the sum of cubes calculator
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...
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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...
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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...
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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 ... p, r.Soforeachfixed p, 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 ... 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...
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Báo cáo toán học: "The sum of degrees in cliques" doc
Ngày tải lên :
07/08/2014, 08:22
... every graph G, let ∆
r
(G) be the maximum of the sum of degrees
of the vertices of an r-clique, as in the abstract. If G has no r-cliques, we set ∆
r
(G)=0.
Furthermore, let
∆
r
(n, m)= min
G=G(n,m)
∆
r
(G) ... r-clique of G
and let ∆
r
(n, m) be the minimum of ∆
r
(G) taken over all graphs of order n and
size m.Writet
r
(n)forthesizeofther-chromatic Tur´an graph of order n.
Improving earlier results of ... grateful to Prof. D. Todorov for pointing out
a fallacy in an earlier version of the proof of Theorem 2 and to the referee for his valuable
suggestions.
Added on July 1st, 2005. The results of this...
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Báo cáo toán học: "COMPOSITION SUM IDENTITIES RELATED TO THE DISTRIBUTION OF COORDINATE VALUES IN A DISCRETE SIMPLEX" pptx
Ngày tải lên :
07/08/2014, 06:20
... bijection between the set of compositions of n −2by{1, 2} and the
set of compositions of n − 1 by odd numbers. The bijection works by prepending a 1 to a
composition of the former type, and then performing ... pictures of weight k.Acard of weight k
is a pair consisting of a picture of weight k and a k-element subset of N that we will call the
label set of the card. A hand of weight n and size l is a set of ... form a partition of {1, 2, ,n} into l disjoint groups. The goal of
labelled counting is to establish a relation between the cardinality of the sets of hands and
the cardinality of the decks.
For...
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Báo cáo toán học: "A note on the number of (k, )-sum-free sets" potx
Ngày tải lên :
07/08/2014, 06:20
... Denote
by SF
n
k,
the number of (k, ) -sum- free subsets of [1,n]. Since the set of
1
the electronic journal of combinatorics 7 (2000), #R30 7
Proof. We need to show that the left-hand side of (4) is ... ) -sum- free sets. Thus, the number of (k, ) -sum- free sets
satisfying neither (1), nor (2) does not exceed
(ϕ + ϕ
r
)2
n/ρ
+2n
2
2
n/(ρ+1)
.
To complete the proof it is sufficient to show that the ... Taylor: Counting sets of integers, no k of which sum
to another, J. Number Theory 57 (1996), 323–327.
[5] N. J. Calkin, J. M. Thomson, Counting generalized sum- free sets,J.
Number Theory 68 (1998),...
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Báo cáo toán học: "Convexly independent subsets of the Minkowski sum of planar point sets" pdf
Ngày tải lên :
07/08/2014, 15:23
... sets
of size m and n in the plane. The Minkowski sum of P and Q is P ⊕ Q = {p + q | p ∈
P, q ∈ Q}.
What is the maximum size of a convexly independent subset of P ⊕ Q ?
the electronic journal of ... vertex set of a convex polygon? In the special case when the elements of X themselves
are convexly independent, they found a linear upper bound, 5n−6, on this quantity. They
asked whether there exists ... X of n points
in the plane, what is the maximum number of pairs that can be selected from X so that
the midpoints of their connecting segments are convexly independent, that is, they form
the...
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Báo cáo toán học: "On sum of powers of the Laplacian and signless Laplacian eigenvalues of graphs" ppt
Ngày tải lên :
08/08/2014, 12:22
... defined
as the sum of singular values of the incidence matrix [9]. The directed incidence energy
DIE(G) is defined as the sum of singular values of the directed incidence matrix [7]. In
other words,
IE(G) ... A not e on sum of powers of the Laplacian
eigenvalues of bipartite graphs, Linear Algebra Appl. 430 (2009), 2503–2510.
[12] B. Zhou, On sum of powers of the Laplacian eigenvalues of graphs, Linear ... terms of powers of ∆ − ℓI and
A, then the terms appearing in both expansions, regardless their signs, are the same. To
prove the claim, we determine the sign of each term in both expansions. In the...
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Báo cáo toán học: "The Enumeration of Sequences with Restrictions on their Partial Sum" docx
Ngày tải lên :
08/08/2014, 12:23
... section t he sum of the crossing numbers about ℓ. Fo r ω ∈ Ω, we
define the upcrossing sum of ω about ℓ as the sum of all upcrossing numbers of ω about ℓ,
and similarly, the downcrossing sum of ω about ... now follows similarly to the proofs of Theorems 3.1 and 3.2 that the maximum
partial sum of ω = ω
+
(ω
−
)
R
equals r + k. If r < 0, then the initial part of ω through the
first downcrossing about ... in the manner of the proofs
of (13) and ( 14). However, since the fo r mulas obtained ar e rather complicated, we shall
not state them explicitly.
Theorem 2.4. Assume that ℓ max(r, 0). Then...
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Báo cáo toán học: "On the number of subsequences with a given sum in a finite abelian grou" pptx
Ngày tải lên :
08/08/2014, 14:23
... the proof of the theorem.
If we further assume that E(S) = {0} in Theorem 12, t he structure of S can be
further restricted.
Corollary 13. If S is a sequence over a finite abelian group G of odd ... 2
m−D(G)+1
.
This completes the proof of the theorem.
Notice that the result in [24] that N
0
(S) ≥ 2
|S|−D(G)+1
for any sequence S over G,
together with the following lemma, also gives Theorem 2.
Lemma ... g of G, the lower bound of N
g
(S) remains undetermined.
In this paper, we determine the best possible lower bound of N
g
(S) for an arbitrary
g of G. We also characterize the structures of the...
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Measures to promote the reform of administrative procedures in the operation of the public universities in vietnam (Sum)
Ngày tải lên :
22/10/2014, 11:11
... contribution to the mission of the reform of
university and education in general which we are carrying out in accordance
with resolution of the 4
th
Congress of The Central Committee of the
Communist ... procedures and the reform of
administrative procedures, we suppose that the reform of administrative
procedures in operation of universities is actually the reform of
implementation of these administrative ... mark of
the term was below 0,08 for the first term of the school year or below 1,00 for
the next terms. However, according to the rules of 2012, for those whose study
result was fallen in the...
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A study on the impact of foreign direct investment on economic development of lao p d r (sum)
Ngày tải lên :
13/11/2014, 21:13
... investor on the management of
9
the enterprise. Ownership of at least 10% of the voting power, representing
the influence by the investor, is the basic criterion used (OECD, 2012).
In the ... leads to the
creation of more opportunities in the sectors of education, healthcare,
employment and the conservation of the environment. It implies an
increase in the per capita income of every ... help
explain the motivations of FDI and identify FDI’s determinants.
- Internalization
- Eclectic paradigm
- Complement Theory of FDI
- The Resource-Based Theory
- The Theory of New Economic...
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