the sum of two cubes rule

Ngày tải lên: 07/08/2014, 08:20

... On the q-analogue of the sum of cubes S. Ole Warnaar ∗ Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australia warnaar@ms.unimelb.edu.au Submitted: ... 2004 2000 Mathematics Subject Classification: 05A19 Abstract Asimpleq-analogue of the sum of cubes is given. This answers a question posed in this journal by Garrett and Hummel. The sum of cubes and ... coefficient. In their paper Garrett and Hummel commiserate the fact that (1) is not as simple as one might have hoped, and ask for a simpler sum of q -cubes. In response to this I propose the identity n  k=1 q 2n−2k (1...

Ngày tải lên: 07/08/2014, 08:20

... yields the claim in (2.80). This completes the proof of Lemma 6. This completes the proof of Proposition 4 and hence of Theorem 1. 3. Proof of Theorem 2 In this section we indicate how the arguments ... second term on the right-hand side of the analogue of (2.93). 4. Proof of Theorem 3 In Sections 4–6 we prove Theorems 3–5. The proof follows the same line of reasoning as in [3, §5], but there are ... and generalization of the proof of Proposition 3 in [3]. We outline the main steps, while skipping the details. Step 1. One of the basic ingredients in the proof in [3] is to approximate the volume of the Wiener...

Ngày tải lên: 14/02/2014, 17:20

... numbers over the diagrams indicate the position of the corresponding nucleotides relative to the translation start site, and the numbers in parentheses below the diagram show the position of the corresponding ... of pGL3–basic vector and was half that of pGL3–promoter vector. These findings demonstrate the promoter activity of the 5¢-flanking region of exon 1 in DNASE1. Deletion of the upstream end of the ... generation of two kinds of gene transcript. However, only the sequences of the 5¢-UTR are differ- ent between them, because these transcripts share all the other exons, exons 2–9, which contain the...

Ngày tải lên: 19/02/2014, 06:20

... higher than the beasts – and unworthy of the designation of honor, and, the title of man. From the founding of the working class. Those on the lowest rung of the socio-economic ladder in the Nort ... and the reason for fighting was the protection of the tribe and the defense of kin. The Civil War, on the other hand, was a modern battle that bore little resemblance to the primal skirmishes of ... philosophical posts are the most important. You should do more of them.” The heart of the debate both for the 19 th century man and for today’s, is whether manhood, and thus the honor code of men, should...

Ngày tải lên: 14/03/2014, 09:31

... anaiyze the applicability of Neuber's rule in the local strain approach. The layout of the thesis is as follows. Chapter 2 contains a review of research regarding the prediction of ... Estimation The cornparison of the fatigue damage at the notch of a component and in the smooth specimen used to generate the test data requires an estimation of the stress and strain at the ... of the problems identified with the nominal stress approach, narnely its empirical nature, and the disregard for the actual stress-strain response of the matenal. The basis for the...

Ngày tải lên: 24/05/2014, 20:30

... our further considerations. Let X be a Banach space, Ω(X) is the collection of all nonempty bounded subsets of X,and W ( X ) is the subset of Ω(X) consisting of al l weak compact subsets of X.LetB r denote ... Since then, there have appeared a large number of articles contributing gener ali- zations or modifications of the Krasnoselskii fixed point theorem and their applications (see [2]-[21]). The study ... fixed point theorem of Krasnoselskii for a sum of contraction and compact mappings in Banach spaces. The cla ss of asymptotically nonexpansive m appi ngs includes properly the class of nonexpansive...

Ngày tải lên: 21/06/2014, 01:20

... solution of 2.11 if and only if u satisfies the operator equation ∇F  u   0. 2.25 3. The Main Theorems Now, we state and prove the following theorem concerning the solution of problem 2.11. Theorem ... 0,π. v 0 t is just a unique solution of 2.12 and u 0 tv 0 tωt is exactly a unique solution of 2.11. The proof of Theorem 3.1 is completed. Now, we assume that there exists a positive integer ... λ 1 ≤ μ 2  λ 2 ≤ ··· ≤ μ n  λ n are the eigenvalues of A  B. The proof of Lemma 2.2 is fulfilled. Let ·, · denote the usual inner product on R n and denote the corresponding norm by |u|  {  n i1 u 2 i } 1/2 ,...

Ngày tải lên: 21/06/2014, 20:20

... new inequalities for the trace of the product of two arbitrary real square matrices. These bounds improve and extend the recent results. Further, we give their application in the algebraic Riccati ... 1979. Journal of Inequalities and Applications 3 Therefore, considering the application of the trace bounds, many scholars pay much attention to estimate the trace bounds for the product of two matrices. Marshall ... particularly as the dimensions of the system matrices increase. Thus, a number of works have been presented by researchers to evaluate the bounds and trace bounds for the solution of the ARE6–12. In...

Ngày tải lên: 22/06/2014, 02:20

... pattern on the weight lattice of the affine Weyl grou p E (1) 8 or one of its degenerations. This proliferation of discrete Painlev ´ e equations raises the question of the indepen- dence of the various ... explicitly derived the equation at hand from the elementary Miura transformations of D (1) 5 . Further, we have clarified these relations from the point of view of Weyl group theory and of rational surfaces. References [1] ... present explicit relations of these discrete Painlev ´ e equations to the extended Weyl group of type D (1) 5 , and discuss their space of initial conditions in the spirit of the Okamoto-Sakai approach...

Ngày tải lên: 22/06/2014, 22:20

... promoter (the promoter located in front of the AtAOS2 gene), and the AtAOS2 gene were clearly detected in the transgenic rice genomes. These results strongly confirmed the appearance of the transgene ... genome. The results of PCRs would confirm the appearance of the AtAOS2 gene in the transgenic rice genome. 2. MATERIALS AND METHODS 2.1. Extraction of rice DNA The leaves of one-month-old wild-type ... of PCRs was carried out to detect the hygromycin resistance gene, CaMV35S promoter (the promoter located in front of the AtAOS2 gene), and the AtAOS2 gene in the transgenic rice genome. The...

Ngày tải lên: 06/08/2014, 18:21

... 2r)  p r  . But then one can verify that the coefficient of the first power of k in the polynomial on the left of (3) is (−1) p−1 (p − r)/(p − 2r), while the corresponding coefficient on the right is ... completing the proofofTheorem1. To prove Theorem 2, we see that if r =0thenS p,r,s (n)=f p,s (n), and if s = p then S p,r,s (n)=2 pn − f p,r (n)+  pn rn  , so in these two cases the assertion ... immediately from Theorem 1. If r =0ands=pthen write S p,r,s (n)=f p,s (n) − f p,r (n)+  pn rn  . As in the proof of Theorem 1, f p,s (n) − f p,r (n) can be written as the indefinite sum of two hypergeometric...

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