factor the sum of two cubes example
Báo cáo toán học: "A Combinatorial Proof of the Sum of q-Cubes" pptx
Ngày tải lên :
07/08/2014, 08:20
... number of boxes in the first row is equal to the
size of the first part, the number of boxes in the second row is equal to the size of the
second part, etc. For example, the Ferrers diagram for the ... 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...
- 6
- 299
- 1
Báo cáo khoa học:On the q-analogue of the sum of cubes ppsx
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...
- 2
- 211
- 0
Tài liệu Đề tài " On the volume of the intersection of two Wiener sausages " ppt
Ngày tải lên :
14/02/2014, 17:20
... 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 ... 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 ... 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, Đ5], but there are...
- 44
- 331
- 0
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
Ngày tải lên :
19/02/2014, 06:20
... 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 which the 3Â 380 bp were identical to
exons 13 from the 5Â-end of the reverse primer
DN+231 to the 5Â-end of the region where the
sequence of exon 1 reported previously [15] matched
with the ... generation of two kinds of gene transcript.
However, only the sequences of the 5Â-UTR are differ-
ent between them, because these transcripts share all
the other exons, exons 2–9, which contain the...
- 12
- 609
- 0
The Gentlemen and the Roughs: The Collision of Two Honor Codes in the American North
Ngày tải lên :
14/03/2014, 09:31
... 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...
- 14
- 365
- 0
Báo cáo hóa học: " On the Krasnoselskii-type fixed point theorems for the sum of continuous and asymptotically nonexpansive mappings in Banach spaces" pptx
Ngày tải lên :
21/06/2014, 01:20
... 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...
- 11
- 457
- 0
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
Ngày tải lên :
21/06/2014, 20:20
... 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
tv
0
tωt 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
i1
u
2
i
}
1/2
,...
- 12
- 198
- 0
Báo cáo hóa học: " Research Article New Trace Bounds for the Product of Two Matrices and Their Applications in the Algebraic Riccati Equation" potx
Ngày tải lên :
22/06/2014, 02:20
... 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 ... 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 ARE6–12.
In...
- 18
- 456
- 0
Báo cáo hóa học: "ON THE IDENTITY OF TWO q-DISCRETE PAINLEVÉ EQUATIONS AND THEIR GEOMETRICAL DERIVATION" docx
Ngày tải lên :
22/06/2014, 22:20
... 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...
- 11
- 304
- 0
Scientific report: "Application of PCR to detect the presence of two genes in the genome of rice AtAOS has been genetically modified" pdf
Ngày tải lên :
06/08/2014, 18:21
... 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...
- 8
- 406
- 0
Báo cáo toán học: "When can the sum of (1/p)th of the binomial coefficients have closed form" pot
Ngày tải lên :
07/08/2014, 06:22
... 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...
- 7
- 314
- 0
