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1.
Crossing numbers of graphs are in general very difficult to compute. There are several known exact results on the crossing
number of the Cartesian products of paths, cycles or stars with small graphs. In this paper we study cr(Km □ Pn), the crossing number of the Cartesian product Km □ Pn. We prove that
for m ≥ 3,n ≥ 1 and cr(Km □ Pn)≥ (n − 1)cr(Km+2 − e) + 2cr(Km+1). For m≤ 5, according to Klešč, Jendrol and Ščerbová, the equality holds. In this paper, we also prove that the equality holds for
m = 6, i.e., cr(K6 □ Pn) = 15n + 3.
Research supported by NFSC (60373096, 60573022). 相似文献
2.
E. G. Goluzina 《Journal of Mathematical Sciences》2007,143(3):3023-3029
The paper studies the region of values Dm,n(T) of the system {f(z1), f(z2),..., f(zm), f(r1), f(r2),..., f(rn)}, where m ≥ 1; n > 1; zj, j = 1, ... m, are arbitrary fixed points of the disk U = {z: |z| < 1} with Im zj ≠ 0, j = 1, 2, ..., m; rj, 0 < rj < 1, j = 1, 2, ..., n, are fixed; f ∈ T, and the class T consists of functions f(z) = z + c2z2 + ... regular in the disk U and satisfying the condition Im f(z) · Im z > 0 for Im z ≠= 0, z ∈ U. An algebraic characterization of the set Dm,n(T) in terms of nonnegative-definite Hermitian forms is provided, and all the boundary functions are described. As an implication,
the region of values of f(z1) in the subclass of functions f ∈ T with prescribed values f(rj) (j = 1, 2, 3) is determined. Bibliography: 12 titles.
Dedicated to the 100th anniversary of my father’s birthday
__________
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 337, 2006, pp. 23–34. 相似文献
3.
4.
Emília Draženská 《Mathematica Slovaca》2011,61(5):675-686
The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The
crossing numbers of G□C
n
for some graphs G on five and six vertices and the cycle C
n
are also given. In this paper, we extend these results by determining the crossing number of the Cartesian product G □ C
n
, where G is a specific graph on six vertices. 相似文献
5.
V. A. Belonogov 《Algebra and Logic》2008,47(2):77-90
Previously, we dubbed the conjecture that the alternating group An has no semiproportional irreducible characters for any natural n [1]. This conjecture was then shown to be equivalent to
the following [3]. Let α and β be partitions of a number n such that their corresponding characters χα and χβ in the group Sn are semiproportional on An. Then one of the partitions α or β is self-associated. Here, we describe all pairs (α, β) of partitions satisfying the hypothesis
and the conclusion of the latter conjecture.
Supported by RFBR (grant No. 07-01-00148) and by RFBR-NSFC (grant No. 05-01-39000).
__________
Translated from Algebra i Logika, Vol. 47, No. 2, pp. 135–156, March–April, 2008. 相似文献
6.
Marián Klešč 《Graphs and Combinatorics》2001,17(2):289-294
There are several known exact results on the crossing numbers of Cartesian products of paths or cycles with “small” graphs.
In this paper we extend these results to the Cartesian products of two specific 5-vertex graphs with the star K
1,
n
. In addition, we give the crossing number of the graph obtained by adding two edges to the graph K
1,4,
n
in such a way that these new edges join a vertex of degree n+1 of the graph K
1,4,
n
with two its vertices of the same degree.
Received: December 8, 1997 Final version received: August 14, 1998 相似文献
7.
V. A. Belonogov 《Siberian Mathematical Journal》2008,49(5):784-795
In studying the pairs of irreducible characters of the symmetric group S
n
with the same zero set on A
n
or S
n
∖ A
n
(as well as the pairs of irreducible characters with the same zero set on the alternating group A
n
), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem
that considerably generalizes two previous results of frequent use in this direction.
Original Russian Text Copyright ? 2008 Belonogov V. A.
The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00148) and the RFBR-NSFC (Grant 05-01-39000).
__________
Ekaterinburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 5, pp. 992–1006, September–October, 2008. 相似文献
8.
The decomposition of the complete graph Kv into Kr×Kc's, the products of Kr and Kc,is originated from the use of DNA library screening. In this paper, we consider the case where r=2 and c = 5, and show that such a decomposition exists if and only if v ≡ 1 (mod 25). 相似文献
9.
Let λK
m,n be a bipartite multigraph with two partite sets having m and n vertices, respectively. A P
v-factorization of λK
m,n is a set of edge-disjoint P
v
-factors of λK
m,n which partition the set of edges of λK
m,n. When v is an even number, Ushio, Wang and the second author of the paper gave a necessary and sufficient condition for the existence
of a P
v
-factorization of λK
m,n. When v is an odd number, we proposed a conjecture. However, up to now we only know that the conjecture is true for v = 3. In this paper we will show that the conjecture is true when v = 4k − 1. That is, we shall prove that a necessary and sufficient condition for the existence of a P
4k−1-factorization of λK
m,n is (1) (2k − 1)m ⩽ 2kn, (2) (2k − 1)n ⩽ 2km, (3) m + n ≡ 0 (mod 4k − 1), (4) λ(4k − 1)mn/[2(2k − 1)(m + n)] is an integer. 相似文献
10.
We continue the study of small cycle double covers of products of graphs that began in [7], concentrating here on the categorical
product and the strong product. Under the assumption that G has an SCDC, we show that G × P
m
has an SCDC for all m ≠ 3, and that G × C
m
has an SCDC for all m ≥ 3. For the strong product we use results about the categorical product and the Cartesian product [7] to show that if G has an SCDC, then so does G ⊠ C
m
, m ≥ 5. Some results are also given for G ⊠ P
m
, but require additional assumptions about the SCDC of G.
The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada. 相似文献
11.
Ming-Chang Kang 《Israel Journal of Mathematics》2005,146(1):77-92
LetK be any field which may not be algebraically closed,K(x
1
,x
2
,x
3
) be the rational function field of three variables overK, and σ:K(x
1
,x
2
,x
3
) → K(x
1
,x
2
,x
3
) be aK-automorphism defined by
wherea
i
,b
i
,c
i
,d
i
∈K anda
i
d
i
−b
i
c
i
≠0. Let
,f
i
(T)=T
2
−(a
i
+d
i
)T+(a
i
d
i
−b
i
c
i
)∈K[T] be the “characteristic polynomial” of σ
i
.
Theorem:Assume that charK≠2.Then the fixed field K(x
1
,x
2
,x
3
)
<σ>
is not rational (=purely transcendental) over K if and only if (i) for each 1≤i≤3, f
i
(T) is irreducible; (ii) the Galois group of f
1
(T)f
2
(T)f
3
(T) over K is of order 8; and (iii) for each 1≤i≤3,ord (σ
[itn]
)is an even integer. 相似文献
12.
Representations of quantum superalgebras provide a natural framework in which to model supersymmetric quantum systems. Each
quantum superalgebra, belonging to the class of quasi-triangular Hopf superalgebras, contains a universal R-matrix which automatically satisfies the Yang–Baxter equation. Applying the vector representation π, which acts on the vector module V, to the left-hand side of a universal R-matrix gives a Lax operator. In this article a Lax operator is constructed for the quantised orthosymplectic superalgebras
U
q
[osp(m|n)] for all m > 2, n ≥ 0 where n is even. This can then be used to find a solution to the Yang–Baxter equation acting on V ⊗ V ⊗ W, where W is an arbitrary U
q
[osp(m|n)] module. The case W = V is studied as an example.
Presented by A. Verschoren. 相似文献
13.
In this paper we discuss a relatively general kind of iterative functional equation G(x,f(x), ...,f
n
(x)) = 0 (for allx ∈J), whereJ is a connected closed subset of the real number axis ℝ,G∈C
m
(J
n+1, ℝ) andn ≥ 2. Using the method of approximating fixed points by small shift of maps, choosing suitable metrics on functional spaces
and finding a relation between uniqueness and stability of fixed points of maps of general spaces, we prove the existence,
uniqueness and stability ofCm solutions of the above equation for any integer m ≥ 0 under relatively weak conditions, and generalize related results in
reference in different aspects. 相似文献
14.
V. Chernousov 《Mathematische Annalen》2003,326(2):297-330
We prove that for a simple simply connected quasi-split group of type 3,6
D
4
,E
6
,E
7
defined over a perfect field F of characteristic ≠=2,3 the Rost invariant has trivial kernel. In certain cases we give a formula for the Rost invariant.
It follows immediately from the result above that if cd F≤2 (resp. vcd F≤2) then Serre's Conjecture II (resp. the Hasse principle) holds for such a group. For a (C
2
)-field, in particular ℂ(x,y), we prove the stronger result that Serre's Conjecture II holds for all (not necessary quasi-split) exceptional groups of
type 3,6
D
4
,E
6
,E
7
.
Received: 27 March 2002 /
Published online: 28 March 2003
The author gratefully acknowledge the support of TMR ERB FMRX CT-97-0107 and Forschungsinstitut für Mathematik, ETH in Zürich 相似文献
15.
We prove that if ma = mK*da*mK{\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaK\subseteq SU(n)} , for K = SO(n), then mak{\mu _{a}^{k}} is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ
a
, supported on the minimal dimension double coset has the property that man-1{\mu _{a}^{n-1}} is singular to the Haar measure. 相似文献
16.
Xiao Hong Fu 《数学学报(英文版)》2008,24(9):1475-1482
This paper considers the isometric extension problem concerning the mapping from the unit sphere S
1(E) of the normed space E into the unit sphere S
1(l
∞(Γ)). We find a condition under which an isometry from S
1(E) into S
1(l
∞(Γ)) can be linearly and isometrically extended to the whole space. Since l
∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are
solved. More precisely, if E and F are two normed spaces, and if V
0: S
1(E) → S
1(F) is a surjective isometry, where c
00(Γ) ⊆ F ⊆ l
∞(Γ), then V
0 can be extended to be an isometric operator defined on the whole space.
This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614) 相似文献
17.
We show that every (possibly unbounded) convex polygon P in
\mathbbR2{\mathbb{R}^2} with m edges can be represented by inequalities p
1 ≥ 0, . . ., p
n
≥ 0, where the p
i
’s are products of at most k affine functions each vanishing on an edge of P and n = n(m, k) satisfies s(m, k) £ n(m, k) £ (1+em) s(m, k){s(m, k) \leq n(m, k) \leq (1+\varepsilon_m) s(m, k)} with s(m,k) ≔ max {m/k, log2
m} and em ? 0{\varepsilon_m \rightarrow 0} as m ? ¥{m \rightarrow \infty}. This choice of n is asymptotically best possible. An analogous result on representing the interior of P in the form p
1 > 0, . . ., p
n
> 0 is also given. For k ≤ m/log2
m these statements remain valid for representations with arbitrary polynomials of degree not exceeding k. 相似文献
18.
Let Γ
g, n
be the mapping class group of a compact Riemann surface of genusg withn points preserved (2−2g−n<0,g≥1,n≥0). The Torelli subgroup of Γ
g, n
has a natural weight filtration {Γg, n(m)}
m≥1. Each graded quotient gr
m
Γ
g, n
⊗ ℚ (m≥1) is a finite dimensional vector space over ℚ on which the group Sp(2g, ℚ)×S
n
naturally acts.
In this paper, we have determined the Sp(2g, ℚ)×S
n
module structure of gr
m
Γ
g, n
⊗ ℚ for 1≤m≤3. This includes a verification of an expectation by S. Morita. Also, for generalm, we have identified a certain Sp(2g, ℚ)-irreducible component of gr
m
Γ
g, n
⊗ ℚ by constructing explicitly elements in these modules. 相似文献
19.
László Verhóczki 《Monatshefte für Mathematik》2004,109(1):323-335
In the present paper we discuss in detail the cohomogeneity one isometric actions of the Lie groups SU(3) × SU(3) and SU(3) on the exceptional compact symmetric spaces G2 and G2/SO(4), respectively. We show that the principal orbits coincide with the tubular hypersurfaces around the totally geodesic singular orbits, and the symmetric spaces G2 and G2/SO(4) can be thought of as compact tubes around SU(3) and P2, respectively. Moreover, we determine the radii of these tubes and describe the shape operators of the principal orbits. Finally, we apply these results to compute the volumes of the two symmetric spaces. 相似文献
20.
A. S. Kondratev 《Siberian Mathematical Journal》2007,48(6):1001-1018
We prove that if L is one of the simple groups E
6(q) and 2
E
6(q) and G is some finite group with the same spectrum as L, then the commutant of G/F(G) is isomorphic to L and the quotient G/G′ is a cyclic {2,3}-group.
Original Russian Text Copyright ? 2007 Kondrat’ev A. S.
The author was supported by the Russian Foundation for Basic Research (Grant 04-01-00463) and the RFBR-NSFC (Grant 05-01-39000).
__________
Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 6, pp. 1250–1271, November–December, 2007. 相似文献