一类优美图

设u、ν是两个固定顶点.用b条内部互不相交且长度皆为a的道路连接u、ν所得的图用P_a,b表示.KM.Kathiresan证实P_2,2m-1(r,m皆为任意正整数)是优美的,且猜想:除了(a,b)=(2r+1,4s+2)外,所有的P_a,b都是优美的.杨元生已证实P_2r+1,2m-1是优美的,并且证实了当r=1,2,3,4时的P_2r,2m也是优美的.本文证实r=5,6,7时P_{2r,2m相似文献}   

A combinatorial characterization of normalizations of Boolean algebras

In this paper we prove that if a groupoid has exactly distinct n-ary term operations for n=1, 2, 3 and the same number of constant unary term operations for n=0, then it is a normalization of a nontrivial Boolean algebra. This, together with some general facts concerning normalizations of algebras, which we recall, yields a clone characterization of normalizations of nontrivial Boolean algebras: A groupoid (G;·) is clone equivalent to a normalization of a nontrivial Boolean algebra if and only if the value of the free spectrum for (G;·) is for n = 0, 1, 2, 3. In the last section the Minimal Extension Property for the sequence (2, 3) in the class of all groupoids is derived. Received September 15, 2004; accepted in final form October 4, 2005.     

Chromaticity of two-trees

The graphs called 2-trees are defined by recursion. The smallest 2-tree is the complete graph on 2 vertices. A 2-tree on n + 1 vertices (where n ≥ 2) is obtained by adding a new vertex adjacent to each of 2 arbitrarily selected adjacent vertices in a 2-tree on n vertices. A graph G is a 2-tree on n(≥2) vertices if and only if its chromatic polynomial is equal to γ(γ - 1)(γ - 2)^n—2.     

Distribution of Selmer groups of quadratic twists of a family of elliptic curves

We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in Q. We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate-Shafarevich group and a large Tate-Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1-2) (2005) 139-154 (2006)], this shows that the mean value of the 2-rank of the large Tate-Shafarevich group for square-free positive integers n less than X is , as X→∞.     

New proof of a theorem of Szekeres

A Hadamard matrix H of order 16t² is constructed for all t for which there is a Hadamard matrix of order 4t, in such a way that each row of H contains exactly 8t² + 2t ones. As a consequence a new method of constructing the symmetric block designs with parameters (16t², 8t² + 2t, 4t² + 2t) for all t for which there is a Hadamard matrix of order 4t is given.     

Semigroup identities in the monoid of two-by-two tropical matrices

We show that the monoid $M_{2}(\mathbb {T})$ of 2×2 tropical matrices is a regular semigroup satisfying the semigroup identity $$A^2B^4A^2A^2B^2A^2B^4A^2=A^2B^4A^2B^2A^2A^2B^4A^2.$$ Studying reduced identities for subsemigroups of $M_{2}(\mathbb {T})$ , and introducing a faithful semigroup representation for the bicyclic monoid by 2×2 tropical matrices, we reprove Adjan’s identity for the bicyclic monoid in a much simpler way.     

Bireflectionality of orthogonal and symplectic groups of characteristic 2

Let V be a finite dimensional vector space over a field K of characteristic 2. Let O(V) be the orthogonal group defined by a nondegenerate quadratic form. Then every element in O(V) is a product of two elements of order 2, unless all nonsingular subspaces of V are at most 2-dimensional. If V is a nonsingular symplectic space, then every element in the symplectic group Sp (V) is a product of two elements of order 2, except if dim V = 2 and |K| = 2.     

A Construction of Difference Sets in High Exponent 2-Groups Using Representation Theory

Nontrivial difference sets in groups of order a power of 2 are part of the family of difference sets called Menon difference sets (or Hadamard), and they have parameters (2^2d+2, 2^2d+1±2 ^d , 2^2d ±2 ^d ). In the abelian case, the group has a difference set if and only if the exponent of the group is less than or equal to 2 ^d+2. In [14], the authors construct a difference set in a nonabelian group of order 64 and exponent 32. This paper generalizes that result to show that there is a difference set in a nonabelian group of order 2^2d+2 with exponent 2 ^d+3. We use representation theory to prove that the group has a difference set, and this shows that representation theory can be used to verify a construction similar to the use of character theory in the abelian case.     

Endomorphisms of automorphism groups of free groups

It is proved that any non-trivial endomorphism of an automorphism group AutF_n of a free group F_n, for n 3, either is an automorphism or factorization over a proper automorphism subgroup. An endomorphism of AutF₂ is an automorphism, or else a homomorphism onto one of the groups S₃, D₈, Z₂ × Z₂, Z₂, or (Z₂ × Z₂). A non-trivial homomorphism of AutF_n into AutF_m, for n 3, m 2, and n > m, is a homomorphism onto Z₂ with kernel SAutF_n. As a consequence, we obtain that AutF_n is co-Hopfian.Supported by RFBR grant No. 02-01-00293 and by the Council for Grants (under RF President) and State Aid of Fundamental Science Schools, project NSh-2069.2003.1.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 211–237, March–April, 2005.     

Triangle in a triangle: On a problem of Steinhaus

A necessary and sufficient condition on the sidesp, q, r of a trianglePQR and the sidesa, b, c of a triangleABC in order thatABC contains a congruent copy ofPQR is the following: At least one of the 18 inequalities obtained by cyclic permutation of {a, b, c} and arbitrary permutation of {itp, q, r} in the formula $$\begin{array}{l} Max\{ F(q^2 + r^2 - p^2 ), F'(b^2 + c^2 - a^2 )\} \\ + Max\{ F(p^2 + r^2 - q^2 ), F'(a^2 + c^2 - b^2 )\} \le 2Fcr \\ \end{array}$$ is satisfied. In this formulaF andF′ denote the surface areas of the triangles, i.e. $$\begin{array}{l} F = {\textstyle{1 \over 4}}(2a^2 b^2 + 2b^2 c^2 + 2c^2 a^2 - a^4 - b^4 - c^4 )^{1/2} \\ F' = {\textstyle{1 \over 4}}(2p^2 q^2 + 2q^2 r^2 + 2r^2 p^2 - p^4 - q^4 - r^4 )^{1/2} . \\ \end{array}$$      

Characterizations of Buekenhout-Metz unitals

We give a characterization of the Buekenhout-Metz unitals in PG(2, q ²), in the cases that q is even or q=3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q ²), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q ²) is a Buekenhout-Metz unital.     

Non-separating 2-factors of an even-regular graph

For a 2-factor F of a connected graph G, we consider G−F, which is the graph obtained from G by removing all the edges of F. If G−F is connected, F is said to be a non-separating 2-factor. In this paper we study a sufficient condition for a 2r-regular connected graph G to have such a 2-factor. As a result, we show that a 2r-regular connected graph G has a non-separating 2-factor whenever the number of vertices of G does not exceed 2r²+r.     

Existence of HSOLSSOMs of type

In this paper we investigate the existence of holey self-orthogonal Latin squares with a symmetric orthogonal mate of type 2ⁿu¹ (HSOLSSOM(2ⁿu¹)). For u2, necessary conditions for existence of such an HSOLSSOM are that u must be even and n3u/2+1. Xu Yunqing and Hu Yuwang have shown that these HSOLSSOMs exist whenever either (1) n9 and n3u/2+1 or (2) n263 and n2(u-2). In this paper we show that in (1) the condition n9 can be extended to n30 and that in (2), the condition n263 can be improved to n4, except possibly for 19 pairs (n,u), the largest of which is (53,28).     

Asymptotics of the spectrum of nonsmooth perturbations of differential operators of order 2m

On a finite closed interval, we obtain the asymptotics of the eigenvalues of a differential operator of order 2m perturbed by a differential operator of order 2m ? 2 given by a quasidifferential expression. We also consider the case of multiple eigenvalues.     

On the free boundary problem of magnetohydrodynamics

In the paper, the solvability of the free boundary problem of magnetohydrodynamics for a viscous incompressible fluid in a simply connected domain is proved. The solution is obtained in the Sobolev–Slobodetskii spaces W₂^{2 + l,1 + l/2},1/2 < l < 1 W_2^{2 + l,1 + l/2},1/2 < l < 1 . Bibliography: 15 titles.     

An Alternate Characterization of the Bentness of Binary Functions, with Uniqueness

In a previous paper, we have obtained a characterization of the binary bent functions on (GF(2))ⁿ (n even) as linear combinations modulo , with integral coefficients, of characteristic functions (indicators) of -dimensional vector-subspaces of (GF(2))ⁿ. There is no uniqueness of the representation of a given bent function related to this characterization. We obtain now a new characterization for which there is uniqueness of the representation.     

The functional equation for Poincaré series of trace rings of generic 2×2 matrices

In this note we give a rational expression for the Poincaré series of Π_m,2, the trace ring ofm generic 2×2 matrices. This result extends the computations of E. Formanek form⩽4. As a consequence, we prove that the Poincaré series satisfies the functional equation(II_m,2;1/t)=-t^4m.P(II_m,2,t) (m>2) supporting the conjecture that Π_m,2 is a Gorenstein ring. Work supported by an NFWO/FNRS grant.     

The strength of weak proximity

This paper studies weak proximity drawings of graphs and demonstrates their advantages over strong proximity drawings in certain cases. Weak proximity drawings are straight line drawings such that if the proximity region of two points p and q representing vertices is devoid of other points representing vertices, then segment (p,q) is allowed, but not forced, to appear in the drawing. This differs from the usual, strong, notion of proximity drawing in which such segments must appear in the drawing.Most previously studied proximity regions are associated with a parameter β, 0β∞. For fixed β, weak β-drawability is at least as expressive as strong β-drawability, as a strong β-drawing is also a weak one. We give examples of graph families and β values where the two notions coincide, and a situation in which it is NP-hard to determine weak β-drawability. On the other hand, we give situations where weak proximity significantly increases the expressive power of β-drawability: we show that every graph has, for all sufficiently small β, a weak β-proximity drawing that is computable in linear time, and we show that every tree has, for every β less than 2, a weak β-drawing that is computable in linear time.     

Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

Let Ω be a bounded, smooth domain in ?²ⁿ, n ≥ 2. The well‐known Moser‐Trudinger inequality ensures the nonlinear functional J_ρ(u) is bounded from below if and only if ρ ≤ ρ_2n := 2²ⁿn!(n ? 1)!ω_2n, where in , and ω_2n is the area of the unit sphere ??^{2n ? 1} in ?²ⁿ. In this paper, we prove the inf_u∈X J_ρ(u) is always attained for ρ ≤ ρ_2n. The existence of minimizers of J_ρ at the critical value ρ = ρ_2n is a delicate problem. The proof depends on the blowup analysis for a sequence of bubbling solutions. Here we develop a local version of the method of moving planes to exclude the boundary bubbling. The existence of minimizers for J_ρ at the critical value ρ = ρ_2n is in contrast to the case of two dimensions. © 2003 Wiley Periodicals, Inc.