Nonorientable triangular embeddings of complete graphs with arbitrarily large looseness

The looseness of a triangular embedding of a complete graph in a closed surface is the minimum integer m such that for every assignment of m colors to the vertices of the embedding (such that all m colors are used) there is a face incident with vertices of three distinct colors. In this paper we show that for every p?3 there is a nonorientable triangular embedding of a complete graph with looseness at least p.     

Circulant graphs with det(−A(G))=−deg(G): codeterminants with Kn

Circulant graphs satisfying det(−A(G))=−deg(G) are used to construct arbitrarily large families of graphs with determinant equal to that of the complete graph K_n.     

A new approach to constructing exponentially many nonisomorphic nonorientable triangular embeddings of complete graphs

We prove a theorem that for an integer s?0, if 12s+7 is a prime number, then the number of nonisomorphic face 3-colorable nonorientable triangular embeddings of K_n, where n=(12s+7)(6s+7), is at least . By some number-theoretic arguments there are an infinite number of integers s satisfying the hypothesis of the theorem. The theorem is the first known example of constructing at least 2^αn?+o(n?), ?>1, nonisomorphic nonorientable triangular embeddings of K_n for n=6t+1, . To prove the theorem, we use a new approach to constructing nonisomorphic triangular embeddings of complete graphs. The approach combines a cut-and-paste technique and the index one current graph technique. A new connection between Steiner triple systems and constructing triangular embeddings of complete graphs is given.     

The lower bound for the volume product Vn(K)Vn(K)

We demonstrate by construction that the (conjectured) lower bound for the volume product holds with equality for a simplex.     

A combinatorial interpretation of the recurrence fn+1 = 6fn − fn−1

Bonin et al. (1993) recalled an open problem related to the recurrence relation verified by NSW numbers. The recurrence relation is the following: f_n+1 = 6f_n − f_n−1, with f₁ = 1 and f₂ = 7, and no combinatorial interpretation seems to be known. In this note, we define a regular language L whose number of words having length n is equal to f_n+1. Then, by using L we give a direct combinatorial proof of the recurrence.     

A lower bound for the number of triangular embeddings of some complete graphs and complete regular tripartite graphs

We prove that, for a certain positive constant a and for an infinite set of values of n, the number of nonisomorphic triangular embeddings of the complete graph K_n is at least n^an2. A similar lower bound is also given, for an infinite set of values of n, on the number of nonisomorphic triangular embeddings of the complete regular tripartite graph K_n,n,n.     

The Crossing Number of Cartesian Products of Complete Bipartite Graphs K 2, m with Paths P n

Most results on the crossing number of a graph focus on the special graphs, such as Cartesian products of small graphs with paths P_n, cycles C_n or stars S_n. In this paper, we extend the results to Cartesian products of complete bipartite graphs K_2,m with paths P_n for arbitrary m ≥ 2 and n ≥ 1. Supported by the NSFC (No. 10771062) and the program for New Century Excellent Talents in University.     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

Computation of cohomology groups of the Lie algebras of type Bn and Cn

We adduce the results of the numerical experiment in calculating the spaces of local deformations of classical Lie algebras of type B _n and C _n in characteristic p = 2. Original Russian Text ? D.V. Reshetnikov, 2009, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2009, No. 8, pp. 71–72.     

The decomposition of Kv into K2×K5''''s

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).     

On difference between the spectra of the simple groups Bn(q) and Cn(q)

We prove that the nonisomorphic simple groups B _n (q) and C _n (q) have different sets of element orders. Original Russian Text Copyright ? 2007 Grechkoseeva M. A. __________ Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 1, pp. 89–92, January–February, 2007.     

Zeros in tables of characters for the groups Sn and An

In the representation theory of symmetric groups, for each partition of a natural number n, the partition h() of n is defined so as to obtain a certain set of zeros in the table of characters for S_n. Namely, h() is the greatest (under the lexicographic ordering ) partition among P(n) such that (g) 0. Here, is an irreducible character of S_n, indexed by a partition , and g is a conjugacy class of elements in S_n, indexed by a partition . We point out an extra set of zeros in the table that we are dealing with. For every non self-associated partition P(n), the partition f() of n is defined so that f() is greatest among the partitions of n which are opposite in sign to h() and are such that (g) 0 (Thm. 1). Also, for any self-associated partition of n > 1, we construct a partition () P(n) such that () is greatest among the partitions of n which are distinct from h() and are such that (g) 0 (Thm. 2).Supported by RFBR grant No. 04-01-00463 and by RFBR-BRFBR grant No. 04-01-81001.Translated from Algebra i Logika, Vol. 44, No. 1, pp. 24–43, January–February, 2005.     

Cut elimination for S4n and K4n with the central agent axiom

We analyze the multimodal logic S4 _n with the central agent axiom. We present a Hilbert-type calculus, then derive a Gentzen-type calculus with cut, and prove a cut-elimination theorem. The work shows that it is possible to construct a cut-free Gentzen-type calculus for this logic. Moreover, it also provides analogous results for the multimodal logic K4 _n with the central agent axiom.     

Non-neutrality of the Stiefel manifolds Vn,k II

The Stiefel manifolds V_2^m−1,k are shown to be non-neutral for m5, 2^m−1+2k=2ℓ<2^m−2.     

S(Hn)上的满数值半径等距

本文研究了矩阵空间到自身的满数值半径等距问题. 利用等距嵌入方法, 获得了自共轭矩阵空间单位球面到自身的满数值半径等距可实线性延拓至全空间上的满数值半径等距, 为Tingley等距延拓问题提供了一种方法.     

Inner Functions and Cyclic Composition Operators on H(Bn)

The present paper shows that the algebra generated by {C|  Aut(B_n)} is cyclic on H²(B_n), and any nonconstant function f  H²(B_n) is a cyclic vector of . In addition, the hypercyclic and cyclic composition operators will be discussed.     

On convex embeddings of planar 3‐connected graphs

A well‐known Tutte's theorem claims that every 3‐connected planar graph has a convex embedding into the plane. Tutte's arguments also show that, moreover, for every nonseparating cycle C of a 3‐connected graph G, there exists a convex embedding of G such that C is a boundary of the outer face in this embedding. We give a simple proof of this last result. Our proof is based on the fact that a 3‐connected graph admits an ear assembly having some special properties with respect to the nonseparating cycles of the graph. This fact may be interesting and useful in itself. © 2000 John Wiley & Sons, Inc. J. Graph Theory 33: 120–124, 2000     

On the distributions of the form ∑i (δpi−δni)

We present some properties of the distributions T of the form ∑_i (δ_pi−δ_ni), with ∑_i d(p_i,n_i)<∞, which arise in the study of the 3-d Ginzburg–Landau problem; see Bourgain et al. (C. R. Acad. Sci. Paris, Ser. I 331 (2000) 119–124). We show that there always exists an irreducible representation of T. We also extend a result of Smets (C. R. Acad. Sci. Paris, Ser. I 334 (2002) 371–374) which says that T is a measure iff T can be written as a finite sum of dipoles.