ε 0 uniformly for (t, δ ) [r 1 δ 1, r 2 δ 1 ] V. This lemma is analogous to Lemma 2.1 in [1, p. analogously. (2.5) 21] and it is proved Lemma 2.2. Let x(t), t [τ, t 1 ] be a solution corresponding to the element A, t i (a, b), i = 0, 1; let K 1 O be a compact which contains some neighborhood of the set cl ϕ(j 1 ) x([, t 1 ]) and let M 1 G be a compact which contains some neighborhood of the set cl ũ(j ). Then there exist numbers ε 1 > 0, δ 1 > 0 such that for any (ε, δ ) [0, ε 1 ] V, to the element εδ A there corresponds the solution x(t; εδ ), t [τ, t 1 δ 1 ] J 1. Moreover, x(t; εδ ) K 1, t [τ, t 1 δ 1 ], u(t) = ũ(t) εδu(t) M 1, t J. (2.6) It is easy to see that if in Lemma 2.1 r 1 =, r 2 = t 1, then ỹ(t) = x(t), t [, t 1 ]; x(t; εδ ) = h(, ϕ, y( ; εδ ))(t), (t, ε, δ ) [τ, t 1 δ 1 ] [0, ε 1 ] V (see (2.4)). Thus Lemma 2.2 is a simple corollary (see (2.5)) of Lemma 2.1.

7 THE FORMULAS OF VARIATION OF SOLUTION 89 Due to uniqueness, the solution y(t; ) on the interval [r 1 δ 1, r 2 δ 1 ] is a continuation of the solution ỹ(t); therefore the solution ỹ(t) in the sequel is assumed to be defined on the whole interval [r 1 δ 1, r 2 δ 1 ]. Let us define the increment of the solution ỹ(t) = y(t; ), y(t) = y(t; εδ ) = y(t; εδ ) ỹ(t), It is obvious (see Lemma 2.1) that (t, ε, δ ) [r 1 δ 1, r 2 δ 1 ] [0, ε 1 ] V. (2.7) lim y(t; εδ ) = 0 uniformly for (t, δ ) [r 1 δ 1, r 2 δ 1 ] V. (2.8) ε 0 Lemma 2.3 ([1, p. 35]). For arbitrary compacts K O, M G there exists a function L K,M ( ) L(J, R ) such that for an arbitrary x i, x i K, i = 1,..., s, u, u M and for almost all t J, the inequality f(t, x 1,..., x s, u ) f(t, x 1,..., x s, u ) ( ) L K,M (t) x i x i u u (2.9) is valid. Lemma 2.4. Let the conditions of Theorem 1.1 be fulfilled. Then there exist numbers ε 2 > 0, δ 2 > 0 such that for any (ε, δ ) [0, ε 2 ] V the inequality max y(t) O(εδ ) (2.10) t [,r 2δ 2] is valid. Moreover, y( ) = ε [ δϕ( ) ( ϕ f )δ ] o(εδ ). (2.11) Proof. Let us define the following sets I 1 = { i {1,..., s} : τ i ( ) <, γ i ( ) > r 2 }, I 2 = { i {1,..., s} : τ i ( ) <, γ i ( ) r 2 }, I 3 = { i {1,..., s} : τ i ( ) = }. Let ε 2 (0, ε 1 ], δ 2 (0, δ 1 ] be so small that for any (ε, δ ) [0, ε 2 ] V the following relations are fulfilled: γ i ( ) > r 2 δ 2, i I 1, τ i ( ) <, i I 2. The function y(t) on the interval [, r 2 δ 1 ] satisfies the equation y(t) = a(t; εδ ), (2.12) where a(t; εδ ) = f(, ϕ, u, ỹ y)(t) f(, ϕ, ũ, ỹ)(t). (2.13)

8 90 T. Tadumadze and L. Alkhazishvili Now let us rewrite the equation (2.13) in the integral form y(t) = y( ) Hence y(t) y( ) a(ξ, εδ ) dξ, t [, r 2 δ 1 ]. a(ξ; εδ ) dξ = y( ) a 1 (t; εδ ). (2.14) Now let us prove the equality (2.11). It is easy to see that y( ) = y( ; εδ ) ỹ( ) = ϕ( ) Next, So, f(, ϕ, u, ỹ y)(t) dt ϕ( ). lim δϕ() = δϕ( ), uniformly for δ V. ε 0 (2.15) ϕ( ) ϕ( ) = It is clear that 0 ϕ(t) dt εδϕ( ) ε(δϕ( ) δϕ( )) = = ε[ ϕ δ δϕ( )] o(εδ ). lim h(, ϕ, ỹ y)(τ i (t)) = ϕ(τ i ( )), i = 1,..., s, (ε,t) (0, ) (see (2.3), (2.8)). Consequently, lim sup ε 0 t [,] From (2.17) it follows that (2.16) f(, ϕ, ỹ y)(t) f = 0 (2.17) f(t0, ϕ, u, ỹ y)(t) dt = εf δ [ f(, ϕ, u, ỹ y)(t) f ] dt = εf δ o(εδ ). (2.18) From (2.15), by virtue of (2.16), (2.18), we obtain (2.11). Now, to prove the inequality (2.10), let us estimate the function a 1 (t, εδ ), t [, r 2 δ 2 ]. We

9 THE FORMULAS OF VARIATION OF SOLUTION 91 have a 1 (t; εδ ) = ( ) L K1,M 1 (ξ) h(, ϕ, ỹ y)(τ i (ξ)) h(, ϕ, ỹ)(τ i (ξ)) ε δu(ξ) dξ i I 1 r 2δ 2 L K1,M 1 (ξ)δϕ(τ i (ξ)) dξ i I 2 τ i() L K1,M 1 (γ i (ξ)) h(, ϕ, ỹ y)(ξ) h(, ϕ, ỹ)(ξ) γ i (ξ) dξ L K1,M 1 (ξ) y(ξ) γ i (ξ) dξ O(εδ ) O(εδ ) i I 3 i I 2 τ i() L K1,M 1 (γ i (t)) h(, ϕ, ỹ y)(t) ϕ(t) γ i (t) dt L(ξ) y(ξ) dξ, where L(ξ) = χ(γ i (ξ))l K1,M 1 (γ i (ξ)) γ i (ξ), and χ(t) is the characteristic function of the interval J. If I k =, we assume that i I k α i = 0. Let I 2, and i I 2, β i = τ i() L K1,M 1 (γ i (t)) h(, ϕ, ỹ y)(t) ϕ(t) γ i (t) dt = = ε 0 L K1,M 1 (γ i (t)) δϕ(t) γ i (t) dt τ i() L K1,M 1 (γ i (t)) y(t, εδ ) ϕ(t) γ i (t) dt.

10 92 T. Tadumadze and L. Alkhazishvili It is obvious that when t [, ], y(t, εδ ) ϕ(t) = ϕ( ) O(εδ ) ϕ(t) dt f(, ϕ, u, ỹ y)(ξ) dξ ϕ(t) f(, ϕ, u, ỹ y)(t) dt. (2.19) For sufficiently small ε 2 the integrands are bounded, hence β i = O(εδ ) (see (2.18)). Thus, a 1 (t; εδ ) O(εδ ) L(ξ) y(ξ) dξ. (2.20) From (2.14), on account of (2.11), (2.20), on the interval t [, r 2 δ 2 ] we have y(t) O(εδ ) L(ξ) y(ξ) dξ. From which, by virtue of Gronwall s lemma, we obtain (2.10). Lemma 2.5. Let the conditions of Theorem 2 be fulfilled. Then there exist numbers ε 2 > 0, δ 2 > 0 such that for any (ε, δ ) [0, ε 2 ] V the inequality max y(t) O(εδ ) (2.21) t [,r 2δ 2] is valid. Moreover, ] y( ) = ε [δϕ( ) ( ϕ f )δ o(εδ ). (2.22) This lemma is proved analogously to lemma 2.4. Let r 1 =, r 2 = t 1, A, then 3. Proof of Theorem 1.1 ỹ(t) = x(t), t [, t 1 δ 1 ], y(t; εδ ) = x(t; εδ ), t [, t 1 δ 1 ], (εδ ) [0, ε 1 ] V (see Lemmas 2.1 and 2.2). Thus εδϕ(t), t [τ, ), x(t) = y(t; εδ ) ϕ(t), t [, ], y(t), t [, t 1 δ 1 ] (see (1.4), (2.7)).

11 THE FORMULAS OF VARIATION OF SOLUTION 93 By virtue of Lemma 2.4 and (2.19), there exist numbers ε 2 (0, ε 1 ], δ 2 (0, δ 1 ] such that x(t) O(εδ ) (t, ε, δ ) [τ, t 1 δ 2 ] [0, ε 2 ] V, (3.1) } x( ) = ε {δϕ( ) ( ϕ f )δ o(εδ ). (3.2) The function x(t) on the interval [, t 1 δ 2 ] satisfies the following equation x(t) = f xi [t] x(τ i (t)) ε f u [t]δu(t) R(t; εδ ), (3.3) where R(t; εδ )=f(t, x(τ 1 (t)) x(τ 1 (t)),..., x(τ s (t)) x(τ s (t)), ũ(t)εδu(t)) f[t] f xi [t] x(τ i (t)) ε f u [t]δu(t). (3.4) By means of the Cauchy formula, the solution of the equation (3.3) can be represented in the form where x(t) = Y ( ; t) x( ) ε Y (ξ; t) f u [ξ]δu(ξ) dξ 1 h i (t;, εδ ), t [, t 1 δ 1 ], (3.5) i=0 h 0 (t;, εδ ) = h 1 (t;, εδ ) = τ i() Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ) x(ξ) dξ, Y (ξ; t)r(ξ; εδ ) dξ. (3.6) The matrix function Y (ξ; t) satisfies the equation (1.7) and the condition (1.8). By virtue of Lemma 3.4 [1, p. 37], the function Y (ξ; t) is continuous on the set Π = {(ξ, t) : a ξ t, t J }. Hence } Y ( ; t) x( ) = εy ( ; t) {δϕ( ) ( ϕ f )δ o(t; εδ ) (3.7) (see (3.3)).

12 94 T. Tadumadze and L. Alkhazishvili Now we transform h 0 (t;, εδ ). We have h 0 (t;, εδ ) = [ ε 0 Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ i I 1 I 2 τ i() ] Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ) x(ξ) dξ = [ ] εα i (t) β i (t). It is easy to see that i I 1 I 2 α i (t) = τ i() Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ, (see (2.19). Thus, β i (t) = o(t; εδ ) h 0 (t;, εδ ) = = ε τ i() Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ o(t; εδ ). (3.8) Finally we estimate h 1 (t;, εδ ). We have: t 1δ 2 1 [ h 1 (t;, εδ ) Y d dξ f(t, x(τ 1(t)) ξ x(τ 1 (t)),..., Y 0 x(τ s (t)) ξ x(τ s (t)), u(t) ξεδu(t)) ] f xi [t] x(τ i (t)) ε f u [t]δu(t) dξ dt t 1δ 2 { 1 0 [ f xi (t, x(τ 1 (t)) ξ x(τ 1 (t)),..., ) f xi [t] x(τ i (t)) ] } ε f u (t, x(τ 1 (t)) ξ x(τ 1 (t)),..., ) f u [t] δu(t) dξ dt ( Y O(εδ ) ) σ i (εδ ) εασ 0 (εδ ), (3.9)

13 THE FORMULAS OF VARIATION OF SOLUTION 95 where Thus σ i (εδ ) = σ 0 (εδ ) = t 1δ 2 [ 1 0 t 1δ 2 [ 1 By Lebesgue s theorem 0 Y = sup Y (ξ; t), (ξ,t) Π f xi (t, x(τ 1 (t)) ξ x(τ 1 (t)),... ) f xi [t] ] dξ dt, fu (t, x(τ 1 (t)) ξ x(τ 1 (t)),... ) f u [t] dξ ]dt. lim σ i(εδ ) = 0, i = 0,..., s, uniformly for δ V. ε 0 h 1 (t;, εδ ) o(t; εδ ). (3.10) From (3.5), taking into account (3.7), (3), (3.10), we obtain (1.5), where δx(t; δ ) has the form (1.6). 4. Proof of Theorem 1.2 Analogously to the proof of Theorem 1.1, by virtue of Lemma 2.5 we obtain x(t) O(εδ ) (t, εδ ) [τ, t 1 δ 2 ] (0, ε 2 ] V, (4.1) x( ) = ε[δϕ( ) ( ϕ f )δ ] o(εδ ). (4.2) Let s 0 (, t 1 ) be the fixed point and let a number ε 2 (0, ε 1 ] be so small that for an arbitrary(ε, δ ) [0, ε 2 ] V the inequalities < s 0, τ i ( ) <, i I 1 I 2 are valid. The function x(t) satisfies the equation (3.3) on the interval [, t 1 δ 2 ], which by means of the Cauchy formula can be represented in the form x(t) = Y ( ; t) x( ) ε Y (ξ, t) f u [ξ]δu(ξ) dξ 1 h i (t;, εδ ), t [, t 1 δ 2 ], (4.3) i=0 where the functions h i (t;, εδ ), i = 0, 1, have the form (3.6) By virtue of Lemma 3.4 [1, p. 37], the function Y (ξ; t) is continuous on the set [, s 0 ] [s 0, t 1 δ 2 ]. Hence Y ( ; t) x( ) = εy ( ; t)[δϕ( ) ( ϕ f )δ ] o(t; εδ ). (4.4)

14 96 T. Tadumadze and L. Alkhazishvili Now we transform h 0 (t;, εδ ): h 0 (t;, εδ ) = [ ε Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ 0 i I 1 I 2 τ i() ] Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ) x(ξ) dξ i I 3 0 Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ) x(ξ) dξ = τ i() = (εα i (t) β i (t)) σ i (t). i I 1 I 2 i I 3 It is obvious that β i (t) = o(t; εδ ), σ i (t) = o(t; εδ ). Next, α i (t) = Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ τ i() τ i() Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ. τ i() Consequently = ε τ i() h 0 (t;, εδ ) = Y (γ i (ξ); t) f xi [γ i (ξ)] γ i (ξ)δϕ(ξ) dξ o(t; εδ ). (4.5) The following estimation is proved analogously (see (3.9)). Finally we note that ε Y (ξ; t) f u [ξ]δu(ξ)dξ = ε h 1 (t;, εδ ) o(t; εδ ) (4.6) Y (ξ; t) f u [ξ]δu(ξ)dξ o(t; εδ ) (4.7) From (4.3), taking into account (4.4) (4.7), we obtain (1.5), where δx(t; δ ) has the form (1.9).

### Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit

Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal

Διαβάστε περισσότερα

### 2 Composition. Invertible Mappings

Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,

Διαβάστε περισσότερα

### Chapter 6: Systems of Linear Differential. be continuous functions on the interval

Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations

Διαβάστε περισσότερα

### Section 8.3 Trigonometric Equations

99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.

Διαβάστε περισσότερα

### A Note on Intuitionistic Fuzzy. Equivalence Relation

International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com

Διαβάστε περισσότερα

### 2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)

Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok

Διαβάστε περισσότερα

### Coefficient Inequalities for a New Subclass of K-uniformly Convex Functions

International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for

Διαβάστε περισσότερα

### Tridiagonal matrices. Gérard MEURANT. October, 2008

Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,

Διαβάστε περισσότερα

### D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA

GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 1, 1996, 11-26 PARTIAL AVERAGING FOR IMPULSIVE DIFFERENTIAL EQUATIONS WITH SUPREMUM D. BAINOV, YU. DOMSHLAK, AND S. MILUSHEVA Abstract. Partial averaging for

Διαβάστε περισσότερα

### C.S. 430 Assignment 6, Sample Solutions

C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order

Διαβάστε περισσότερα

### A Two-Sided Laplace Inversion Algorithm with Computable Error Bounds and Its Applications in Financial Engineering

Electronic Companion A Two-Sie Laplace Inversion Algorithm with Computable Error Bouns an Its Applications in Financial Engineering Ning Cai, S. G. Kou, Zongjian Liu HKUST an Columbia University Appenix

Διαβάστε περισσότερα

### Finite Field Problems: Solutions

Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The

Διαβάστε περισσότερα

### MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS

MINIMAL CLOSED SETS AND MAXIMAL CLOSED SETS FUMIE NAKAOKA AND NOBUYUKI ODA Received 20 December 2005; Revised 28 May 2006; Accepted 6 August 2006 Some properties of minimal closed sets and maximal closed

Διαβάστε περισσότερα

### 4.6 Autoregressive Moving Average Model ARMA(1,1)

84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this

Διαβάστε περισσότερα

### n=2 In the present paper, we introduce and investigate the following two more generalized

MATEMATIQKI VESNIK 59 (007), 65 73 UDK 517.54 originalni nauqni rad research paper SOME SUBCLASSES OF CLOSE-TO-CONVEX AND QUASI-CONVEX FUNCTIONS Zhi-Gang Wang Abstract. In the present paper, the author

Διαβάστε περισσότερα

### Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.

Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action

Διαβάστε περισσότερα

### Areas and Lengths in Polar Coordinates

Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the

Διαβάστε περισσότερα

### EE512: Error Control Coding

EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3

Διαβάστε περισσότερα

### Second Order RLC Filters

ECEN 60 Circuits/Electronics Spring 007-0-07 P. Mathys Second Order RLC Filters RLC Lowpass Filter A passive RLC lowpass filter (LPF) circuit is shown in the following schematic. R L C v O (t) Using phasor

Διαβάστε περισσότερα

### CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS

CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =

Διαβάστε περισσότερα

### A General Note on δ-quasi Monotone and Increasing Sequence

International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in

Διαβάστε περισσότερα

### Homework 3 Solutions

Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For

Διαβάστε περισσότερα

### 1. Introduction and Preliminaries.

Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.yu/filomat Filomat 22:1 (2008), 97 106 ON δ SETS IN γ SPACES V. Renuka Devi and D. Sivaraj Abstract We

Διαβάστε περισσότερα

### Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.

Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given

Διαβάστε περισσότερα

### Eigenvalues and eigenfunctions of a non-local boundary value problem of Sturm Liouville differential equation

Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 5 (2016, pp. 3885 3893 Research India Publications http://www.ripublication.com/gjpam.htm Eigenvalues and eigenfunctions

Διαβάστε περισσότερα

### 6.3 Forecasting ARMA processes

122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear

Διαβάστε περισσότερα

### Second Order Partial Differential Equations

Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y

Διαβάστε περισσότερα

### Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013

Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering

Διαβάστε περισσότερα

### w o = R 1 p. (1) R = p =. = 1

Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:

Διαβάστε περισσότερα

### Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------

Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin

Διαβάστε περισσότερα

### Solution Series 9. i=1 x i and i=1 x i.

Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x

Διαβάστε περισσότερα

### Problem Set 3: Solutions

CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C

Διαβάστε περισσότερα

### ORDINAL ARITHMETIC JULIAN J. SCHLÖDER

ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.

Διαβάστε περισσότερα

### forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with

Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We

Διαβάστε περισσότερα

### ω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω

0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +

Διαβάστε περισσότερα

### Bounding Nonsplitting Enumeration Degrees

Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,

Διαβάστε περισσότερα

### Homework 8 Model Solution Section

MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx

Διαβάστε περισσότερα

### g-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King

Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i

Διαβάστε περισσότερα

### On a four-dimensional hyperbolic manifold with finite volume

BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In

Διαβάστε περισσότερα

### derivation of the Laplacian from rectangular to spherical coordinates

derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used

Διαβάστε περισσότερα

### Strain gauge and rosettes

Strain gauge and rosettes Introduction A strain gauge is a device which is used to measure strain (deformation) on an object subjected to forces. Strain can be measured using various types of devices classified

Διαβάστε περισσότερα

### Roman Witu la 1. Let ξ = exp(i2π/5). Then, the following formulas hold true [6]:

Novi Sad J. Math. Vol. 43 No. 1 013 9- δ-fibonacci NUMBERS PART II Roman Witu la 1 Abstract. This is a continuation of paper [6]. We study fundamental properties applications of the so called δ-fibonacci

Διαβάστε περισσότερα

### ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007

Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο

Διαβάστε περισσότερα

### Approximation of distance between locations on earth given by latitude and longitude

Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth

Διαβάστε περισσότερα

### Lecture 15 - Root System Axiomatics

Lecture 15 - Root System Axiomatics Nov 1, 01 In this lecture we examine root systems from an axiomatic point of view. 1 Reflections If v R n, then it determines a hyperplane, denoted P v, through the

Διαβάστε περισσότερα

### Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1

Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test

Διαβάστε περισσότερα

### k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +

Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b

Διαβάστε περισσότερα

### Trigonometric Formula Sheet

Trigonometric Formula Sheet Definition of the Trig Functions Right Triangle Definition Assume that: 0 < θ < or 0 < θ < 90 Unit Circle Definition Assume θ can be any angle. y x, y hypotenuse opposite θ

Διαβάστε περισσότερα

### On a Subclass of k-uniformly Convex Functions with Negative Coefficients

International Mathematical Forum, 1, 2006, no. 34, 1677-1689 On a Subclass of k-uniformly Convex Functions with Negative Coefficients T. N. SHANMUGAM Department of Mathematics Anna University, Chennai-600

Διαβάστε περισσότερα

### Nonlinear Fourier transform for the conductivity equation. Visibility and Invisibility in Impedance Tomography

Nonlinear Fourier transform for the conductivity equation Visibility and Invisibility in Impedance Tomography Kari Astala University of Helsinki CoE in Analysis and Dynamics Research What is the non linear

Διαβάστε περισσότερα

### DIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS

GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University

Διαβάστε περισσότερα

### TMA4115 Matematikk 3

TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet

Διαβάστε περισσότερα

### Bessel functions. ν + 1 ; 1 = 0 for k = 0, 1, 2,..., n 1. Γ( n + k + 1) = ( 1) n J n (z). Γ(n + k + 1) k!

Bessel functions The Bessel function J ν (z of the first kind of order ν is defined by J ν (z ( (z/ν ν Γ(ν + F ν + ; z 4 ( k k ( Γ(ν + k + k! For ν this is a solution of the Bessel differential equation

Διαβάστε περισσότερα

### Lecture 13 - Root Space Decomposition II

Lecture 13 - Root Space Decomposition II October 18, 2012 1 Review First let us recall the situation. Let g be a simple algebra, with maximal toral subalgebra h (which we are calling a CSA, or Cartan Subalgebra).

Διαβάστε περισσότερα

### GAUGES OF BAIRE CLASS ONE FUNCTIONS

GAUGES OF BAIRE CLASS ONE FUNCTIONS ZULIJANTO ATOK, WEE-KEE TANG, AND DONGSHENG ZHAO Abstract. Let K be a compact metric space and f : K R be a bounded Baire class one function. We proved that for any

Διαβάστε περισσότερα

### CRASH COURSE IN PRECALCULUS

CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter

Διαβάστε περισσότερα

### Homomorphism and Cartesian Product on Fuzzy Translation and Fuzzy Multiplication of PS-algebras

Annals of Pure and Applied athematics Vol. 8, No. 1, 2014, 93-104 ISSN: 2279-087X (P), 2279-0888(online) Published on 11 November 2014 www.researchmathsci.org Annals of Homomorphism and Cartesian Product

Διαβάστε περισσότερα

### ON THE STABILITY OF THE SOLUTIONS OF CERTAIN FIFTH ORDER NON-AUTONOMOUS DIFFERENTIAL EQUATIONS

ARCHIVUM MATHEMATICUM (BRNO) Tomus 41 (25), 93 16 ON THE STABILITY OF THE SOLUTIONS OF CERTAIN FIFTH ORDER NON-AUTONOMOUS DIFFERENTIAL EQUATIONS A. I. SADEK Our aim in this paper is to present sufficient

Διαβάστε περισσότερα

### Relative order and type of entire functions represented by Banach valued Dirichlet series in two variables

Int J Nonlinear Anal Appl 7 (2016) No 1, 1-14 ISSN: 2008-6822 (electronic) http://dxdoig/1022075/ijnaa2016288 Relative der type of entire functions represented by Banach valued Dirichlet series in two

Διαβάστε περισσότερα

### arxiv: v1 [math.ra] 19 Dec 2017

TWO-DIMENSIONAL LEFT RIGHT UNITAL ALGEBRAS OVER ALGEBRAICALLY CLOSED FIELDS AND R HAHMED UBEKBAEV IRAKHIMOV 3 arxiv:7673v [mathra] 9 Dec 7 Department of Math Faculty of Science UPM Selangor Malaysia &

Διαβάστε περισσότερα

### SPECIAL FUNCTIONS and POLYNOMIALS

SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195

Διαβάστε περισσότερα

### Appendix S1 1. ( z) α βc. dβ β δ β

Appendix S1 1 Proof of Lemma 1. Taking first and second partial derivatives of the expected profit function, as expressed in Eq. (7), with respect to l: Π Π ( z, λ, l) l θ + s ( s + h ) g ( t) dt λ Ω(

Διαβάστε περισσότερα

### 9.09. # 1. Area inside the oval limaçon r = cos θ. To graph, start with θ = 0 so r = 6. Compute dr

9.9 #. Area inside the oval limaçon r = + cos. To graph, start with = so r =. Compute d = sin. Interesting points are where d vanishes, or at =,,, etc. For these values of we compute r:,,, and the values

Διαβάστε περισσότερα

### CHAPTER 101 FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD

CHAPTER FOURIER SERIES FOR PERIODIC FUNCTIONS OF PERIOD EXERCISE 36 Page 66. Determine the Fourier series for the periodic function: f(x), when x +, when x which is periodic outside this rge of period.

Διαβάστε περισσότερα

### The kinetic and potential energies as T = 1 2. (m i η2 i k(η i+1 η i ) 2 ). (3) The Hooke s law F = Y ξ, (6) with a discrete analog

Lecture 12: Introduction to Analytical Mechanics of Continuous Systems Lagrangian Density for Continuous Systems The kinetic and potential energies as T = 1 2 i η2 i (1 and V = 1 2 i+1 η i 2, i (2 where

Διαβάστε περισσότερα

### ( ) 2 and compare to M.

Problems and Solutions for Section 4.2 4.9 through 4.33) 4.9 Calculate the square root of the matrix 3!0 M!0 8 Hint: Let M / 2 a!b ; calculate M / 2!b c ) 2 and compare to M. Solution: Given: 3!0 M!0 8

Διαβάστε περισσότερα

### Homework 3 Solutions

Homework 3 Solutions Differential Topology (math876 - Spring2006 Søren Kold Hansen Problem 1: Exercise 3.2 p. 246 in [MT]. Let {ɛ 1,..., ɛ n } be the basis of Alt 1 (R n dual to the standard basis {e 1,...,

Διαβάστε περισσότερα

### Testing for Indeterminacy: An Application to U.S. Monetary Policy. Technical Appendix

Testing for Indeterminacy: An Application to U.S. Monetary Policy Technical Appendix Thomas A. Lubik Department of Economics Johns Hopkins University Frank Schorfheide Department of Economics University

Διαβάστε περισσότερα

### Solution to Review Problems for Midterm III

Solution to Review Problems for Mierm III Mierm III: Friday, November 19 in class Topics:.8-.11, 4.1,4. 1. Find the derivative of the following functions and simplify your answers. (a) x(ln(4x)) +ln(5

Διαβάστε περισσότερα

### ( )( ) ( ) ( )( ) ( )( ) β = Chapter 5 Exercise Problems EX α So 49 β 199 EX EX EX5.4 EX5.5. (a)

hapter 5 xercise Problems X5. α β α 0.980 For α 0.980, β 49 0.980 0.995 For α 0.995, β 99 0.995 So 49 β 99 X5. O 00 O or n 3 O 40.5 β 0 X5.3 6.5 μ A 00 β ( 0)( 6.5 μa) 8 ma 5 ( 8)( 4 ) or.88 P on + 0.0065

Διαβάστε περισσότερα

### Forced Pendulum Numerical approach

Numerical approach UiO April 8, 2014 Physical problem and equation We have a pendulum of length l, with mass m. The pendulum is subject to gravitation as well as both a forcing and linear resistance force.

Διαβάστε περισσότερα

### AN APPLICATION OF THE SUBORDINATION CHAINS. Georgia Irina Oros. Abstract

AN APPLICATION OF THE SUBORDINATION CHAINS Georgia Irina Oros Abstract The notion of differential superordination was introduced in [4] by S.S. Miller and P.T. Mocanu as a dual concept of differential

Διαβάστε περισσότερα

### Section 9.2 Polar Equations and Graphs

180 Section 9. Polar Equations and Graphs In this section, we will be graphing polar equations on a polar grid. In the first few examples, we will write the polar equation in rectangular form to help identify

Διαβάστε περισσότερα

### Math 248 Homework 1. Edward Burkard. Exercise 1. Prove the following Fourier Transforms where a > 0 and c R: f (x) = b. f(x c) = e.

Math 48 Homework Ewar Burkar Exercise. Prove the following Fourier Transforms where a > an c : a. f(x) f(ξ) b. f(x c) e πicξ f(ξ) c. eπixc f(x) f(ξ c). f(ax) f(ξ) a e. f (x) πiξ f(ξ) f. xf(x) f(ξ) πi ξ

Διαβάστε περισσότερα

### Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type

Noname manuscript No. will be inserted by the editor Evaluation of some non-elementary integrals of sine, cosine and exponential integrals type Victor Nijimbere Received: date / Accepted: date Abstract

Διαβάστε περισσότερα

### If we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### MATRIX INVERSE EIGENVALUE PROBLEM

English NUMERICAL MATHEMATICS Vol.14, No.2 Series A Journal of Chinese Universities May 2005 A STABILITY ANALYSIS OF THE (k) JACOBI MATRIX INVERSE EIGENVALUE PROBLEM Hou Wenyuan ( ΛΠ) Jiang Erxiong( Ξ)

Διαβάστε περισσότερα

### L 2 -boundedness of Fourier integral operators with weighted symbols

Functional Analysis, Approximation and Computation 8 (2) (2016), 23 29 Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/faac L 2 -boundedness

Διαβάστε περισσότερα

### Strukturalna poprawność argumentu.

Strukturalna poprawność argumentu. Marcin Selinger Uniwersytet Wrocławski Katedra Logiki i Metodologii Nauk marcisel@uni.wroc.pl Table of contents: 1. Definition of argument and further notions. 2. Operations

Διαβάστε περισσότερα

### Lecture 2. Soundness and completeness of propositional logic

Lecture 2 Soundness and completeness of propositional logic February 9, 2004 1 Overview Review of natural deduction. Soundness and completeness. Semantics of propositional formulas. Soundness proof. Completeness

Διαβάστε περισσότερα

### On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University)

On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University) 1 1 Introduction (E) {1+x 2 +β(x,y)}y u x (x,y)+{x+b(x,y)}y2 u y (x,y) +u(x,y)=f(x,y)

Διαβάστε περισσότερα

### If we restrict the domain of y = sin x to [ π 2, π 2

Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the

Διαβάστε περισσότερα

### Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University

Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of

Διαβάστε περισσότερα

### Nonlinear Fourier transform and the Beltrami equation. Visibility and Invisibility in Impedance Tomography

Nonlinear Fourier transform and the Beltrami equation Visibility and Invisibility in Impedance Tomography Kari Astala University of Helsinki Beltrami equation: z f(z) = µ(z) z f(z) non linear Fourier transform

Διαβάστε περισσότερα

### Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1

Main source: "Discrete-time systems and computer control" by Α. ΣΚΟΔΡΑΣ ΨΗΦΙΑΚΟΣ ΕΛΕΓΧΟΣ ΔΙΑΛΕΞΗ 4 ΔΙΑΦΑΝΕΙΑ 1 A Brief History of Sampling Research 1915 - Edmund Taylor Whittaker (1873-1956) devised a

Διαβάστε περισσότερα

### Appendix to Technology Transfer and Spillovers in International Joint Ventures

Appendix to Technology Transfer and Spillovers in International Joint Ventures Thomas Müller Monika Schnitzer Published in Journal of International Economics, 2006, Volume 68, 456-468 Bundestanstalt für

Διαβάστε περισσότερα

### Two generalisations of the binomial theorem

39 Two generalisations of the binomial theorem Sacha C. Blumen Abstract We prove two generalisations of the binomial theorem that are also generalisations of the q-binomial theorem. These generalisations

Διαβάστε περισσότερα

### Trigonometry 1.TRIGONOMETRIC RATIOS

Trigonometry.TRIGONOMETRIC RATIOS. If a ray OP makes an angle with the positive direction of X-axis then y x i) Sin ii) cos r r iii) tan x y (x 0) iv) cot y x (y 0) y P v) sec x r (x 0) vi) cosec y r (y

Διαβάστε περισσότερα

### Intuitionistic Supra Gradation of Openness

Applied Mathematics & Information Sciences 2(3) (2008), 291-307 An International Journal c 2008 Dixie W Publishing Corporation, U. S. A. Intuitionistic Supra Gradation of Openness A. M. Zahran 1, S. E.

Διαβάστε περισσότερα

### UNIT - I LINEAR ALGEBRA. , such that αν V satisfying following condition

UNIT - I LINEAR ALGEBRA Definition Vector Space : A non-empty set V is said to be vector space over the field F. If V is an abelian group under addition and if for every α, β F, ν, ν 2 V, such that αν

Διαβάστε περισσότερα

### STAT200C: Hypothesis Testing

STAT200C: Hypothesis Testing Zhaoxia Yu Spring 2017 Some Definitions A hypothesis is a statement about a population parameter. The two complementary hypotheses in a hypothesis testing are the null hypothesis

Διαβάστε περισσότερα

### The Probabilistic Method - Probabilistic Techniques. Lecture 7: The Janson Inequality

The Probabilistic Method - Probabilistic Techniques Lecture 7: The Janson Inequality Sotiris Nikoletseas Associate Professor Computer Engineering and Informatics Department 2014-2015 Sotiris Nikoletseas,

Διαβάστε περισσότερα

### 2. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν.

Experiental Copetition: 14 July 011 Proble Page 1 of. Μηχανικό Μαύρο Κουτί: κύλινδρος με μια μπάλα μέσα σε αυτόν. Ένα μικρό σωματίδιο μάζας (μπάλα) βρίσκεται σε σταθερή απόσταση z από το πάνω μέρος ενός

Διαβάστε περισσότερα

### Jordan Form of a Square Matrix

Jordan Form of a Square Matrix Josh Engwer Texas Tech University josh.engwer@ttu.edu June 3 KEY CONCEPTS & DEFINITIONS: R Set of all real numbers C Set of all complex numbers = {a + bi : a b R and i =

Διαβάστε περισσότερα

### Integrals in cylindrical, spherical coordinates (Sect. 15.7)

Integrals in clindrical, spherical coordinates (Sect. 5.7 Integration in spherical coordinates. Review: Clindrical coordinates. Spherical coordinates in space. Triple integral in spherical coordinates.

Διαβάστε περισσότερα

### Spherical Coordinates

Spherical Coordinates MATH 311, Calculus III J. Robert Buchanan Department of Mathematics Fall 2011 Spherical Coordinates Another means of locating points in three-dimensional space is known as the spherical

Διαβάστε περισσότερα

### Multistring Solutions of the Self-Graviting Massive W Boson

Multistring Solutions of the Self-Graviting Massive W Boson Dongho Chae Department of Mathematics Sungkyunkwan University Suwon 44-746, Korea e-mail: chae@skku.edu Abstract We consider a semilinear elliptic

Διαβάστε περισσότερα

### arxiv: v1 [stat.me] 13 Oct 2015

A note on the best attainable rates of convergence for estimates of the shape parameter of regular variation Meitner Cadena arxiv:151.3617v1 [stat.me] 13 Oct 15 October 14, 15 Abstract Hall Welsh gave

Διαβάστε περισσότερα

### The k-fractional Hilfer Derivative

Int. Journal of Math. Analysis, Vol. 7, 213, no. 11, 543-55 The -Fractional Hilfer Derivative Gustavo Abel Dorrego and Rubén A. Cerutti Faculty of Exact Sciences National University of Nordeste. Av. Libertad

Διαβάστε περισσότερα

### Example of the Baum-Welch Algorithm

Example of the Baum-Welch Algorithm Larry Moss Q520, Spring 2008 1 Our corpus c We start with a very simple corpus. We take the set Y of unanalyzed words to be {ABBA, BAB}, and c to be given by c(abba)

Διαβάστε περισσότερα