the sum of two cubes rule

Báo cáo toán học: "A Combinatorial Proof of the Sum of q-Cubes" pptx

Báo cáo khoa học:On the q-analogue of the sum of cubes ppsx

... On the q-analogue of the sum of cubes S. Ole Warnaar∗Department of Mathematics and Statistics, The University of Melbourne, VIC 3010, Australiawarnaar@ms.unimelb.edu.auSubmitted: ... 20042000 Mathematics Subject Classiﬁcation: 05A19AbstractAsimpleq-analogue of the sum of cubes is given. This answers a question posedin 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 asone might have hoped, and ask for a simpler sum of q -cubes. In response to this I propose the identitynk=1q2n−2k(1...

Tài liệu Đề tài " On the volume of the intersection of two Wiener sausages " ppt

Danh mục: Thạc sĩ - Cao học

... 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 2In this section we indicate how the arguments ... secondterm on the right-hand side of the analogue of (2.93).4. Proof of Theorem 3In Sections 4–6 we prove Theorems 3–5. The proof follows the same line of reasoning as in [3, §5], but there are ... andgeneralization 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...

Tài liệu Báo cáo khoa học: Characterization of human deoxyribonuclease I gene (DNASE1) promoters reveals the utilization of two transcription-starting exons and the involvement of Sp1 in its transcriptional regulation ppt

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... numbers over the diagrams indicate the position of the corresponding nucleotides relative to the translation start site, and the numbersin parentheses below the diagram show the position of the corresponding ... of pGL3–basic vector andwas half that of pGL3–promoter vector. These ﬁndingsdemonstrate the promoter activity of the 5¢-ﬂankingregion 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...

The Gentlemen and the Roughs: The Collision of Two Honor Codes in the American North

Danh mục: Tâm lý - Nghệ thuật sống

... 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 amodern 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 19th century man and for today’s, is whether manhood, andthus the honor code of men, should...

A study on the use of neuber's rule in fatigue crack

Danh mục: Kiến trúc - Xây dựng

... 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...

Báo cáo hóa học: " On the Krasnoselskii-type fixed point theorems for the sum of continuous and asymptotically nonexpansive mappings in Banach spaces" pptx

Danh mục: Hóa học - Dầu khí

... our furtherconsiderations. Let X be a Banach space, Ω(X) is the collection of all nonemptybounded subsets of X,andW(X)is the subset of Ω(X) consisting of al l weak compactsubsets of X.LetBrdenote ... 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 asymptoticallynonexpansive m appi ngs includes properly the class of nonexpansive...

Báo cáo hóa học: "Research Article The Solution of Two-Point Boundary Value Problem of a Class of Dufﬁng-Type Systems with Non-C1 Perturbation Term" pdf

Danh mục: Hóa học - Dầu khí

... solution of 2.11 if and only if u satisﬁes the operator equation∇Fu 0. 2.253. The Main TheoremsNow, we state and prove the following theorem concerning the solution of problem 2.11.Theorem ... 0,π.v0t is just a unique solution of 2.12 and u0tv0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 ... λ1≤ μ2 λ2≤ ··· ≤ μn λnare the eigenvalues of A  B. The proof of Lemma 2.2 is fulﬁlled.Let ·, · denote the usual inner product on Rnand denote the corresponding normby |u|  {ni1u2i}1/2,...

Báo cáo hóa học: " Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation" potx

Danh mục: Hóa học - Dầu khí

... new inequalitiesfor 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 3Therefore, considering the application of the trace bounds, many scholars pay muchattention 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 presentedby researchers to evaluate the bounds and trace bounds for the solution of the ARE6–12.In...

Báo cáo hóa học: "ON THE IDENTITY OF TWO q-DISCRETE PAINLEVÉ EQUATIONS AND THEIR GEOMETRICAL DERIVATION" docx

Danh mục: Báo cáo khoa học

... pattern on the weight lattice of the affineWeyl grou p E(1)8or one of its degenerations.This proliferation of discrete Painlevé 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 Painlevé 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...

Scientific report: "Application of PCR to detect the presence of two genes in the genome of rice AtAOS has been genetically modified" pdf

Danh mục: Báo cáo khoa học

... 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...

Báo cáo toán học: "When can the sum of (1/p)th of the binomial coeﬃcients have closed form" pot

Danh mục: Báo cáo khoa học

... 2r)pr.But then one can verify that the coefficient of the first power of k in the polynomialon 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 =0thenSp,r,s(n)=fp,s(n), and if s = p thenSp,r,s(n)=2pn− fp,r(n)+pnrn, so in these two cases the assertion ... immediatelyfrom Theorem 1.If r =0ands=pthen writeSp,r,s(n)=fp,s(n) − fp,r(n)+pnrn.As in the proof of Theorem 1, fp,s(n) − fp,r(n) can be written as the indefinite sum of two hypergeometric...