Let Γ be a distance-regular graph of diameter d ≥ 3 with c2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
(SC)m: For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them.
(BB)m: Let (x, y, z) be a triple of vertices with ∂Γ(x, y) = 1 and ∂Γ(x, z) = ∂Γ(y, z) = m. Then B(x, z) = B(y, z).
(CA)m: Let (x, y, z) be a triple of vertices with and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z).
In [12] we have shown that the condition (SC)m holds if and only if both of the conditions (BB)i and (CA)i hold for i = 1,...,m. In this paper we show that if a1 = 0 < a2 and the condition (BB)i holds for i = 1,...,m, then the condition (CA)i holds for i = 1,...,m. In particular, the condition (SC)m holds. Applying this result we prove that a distance-regular graph with classical parameters (d, b, α, β) such that c2 > 1 and a1 = 0 < a2 satisfies the condition (SC)i for i = 1,...,d − 1. In particular, either (b, α, β) = (− 2, −3, −1 − (−2)d) or holds. 相似文献
Let (m, n) ∈ ℕ2, Ω an open bounded domain in ℝm, Y = [0, 1]m; uε in (L2(Ω))n which is two-scale converges to some u in (L2(Ω × Y))n. Let φ: Ω × ℝm × ℝn → ℝ such that: φ(x, ·, ·) is continuous a.e. x ∈ Ω φ(·, y, z) is measurable for all (y, z) in ℝm × ℝn, φ(x, ·, z) is 1-periodic in y, φ(x, y, ·) is convex in z. Assume that there exist a constant C1 > 0 and a function C2 ∈ L2(Ω) such that
We find that some union of two star-configurations in ?2 has generic Hilbert function. Applying the result, we prove that some Artinian quotients of a coordinate ring of a star-configuration in ?n satisfy the weak-Lefschetz property. More precisely, let 𝕏 and 𝕐 be star-configurations in ?2 of type (2, s) and (2, s + 1) defined by forms F1,…, Fs, and G1,…, Gs, L, respectively, with deg(Fi) = deg(Gi) ≤2 for i = 1,…, s and s ≥ 3. If L is a general linear form in R = 𝕜[x0, x1, x2], then R/(I𝕏 + I𝕐) has the weak-Lefschetz property with a Lefschetz element L, which extends the result of [21Shin, Y. S. (2012). Star-configurations in ?2 having generic Hilbert functions and the weak-Lefschetz property. Comm. in Algebra 40:2226–2242.[Taylor &; Francis Online], [Web of Science ®], [Google Scholar]]. 相似文献
Let Γ be a distance-regular graph of diameter d ≥ 3 with c2 > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:
(SC)m : For any pair of vertices at distance m there exists a strongly closed subgraph of diameter m containing them.
(BB)m : Let (x, y, z) be a triple of vertices with ∂Γ (x, y) = 1 and ∂Γ (x, z) = ∂Γ (y, z) = m. Then B(x, z) = B(y, z).
(CA)m : Let (x, y, z) be a triple of vertices with ∂Γ (x, y) = 2, ∂Γ (x, z) = ∂Γ (y, z) = m and |C(z, x) ∩ C(z, y)| ≥ 2. Then C(x, z) ∪ A(x, z) = C(y, z) ∪ A(y, z).
Suppose that the condition (SC)m holds. Then it has been known that the condition (BB)i holds for all i with 1 ≤ i ≤ m. Similarly we can show that the condition (CA)i holds for all i with 1 ≤ i ≤ m. In this paper we prove that if the conditions (BB)i and (CA)i hold for all i with 1 ≤ i ≤ m, then the condition (SC)m holds. Applying this result we give a sufficient condition for the existence of a dual polar graph as a strongly closed subgraph
in Γ. 相似文献
A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a
Brenner , J. L. ,
Wiegold , J. ( 1975a ). Two-generator groups. I . Michigan Math. J. 22 : 53 – 64 .[Crossref], [Web of Science ®], [Google Scholar]) defined a finite group G to have spread r if for every set {x1, x2,…, xr} of distinct nontrivial elements of G, there exists an element y ? G such that G = 〈 xi, y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996
Ganief , M. S. ( 1996 ). 2-Generations of the Sporadic Simple Groups , Ph.D thesis , University of Natal .[Google Scholar]) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001
Ganief , M. S. ,
Moori , J. ( 2001 ). On the spread of the sporadic simple groups . Comm. Algebra 29 : 3239 – 3255 .[Taylor &; Francis Online], [Web of Science ®], [Google Scholar]) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups. 相似文献
Let R be an integral domain, and let x ∈ R be a nonzero nonunit that can be written as a product of irreducibles. Coykendall and Maney (to appear), defined the irreducible divisor graph of x, denoted G(x), as follows. The vertices of G(x) are the nonassociate irreducible divisors of x (each from a pre-chosen coset of the form π U(R) for π ∈ R irreducible). Given distinct vertices y and z, we put an edge between y and z if and only if yz|x. Finally, if yn|x but yn+1 ? x, then we put n ? 1 loops on the vertex y. In this article, inspired by the approach of the authors from Akhtar and Lee (to appear
Akhtar , R. ,
Lee , L.Homology of zero divisors . To appear in Rocky Mountain J. Math.[Google Scholar]), we study G(x) using homology. A connection is found between H1 and the cycle space of G(x). We also characterize UFDs via these homology groups. 相似文献
It is proved that if an entire function f: ? → ? satisfies an equation of the form α1(x)β1(y) + α2(x)β2(y) + α3(x)β3(y), x,y ∈ C, for some αj, βj: ? → ? and there exist no \({\widetilde \alpha _j}\) and ?\({\widetilde \beta _j}\) for which \(f\left( {x + y} \right)f\left( {x - y} \right) = {\overline \alpha _1}\left( x \right){\widetilde \beta _1}\left( y \right) + {\overline \alpha _2}\left( x \right){\widetilde \beta _2}\left( y \right)\), then f(z) = exp(Az2 + Bz + C) ? σΓ(z - z1) ? σΓ(z - z2), where Γ is a lattice in ?; σΓ is the Weierstrass sigma-function associated with Γ; A,B,C, z1, z2 ∈ ?; and \({z_1} - {z_2} \notin \left( {\frac{1}{2}\Gamma } \right)\backslash \Gamma \). 相似文献
In this paper, we consider the partial difference equation with continuous variables of the form P1z(x + a, y + b) + p2z (x + a, y) + p3z (x, y + b) − p4z (x, y) + P (x, y) z (x − τ, y − σ) = 0, where PϵC(R+ × R+, R+ − {0}), a, b, τ, σ are real numbers and pi (i = 1, 2, 3, 4) are nonnegative constants. Some sufficient conditions for all solutions of this equation to be oscillatory are obtained. 相似文献
Summary Letxi=yi+zi,i=1, ...n, and writex(1)≦...≦x(n), with corresponding notation for the orderedyi andzi. It is shown, for example, that
,r=1, ...n. Inequalities are also obtained for convex (or concave) functions of thex(i). The results lead immediately to bounds for the expected values of order statistics in nonstandard situations in terms of
simpler expectations. A small numerical example illustrates the method.
Research supported by U.S. Army Research Office. 相似文献
Solutions are obtained for the boundary value problem, y(n) + f(x,y) = 0, y(i)(0) = y(1) = 0, 0 in – 2, where f(x,y) is singular at y = 0. An application is made of a fixed point theorem for operators that are decreasing with respect to a cone. 相似文献
Divided differences forf (x, y) for completely irregular spacing of points (xi,yi) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (xi,yi) to be at corners of rectangles, or give polynomials Σajkxjyk having more coefficients than interpolation conditions. Here the generalizednth divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (xi,yi), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inxijyik. The generalization of Newton's div. diff. formula is (5)
The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line K_1(y)u_(xx) |K_2(x)|u_(yy) a(x,y)u_x b(x, y)u_y c(x,y)u=-d(x,y) in any plane domain D with the boundary D=Γ∪L_1∪L_2∪L_3∪L_4, whereΓ(■{y>0})∈C_μ~2 (0<μ<1) is a curve with the end points z=-1,1. L_1, L_2, L_3, L_4 are four characteristics with the slopes -H_2(x)/H_1(y), H_2(x)/H_1(y),-H_2(x)/H_1(y), H_2(x)/H_1(y)(H_1(y)=|k_1(y)|~(1/2), H_2(x)=|K_2(x)|~(1/2) in {y<0}) passing through the points z=x iy=-1,0,0,1 respectively. And the boundary condition possesses the form 1/2 u/v=1/H(x,y)Re[λuz]=r(z), z∈Γ∪L_1∪L_4, Im[λ(z)uz]|_(z=z_l)=b_l, l=1,2, u(-1)=b_0, u(1)=b_3, in which z_1, z_2 are the intersection points of L_1, L_2, L_3, L_4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations K_1(y)(M_2(x)u_x)_x M_1(x)(K_2(y)u_y)_y r(x,y)u=f(x,y), in D as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u_(xx) u_(yy)=0 with the boundary condition u(z)=φ(z) onΓ∪L_1∪L_4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin- Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z)=W(x iy)=u_z=[H_1(y)u_x-iH_2(x)u_y]/2 in the elliptic domain and W(z)=W(x jy)=u_z=[H_1(y)u_x-jH_2(x)u_y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied. 相似文献
Suppose that G is a finite group and f is a complex-valued function on G. f induces a (left) convolution operator from L2(G) to L2(G) by g ? f *g{g \mapsto f \ast g} where
f *g(z) : = \mathbbExy=zf(x)g(y) for all z ? G.f \ast g(z) := \mathbb{E}_{xy=z}f(x)g(y)\,\, {\rm for\,\,all} \, z \in G. 相似文献
The following results for proper quasi‐symmetric designs with non‐zero intersection numbers x,y and λ > 1 are proved.
(1) Let D be a quasi‐symmetric design with z = y ? x and v ≥ 2k. If x ≥ 1 + z + z3 then λ < x + 1 + z + z3.
(2) Let D be a quasi‐symmetric design with intersection numbers x, y and y ? x = 1. Then D is a design with parameters v = (1 + m) (2 + m)/2, b = (2 + m) (3 + m)/2, r = m + 3, k = m + 1, λ = 2, x = 1, y = 2 and m = 2,3,… or complement of one of these design or D is a design with parameters v = 5, b = 10, r = 6, k = 3, λ = 3, and x = 1, y = 2.
(3) Let D be a triangle free quasi‐symmetric design with z = y ? x and v ≥ 2k, then x ≤ z + z2.
(4) For fixed z ≥ 1 there exist finitely many triangle free quasi‐symmetric designs non‐zero intersection numbers x, y = x + z.
(5) There do not exist triangle free quasi‐symmetric designs with non‐zero intersection numbers x, y = x + 2.
Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f1, f2, f3, f4, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f1(x + t, y + s) + f2(x - t, y - s) = f3(x + s, y - t) + f4(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f1, f2, f3, and f4 are given by f1 = w + h, f2 = w - h, f3 = w + k, f4 = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of Dy,t3g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and Dx,t3g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G . 相似文献
We prove the following theorem: Let φ(x) be a formula in the language of the theory PA? of discretely ordered commutative rings with unit of the form ?yφ′(x,y) with φ′ and let ∈ Δ0 and let fφ: ? → ? such that fφ(x) = y iff φ′(x,y) & (?z < y) φ′(x,z). If I ∏ ∈(?x ≥ 0), φ then there exists a natural number K such that I ∏ ? ?y?x(x > y ? ?φ(x) < xK). Here I ∏1? denotes the theory PA? plus the scheme of induction for formulas φ(x) of the form ?yφ′(x,y) (with φ′) with φ′ ∈ Δ0. 相似文献