Browsing by Author "Weng, CW"

Now showing 1 - 12 of 12

A construction of pooling designs with some happy surprises
(2005-10-01)
D'yachkov, A
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Hwang, F
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Macula, A
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Vilenkin, P
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Weng, CW
The screening of data sets for "positive data objects" is essential to modern technology. A ( group) test that indicates whether a positive data object is in a specific subset or pool of the dataset can greatly facilitate the identification of all the positive data objects. A collection of tested pools is called a pooling design. Pooling designs are standard experimental tools in many biotechnical applications. In this paper, we use the ( linear) subspace relation coupled with the general concept of a "containment matrix" to construct pooling designs with surprisingly high degrees of error correction ( detection.) Error-correcting pooling designs are important to biotechnical applications where error rates often are as high as 15%. What is also surprising is that the rank of the pooling design containment matrix is independent of the number of positive data objects in the dataset.
A note on decoding of superimposed codes
(2003-12-01)
Huang, T
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Weng, CW
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應用數學系
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Department of Applied Mathematics
A superimposed code with general distance D can be used to construct a non-adaptive pooling design. It can then be used to identify a fewunknown positives from a large set of items by associating naturally an outcome vector u. A simple method for decoding the outcome vector u is given whenever there are at most [D-1/2] errors occuring in the outcome vector u. Moreover, another simple method of detecting whether there is any error occuring in the outcome vector u is also given whenever there are at most D - 1 errors in u. Our method is a generalization of the classical result of Kautz and Singleton (Nonadaptive binary superimposed codes, IEEE Trans. Inform. Theory, vol. 10, pp. 363 - 377, 1964).
A typical vertex of a tree
(2001-01-06)
Wei, PH
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Weng, CW
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應用數學系
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Department of Applied Mathematics
Let T denote a tree with at least three vertices. Observe that T contains a vertex which has at least two neighbors of degree one or two. A class of algorithms on trees related to the observation are discussed and characterized. One of the example is an algorithm to compute the minimum rank m(T) of the symmetric matrices with prescribed graph T, which is easier to process than the algorithm previous found by Nylen [Linear Algebra Appl. 248 (1996) 303-316]. Two interpretations of the number m(T) in terms of some combinatorial properties on trees are given. (C) 2001 Elsevier Science B.V. All rights reserved. MSC: 05C05.
An inequality for regular near polygons
(2005-02-01)
Terwilliger, P
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Weng, CW
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應用數學系
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Department of Applied Mathematics
Let Gamma denote a near polygon distance-regular graph with diameter d greater than or equal to 3, valency k and intersection numbers a(1) > 0, c(2) > 1. Let theta(1) denote the second largest eigenvalue of Gamma. We show theta(1) less than or equal to k - a(1) - c(2)/ c(2) - 1. We show the following (i)-(iii) are equivalent. (i) Equality is attained above; (ii) Gamma is Q-polynomial with respect to theta(1); (iii) Gamma is a dual polar graph or a Hamming graph. (C) 2004 Elsevier Ltd. All rights reserved.
Classical distance-regular graphs of negative type
(1999-05-01)
Weng, CW
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應用數學系
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Department of Applied Mathematics
We prove the following theorem. Theorem. Let Gamma = (X, R) denote a distance-regular graph with classical parameters (d, b, alpha, beta) and d greater than or equal to 4. Suppose b < -1, and suppose the intersection numbers a(1) not equal 0, c(2) > 1. Then precisely one of the following (i) (iii) holds. (i) Gamma is the dual polar graph (2)A(dd-1)(-b). (ii) Gamma is the Hermitian forms graph Her(-b)(d). (iii) alpha = (b - 1)/2, beta = -(1 + b(d))/2, and -b is a power of an odd prime. (C) 1999 Academic Press.
Cycle-symmetric matrices and convergent neural networks
(2000-11-15)
Shih, CW
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Weng, CW
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應用數學系
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Department of Applied Mathematics
This work investigates a class of neural networks with cycle-symmetric connection strength. We shall show that, by changing the coordinates, the convergence of dynamics by Fiedler and Gedeon [Physica D 111 (1998) 288] is equivalent to the classical results. This presentation also addresses the extension of the convergence theorem to other classes of signal functions with saturations. In particular, the result of Cohen and Grossberg [IEEE Trans. Syst. Man Cybernet. SMC-13 (1983) 815] is recast and extended with a more concise verification. (C) 2000 Elsevier Science B.V. All rights reserved.
Distance-regular graphs, pseudo primitive idempotents, and the Terwilliger algebra
(2004-02-01)
Terwilliger, P
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Weng, CW
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應用數學系
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Department of Applied Mathematics
Let Gamma denote a distance-regular graph with diameter D greater than or equal to 3, intersection numbers a(i), b(i), c(i) and Bose-Mesner algebra M. For theta is an element of C boolean OR infinity we define a one-dimensional subspace of M which we call M(theta). If theta is an element of C then M(theta) consists of those Y in M such that (A - thetaI)Y is an element of CA(D), where A (resp. A(D)) is the adjacency matrix (resp. Dth distance matrix) of Gamma. If theta = infinity then M(theta) = CA(D). By a pseudo primitive idempotent for theta we mean a nonzero element of M(theta). We use these as follows. Let X denote the vertex set of Gamma and fix x is an element of X. Let T denote the subalgebra. of Mat(X) (C) generated by A, E-0(*), E-1(*),..., E-D(*), where E-i(*) denotes the projection onto the ith subconstituent of Gamma with respect to x. T is called the Terwilliger algebra. Let W denote an irreducible T-module. By the end point of W we mean min{i (EiW)-W-* not equal 0}. W is called thin whenever dim(E-i(*) W) less than or equal to 1 for 0 less than or equal to i less than or equal to D. Let V = C-X denote the standard T-module. Fix 0 not equal v is an element of (E1V)-V-* with v orthogonal to the all ones vector. We define (M; v) := {P is an element of M Pv is an element of (EDV)-V-*}. We show the following are equivalent: (i) dim(M; v) greater than or equal to 2; (ii) v is contained in a thin irreducible T-module with endpoint 1. Suppose (i), (ii) hold. We show (M; v) has a basis J, E where J has all entries I and E is defined as follows. Let W denote the T-module which satisfies (ii). Observe (E1W)-W-* is an eigenspace for E(1)(*)AE(1)(*); let 17 denote the corresponding eigenvalue. Define = (η) over tilde = -1 - b(1) (1 + eta)(-1) if eta not equal -1 and infinity if eta = -1. Then E is a pseudo primitive idempotent for. (C) 2003 Elsevier Ltd. All rights reserved.
On the templates corresponding to cycle-symmetric connectivity in cellular neural networks
(2002-12-01)
Shih, CW
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Weng, CW
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應用數學系
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Department of Applied Mathematics
In the architecture of cellular neural networks (CNN), connections among cells are built on linear coupling laws. These laws are characterized by the so-called templates which express the local interaction weights among cells. Recently, the complete stability for CNN has been extended from symmetric connections to cycle-symmetric connections. In this presentation, we investigate a class of two-dimensional space-invariant templates. We find necessary and sufficient conditions for the class of templates to have cycle-symmetric connections. Complete stability for CNN with several interesting templates is thus concluded.
Parallelogram-free distance-regular graphs
(1997-11-01)
Liang, YJ
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Weng, CW
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交大名義發表
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應用數學系
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National Chiao Tung University
Let Gamma=(X, R) denote a distance-regular graph with distance function partial derivative and diameter d greater than or equal to 4. By a parallelogram of length i (2 less than or equal to i less than or equal to d), we mean a 4-tuple xyzu of vertices in X such that partial derivative(x, y) = partial derivative(z, u) = 1, partial derivative(x,u) = i, and partial derivative(x,z) = partial derivative(y, z) = partial derivative(y, u)=i-1. We prove the following theorem. THEOREM. Let Gamma denote a distance-regular graph with diameter d greater than or equal to 4, and intersection numbers a(1) = 0, a(2) not equal 0. Suppose Delta is Q-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. Then Gamma has classical parameters (d, b, alpha, beta) with b < -1. By including results in [3]; [9], we have the following corollary. COROLLARY. Let Gamma denote a distance-regular graph with the Q-polynomial property. Suppose the diameter d greater than or equal to 4. Then the following (i)-(ii) are equivalent. (i) Gamma contains no parallelograms of any length. (ii) One of the following (iia)-(iic) holds. (iia) Gamma is bipartite. (iib) Gamma is a generalized odd graph. (iic) Gamma has classical parameters (d b, alpha, beta) and either b < -1 or Gamma is a Hamming graph or a dual polar graph. (C) 1997 Academic Press.
Parallelogram-free distance-regular graphs
(1997-11-01)
Liang, YJ
;
Weng, CW
;
應用數學系
;
Department of Applied Mathematics
Let Gamma=(X, R) denote a distance-regular graph with distance function partial derivative and diameter d greater than or equal to 4. By a parallelogram of length i (2 less than or equal to i less than or equal to d), we mean a 4-tuple xyzu of vertices in X such that partial derivative(x, y) = partial derivative(z, u) = 1, partial derivative(x,u) = i, and partial derivative(x,z) = partial derivative(y, z) = partial derivative(y, u)=i-1. We prove the following theorem. THEOREM. Let Gamma denote a distance-regular graph with diameter d greater than or equal to 4, and intersection numbers a(1) = 0, a(2) not equal 0. Suppose Delta is Q-polynomial and contains no parallelograms of length 3 and no parallelograms of length 4. Then Gamma has classical parameters (d, b, alpha, beta) with b < -1. By including results in [3]; [9], we have the following corollary. COROLLARY. Let Gamma denote a distance-regular graph with the Q-polynomial property. Suppose the diameter d greater than or equal to 4. Then the following (i)-(ii) are equivalent. (i) Gamma contains no parallelograms of any length. (ii) One of the following (iia)-(iic) holds. (iia) Gamma is bipartite. (iib) Gamma is a generalized odd graph. (iic) Gamma has classical parameters (d b, alpha, beta) and either b < -1 or Gamma is a Hamming graph or a dual polar graph. (C) 1997 Academic Press.
Pooling spaces and non-adaptive pooling designs
(2004-05-06)
Huang, TY
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Weng, CW
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應用數學系
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Department of Applied Mathematics
A pooling space is defined to be a ranked partially ordered set with atomic intervals. We show how to construct non-adaptive pooling designs from a pooling space. Our pooling designs are e-error detecting for some e; moreover, e can be chosen to be very large compared with the maximal number of defective items. Eight new classes of non-adaptive pooling designs are given, which are related to the Hamming matroid, the attenuated space, and six classical polar spaces. We show how to construct a new pooling space from one or two given pooling spaces. (C) 2003 Elsevier B.V. All rights reserved.
The quantum algebra U-q(sl(2)) and its equitable presentation
(2006-04-01)
Ito, T
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Terwilliger, P
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Weng, CW
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應用數學系
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Department of Applied Mathematics
We show that the quantum algebra U-q(sl(2)) has a presentation with generators x(+/- 1), y, Z and relations xx(-1) = x(-1)x = 1, [GRAPHICS] We call this the equitable presentation. We show that y (respectively z) is not invertible in U-q (sl(2)) by displaying an infinite-dimensional U-q (sl(2))-module that contains a nonzero null vector for y (respectively z). We consider finite-dimensional Uq (sl(2))-modules under the assumption that q is not a root of 1 and char(K) not equal 2, where K is the underlying field. We show that y and z are invertible on each finite-dimensional Uq(sl(2))-module. We display a linear operator Omega that acts on finite-dimensional U-q(sl(2))-modules, and satisfies Omega(-1) x Omega = y, Omega(-1) y Omega = z, Omega(-1) z Omega = x on these modules. We define Omega using the q-exponential function. (c) 2005 Elsevier Inc. All rights reserved.