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1.
Brian Curtin 《Graphs and Combinatorics》1999,15(2):143-158
Let Γ=(X,E) denote a bipartite distance-regular graph with diameter D≥4, and fix a vertex x of Γ. The Terwilliger algebra
T=T(x) is the subalgebra of Mat
X(C) generated by A, E
*
0, E
*
1,…,E
*
D, where A denotes the adjacency matrix for Γ and E
*
i denotes the projection onto the i
TH subconstituent of Γ with respect to x. An irreducible T-module W is said to be thin whenever dimE
*
i
W≤1 for 0≤i≤Di. The endpoint of W is min{i|E
*
i
W≠0}.
We determine the structure of the (unique) irreducible T-module of endpoint 0 in terms of the intersection numbers of Γ. We show that up to isomorphism there is a unique irreducible
T-module of endpoint 1 and it is thin. We determine its structure in terms of the intersection numbers of Γ. We determine the
structure of each thin irreducible T-module W of endpoint 2 in terms of the intersection numbers of Γ and an additional real parameter ψ=ψ(W), which we refer to as the type of W.
We now assume each irreducible T-module of endpoint 2 is thin and obtain the following two-fold result. First, we show that the intersection numbers of Γ
are determined by the diameter D of Γ and the set of ordered pairs
where Φ2 denotes the set of distinct types of irreducible T-modules with endpoint 2, and where mult(ψ) denotes the multiplicity with which the module of type ψ appears in the standard
module. Secondly, we show that the set of ordered pairs {(ψ,mult(ψ)) |ψ∈Φ2} is determined by the intersection numbers k, b
2, b
3 of Γ and the spectrum of the graph , where
and where ∂ denotes the distance function in Γ. Combining the above two results, we conclude that if every irreducible T-module of endpoint 2 is thin, then the intersection numbers of Γ are determined by the diameter D of Γ, the intersection numbers k, b
2, b
3 of Γ, and the spectrum of Γ2
2.
Received: November 13, 1995 / Revised: March 31, 1997 相似文献
2.
Let Γ be a distance-regular graph of diameter D. Let X denote the vertex set of Γ and let Y be a nonempty subset of X. We define an algebra τ = τ(Y). This algebra is finite dimensional and semisimple. If Y consists of a single vertex then τ is the corresponding subconstituent algebra defined by P. Terwilliger. We investigate the irreducible τ-modules. We define endpoints and thin condition on irreducible τ-modules as a generalization of the case when Y consists of a single vertex. We determine when an irreducible module is thin. When the module is generated by the characteristic vector of Y, it is thin if and only if Y is a completely regular code of Γ. By considering a suitable subset Y, every irreducible τ(x)-module of endpoint i can be regarded as an irreducible τ(Y)-module of endpoint 0.This research was partially supported by the Grant-in-Aid for Scientific Research (No. 12640039), Japan Society of the Promotion of Science. A part of the research was done when the author was visiting the Ohio State University. 相似文献
3.
Kenichiro Tanabe 《Journal of Algebraic Combinatorics》1997,6(2):173-195
Let Y be any commutative association scheme and we fix any vertex x of Y. Terwilleger introduced a non-commutative, associative, and semi-simple C-algebraT=T(x) for Y and x in [4]. We call T the Terwilliger (or subconstituent) algebra ofY with respect to x.Let
be an irreducible T(x)-module. W is said to be thin if W satisfies a certain simple condition.Y is said to be thin with respect to x if each irreducible T(x) -module is thin. Y is said to be thin if Y is thin with respect to each vertex in X.The Doob schemes are direct product of a number of Shrikhande graphs and some complete graphsK
4 . Terwilliger proved in [4] that Doob scheme is not thin if the diameter is greater than two. I give the irreducible T(x)-modules of Doob schemes. 相似文献
4.
Benjamin V. C. Collins 《Graphs and Combinatorics》1997,13(1):21-30
Let Γ be a distance-regular graph of diameterd≥3. For each vertexx of Γ, letT(x) denote the Terwilliger algebra for Γ with respect tox. An irreducibleT(x)-moduleW is said to bethin if dimE i * (x)W≤1 for 0≤i≤d, whereE i * (x) is theith dual idempotent for Γ with respect tox. The graph Γ isthin if for each vertexx of Γ, every irreducibleT(x)-module is thin. Aregular generalized quadrangle is a bipartite distance-regular graph with girth 8 and diameter 4. Our main results are as follows: Theorem. Let Γ=(X,R) be a distance-regular graph with diameter d≥3 and valency k≥3. Then the following are equivalent:
- Γis a regular generalized quadrangle.
- Γis thin and c 3=1.
5.
Vlad Timofte 《Journal of Approximation Theory》2002,119(2):291-299
In this paper, we give special uniform approximations of functions u from the spaces CX(T) and C∞(T,X), with elements
of the tensor products CΓ(T)X, respectively C0(T,Γ)X, for a topological space T and a Γ-locally convex space X. We call an approximation special, if
satisfies additional constraints, namely supp vu−1(X\{0}) and
(T) co(u(T)) (resp. co(u(T){0})). In Section 3, we give three distinct applications, which are due exactly to these constraints: a density result with respect to the inductive limit topology, a Tietze–Dugundji's type extension new theorem and a proof of Schauder–Tihonov's fixed point theorem. 相似文献
6.
The necessary and sufficient conditions of outer conjugation for automorphisms from the normalizer of approximated III type groups are found. Let T be an automorphism of a Lebesgue space (X, μ) of the III0 type, [T] the full group generated by T, N[T] its normalizer, {Wt(T)} the flow associated with T and α → mod α the homomorphism from N[T] to C{W} the centralizer of the associated flow. The following results are obtained:
such that mod
ga = α; automorphisms α1, and α2 from N[T] are outer conjugate if and only if p(α1) = p(α2), mod α1 = γ mod α2γ−1, where γ C{W} and p(·) is the outer period; the canonical form of the elements from N[T] is found. The case is also considered where T is types IIIλ (0 < λ < 1) and III1. 相似文献
7.
Bimal Kumar Sinha 《Journal of multivariate analysis》1976,6(4):617-625
Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p Wp(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ Ik) with ξ(p × k) unknown. Denote the columns of X by X(1) ,…, X(k) and set ψ(0)(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ(i)(X, X) = min[ψ(i−1)(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X(1) X′(1) + + X(i) X′(i) |] and Ψ(i)(S, X) = min[ψ(0)(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X(1) X′(1) + + X(i) X′(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ(0)(S, X) is dominated by each of Ψ(i)(S, X), i = 1,…,k and for every i, ψ(i)(S, X) is better than ψ(i−1)(S, X). In particular, ψ(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X(1)X′(1)|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X(1)X′(1) + + X(k)X′(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance. 相似文献
8.
Let (T,
, P) be a probability space,
a P-complete sub-δ-algebra of
and X a Banach space. Let multifunction t → Γ(t), t T, have a
(X)-measurable graph and closed convex subsets of X for values. If x(t) ε Γ(t) P-a.e. and y(·) ε Ep
x(·), then y(t) ε Γ(t) P-a.e. Conversely, x(t) ε F(Γ(t), y(t)) P-a.e., where F(Γ(t), y(t)) is the face of point y(t) in Γ(t). If X =
, then the same holds true if Γ(t) is Borel and convex, only. These results imply, in particular, extensions of Jensen's inequality for conditional expectations of random convex functions and provide a complete characterization of the cases when the equality holds in the extended Jensen inequality. 相似文献
9.
Dieter Gaier 《Journal of Approximation Theory》1999,101(2):567
Let G be a domain bounded by a Jordan curve Γ, and let A(G) be the Banach space of functions continuous on G and holomorphic in G. The Faber operator T is a linear mapping from A(
) to A(G) mapping wn onto the nth Faber polynomial Fn(z) (n=0, 1, 2, …). We show that T<∞ if Γ is piecewise Dini-smooth, and give an example of a quasicircle Γ for which T=∞. 相似文献
10.
Let (X, Y) be an
d ×
-valued random vector and let (X1, Y1),…,(XN, YN) be a random sample drawn from its distribution. Divide the data sequence into disjoint blocks of length l1, …, ln, find the nearest neighbor to X in each block and call the corresponding couple (Xi*, Yi*). It is shown that the estimate mn(X) = Σi = 1n wniYi*/Σi = 1n wni of m(X) = E{Y|X} satisfies E{|mn(X) − m(X)|p}
0 (p ≥ 1) whenever E{|Y|p} < ∞, ln
∞, and the triangular array of positive weights {wni} satisfies supi ≤ nwni/Σi = 1n wni
0. No other restrictions are put on the distribution of (X, Y). Also, some distribution-free results for the strong convergence of E{|mn(X) − m(X)|p|X1, Y1,…, XN, YN} to zero are included. Finally, an application to the discrimination problem is considered, and a discrimination rule is exhibited and shown to be strongly Bayes risk consistent for all distributions. 相似文献
11.
Let (Ω,
, μ) be a measure space,
a separable Banach space, and
* the space of all bounded conjugate linear functionals on
. Let f be a weak* summable positive B(
*)-valued function defined on Ω. The existence of a separable Hilbert space
, a weakly measurable B(
)-valued function Q satisfying the relation Q*(ω)Q(ω) = f(ω) is proved. This result is used to define the Hilbert space L2,f of square integrable operator-valued functions with respect to f. It is shown that for B+(
*)-valued measures, the concepts of weak*, weak, and strong countable additivity are all the same. Connections with stochastic processes are explained. 相似文献
12.
Given a graph G=(V,E) with strictly positive integer weights ωi on the vertices iV, a k-interval coloring of G is a function I that assigns an interval I(i){1,…,k} of ωi consecutive integers (called colors) to each vertex iV. If two adjacent vertices x and y have common colors, i.e. I(i)∩I(j)≠0/ for an edge [i,j] in G, then the edge [i,j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi=1 for all vertices iV).Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1,…,k such that ℓI1(i) if and only if π(ℓ)I2(i) for all vertices iV. As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings, assuming that they can be identified very quickly. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions. 相似文献
13.
Paul Terwilliger 《Journal of Algebraic Combinatorics》2004,19(2):143-172
Let denote a distance-regular graph with diameter D 3, valency k, and intersection numbers a
i, b
i, c
i. Let X denote the vertex set of and fix x X. Let denote the vertex-subgraph of induced on the set of vertices in X adjacent X. Observe has k vertices and is regular with valency a
1. Let 1 2 ···
k
denote the eigenvalues of and observe 1 = a
1. Let denote the set of distinct scalars among 2, 3, ...,
k
. For let mult denote the number of times appears among 2, 3,...,
k
. Let denote an indeterminate, and let p
0, p1, ...,p
D denote the polynomials in
[] satisfying p
0 = 1 andp
i = c
i+1
p
i+1 + (a
i – c
i+1 + c
i)p
i + b
i
p
i–1 (0 i D – 1),where p
–1 = 0. We show
where we abbreviate
= –1 – b
1(1+)–1. Concerning the case of equality we obtain the following result. Let T = T(x) denote the subalgebra of Mat
X
(
) generated by A, E*0, E*1, ..., E*
D
, where A denotes the adjacency matrix of and E*
i
denotes the projection onto the ith subconstituent of with respect to X. T is called the subconstituent algebra or the Terwilliger algebra. An irreducible T-module W is said to be thin whenever dimE*
i
W 1 for 0 i D. By the endpoint of W we mean min{i|E*
i
W 0}. We show the following are equivalent: (i) Equality holds in the above inequality for 1 i D – 1; (ii) Equality holds in the above inequality for i = D – 1; (iii) Every irreducible T-module with endpoint 1 is thin. 相似文献
14.
In this paper we study the asymptotic behaviors of the likelihood ratio criterion (TL(s)), Watson statistic (TW(s)) and Rao statistic (TR(s)) for testing H0s: μ
(a given subspace) against H1s: μ
, based on a sample of size n from a p-variate Langevin distribution Mp(μ, κ) when κ is large. For the case when κ is known, asymptotic expansions of the null and nonnull distributions of these statistics are obtained. It is shown that the powers of these statistics are coincident up to the order κ−1. For the case when κ is unknown, it is shown that TR(s) TL(s) TW(s) in their powers up to the order κ−1. 相似文献
15.
A. G. Ramm 《Applied Mathematics Letters》1988,1(3)
Let·(σ(x)u)= 0 in D R3, where D is a bounded domain with a smooth boundary. Suppose that σ ≥ m> 0, σ H3(D), where Hℓ is the Sobolev space. Let the set {u, σuN} be given on Γ for all u H3/2(Γ), where uN is the normal derivative of u on Γ. 相似文献
16.
17.
In this paper we define the vertex-cover polynomial Ψ(G,τ) for a graph G. The coefficient of τr in this polynomial is the number of vertex covers V′ of G with |V′|=r. We develop a method to calculate Ψ(G,τ). Motivated by a problem in biological systematics, we also consider the mappings f from {1, 2,…,m} into the vertex set V(G) of a graph G, subject to f−1(x)f−1(y)≠ for every edge xy in G. Let F(G,m) be the number of such mappings f. We show that F(G,m) can be determined from Ψ(G,τ). 相似文献
18.
Let denote a distance-regular graph with vertex set X, diameter D 3, valency k 3, and assume supports a spin model W. Write W = i = 0D ti Ai where Ai is the ith distance-matrix of . To avoid degenerate situations we assume is not a Hamming graph and ti {t0, –t0 } for 1 i D. In an earlier paper Curtin and Nomura determined the intersection numbers of in terms of D and two complex parameters and q. We extend their results as follows. Fix any vertex x X and let T = T(x) denote the corresponding Terwilliger algebra. Let U denote an irreducible T-module with endpoint r and diameter d. We obtain the intersection numbers ci(U), bi(U), ai(U) as rational expressions involving r, d, D, and q. We show that the isomorphism class of U as a T-module is determined by r and d. We present a recurrence that gives the multiplicities with which the irreducible T-modules appear in the standard module. We compute these multiplicites explicitly for the irreducible T-modules with endpoint at most 3. We prove that the parameter q is real and we show that if is not bipartite, then q > 0 and is real.AMS 2000 Subject Classification: Primary 05E30 相似文献
19.
Let F be a Banach space with a sufficiently smooth norm. Let (Xi)i≤n be a sequence in LF2, and T be a Gaussian random variable T which has the same covariance as X = Σi≤nXi. Assume that there exists a constant G such that for s, δ≥0, we have P(sTs+δ)Gδ. (*) We then give explicit bounds of Δ(X) = supi|P(|X|≤t)−P(|T|≤t)| in terms of truncated moments of the variables Xi. These bounds hold under rather mild weak dependence conditions of the variables. We also construct a Gaussian random variable that violates (*). 相似文献
20.
Given a set function, that is, a map ƒ:
(E) →
{−∞} from the set
(E) of subsets of a finite set E into the reals including −∞, the standard greedy algorithm (GA) for optimizing ƒ starts with the empty set and then proceeds by enlarging this set greedily, element by element. A set function ƒ is said to be tractable if in this way a sequence x0 , x1, . . ., xN E (N #E) of subsets with max(ƒ) {ƒ(x0), ƒ(x1), . . ., ƒ(xN)} will always be found. In this note, we will reinterpret and transcend the traditions of classical GA-theory (cf., e.g., [KLS]) by establishing necessary and sufficient conditions for a set function ƒ not just to be tractable as it stands, but to give rise to a whole family of tractable set functions ƒ(η) :
(E) →
: x ƒ(x) + Σe xη(e), where η runs through all real valued weighting schemes η : E →
, in which case ƒ will be called rewarding. In addition, we will characterize two important subclasses of rewarding maps, viz. truncatably rewarding (or well-layered) maps, that is, set functions ƒ such that [formula] is rewarding for every i = 1, . . ., N, and matroidal maps, that is, set functions ƒ such that for every η : E →
and every ƒeta-greedy sequence x0, x1, . . ., xN as above, one has max(ƒη) = ƒη(xi) for the unique i {0, . . ., N} with ƒη(x0) < ƒη(x1) < ··· < ƒη(xi) ≥ ƒη(xi + 1). 相似文献