Linking (n − 2)-dimensional panels inn-space I: (k − 1,k)-graphs and (k − 1,k)-frames

A (k – 1,k)-graph is a multi-graph satisfyinge (k – 1)v – k for every non-empty subset ofe edges onv vertices, with equality whene = |E(G)|. A (k – 1,k)-frame is a structure generalizing an (n – 2, 2)-framework inn-space, a structure consisting of a set of (n – 2)-dimensional bodies inn-space and a set of rigid bars each joining a pair of bodies using ball joints. We prove that a graph is the graph of a minimally rigid (with respect to edges) (k – 1,k)-frame if and only if it is a (k – 1,k)-graph. Rigidity here means infinitesimal rigidity or equivalently statical rigidity.     

On the Monotonicity of (κ; g, h)-graphs

A (k;g)-graph is a k-regular graph with girth g. A (k;g)-cage is a (k;g)-graph with the least possible number of vertices. Let f(k;g) denote the number of vertices in a (k;g)-cage. The girth pair of a graph gives the length of a shortest odd and a shortest even cycle. A fc-regular graph with girth pair (g,h) is called a (k;g,h)-graph. A (k;g,h)-cage is a (k;g,h)-graph with the least possible number of vertices. Let f(k;g,h) denote the number of vertices in a (k;g,h)-cage. In this paper, we prove the following strict inequality f(k;h-1,h)相似文献   

Ramsey Number of K_(2,s 1) vs.K_(1,n)

It is shown that the Ramsey number r(K_(2,s 1),K_(1,n))(?)n sn~(1/2) (s 3)/2 o(1) for large n,and r(K_(2,s 1),K_(1,n))∈{((q-1)~2/s) 1,((q-1)~2/s) 2},where n=((q-1)~2/s)-q 2 and q is a prime power such that s|(q-1).     

Rigid Sets of Dimension n – 1 in ℝ n

We give conditions allowing an intrinsic isometry on a dense subset to be extended to an isometry of the whole set. This enables us to find examples of (n-1)-dimensional sets rigid in ⁿ .     

Overgroups of SU(n,K, f) or Ω(n,K, Q) in GL(n,K)

All the overgroups of SU(n, K, f) or (n, K, Q) in GL(n, K) are determined, for arbitrary division rings K and arbitrary forms f or Q with positive Witt indices.     

从(n 1)/n×(n 1)=(n 1)/n (n 1)谈起

一问题的提出本刊2003年第5期刊载了《运用发现法解题》(以下简称《解题》)一文,文章在谈到“归纳发现法”时,提到这样一个例子: 1.观察下列各式     

Intriguing Sets of Points of Q(2n, 2) \Q +(2n ? 1, 2)

Intriguing sets of vertices have been studied for several classes of strongly regular graphs. In the present paper, we study intriguing sets for the graphs Γ _n , n ≥ 2, which are defined as follows. Suppose Q(2n, 2), n?≥ 2, is a nonsingular parabolic quadric of PG(2n, 2) and Q ⁺(2n ? 1, 2) is a nonsingular hyperbolic quadric obtained by intersecting Q(2n, 2) with a suitable nontangent hyperplane. Then the collinearity relation of Q(2n, 2) defines a strongly regular graph Γ _n on the set Q(2n, 2) \ Q ⁺(2n ? 1, 2). We describe some classes of intriguing sets of Γ _n and classify all intriguing sets of Γ₂ and Γ₃.     

(g,f)-factors with Special Properties in(mg,mf)-graphs

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Minimal Covers of Q +(2n + 1, q) by (n − 1)-Dimensional Subspaces

A t-cover of a quadric is a set C of t-dimensional subspaces contained in such that every point of is contained in at least one element of C.We consider (n – 1)-covers of the hyperbolic quadric Q ⁺(2n + 1, q). We show that such a cover must have at least q ^{n + 1} + 2q + 1 elements, give an example of this size for even q and describe what covers of this size should look like.     

Small representations of the relation algebra ɛn+1(1, 2, 3)

Applying combinatorial methods, we prove that the symmetric relation algebra _n+1(1, 2, 3) ofn+1 atoms is finitely representable for alln 1, on at most (2+o(1))n² elements asn . We explicitly construct a representation of size 4.5n², for every n >1.Presented by B. Jonsson.     

Isomorphic embedding of ℓ p n , 1<p<2, into ℓ 1 (1+ε)n

In this paper we partially answer a question posed by V. Milman and G. Schechtman by proving that ℓ _p ⁿ , (C logn)^1/q(1+1/ε)-embeds into ℓ ₁ ^(1+ε)n , where 1<p<2 and 1/p+1/q=1. Supported by ISF.     

Maximal subalgebras of rank n −1 of the algebra AP(1, n) and reduction of nonlinear wave equations. I

The concept of canonical decomposition of an arbitrary subalgebra of the algebraAO(1,n) is introduced. With the help of this decomposition all maximal subalgebras L of rankn–1 of the algebraAP(1,n), satisfying the conditionL V=1,...;P _n>, whereV=<P ₀,P ₁,...,P ₁> is the space of translation are described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1552–1559, November, 1990.     

A characterization of a class of hyperplanes of DW(2n−1,F)

A hyperplane of the symplectic dual polar space $DW(2n?1,\mathbb{F})$, $n\ge 2$, is said to be of subspace-type if it consists of all maximal singular subspaces of $W(2n?1,\mathbb{F})$ meeting a given $(n?1)$-dimensional subspace of $\text{PG}(2n?1,\mathbb{F})$. We show that a hyperplane of $DW(2n?1,\mathbb{F})$ is of subspace-type if and only if every hex $F$ of $DW(2n?1,\mathbb{F})$ intersects it in either $F$, a singular hyperplane of $F$ or the extension of a full subgrid of a quad. In the case $\mathbb{F}$ is a perfect field of characteristic 2, a stronger result can be proved, namely a hyperplane $H$ of $DW(2n?1,\mathbb{F})$ is of subspace-type or arises from the spin-embedding of $DW(2n?1,\mathbb{F})?DQ(2n,\mathbb{F})$ if and only if every hex $F$ intersects it in either $F$, a singular hyperplane of $F$, a hexagonal hyperplane of $F$ or the extension of a full subgrid of a quad.     

SO(n)-Invariant Special Lagrangian Submanifolds of C~(n 1) with Fixed Loci

Let SO(n) act in the standard way on Cn and extend this action in the usual way to Cn 1 =C Cn. It is shown that a nonsingular special Lagrangian submanifold L (?) Cn 1 that is invariant under this SO(n)-action intersects the fixed C (?) Cn 1 in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A(?)C lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear pde and ultimately calls on the work of Gerard and Tahara to prove the existence of the extension.     

n~(n 1)与(n 1)~n谁大?

学生曾问我“20022003与20032002谁大?”,他钻研后认为n(n 1)与(n 1)n(n∈N)当n取较小的值时,有时前者比后者小,有时前者比后者大,当n取较大的值时,又不好证明,这引起了我们的研究兴趣。     

On the stability of quasidouble step spline approximations for solutions ofy (n) = f(x,y, ⋯, y (n−1) )

In this paper, we continue the study [4] of the stability question of the quasidouble step spline function approximations,s(x) C ^m–2 , to the initial value problemy ⁽ⁿ⁾ = f(x, y, , y ^(n–1) ). It will be shown that the method is unstable and hence divergent form n + 4.