factoring the sum of two cubes calculator

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

... For this reason it may be of interest to investigate other q-analogues of the sum of cubes formula In addition, as there are classical results for sums of other powers of integers, it would be ... that the set S consists of pairs of partitions with no primed parts In the definition of φ−1 when there is a primed integer in the image, we consider only the value of the integer and ignore the ... proof described to him by N.M Ferrers The Ferrers shape of a partition is an array of boxes, left justified, in which the number of boxes in the first row is equal to the size of the first part, the...

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

Danh mục: Báo cáo khoa học

... q)(1 − q )(1 − q ) and the q-analogue of n k= k=1 is n k=1 q 2n−2k n+1 n+1 (1 − q k ) = (1 − q) References [1] K C Garrett and K Hummel, A combinatorial proof of the sum of q -cubes, Electron J Combin ... on n The form of (2) should not really come as a surprise in view of the fact that the q-analogue of the sum of squares n k = n(n + 1)(2n + 1) k=1 is given by n q 2n−2k k=1 (1 − q k )(1 − q 3k ... Proof Since n+1 2 − q2 n 2 = (1 − q n )2 (1 − q 2n ) (1 − q)2 (1 − q ) equation (2) immediately follows by induction on n The form of (2) should not really come as a surprise in view of the...

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

Danh mục: Thạc sĩ - Cao học

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

Danh mục: Báo cáo khoa học

... fragments F and I The number at the right of each RT-PCR product represents the length of the product The scheme also shows the location of the reverse primer DN+89 used in another starting exon-speciﬁc ... 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 ... is done from +1 of exon )11871 above the sequence indicates the 5¢-end of exon 1a The numbers in parentheses demonstrate the position of the corresponding nucleotides relative to the transcription...

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 From the founding of the country, American leaders had argued that the strength ... and hard work The watchword of Northern honor, as it was for the English, was self-restraint This was the virtue that tied the others together; the man who had mastered himself had the discipline ... court-martialed for the use of profanity during the war, in violation of the 83rd Article of War requiring conduct appropriate to one’s status as an “officer and a gentleman.”) On the other hand, you...

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 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 all weak compact subsets of X Let ... generalized forms of the Krasnoselskii theorem on fixed points for the sum A + B of a weakly-strongly continuous mapping and an asymptotically nonexpansive mapping in Banach spaces These results ... contributions The work presented here was carried out in collaboration between all authors SP and AA defined the research theme SP designed theorems and methods of proof and interpreted the results...

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 2.25 The Main Theorems Now, we state and prove the following theorem concerning the solution of problem 2.11 Theorem 3.1 Assume ... v0 t ω t is exactly a unique solution of v0 t is just a unique solution of 2.12 and u0 t 2.11 The proof of Theorem 3.1 is completed Now, we assume that there exists a positive integer N such ... diagonalization matrix and μ1 λ1 ≤ μ2 B The proof of Lemma 2.2 is fulﬁlled λ2 ≤ · · · ≤ μn λn are the eigenvalues of Let ·, · denote the usual inner product on Rn and denote the corresponding norm 1/2 by...

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

... Journal of Inequalities and Applications 11 Applications of the Results Wang et al in have obtained the following: let K be the positive semideﬁnite solution of the ARE 1.4 Then the trace of matrix ... including 3.1 Some of our results and 3.1 cannot contain each other Theorem 3.1 If 1/p and K is the positive semideﬁnite solution of the ARE 1.4 , then 1/q the trace of matrix K has the lower and ... Journal of Inequalities and Applications Therefore, considering the application of the trace bounds, many scholars pay much attention to estimate the trace bounds for the product of two matrices...

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 affine Weyl group E(1) or one of its degenerations This proliferation of discrete Painlev´ equations raises the question of the indepene dence of the various ... that the geometry of the evolution of this extended equation, together with its Schlesinger transformations, can be described by the affine Weyl group D(1) By using the freedom of the origin of ... have pointed out in [11], the geometry of the transformations of this equation is related to the affine Weyl group D(1) , just as in the case of (2.1) On the identity of two q-discrete Painlev´ equations...

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

... set 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 ... rice genome The results of PCRs would confirm the appearance of the AtAOS2 gene in the transgenic rice genome MATERIALS AND METHODS 2.1 Extraction of rice DNA The leaves of one-month-old wild-type ... thaliana AOS2 gene in the rice genome CONCLUSION From the above results, the hygromycin resistance gene, CaMV35S promoter (the promoter located in front of the AtAOS2 gene), and the AtAOS2 gene were...

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

... r the argument would work, but without further human input it could not produce a general proof, i.e., a proof for all p, r This is somewhat analogous to the sums of the pth powers of all of the ... 2r, completing the proof of Theorem To prove Theorem 2, we see that if r = then Sp,r,s (n) = fp,s (n), and if s = p then Sp,r,s (n) = 2pn − fp,r (n) + pn , so in these two cases the assertion ... immediately rn from Theorem If r = and s = p then write Sp,r,s (n) = fp,s (n) − fp,r (n) + pn rn As in the proof of Theorem 1, fp,s (n) − fp,r (n) can be written as the indeﬁnite sum of two hypergeometric...