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OD-Characterization of certain four dimensional linear groups with related results concerning degree patterns
B. AKBARI,A. R. MOGHADDAMFAR
Front. Math. China. 2015, 10 (1): 1-31.
https://doi.org/10.1007/s11464-014-0430-2
The prime graph of a finite group G, which is denoted by GK(G), is a simple graph whose vertex set is comprised of the prime divisors of |G| and two distinct prime divisors p and q are joined by an edge if and only if there exists an element of order pq in G. Let p1<p2<?<pk be all prime divisors of |G|. Then the degree pattern of G is defined as D(G) = (degG(p1), degG(p2), ? , degG(pk)), where degG(p) signifies the degree of the vertex p in GK(G). A finite group H is said to be OD-characterizable if G? H for every finite group G such that |G| = |H| and D(G) = D(H). The purpose of this article is threefold. First, it finds sharp upper and lower bounds on ?(G), the sum of degrees of all vertices in GK(G), for any finite group G (Theorem 2.1). Second, it provides the degree of vertices 2 and the characteristic p of the base field of any finite simple group of Lie type in their prime graphs (Propositions 3.1-3.7). Third, it proves the linear groups L4(q), q = 19, 23, 27, 29, 31, 32, and 37, are OD-characterizable (Theorem 4.2).
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Algorithms for enumeration problem of linear congruence modulo m as sum of restricted partition numbers
Tian-Xiao HE, Peter J. -S. SHIUE
Front. Math. China. 2015, 10 (1): 69-89.
https://doi.org/10.1007/s11464-014-0394-2
We consider the congruence x 1 + x 2 + + x r ≡ c (mod m), where m and r are positive integers and . Recently, W. -S. Chou, T. X. He, and Peter J. -S. Shiue considered the enumeration problems of this congruence, namely, the number of solutions with the restriction , and got some properties and a neat formula of the solutions. Due to the lack of a simple computational method for calculating the number of the solution of the congruence, we provide an algebraic and a recursive algorithms for those numbers. The former one can also give a new and simple approach to derive some properties of solution numbers.
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Nagata rings
Pascual JARA
Front. Math. China. 2015, 10 (1): 91-110.
https://doi.org/10.1007/s11464-014-0388-0
Let A be a commutative ring. For any set p of prime ideals of A, we define a new ring Na(A, p): the Nagata ring. This new ring has the particularity that we may transform certain properties relative to p to properties on the whole ring Na(A, p); some of these properties are: ascending chain condition, Krull dimension, Cohen-Macaulay, Gorenstein. Our main aim is to show that most of the above properties relative to a set of prime ideals p(i.e., local properties) determine and are determined by the same properties on the Nagata ring (i.e., global properties). In order to look for new applications, we show that this construction is functorial, and exhibits a functorial embedding from the localized category (A, p)-Mod into the module category Na(A, p)-Mod.
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Wintgen ideal submanifolds with a low-dimensional integrable distribution
Tongzhu LI,Xiang MA,Changping WANG
Front. Math. China. 2015, 10 (1): 111-136.
https://doi.org/10.1007/s11464-014-0383-5
Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of M?bius geometry. We classify Wintgen ideal submanfiolds of dimension m≥3 and arbitrary codimension when a canonically defined 2-dimensional distribution ?2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if ?2 generates a k-dimensional integrable distribution ?k<?Pub Caret?>and k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.
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Embedding of circulant graphs and generalized Petersen graphs on projective plane
Yan YANG,Yanpei LIU
Front. Math. China. 2015, 10 (1): 209-220.
https://doi.org/10.1007/s11464-014-0428-9
Both the circulant graph and the generalized Petersen graph are important types of graphs in graph theory. In this paper, the structures of embeddings of circulant graph C(2n + 1; {1, n}) on the projective plane are described, the number of embeddings of C(2n + 1; {1, n}) on the projective plane follows, then the number of embeddings of the generalized Petersen graph P(2n +1, n) on the projective plane is deduced from that of C(2n +1; {1, n}), because C(2n + 1;{1, n}) is a minor of P(2n + 1, n), their structures of embeddings have relations. In the same way, the number of embeddings of the generalized Petersen graph P(2n, 2) on the projective plane is also obtained.
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