the analytic continuation theorem and the generalized liouville theorem

... Difference Equations r r In the special case x 0, Gn,χ,q Gn,χ,q are called the nth generalized q-Genocchi numbers of order r attached to χ Therefore, we obtain the following theorem Theorem 2.1 For r ∈ ... relation on the analogue of the Apostol-Bernoulli and the analogue of the Apostol-Genocchi polynomials,” Applied Mathematical Sciences, vol 3, no 53–56, pp 2757–2764, 2009 T Kim, “On the q-extension ... polynomials and q-Genocchi polynomials, which are treated in this paper In this paper, we first consider the q-extension of the generating function for the higher-order generalized Genocchi numbers and...

Ngày tải lên: 21/06/2014, 05:20

... r In the special case, x 0, the sequence βn,χ,q numbers of order r attaches to χ βn,χ,q are called the n th generalized q-Bernoulli By 2.13 and 2.14 , one obtains the following theorem Theorem ... , and 2.26 , one obtains the following theorem Theorem 2.5 For h ∈ Z, r ∈ N, and n ∈ Z , one has h,r βn,q n n l l h−1,r 1,q q − βn h−1,r βn,q n−1 q−1 l 0,r βl,q r n−1 r q , r! r q! 2.27 Furthermore, ... r−1 r lw1 δ1 r−1 r q r! r q! 2.61 Therefore, one obtains the following theorem 16 Journal of Inequalities and Applications Theorem 2.11 For w1 ∈ Zp , r ∈ N, and δ1 ∈ Z, one has ⎛ ⎞ ⎜ r ⎜ βn,q...

Ngày tải lên: 21/06/2014, 07:20

... Cauchy residue theorem in 3.1 and 3.3 , we obtain the following theorem Theorem 3.2 Let χ be Dirichlet’s character with odd conductor d ∈ N, and let ζ h, s ∈ C, x / 0, −1, −2, , r ∈ N, and n ∈ Z ... systemic study of some formulas of the twisted q-extension of the generalized Euler numbers and polynomials of order k attached to χ On the Twisted q-Extension of the Generalized Euler Polynomials ... k−1 × −ζ m m ex a1 ··· ak md q t 2.5 e n1 χ ni n1 , ,nk dm ··· nk x q t i Therefore, we obtain the following theorem Theorem 2.1 For k ∈ N, n ≥ 0, one has k ∞ 2k En,χ,ζ,q x −ζ k n1 ··· nk n1...

Ngày tải lên: 21/06/2014, 16:20

... shown In order to obtain the existence of periodic solutions, many authors use Horn fixed point theorem or Banach fixed point theorem However, the conditions for Horn fixed point theorem are not easy ... sometimes and the conditions for Banach fixed point theorem are too strong Our method is much different from others’, and we give a new way to show the existence of periodic solutions In addition, the ... P and give the e relation between T0 -periodic P C-mild solution and the fixed point of P After showing the compactness of the Poincar´ operator P and obtaining the boundedness of the fixed point...

Ngày tải lên: 22/06/2014, 03:20

... “Direct theorems for generalized Favard-Kantorovich and Favard-Durrmeyer operators in exponential function spaces,” to appear in Ukrainian Mathematical Journal [10] W Kratz and U Stadtm¨ ller, “On the ... simply from properties of finite differences since the left-hand side of the equation is a half of the finite difference of the polynomial (r1 − x)2s Therefore, 2r1 Sn,1 f (x) = l =r (−1)l 22 j+1−l ... denotes the Favard-Kantorovich operator or the Favard-Durrmeyer operator Throughout the paper, the symbols K(σ,σ , ), K j (σ,σ , ) ( j = 1,2, ) will mean some positive constants, not necessarily the...

Ngày tải lên: 22/06/2014, 06:20

... So the generalized convolution (11) is γ γ γ γ not associative and satisfies the equality f ∗ (g ∗ h) = g ∗ (f ∗ h) The proof for b) is similar to that of a) The theorem is proved Theorem In the ... (22) The equalities (21) and (22) yield (20) The proof is complete Theorem In the space of functions belonging to L(R+ ) the generalized convolution (11) is not associative and satisfies the following ... R γ Therefore (e1 ∗g)(x) = g(x), to L(R+ ) The theorem is proved Set L(ex , R+ ) = f : +∞ ∀x > Thus, e1 is the left unit element belonging ex |f (x)|dx < +∞ γ Theorem (Titchmarch type - Theorem) ...

Ngày tải lên: 06/08/2014, 05:20

... isomorphisms, it is then appropriate to state the isomorphism theorems Theorem 2.2.15 (First Isomorphism Theorem) Let G and H be groups and let ϕ : G −→ H be a homomorphism Then G/ker(ϕ) ∼ Im(ϕ) = Theorem ... on A and β on B In the case H is cyclic, we call A ∗ B the generalized free product of A H and B with cyclic amalgam or simply the cyclically pinched generalized free product of A and B In the ... implies ⇐⇒ if and only if Cn the cyclic group of order n C∞ the infinite cyclic group Q+ the additive group of the rational numbers |G| the order of the group G τ (A) the torsion subgroup of the group...

Ngày tải lên: 13/11/2014, 09:08

... items in the HADS Anxiety subscale, and the other containing the items in the HADS Depression subscale The comparison of the GAD-7 scores of the four severity groups of anxiety diagnosed by the HADS-A ... into quartiles based on the GAD-7 total score and comparing the upper to the lower quartile, both for the mean scores in individual items and for the overall score For other clinical criteria, ... three other instruments: the Hamilton Anxiety Scale (HAM-A) [14], the Hospital Anxiety and Depression Scale (HADS) [15], and the WHO-DAS II disability questionnaire [27,28], all of them in their...

Ngày tải lên: 18/06/2014, 19:20

... BVP (1.1) The proof of Theorem 3.1 is complete Theorem 3.2 Suppose that conditions (H1 ), (H2 ) and (A1 ) in Theorem 3.1 hold Assume that f also satisfy (A3 ): f0 = 0; (A4 ): f∞ = Then, the boundary ... The Sturm Liouville BVP in Banach space Hua Su∗1 , Lishan Liu2 and Xinjun Wang3 School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan Shandong ... u Therefore, we can have Tu ≤ u , ∀ u ∈ ∂Ωρ∗ Then by Theorem 2.2, we have i(T, Ωρ∗ , K) = (3.6) Next, by condition (A4 ), (2.4) and the property of limits, we can have lim u →ı f (Su) u = Then,...

Ngày tải lên: 18/06/2014, 22:20

... BVP (1.1) The proof of Theorem 3.1 is complete Theorem 3.2 Suppose that conditions (H1 ), (H2 ) and (A1 ) in Theorem 3.1 hold Assume that f also satisfy (A3 ): f0 = 0; (A4 ): f∞ = Then, the boundary ... The Sturm Liouville BVP in Banach space Hua Su∗1 , Lishan Liu2 and Xinjun Wang3 School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics, Jinan Shandong ... u Therefore, we can have Tu ≤ u , ∀ u ∈ ∂Ωρ∗ Then by Theorem 2.2, we have i(T, Ωρ∗ , K) = (3.6) Next, by condition (A4 ), (2.4) and the property of limits, we can have lim u →ı f (Su) u = Then,...

Ngày tải lên: 20/06/2014, 04:20

... one, the general 2.2 The analysis if the recurrent potential function of the type zυ The HC, besides the gamma function, also has the potential component rk+n-3 The generalized equation of the ... sense of the analytical expanding of the HCs (Figure 3) and in the left half-plane (k ≤ 0), it relies on the characteristics of the gamma (Γ) and digamma (ψ0) functions, which make the HC The derivation ... Figure The HC and its derivative with the HC maximum location and optimum for the value of the degree of freedom n0 ≈ 0,46163 for n Î N, for the degree of freedom k = and the common half-edge 2.8 The...

Ngày tải lên: 20/06/2014, 21:20

... of the deviation between the arbitrary random fields and the Markov chain fields on the generalized Bethe tree Main Results Theorem 2.1 Let X {Xt , t ∈ T} be an arbitrary random field on the generalized ... In the recent years, with the development of the information theory scholars get to study the Shannon-McMillan theorems for the random field on the tree graph see The tree models have recently ... the generalized Bethe tree by comparison between the arbitrary measure and Markov measure on the generalized Bethe tree As corollaries, a class of Shannon-McMillan theorems for arbitrary random...

Ngày tải lên: 21/06/2014, 05:20

... , then b ∈ BMO Rn Remark 1.5 It is easy to check that b, μS f x ≤ 2λn b, μ∗, f x see, e.g., the proof λ of 19 in 15, page 89 , we therefore give only the proofs of Theorem 1.2 for b, μΩ and Theorem ... μS , and μ∗, on the generalized Morrey space λ Lp,ϕ Rn and bB Definition 1.1 Let < p < ∞ Suppose that ϕ : 0, ∞ → 0, ∞ be such that ϕ t is nonincreasing and t1/p ϕ t is nondecreasing The generalized ... studied the commutator formed by Riesz potential and BMO and gave another characterization of BMO space The purpose of this paper is to give a characterization of BMO space by the boundedness of the...

Ngày tải lên: 21/06/2014, 07:20

... χ, w2 d − Bn−j,χ w1 x ⎤ j k 2.18 By comparing the coefficients on the both sides of 2.17 and 2.18 , we see the following theorem Theorem 2.1 For d, w1 , w2 ∈ N, n ≥ 0, m ≥ 1, one has n j n j n−j−1 ... called the generalized Bernoulli numbers attached to χ of order k When k 1, the polynomials of numbers are called the generalized Bernoulli polynomials or numbers attached to χ Let w1 , w2 ∈ N Then ... Acknowledgment The present research has been conducted by the research Grant of the Kwangwoon University in 2009 Journal of Inequalities and Applications References M Cenkci, The p-adic generalized...

Ngày tải lên: 22/06/2014, 02:20

... w1 i−1 l−i w2 w1 tl l! Comparing the coefficients on the both sides of 2.18 and 2.19 , we have the following theorem Theorem 2.3 Let ξ ∈ Tp and d, w1 , w2 ∈ N Then one has l l i i i−1 l−i Bi,χ,ξw1 ... i Bk,χ,ξw2 w1 x tk k! Comparing the coefficients on the both sides of 2.22 and 2.23 , we obtain the following theorem Theorem 2.5 Let ξ ∈ Tp and d, w1 , w2 ∈ N Then one has dw1 −1 χ i ξw2 i Bk,χ,ξw1 ... polynomials and studied the symmetry of power sum and twisted Bernoulli polynomials see 11–13 The purpose of this paper is to study the symmetry for the generalized twisted Bernoulli polynomials and...

Ngày tải lên: 22/06/2014, 03:20

... particular case of Theorem 4.1 see also 39 We also show that the generalized stability of 3.1 can be obtained as consequence of the following theorem 12 Fixed Point Theory and Applications Theorem 5.1 ... 5.11 i If ψ x δ > in the above Theorem 5.1, then we will obtain the Ulam-Hyers stability result of Baker 31, Theorem see also 37, Theorem for the linear equation 5.2 L C˘ dariu and V Radu a 13 Corollary ... 5.8 with g, η, h as in hypothesis of Theorem 5.1 The relation 4.2 holds with equality Applying Theorem 4.1, there exists a unique mapping c which satisfies 5.2 and the estimation 5.7 Moreover,...

Ngày tải lên: 22/06/2014, 06:20

... contradiction and completes the proof Proposition 2.9 Assume that λ = and f : X → K is an unbounded function hemicontinuous at the origin Then f satisfies (2.1) if and only if f (X) ⊂ R or k = 1, and (i) ... hemicontinuous at the origin function satisfying (2.1) Then, applying Theorem 2.5 and Remark 2.2, we get f x + f (x)k y = − f (x) f (y) for x, y ∈ X (2.45) and f (0) = −1, respectively Therefore, if ... hemicontinuous at the origin, this means that the origin is an algebraically interior point of the set X \ f −1 ({0}) (see [4, Definition 1]) Thus, applying [4, Theorem 3], we get the assertion Proposition...

Ngày tải lên: 22/06/2014, 22:20

... vector x and Wi j is the value of the interconnection weight between the ith neuron in the RBF layer and the jth neuron in the output layer x − ci represents the Euclidean distance and ϕi is the stimulation ... of the proposed method 3.1 The generalized RBF neural network The relation between the output and the input is given in N In this paper, the generalized neural network is used for modeling the ... group are more than other centers are attributed to that cluster and their standard deviations are considered as the Gaussian kernel standard deviation The result of executing the generalized RBF...

Ngày tải lên: 22/06/2014, 23:20

... Thus if |D| = 2, then D ∈ D2 and D = D2 If |D| = p + 1, then D consists of blocks of size p + and D is the union of two orbits of StabSL(2,p) (v), and these orbits have size and p We conclude ... and O2 = {w}, and two of size p, say O3 and O4 By [11, Theorem 5.5 (ii)] the sub-orbits of SL(2, p)(2) relative to v are the same as the sub-orbits of SL(2, p) relative to v Thus the neighbors ... 2-groups, contradicting [4, Theorem 8.3] Theorem 2.2 Let p ≥ be an odd prime such that | (p − 1) Then there exist exactly two digraphs Γi , i = 1, of order 2(p + 1) such that the following properties...

Ngày tải lên: 07/08/2014, 07:21

... · · , n} to the right of the d new N steps, and place cars {n − d, · · · , 2, 1} to the right of the other n − d D steps in the uniquely right order so that the m-reading word of the m-parking ... corollary of the q-binomial theorem Lemma 3.3 [7] The q-Vandermonde Convolution h j=0 q (n−j)(h−j) n j m m+n = h−j h the electronic journal of combinatorics 12 (2005), #R53 Proof Proof of Theorem ... following q-identity Theorem 3.2 k≥d k n d [n] k mn (k−d)(k−1) mn + n − d = q k−1 [mn − d + 1] d, n − d, mn − d Before we proceed to the proof of Theorem 3.2, we cite the q-Vandermonde Convolution,...

Ngày tải lên: 07/08/2014, 13:21